Symplectic q-Schur Algebras

Sebastian Oehms, Mathematisches Institut B, Universität Stuttgart ^{1}

1 Introduction

The general linear group GL_{n}(K) operates on V^{⊗r} the r-fold tensor space of its natural module V. Its group algebra factored by the kernel of this operation is called the Schur algebra and denoted S(n, r). By place permutation the symmetric group S_{r} operates on V^{⊗r}, too. Moreover, both actions centralize each other. This fact is known as Schur-Weyl-Duality.

This situation admits a q-analogue which has been introduced by R. Dipper and G. James in [DJ]. Here, instead of the symmetric group you have to take the Iwahori-Hecke algebra of type A. Its centralizer is called the q-Schur algebra. There are various generalizations of this theory for instance by Dipper, James and A. Mathas [DJM] who replaced the Iwahori-Hecke algebras by Ariki-Koike algebras leading to so called cyclotomic q-Schur algebras. On the other hand the original q-Schur algebra can be obtained (up to Morita equivalence, cf. [DJ2]) using constructions from the theory of quantum groups [DD]. In this paper we will apply these constructions to obtain q-Schur algebras which are related to the symplectic groups. We will denote them by S_{q}^{s}(n, r). Setting the deformation parameter q = 1, we obtain classical symplectic Schur algebras in the sense of S. Donkin [Do1]. The main result in this paper is that the symplectic q-Schur algebras are cellular in the sense of J. Graham and G. Lehrer [GL] and integrally quasi-hereditary as algebras over the ring of integer Laurent polynomials. Furthermore we show that they are the centralizers of the Birman-Murakami-Wenzl algebras acting on V^{⊗r}. In order to obtain the cellular basis we introduce a quantum symplectic version of bideterminants. In [O2] the author has presented a symplectic version of the famous straightening formula for bideterminants in the classical case. Here, we will develop the fundamental calculus for quantum symplectic bideterminants and give a quantized version of that straightening formula. This formula is powerful enough to imply almost all results of the paper.

The standard modules (or cell representations) of S_{q}^{s}(n, r) are indexed by pairs (λ, l) consisting of an integer 0 ≤ l ≤ \frac{r}{2} and a partition λ ∈ Λ^{+}(m, r −2l) of r −2l in not more than m-parts. Here n = 2m is the dimension of the natural module of the symplectic group. The part of the basis corresponding to (λ, l) is labelled by pairs of λ-symplectic standard tableaux in the sense of R.C. King [Ki], or more precisely by a reversed version of them.

The material of this paper is taken from my doctoral thesis [O1] arranged in a completely reorganized form. Further it contains some improvements. For example the second part of Theorem 6.3 has been proved in [O1, 4.1.2] under restrictions only and the proof of Proposition 11.1 is more direct and shortened, here.

1E-Mail: seba@mathematik.uni-stuttgart.de

2 Quantum Symplectic Monoids

Let R be a noetherian integral domain and q ∈ R an invertible element. Let V be a free R-module. Fixe a basis {v_{1}, …, v_{n}} and let e_{ij} denote the corresponding basis of matrix units for E := End_{R}(V). We define two endomorphisms β and γ on V ⊗ V identifying End_{R}(V ⊗ V) with E ⊗ E (we write simply ⊗ instead of ⊗_{R} if no ambiguity can arise):

β := ∑_{1≤i≤n}(q^{2}e_{ii} ⊗ e_{ii} + e_{ii′} ⊗ e_{i′i}) + q ∑_{1≤i≠j,j′≤n} e_{ij} ⊗ e_{ji}+

+(q^{2} − 1) ∑_{1≤j<i≤n}(e_{ii} ⊗ e_{jj} − q^{ρ_{i}−ρ_{j}}ϵ_{i}ϵ_{j}e_{ij′} ⊗ e_{i′j}),

and

γ := ∑_{1≤i,j≤n} q^{ρ_{i}−ρ_{j}}ϵ_{i}ϵ_{j}e_{ij′} ⊗ e_{i′j},

Here, the exponents are given by (ρ_{1}, …, ρ_{n}) = (m, m − 1, …, 1, −1, …, −(m − 1), −m), whereas ϵ_{i} := sign(ρ_{i}). Further, i′ := n − i + 1 defines an involution on \underline{n} := \{1, …, n\}. Thus

(1′, 2′, …, n′) = (n, n − 1, …, 1).

There are slightly more general versions of these endomorphisms involving additional parameters. We may omit them without loss of generality (see [O1, Satz 2.5.8]). For r ∈ ℕ write \underline{r} := \{1, …, r\}. A multi-index is a map i : \underline{r} → \underline{n} frequently denoted as an r-tuple i = (i_{1}, …, i_{r}) where i_{j} ∈ \underline{n}. The set of all such multi-indices will be denoted by I(n, r). We define

v_{i} := v_{i_{1}} ⊗ v_{i_{2}} ⊗ … ⊗ v_{i_{r}} ∈ \underbrace{V ⊗ V ⊗ … ⊗ V}_{r-times} =: V^{⊗r}.

An endomorphism μ of V^{⊗r} may be given by its coefficients μ_{ij} with respect to the basis {v_{i}| i ∈ I(n, r)} of V^{⊗r}, that is

μ(v_{j}) = ∑_{i∈I(n,r)} μ_{ij}v_{i}.

Let F_{R}(n) := R⟨X_{11}, X_{12}, …, X_{nn}⟩ be the free algebra generated by the n^{2} symbols X_{ij} for i, j ∈ \underline{n}. This is a graded algebra, an R-basis of the r-th homogeneous part F_{R}(n, r) is the set

{X_{ij} := X_{i_{1}j_{1}} … X_{i_{2}j_{2}} … X_{i_{r}j_{r}}| i, j ∈ I(n, r)}.

To simplify notation we introduce a new convention to write down frequently used elements of F_{R}(n) an its quotients in a convenient way. For an endomorphism μ on V^{⊗r} we write

μ ≀ X_{ij} := ∑_{k∈I(n,r)} μ_{ik}X_{kj} and X_{ij} ≀ μ := ∑_{k∈I(n,r)} X_{ik}μ_{kj}.  (1)

We will denote the residue classes of X_{ij} in any quotient of F_{R}(n) by x_{ij}. The residue class x_{ij} of X_{ij} then clearly has a similar expression in the x_{ij} as the X_{ij} do in the X_{ij}. The above introduced convention will be used for x_{ij} accordingly.

The object of our investigations is given by the following definition:

A_{R,q}^{s}(n) := F_{R}(n)/ ⟨β ≀ X_{ij} − X_{ij} ≀ β, γ ≀ X_{ij} − X_{ij} ≀ γ| i, j ∈ I(n, 2)⟩ .

Here the brackets ⟨⟩ denote the ideal generated by the enclosed elements and β, γ are the endomorphisms on V ⊗V defined above. Since this ideal in the definition is homogeneous, the algebra A_{R,q}^{s}(n) = ⨁_{r∈ℕ_{0}} A_{R,q}^{s}(n, r) is again graded. Here, A_{R,q}^{s}(n, r) is the R-linear span of the elements x_{ij} for i, j ∈ I(n, r). The algebra A_{R,q}^{s}(n) can be identified with a generalized F RT -construction with respect to the subset N := {β, γ} ⊆ E ⊗ E denoted M_{R}(N) in [O2, section 5]. It has been pointed out there that it possesses the structure of a bialgebra where comultiplication and augmentation on the generators x_{ij} are given by

Δ(x_{ij}) = ∑_{k∈I(n,r)} x_{ik} ⊗ x_{kj}, ϵ(x_{ij}) = δ_{ij}.  (2)

In particular, the homogeneous summands A_{R,q}^{s}(n, r) are subcoalgebras. Furthermore, the tensor space V^{⊗r} is a A_{R,q}^{s}(n) (resp. A_{R,q}^{s}(n, r))-(right-)comodule. The structure map τ_{r} : V^{⊗r} → V^{⊗r} ⊗ A_{R,q}^{s}(n, r) is defined by

τ_{r}(v_{j}) = ∑_{i∈I(n,r)} v_{i} ⊗ x_{ij}.

Now, if q^{2} − 1 is an invertible element in R the endomorphism γ is known to be in the algebraic span of β, explicitly one has γ = \frac{q^{2}β^{−1}−β}{q^{2}−1} + 1. Thus, by [O2, Corollary 2.3] the relations γ ≀ x_{ij} = x_{ij} ≀ γ are redundant in this case. The reader may check that under these circumstances our bialgebra A_{R,q}^{s}(n) is identical to the matrix bialgebra of the usual FRT-construction F_{R}(n)/ ⟨βi ≀ X_{ij} − X_{ij} ≀ β, i, j ∈ I(n, 2)⟩ connected with the symplectic group for example denoted F_{β}(M_{n}) in [CP, 7.3 c].

On the other hand, if q^{2}−1 is not invertible we really need to add the relations γ≀x_{ij} = x_{ij}≀γ. For instance, it has been proved in [O2, Corollary 6.2] that setting q = 1 the bialgebra A_{R,q}^{s}(n) is the coordinate ring of the symplectic monoid scheme SpM_{n}(R) which is defined by

SpM_{n}(R) := {A ∈ M_{n}(R)| ∃ d(A) ∈ R, A^{t}JA = AJA^{t} = d(A)J}.

Here, J is the Gram-matrix of the canonical skew bilinear form, that is J = (J_{ij})_{i,j∈\underline{n}} where J_{ij} := ϵ_{i}δ_{ij′}. The regular function d : SpM_{n}(R) → R is called the coefficient of dilation (cf. [Dt]). On the other hand, in this case the bialgebra of the usual FRTconstruction equals A_{R}(n) = R[x_{11}, x_{12}, …, x_{nn}], the commutative polynomial ring in the x_{ij}, which is just the coordinate ring of the monoid scheme M_{n}(R) of n × n-matrices. Consequently the bialgebra of the usual FRT-construction contains (q^{2} − 1)-torsion elements considered over the ground ring R = ℤ [q, q^{−1}] of integer Laurent polynomials in q.

Let us write down a couple of consequent relations holding in A_{R,q}^{s}(n). For this purpose the algebraic span of the V^{⊗r}-endomorphisms

β_{i} := id_{V^{⊗i−1}} ⊗ β ⊗ id_{V^{⊗r−i−1}} and γ_{i} := id_{V^{⊗i−1}} ⊗ γ ⊗ id_{V^{⊗r−i−1}} i = 1, …, r − 1

in End_{R}(V^{⊗r}) will be denoted by A_{r} (for all r > 1). According to [O2, section 1, 5] in A_{R,q}^{s}(n, r) the following relations hold for all r > 1:

μ ≀ x_{ij} = x_{ij} ≀ μ for all μ ∈ A_{r}, i, j ∈ I(n, r).  (3)

The reader should also note that by [O2, Lemma 2.2] all elements of A_{r} must be morphisms of A_{R,q}^{s}(n, r)-comodules.

3 Quantum Symplectic Bideterminants

Let p, r ∈ ℕ be positive integers and Λ(p, r) denote the set of compositions of r into p parts. These are p-tuples λ = (λ_{1}, …, λ_{p}) of non negative integers λ_{i} ∈ ℕ_{0} summing up to r. To each composition λ ∈ Λ(p, r) there corresponds a parabolic subgroup in the symmetric group S_{r}, called the standard Young subgroup. We will denote it by S_{λ}. It is the subgroup fixing the sets {1, 2, …, λ_{1}}, {λ_{1} + 1, λ_{1} + 2, …, λ_{1} + λ_{2}}, . . .. Now, let w ∈ S_{r} be given by a reduced expression w = s_{i_{1}}s_{i_{2}} … s_{i_{t}}, where the s_{i} = (i, i + 1) are the simple transpositions. We define endomorphisms

β(w) := β_{i_{1}}β_{i_{2}} … β_{i_{t}} ∈ E_{r} := End_{R}(V^{⊗r})

for r > 1 and set β(w) = id_{V} ∈ E for r = 1. It is easy to see that this definition is independent of the choice of the reduced expression for w since any two of them can be transformed into each other using the braid relations. But β satisfies the quantum Yang-Baxter equation which is just the second type braid relation

β_{i}β_{i+1}β_{i} = β_{i+1}β_{i}β_{i+1}

in the case i = 1. The latter one obviously implies the relations for i > 1, whereas the first type braid relations β_{i}β_{j} = β_{j}β_{i} for |i − j| > 1 hold trivially. Observe,

β(ww′) = β(w)β(w′) if l(ww′) = l(w) + l(w′).

where l(w) denotes the length of w, that is the number of transpositions in a reduced expression. Setting y := q^{2} and using our notation (1) we associate a quantum symplectic bideterminant to each triple consisting of a composition λ of r and a pair of multi-indices i, j ∈ I(n, r) by

t_{q}^{λ}(i : j) := ∑_{w∈S_{λ}}(−y)^{−l(w)}β(w) ≀ x_{ij} = ∑_{w∈S_{λ}}(−y)^{−l(w)}x_{ij} ≀ β(w).  (4)

The equality therein follows from (3) applied to μ = β(w). Using the abbreviation κ_{λ} := ∑_{w∈S_{λ}}(−y)^{−l(w)}β(w) ∈ E_{r} we also may write t_{q}^{λ}(i : j) = κ_{λ} ≀ x_{ij} = x_{ij} ≀ κ_{λ}. If q is set to 1, we obtain

x_{ij} ≀ β(w) = x_{i(jw^{−1})} and β(w) ≀ x_{ij} = x_{(iw)j}

since then β(w)_{kj} = δ_{kjw^{−1}} = δ_{kw j}. Therefore, in this case our quantum symplectic bideterminants coincide with ordinary bideterminants which are defined as products of minor λ_{i} × λ_{i}-determinants, one factor for each entry λ_{i} of the composition λ. In keeping with familar notation we set

T_{q}^{λ}(i : j) := t_{q}^{λ′}(i : j)

to each partition λ ∈ Λ^{+}(n, p). By a partition we mean a composition λ = (λ_{1}, …, λ_{p}) ordered decreasingly (λ_{1} ≥ λ_{2} ≥ … ≥ λ_{p} ≥ 0) and by λ′ we denote the dual of the partition λ, that is λ′ = (λ′_{1}, …, λ′_{s}) where s = λ_{1} and λ′_{i} := |{j| λ_{j} ≥ i}|. Using this notation one obtains precisely the classical bideterminant T^{λ}(i : j) (as defined in [Ma, 2.4] for instance) when q is set to 1. Observe that the capital T notation is more restricted since not all compositions occur as duals of partitions. This makes it necessary to consider t_{q}^{λ}(i : j) as well for technical reasons.

It should be remarked that the well known quantum determinants corresponding to the general linear groups (see for example [DD, 4.1.2, 4.1.7], [CP, p. 236], [Tk, p. 152], [Ha, p. 157]) can be defined in a similar way using the quantum Yang-Baxter operator of type A instead of our β. In contrast, explicit expressions for quantum symplectic bideterminants become very complicated for r > 2 (apart from the case λ = α_{r} := (r) ∈ Λ^{+}(1, r) in which the bideterminants T_{q}^{α_{r}}(i : j) just are the monomials x_{ij}). Denoting the fundamental weights by ω_{r} := (1, 1, …, 1) ∈ Λ^{+}(r, r) one obtains a single r × r-minor determinant. If r = 2, explicit expressions are for example

| \begin{matrix}x_{ki} & x_{kj} \\ x_{li} & x_{lj} \end{matrix} |_{q} := T_{q}^{ω_{2}}((k, l) : (i, j)) = x_{ki}x_{lj} − q^{−1}x_{kj}x_{li}

if k < l, i < j, i ≠ j′ = n − j + 1 and

| \begin{matrix}x_{ki} & x_{ki′} \\ x_{li} & x_{li′} \end{matrix} |_{q} := T_{q}^{ω_{2}}((k, l) : (i, i′)) = x_{ki}x_{li′} − q^{−2}x_{ki′}x_{li} − (q^{−2} − 1) ∑_{j=1}^{i−1} q^{j−i}x_{kj′}x_{lj},

in the cases k < l, i ≤ m. The calculation of T_{q}^{ω_{3}}((j, k, l) : (i, i′, i)) for j < k < l, i ≤ m is really hard work. Note that such a bideterminant might be different from zero even though it contains two identical columns.

4 Quantum Coefficient of Dilation

In the definition of the symplectic monoid SpM_{n}(R) we have introduced a function called the coefficient of dilation. This is neccessarily a regular function. Now we will define its quantization which will be called the quantum coefficient of dilation. Using notation (1) we see that

−q^{−ρ_{k}−ρ_{l}}ϵ_{k}ϵ_{l} γ ≀ x_{(k,k′)(l,l′)} = q^{−ρ_{l}}ϵ_{l} ∑_{i=1}^{n} q^{ρ_{i}}ϵ_{i}x_{il}x_{i′l′}.

is independent on k, whereas

−q^{−ρ_{k}−ρ_{l}}ϵ_{k}ϵ_{l} x_{(k,k′)(l,l′)} ≀ γ = −q^{−ρ_{k}}ϵ_{k} ∑_{i=1}^{n} q^{ρ_{i}}ϵ_{i}x_{ki}x_{k′i′}.

is independent of l. But, as γ ≀ x_{(k,k′)(l,l′)} = x_{(k,k′)(l,l′)} ≀ γ according to (3), both expressions coincide and consequently are independent of both k and l. Thus, the element

d_{q} := −q^{−ρ_{k}−ρ_{l}}ϵ_{k}ϵ_{l} γ ≀ x_{(k,k′)(l,l′)} = −q^{−ρ_{k}−ρ_{l}}ϵ_{k}ϵ_{l} x_{(k,k′)(l,l′)} ≀ γ  (5)

is well defined in A_{R,q}^{s}(n). In fact it is a grouplike element of this bialgebra. More precisely it is the coefficient function of the one dimensional subcomodule of V ⊗V that is spanned by the tensor

J^{∗} := ∑_{i=1}^{n} ϵ_{i}q^{ρ_{i}}v_{i} ⊗ v_{i′} ∈ V ⊗ V.

To see this, note that J^{∗} = γ(−q^{−ρ_{l}}ϵ_{l}v_{l} ⊗ v_{l′}) for each l and that γ is a morphism of A_{R,q}^{s}(n)-comodules. One calculates

τ_{2}(J^{∗}) = τ_{2} ∘ γ(−q^{−ρ_{l}}ϵ_{l}v_{l} ⊗ v_{l′}) = γ ⊗ id( ∑_{i,k=1}^{n}(v_{i} ⊗ v_{k}) ⊗ (−q^{−ρ_{l}}ϵ_{l} x_{(i,k)(l,l′)})) =

J^{∗} ⊗ q^{−ρ_{l}}ϵ_{l} ∑_{k=1}^{n} q^{ρ_{k}}ϵ_{k} x_{(k,k′)(l,l′)}) = J^{∗} ⊗ d_{q}.

In order to obtain a formula which connects d_{q} with quantum symplectic 2×2-determinants one easily verifies the following equation using induction on l:

∑_{i=1}^{l} q^{−i}x_{ki}x_{li′} ≀ β = ∑_{i=1}^{l} q^{i−2l}x_{ki′}x_{li}.  (6)

For l = m we deduce the connection formula

∑_{i=1}^{m} q^{−i}T_{q}^{ω_{2}}((k, l) : (i, i′)) = \{ \begin{matrix}q^{−k}d_{q} & k = l′ ≤ m, \\ 0 & k ≠ l′. \end{matrix}  (7)

5 The Symplectic q-Schur algebra

Remember that A_{R,q}^{s}(n, r) is a coalgebra for each r. Therefore, its dual R-module inherits the structure of an R-algebra. We define

S_{R,q}^{s}(n, r) := Hom_{R}(A_{R,q}^{s}(n, r), R)

and call it the symplectic q-Schur algebra. Two linear forms μ, ν ∈ S_{R,q}^{s}(n, r) are multiplied by convolution, that is

μν(a) := (μ ⊗ ν) ∘ Δ(a)

for all a ∈ A_{R,q}^{s}(n, r). The reader may verify that one obtains the symplectic Schur algebra in the classical situation as defined in [O2]. This also is identical to the symplectic Schur algebra in the sense of S. Donkin, respectively S. Doty ([Do2] respectively [Dt]). One aim is to show that the construction is stable under base changes and that it is a free R-module. Both facts follow when we have shown that A_{R,q}^{s}(n, r) is free as an R-module. Further we want to initiate the study of the representation theory of this algebra. The easiest way to do this is to check that the axioms of a cellular algebra given by J. Graham and G. Lehrer in [GL] hold. These axioms are as follows:

Let A be an associative unital algebra over a commutative unital ring R together with a partially ordered finite set Λ and finite sets M(λ) to each λ ∈ Λ (the set of “λ-tableaux”). A is called a cellular algebra if the following properties hold:

(C1) A possesses an R-basis {C_{S,T}^{λ} | λ ∈ Λ, S, T ∈ M(λ)}.

(C2) A posesses an R-linear involution ^{∗} which is an algebra anti-automorphism such that C_{S,T}^{λ}^{∗} = C_{T,S}^{λ} holds for all λ ∈ Λ and S, T ∈ M(λ).

(C3) For all a ∈ A, λ ∈ Λ and S, T ∈ M(λ) the congruence relation

aC_{S,T}^{λ} ≡ ∑_{S′∈M(λ)} r_{a}(S′, S)C_{S′,T}^{λ} mod A(< λ),

holds, where the elements r_{a}(S′, S) ∈ R are independent of T and A(< λ) is defined as the R-linear span of basis elements C_{U,V}^{μ} where μ < λ and U, V ∈ M(μ).

Starting with these axioms the representation theory of A is developed in [GL] along the following lines. To each λ ∈ Λ a standard module W (λ) is defined on a free R-basis {C_{S}^{λ}| S ∈ M(λ)}. An element a ∈ A acts on it via aC_{S}^{λ} = ∑_{S′∈M(λ)} r_{a}(S′, S)C_{S′}^{λ}. Each W (λ) possesses a symmetric bilinear form ϕ_{λ} for which the formula ϕ_{λ}(a^{∗}x, y) = ϕ_{λ}(x, ay) is valid for all a ∈ A and x, y ∈ W (λ). In the case where R is a field and ϕ_{λ} ≠ 0, the radical of W (λ) is the same as the radical of the bilinear form ϕ_{λ}. The simple head L_{λ} of W (λ) then is absolutely irreducible. In this way a complete set of pairwise non-isomorphic simple A-modules {L_{λ}| λ ∈ Λ_{0}} can be obtained. Here we have set Λ_{0} := {λ ∈ Λ| ϕ_{λ} ≠ 0}.

Denoting the multiplicity of L_{μ} in W (λ) by d_{λμ} to each λ ∈ Λ and μ ∈ Λ_{0} Graham and Lehrer show that d_{λμ} = 0 for λ ≤ μ and d_{λλ} = 1. To each order refining the given partial order on Λ the corresponding decomposition matrix D = (d_{λμ})_{λ∈Λ,μ∈Λ_{0}} is unitriangular. The Cartan-matrix C can be calculated as C = D^{t}D. The theory also supplies a criterion to decide whether A is semisimple or quasi-hereditary. In the first case we must have rad(ϕ_{λ}) = (0) for all λ ∈ Λ whereas in the second case Λ_{0} = Λ will do.

Examples of cellular algebras are the Brauer centralizer algebras B_{R,x,r}, Ariki-Koike-Hecke-algebras, Temperley-Lieb and Jones algebras ([GL]). R.M. Green ([GR]) constructs a q-analogue of the codeterminant basis (in the sense of [Gr]) for the classical Schur algebra S_{R}(n, r) which is cellular as well. The corresponding standard modules W (λ) are precisely the q-Weyl modules in the sense of [DJ2] (see [GR], Proposition 5.3.6).

It should be remarked that the finiteness of Λ is not postulated in the original definition. Since this property is valid in our example we impose this restriction to avoid unnecessary trouble (cf. discussion in [KX], section 3).

Since we have defined the symplectic q-Schur algebra as the dual module of a coalgebra we now translate the concept of cellular algebras to coalgebras:

Let K be a coalgebra over a commutative unital ring R, together with a partially ordered finite set Λ and finite sets M(λ) to each λ ∈ Λ. We call K a cellular coalgebra if the following properties hold:

(C1*) K possesses an R-basis {D_{S,T}^{λ} | λ ∈ Λ, S, T ∈ M(λ)}.

(C2*) K possesses an R-linear involution ^{∗} which is an coalgebra anti-automorphism, such that D_{S,T}^{λ}^{∗} = D_{T,S}^{λ} holds for all λ ∈ Λ and S, T ∈ M(λ).

(C3*) For all λ ∈ Λ and S, T ∈ M(λ) the congruence relation

Δ(D_{S,T}^{λ}) ≡ ∑_{S′∈M(λ)} h(S′, S) ⊗ D_{S′,T}^{λ} mod K ⊗ K(> λ)

holds, where the coalgebra elements h(S′, S) ∈ K are independent of T and K(> λ) is defined as the R-linear span of basis elements D_{U,V}^{μ} where μ > λ and U, V ∈ M(μ).

To an arbitrary R-coalgebra the dual algebra is well defined. The dual coalgebra of an algebra is well defined if the algebra is known to be projective as an R-module. In the case of a cellular algebra this is obviously valid. The connection between the above two concepts is given by the following proposition which can be proved straightforwardly using structure constants with respect to the bases (cf. [O1, 4.2.3]).

Proposition 5.1 The dual algebra of a cellular coalgebra is a cellular algebra. The dual coalgebra of a cellular algebra is a cellular coalgebra. In both cases the corresponding bases and involution maps can be constructed dual to each other, i.e. in the former case C_{S,T}^{λ} (D_{U,V}^{μ}) is 1 if λ = μ, S = U and T = V but 0 otherwise and C_{S,T}^{λ} (D_{U,V}^{μ} ^{∗}) = C_{S,T}^{λ}^{∗}(D_{U,V}^{μ}).

According to the proposition our next task is to find a cellular basis for the coalgebra A_{R,q}^{s}(n, r) together with an appropriate involution map such that the axioms of the cellular coalgebra hold. As soon as this is done the representation theory of S_{R,q}^{s}(n, r) is developed to the extent indicated above.

6 Results

We will define a basis for A_{R,q}^{s}(n, r) consisting of quantum symplectic bideterminants and powers of the quantum symplectic coefficient of dilation. Since they are too large in number we have to single out an appropriate subset. This can be done using so called λ-tableaux which will be defined now: To each partition one associates a Young-diagram reading row lengths out of the components λ_{i}. For example is associated to λ = (3, 2, 2, 1) ∈ Λ^{+}(4, 8). An λ-tableau T_{i}^{λ} is constructed from the diagram of λ by inserting the components of a multi-index i ∈ I(n, r) column by column into the boxes. In the above example:

T_{i}^{λ} := i_{1} i_{5} i_{8} ; i_{2} i_{6} ; i_{3} i_{7} ; i_{4} .

If λ is fixed we will sometimes identify multi-indices with their tableaux. We put a new order ≺ on the set \underline{n}, namely

m ≺ m′ ≺ (m − 1) ≺ (m − 1)′ ≺ … ≺ 1 ≺ 1′.

The reason, why we prefer ≺ instead of the order ≪ considered in [O2] will become clear later on. Now, a multi-index i is called λ-column standard if the entries in T_{i}^{λ} are strictly increasing down columns according to this order. It is called λ-row standard if the entries are weakly increasing along rows and λ-standard if it is both at the same time. We write I_{λ} to denote the subset of I(n, r) consisting of all multi-indices being λ-standard. Such a multi-index i ∈ I_{λ} is called λ-reverse symplectic standard if for each index i ∈ \underline{m} the occurences of i as well as i′ in T_{i}^{λ} is limited to the first m − i + 1 rows. The corresponding subset of I_{λ} will be denoted by I_{λ}^{mys}. The reader should observe that even though this set is different from the one of λ-symplectic standard tableaux (as defined in [Ki] and denoted I_{λ}^{sym} in [O2]) it has the same number of elements. For, let σ ∈ S_{n} be the permutation transforming the order ≪ into ≺, that is σ(i) := (m − i + 1)′ for i ≤ m and σ(i) := m − i′ + 1 for i > m. Then there is an induced bijection on I(n, r) sending (i_{1}, …, i_{r}) to (σ(i_{1}), …, σ(i_{r})) and which carries the set of λ-symplectic standard tableaux precisely to the set of λ-reverse symplectic standard tableaux.

Let us first describe the set Λ occuring in the definition of the cellular coalgebra:

Λ := \{\underline{λ} := (λ, l)| 0 ≤ l ≤ \frac{r}{2}, λ ∈ Λ^{+}(m, r − 2l)\}.

According to the definition of a cellular coalgebra to each \underline{λ} = (λ, l) ∈ Λ a set M(\underline{λ}) must be assigned. We take:

M(\underline{λ}) := I_{λ}^{mys}.

Finally the basis elements themselves are defined by

D\underline{λ} ; i,j := d_{q}^{l}T_{q}^{λ}(i : j).

Now, our principal aim is to prove the following

Theorem 6.1 The R-module A_{R,q}^{s}(n, r) has a basis given by

B_{r} := \{D\underline{λ} ; i,j| \underline{λ} ∈ Λ, i, j ∈ M(\underline{λ})\}.

Furthermore, the unique R-linear map ^{∗} with D\underline{λ} ; i,j^{∗} = D\underline{λ} ; j,i is an involutory anti-coalgebra-automorphism and the axioms of a cellular coalgebra are satisfied.

By Proposition 5.1 we may conclude immediately:

Theorem 6.2 The symplectic q-Schur algebra S_{R,q}^{s}(n, r) is a cellular algebra with the basis dual to B_{r} as a cellular basis.

Another direct consequence of 6.1 can be obtained from [O2, Theorems 3.3 and 4.2] together with the observation that the algebraic span A_{r} of the endomorphism β_{i} and γ_{i} equals the image of the Birman-Murakami-Wenzl algebra under its representation on V^{⊗r} that is Ψ(C_{r}(−q^{1−n}, q)) in the notation of [We, Lemma 5.1].

Theorem 6.3 The symplectic q-Schur algebra is stable under base change and it is identical with the centralizer of the Birman-Murakami-Wenzl algebra acting on V^{⊗r}.

At the end of this paper we will improve 6.2 by showing that the bilinear form ϕ_{λ} on the standard modules W_{λ} is nonzero for each λ. By [GL, 3.10] this means

Theorem 6.4 The symplectic q-Schur algebra S_{R,q}^{s}(n, r) is integrally quasi-hereditary.

Let us first see how the involution of Theorem 6.1 arises. It realizes matrix transposition for our quantum monoid. On the generators x_{ij} this transposition map is defined as in the classical case by x_{ij}^{∗} := x_{ji}. Indeed, this gives a well defined algebra map on A_{R,q}^{s}(n), since the coefficient matrices of β and γ are symmetric implying β ≀ x_{ij}^{∗} = x_{ji} ≀ β and γ ≀ x_{ij}^{∗} = x_{ji} ≀ γ and thus, keeping the relations of that algebra fixed. Furthermore, the endomorphisms κ_{λ} ∈ End_{R}(V^{⊗r}) must have symmetric coefficient matrices, as well. We calculate

t_{q}^{λ}(i : j)^{∗} = κ_{λ} ≀ x_{ij}^{∗} = x_{ji} ≀ κ_{λ} = t_{q}^{λ}(j : i)

and in a similar way d_{q}^{∗} = d_{q}. This shows, that ^{∗} factors to an algebra map of A_{R,q}^{s}(n). From the comultiplication rule (2) it directly follows that ^{∗} is an anti-coalgebra map. This implies axiom (C2*) of a cellular coalgebra.

The verification of axiom (C3*) is the second easiest step in the proof of Theorem 6.1, but we will give it at the end of the paper since some additional ingredients are needed. The first statement of this theorem, which is just axiom (C1*), is the really hard one. It is the q-analogue of [O2, Theorem 6.1]. To prove it we will proceed in a similar way as there. The difficulty is to show that B_{r} is a set of generators. For that porpose the most important step is a quantum symplectic version of the famous straightening formula.

7 The Quantum Symplectic Straightening Formula

In the classical case symplectic versions of the straightening formula have already been given in [Co, 2.4] and [O2, section 7]. In principle, we will follow the lines of the latter paper. But there are a lot of additional difficulties, one of which forces us to work with a reversed version of λ-symplectic standard tableaux. Preparing the statement we define the algebra

A_{R,q}^{sh}(n) := A_{R,q}^{s}(n)/ ⟨d_{q}⟩

by factoring out the ideal generated by the quantum coefficient of dilation. Since d_{q} is homogeneous this algebra is graded again. Let us abbreviate its r-th homogeneous summand by K := A_{R,q}^{sh}(n, r). Since d_{q} is grouplike the comultiplication Δ obviously factors to A_{R,q}^{sh}(n) and A_{R,q}^{sh}(n, r). But A_{R,q}^{sh}(n) is not a bialgebra and A_{R,q}^{sh}(n, r) are not coalgebras, because the augmentation map ϵ does not factor. In the classical case if R = K is a field A_{K,q}^{sh}(n) equals the coordinate ring of the symplectic semigroup SpH_{n}(K) := SpM_{n}(K)\GSp_{n}(K) by [O2, remark 7.5]. The missing augmentation map corresponds to the missing unit element in the semigroup.

We put an order on the set Λ^{+}(r) of all partitions of r, writing λ < μ if the dual λ′ occurs before the dual μ′ in the lexicographic order. In this order the fundamental weight ω_{r} := (1, 1, …, 1) ∈ Λ^{+}(r, r) is the largest element, whereas α_{r} := (r) ∈ Λ^{+}(1, r) is the smallest one. We define K(> λ) (resp. K(≥ λ)) to be the R-linear span in K of all bideterminants T_{q}^{μ}(i : j) such that μ > λ (resp. μ ≥ λ). For λ = ω_{r} we set K(> ω_{r}) := 0. Clearly K = K(≥ α_{r}).

Proposition 7.1 (Quantum Symplectic Straightening Formula) Let λ ∈ Λ^{+}(r) be a partition of r and j ∈ I(n, r). Then, to each k ∈ I_{λ}^{mys} there is an element a_{jk} ∈ R, such that in K we have for all i ∈ I(n, r):

T_{q}^{λ}(i : j) ≡ ∑_{k∈I_{λ}^{mys}} a_{jk}T_{q}^{λ}(i : k) mod K(> λ).

Before starting to prove this, let us deduce the fact that B_{r} generates A_{R,q}^{s}(n, r). Two additional ingredients are needed. First, we use the fact that d_{q} is a central element in A_{R,q}^{s}(n), which can be deduced from the identity β_{1}β_{2}γ_{1} = γ_{2}β_{1}β_{2} holding in A_{3} (for details we refer the reader to [Ha, Corollary 6.3] or [O1, Satz 3.4.1]). From this fact we see that multiplication by d_{q} from the left leads to an exact sequence

A_{R,q}^{s}(n, r − 2) →^{·d_{q}} A_{R,q}^{s}(n, r) → A_{R,q}^{sh}(n, r) → 0.  (8)

for r > 1. Therefore, using induction on r we can reduce to showing that

{T_{q}^{λ}(i : j)| λ ∈ Λ^{+}(m, r), i, j ∈ I_{λ}^{mys}}

is a set of generators for K = A_{R,q}^{sh}(n, r). For this claim it is enough to show that

B_{λ} := {T_{q}^{λ}(i : j)| i, j ∈ I_{λ}^{mys}}

is a set of generators of K(≥ λ)/K(> λ) for each partition λ. To get the last claim from the straightening formula 7.1, observe that the involution^{∗} is well defined on A_{R,q}^{sh}(n) since d_{q}^{∗} = d_{q}. Applying ^{∗} to the congruence relation of Proposition 7.1, one obtains another such formula in which the roles of i and j are exchanged. This shows that B_{λ} is indeed a set of generators for K(> λ)/K(≥ λ). Thus the fact that B_{r} is a set of generators for A_{R,q}^{s}(n, r) is reduced to the validation of Proposition 7.1.

In order to prove the quantum symplectic straightening formula we need a corresponding algorithm. Its classical counterpart is [O2, Proposition 7.3]. The reader familiar with that proposition will remember the importance of a certain order on the set I(n, r). A similar, but in some sense reversed version of that order will be needed now. This circumstance forces us to consider reverse symplectic standard tableaux instead of symplectic ones. To introduce that order we define a map f : I(n, r) → ℕ _{0}^{m} by f(i) = (a_{1}, …, a_{m}), where

a_{l} := |\{j ∈ \underline{r}| i_{j} = l or i_{j} = l′\}|,

and order ℕ _{0}^{m} writing (a_{1}, …, a_{m}) < (b_{1}, …, b_{m}) if and only if (b_{1}, b_{2}, …, b_{m}) appears before (a_{1}, a_{2}, …, a_{m}) in the lexicographic order (induced by the ordinary order on ℕ). This is the step where we change directions relative to [O2, 7.3]. Next, we obtain an order ⊲ on ℕ _{0}^{m} × I(n, r) in a lexicographic way as well:

(a, i) ⊲ (b, j) :⟺ a < b or (a = b and i ≺ j).

Here, we have denoted by ≺ the lexicographic order on I(n, r) induced by our special order ≺ on \underline{n}. Finally, we obtain a second, new order ⊲ on I(n, r) via the embedding I(n, r) ↪ ℕ _{0}^{m} × I(n, r) given by i ↦ (f(i), i). Now we are able to state the symplectic straightening algorithm.

Proposition 7.2 (Strong Quantum Symplectic Straightening Algorithm) Let λ ∈ Λ^{+}(r) be a partition of r and j ∈ I(n, r)\I_{λ}^{mys}. Then to each k ∈ I(n, r) satisfying k ⊲ j there is an element a_{jk} ∈ R such that in K the following congruence relation holds for all i ∈ I(n, r):

T_{q}^{λ}(i : j) ≡ ∑_{k⊲j}a_{jk}T_{q}^{λ}(i : k) mod K(> λ).

In principle we will proceed in a similar way as in [O2] to prove this algorithm, but, there is a remarkable difference concerning the role of the classical straightening algorithm (Proposition 7.2 in that paper). One would expect that this role has to be taken over by a q-analogue related to quantum general linear groups as can be found in [HH, section 9]. But this is not true because the embedding of the symplectic group into the general linear group does not extend to quantum groups. Instead of [O2, Proposition 7.2] we have to establish a weak form of the quantum symplectic straightening algorithm in a first step. More precisely we will first prove 7.2, where I_{λ}^{mys} is substituted by I_{λ}.

Clearly, the straightening formula 7.1 is an easy consequence of the above proposition since the set I(n, r) is finite and therefore the elimination of multi-indices j, that are not λ-reverse symplectic standard in an expression T^{λ}(i : j) must terminate.

The proof of the straightening algorithm will take several sections. We start with some technical tools needed for the first step, the above mentioned weak form.

8 Arithmetic of Bideterminants

The calculus of bideterminants is needed inside K = A_{R,q}^{sh}(n, r). Unless otherwise stated the rules hold in A_{R,q}^{s}(n, r), too. Recall the definition of κ_{λ} from section 3 following (4).

Lemma 8.1 Let λ ∈ Λ(p, r) be a composition and y = q^{2}. Then to each i < r such that the simple transposition s_{i} = (i, i + 1) is contained in the standard Young-subgroup S_{λ}, there are endomorphisms μ_{λ,i}, μ′_{λ,i} ∈ E_{r} satisfying

κ_{λ} = (id_{V^{⊗r}} − y^{−1}β_{i})μ_{λ,i} = μ′_{λ,i}(id_{V^{⊗r}} − y^{−1}β_{i}).

Proof: Let us first reduce to the case λ = α_{r} = (r). Setting κ_{r} := κ_{α_{r}}, k_{s} := λ_{1} + … + λ_{s−1} and

κ_{λ}^{s} := id_{V^{⊗ks}} ⊗ κ_{λ_{s}} ⊗ id_{V^{⊗r−λs−ks}},

we can extend the definition to arbitrary λ by using the formula κ_{λ} = κ_{λ}^{1}κ_{λ}^{2} … κ_{λ}^{p} in which the factors commute. Now, using standard reduced expressions for permutations w ∈ S_{r} one easily verifies the following recursion rules for r > 1:

κ_{r} = κ_{r−1}(id_{V^{⊗r}} + ∑_{l=1}^{r−1}(−y)^{l−r}β_{r−1}β_{r−2} … β_{l}) =

(id_{V^{⊗r}} + ∑_{l=1}^{r−1}(−y)^{l−r}β_{l}β_{l+1} … β_{r−1})κ_{r−1}.

We proceed by induction on r, the case r = 2 being clear. The case i < r − 1 can be handled immediately with the help of the above recursion formula. If i = r − 1 we calculate

κ_{r} = κ_{r−1}(id_{V^{⊗r}} − y^{−1}β_{r−1}) + μ′_{r−1,r−2}(id_{V^{⊗r}} − y^{−1}β_{r−2}) ∑_{l=1}^{r−2}(−y)^{l−r}β_{r−1}β_{r−2} … β_{l}.

But by the braid relations we get

(id_{V^{⊗r}} − y^{−1}β_{r−2})(−y)^{l−r}β_{r−1}β_{r−2} … β_{l} = (−y)^{l−r}β_{r−1}β_{r−2} … β_{l}(id_{V^{⊗r}} − y^{−1}β_{r−1}),

yielding the right hand side factorization of κ_{r}. The other formula is obtained similarly. □

Corollary 8.2 Let j ∈ I(n, r) be a multi-index possessing two identical neighbouring indices j_{l} = j_{l+1} and λ ∈ Λ(p, r) such that the transposition s_{l} is contained in S_{λ}. Then t_{q}^{λ}(i : j) = t_{q}^{λ}(j : i) = 0 holds for all i ∈ I(n, r).

Proof: By assumption, v_{j} lies in the kernel of (id_{V^{⊗r}} − y^{−1}β_{l}). Consequently the assertion concerning t_{q}^{λ}(i : j) follows immediately from Lemma 8.1 by the definition of bideterminants. Using the matrix transposition map ^{∗} introduced in section 7, the formula for exchanged multi-indices follows as well. □

Next, we will demonstrate how the transition from A_{R,q}^{s}(n) to its epimorphic image A_{R,q}^{sh}(n) corresponds to the transition from the Birman-Murakami-Wenzl algebra to the Iwahori-Hecke algebra of type A via the epimorphism sending the generator e_{1} to zero. For this purpose denote by G_{r} the ideal generated by G := γ_{1} = γ ⊗ id_{V^{⊗r−2}} in the image A_{r} of the Birman-Murakami-Wenzl algebra in E_{r}. Note that it contains the other endomorphisms γ_{i} for i = 2, …, r − 1, as well (use the relations β_{i}β_{i+1}γ_{i} = γ_{i+1}β_{i}β_{i+1}). The quotient A_{r}/G_{r} is an epimorphic image of the Iwahori-Hecke algebra sending the generator T_{s_{l}} to β_{l} + G_{r} (notation as in [DD]).

Lemma 8.3 Let A, B ∈ A_{r} be endomorphisms of V^{⊗r} such that A ≡ B modulo G_{r}. Then, the equation x_{ij} ≀ A = x_{ij} ≀ B (to be read in the sense of (1) ) holds in K.

Proof: We have to show that x_{ij} ≀ A = 0 for all A ∈ G_{r}. Let F, H ∈ A_{r} be such that A = F GH. From the defining equation of the quantum coefficient of dilation d_{q} from section 4 we have

x_{ij} ≀ G = \{ \begin{matrix}0 & j′_{1} ≠ j_{2} or i′_{1} ≠ i_{2} \\ −q^{ρ_{i_{1}}+ρ_{j_{1}}}ϵ_{i_{1}}ϵ_{j_{1}} d_{q} x_{i_{3}j_{3}} … x_{i_{r}j_{r}} & j′_{1} = j_{2} and i′_{1} = i_{2} \end{matrix} .

This means x_{ij} ≀ G = 0 in K for all i, j ∈ I(n, r). By (3) we have x_{ij} ≀ F = F ≀ x_{ij} and therefore,

x_{ij} ≀ F GH = ∑_{k,l,s∈I(n,r)} x_{ik}f_{kl}g_{ls}h_{sj} = ∑_{k,l,s∈I(n,r)} f_{ik}x_{kl}g_{ls}h_{sj} = ∑_{k,s∈I(n,r)} f_{ik}x_{ks} ≀ Gh_{sj} = 0.

where (f_{ij})_{i,j∈I(n,r)}, (g_{ij})_{i,j∈I(n,r)} and (h_{ij})_{i,j∈I(n,r)} are the coefficient matrices of F, G and H. □

We introduce another notation similar to the one in (1). Let μ ∈ E_{r} be an endomorphism of V^{⊗r}. Set

t_{q}^{λ}(i : j) ≀ μ = ∑_{k∈I(n,r)} t_{q}^{λ}(i : k)μ_{kj} and μ ≀ t_{q}^{λ}(i : j) = ∑_{k∈I(n,r)} μ_{ik}t_{q}^{λ}(k : j).

Similar expressions are used with respect to the capital T notation for bideterminants.

Lemma 8.4 For all i, j ∈ I(n, r) and w ∈ S_{λ} the following equations hold in K.

β(w) ≀ t_{q}^{λ}(i : j) = β(w)^{−1} ≀ t_{q}^{λ}(i : j) = (−1)^{l(w)}t_{q}^{λ}(i : j) = t_{q}^{λ}(i : j) ≀ β(w)^{−1} = t_{q}^{λ}(i : j) ≀ β(w)

Proof: Modulo G_{r} we have

β(w)κ_{λ} = β(w)^{−1}κ_{λ} = (−1)^{l(w)}κ_{λ} = κ_{λ}β(w)^{−1} = κ_{λ}β(w)

since the corresponding equations (where β(w) is replaced by T_{w}) hold in the Iwahori-Hecke algebra of type A. Thus the assertion follows from Lemma 8.3. □

Let J denote the ideal in the tensor algebra T (V) = ⨁_{r∈ℕ_{0}} V^{⊗r} generated by the twofold invariant tensor J^{∗} = ∑_{i=1}^{n} ϵ_{i}q^{ρ_{i}}v_{i} ⊗ v_{i′} ∈ V ⊗ V and let J_{r} := J ∩ V^{⊗r} be its r-th homogeneous summand.

Lemma 8.5 Let a_{j} ∈ R be such that ∑_{j∈I(n,r)} a_{j}v_{j} ∈ J_{r}. Then, for all i ∈ I(n, r) and compositions λ of r we have

∑_{j∈I(n,r)} a_{j}x_{ij} = 0, and ∑_{j∈I(n,r)} a_{j}t_{q}^{λ}(i : j) = 0.

Proof: First, note that the second equation follows from the first one by definition of bideterminants. Again, we use the equation J^{∗} = γ(−q^{−ρ_{k}}ϵ_{k}v_{k}v_{k′}) holding for each k ∈ \underline{n}. It implies that J_{r} is contained in the R-linear span of all elements of the form γ_{l}(v_{k}) for k ∈ I(n, r) and 1 ≤ l < r. On the other hand, it is easily seen that allways γ_{l}(v_{k}) ∈ J_{r}. Thus, we have reduced to the case ∑_{j∈I(n,r)} a_{j}v_{j} = γ_{l}(v_{k}). This means

∑_{j∈I(n,r)} a_{j}x_{ij} = x_{ik} ≀ γ_{l} = 0.

Since γ_{l} is contained in G_{r} the proof is finished by Lemma 8.3. □

Next, we give a quantum symplectic version of Laplace duality. The corresponding classical result can be found in [Ma, 2.5.1], for instance.

Proposition 8.6 (Laplace Duality) Let λ, μ ∈ Λ(p, r) be compositions, Y a set of left coset representatives of S_{λ} ∩ S_{μ} in S_{λ} and X a set of right coset representatives of S_{λ} ∩ S_{μ} in S_{μ}, such that l(vw) = l(v) + l(w) and l(wu) = l(u) + l(w) holds for all v ∈ Y, u ∈ X and w ∈ S_{λ} ∩ S_{μ}. Then for all i, j ∈ I(n, r) the following equation holds:

∑_{u∈X}(−y)^{−l(u)}t_{q}^{λ}(i : j) ≀ β(u) = ∑_{v∈Y}(−y)^{−l(v)}β(v) ≀ t_{q}^{μ}(i : j).

Proof: Using the disjoint union

Z := S_{λ}S_{μ} = ⋃_{u∈X} S_{λ}u = ⋃_{v∈Y} vS_{μ},

we calculate

∑_{u∈X}(−y)^{−l(u)}t_{q}^{λ}(i : j) ≀ β(u) = ∑_{u∈X} ∑_{w∈S_{λ}}(−y)^{−l(u)−l(w)}x_{ij} ≀ β(w)β(u) =

∑_{z∈Z}(−y)^{−l(z)}x_{ij} ≀ β(z) = ∑_{z∈Z}(−y)^{−l(z)}β(z) ≀ x_{ij} =

∑_{v∈Y} ∑_{w∈S_{μ}}(−y)^{−l(v)−l(w)}β(v)β(w) ≀ x_{ij} = ∑_{v∈Y}(−y)^{−l(v)}β(v) ≀ t_{q}^{μ}(i : j).

□

The next result is needed for the transition from t-bideterminants of compositions to T bideterminants of partitions.

Lemma 8.7 Let λ ∈ Λ(p, r) be a composition and i, j ∈ I(n, r). Then the bideterminant t_{q}^{λ}(i : j) can be written as a linear combination of bideterminants T_{q}^{λ′}(k : l).

Proof: First, there is a permutation π ∈ S_{p}, such that λ̄ = (λ_{π(1)}, …, λ_{π(p)}) ∈ Λ^{+}(p, r) is a partition. This λ̄ is uniquely determined by λ (but π only under the restriction to be of minimal length). Clearly the parabolic subgroups S_{λ} and Sλ̄ in S_{r} are conjugate to each other. Thus, there is an element v ∈ S_{r} such that vS_{λ} = Sλ̄v. Furthermore, it is known from the theory of parabolic subgroups that in the left coset vS_{λ} and in the right coset Sλ̄v there are unique representatives w resp. w̄ of minimal length. In fact we have w = w̄, but this is not important for the proof. The facts we really need are l(wu) = l(w) + l(u) for all u ∈ S_{λ} and l(uw̄) = l(u) + l(w̄) for all u ∈ Sλ̄. For, this implies β(w)κ_{λ} = κλ̄β(w̄). By the definition of bideterminants and the relations (3) holding inside A_{R,q}^{s}(n, r) we obtain

β(w) ≀ t_{q}^{λ}(i : j) = tλ̄ ; q(i : j) ≀ β(w̄) = T λ̄′ ; q(i : j) ≀ β(w̄).

Since λ̄′ = λ′ this results in

t_{q}^{λ}(i : j) = β(w)^{−1} ≀ T_{q}^{λ′}(i : j)β(w̄) = ∑_{k,l∈I(n,r)} β(w)_{ik}^{−1}T^{λ′}(k : l)β(w̄)_{lj}.

□

Next, we introduce a calculus for our bideterminants bringing our special order ⊲ on I(n, r) into the picture. First, some new notation has to be explained. The sum of two multi-indices i ∈ I(n, r) and j ∈ I(n, s) is defined by juxtaposition, that is

i + j := (i_{1}, …, i_{r}, j_{1}, …, j_{s}) ∈ I(n, r + s).

Note that the map f : I(n, r) → ℕ _{0}^{m} occuring in the straightening algorithm is additive in the sense f(i + j) = f(i) + f(j). This implies

f(i + j) < f(i + k) resp. f(j + i) < f(k + i) if f(j) < f(k)  (9)

with respect to the lexicographic order < on ℕ _{0}^{m}. To a multi-index i ∈ I(n, r) we consider the following R-spans in V^{⊗r}

W_{i} := ⟨v_{j}| j ∈ I(n, r), f(j) < f(i)⟩ and \overline{W}_{i} := ⟨v_{j}| j ∈ I(n, r), f(j) ≤ f(i)⟩ .

Furthermore, we set

h_{ij} := \{ \begin{matrix}q^{−1} & j ≠ i, i′ \\ 1 & j = i, j = i′ > m \\ q^{−2} & j = i′ ≤ m \end{matrix} .

and denote the simple transpositions by s_{l} = (l, l + 1), as before. The following lemma is the key concerning calculations with bideterminants. Again we set y = q^{2}.

Lemma 8.8 For all i ∈ I(n, r) and l ∈ \underline{r} the following formulas hold in V^{⊗r} modulo the R-module W_{i}:

β_{l}(v_{i}) ≡ \{ \begin{matrix}yh_{i_{l+1}i_{l}}v_{is_{l}} + (y − 1)(id_{V^{⊗r}} − γ_{l})(v_{i}) & i_{l} > i_{l+1} \\ yh_{i_{l+1}i_{l}}v_{is_{l}} & i_{l} ≤ i_{l+1} \end{matrix}

β_{l}^{−1}(v_{i}) ≡ \{ \begin{matrix}h_{i_{l+1}i_{l}}v_{is_{l}} + (y^{−1} − 1)(id_{V^{⊗r}} − γ_{l})(v_{i}) & i_{l} ≤ i_{l+1} \\ h_{i_{l+1}i_{l}}v_{is_{l}} & i_{l} > i_{l+1} \end{matrix} .

Proof: The congruence relation for β_{l}^{−1} follows from the one for β_{l} because yβ^{−1} = β + (y − 1)(γ − id_{V^{⊗2}}). Therefore, it is enough to prove the first assertion.

First, consider the case i_{l} > i_{l+1}. If i_{l} ≠ i′_{l+1}, the asserted congruence relation is also an equation, as can be seen directly from the definition of β. Turning to the case i_{l} = i′_{l+1} =: j ≤ m, we split i into three summands

i^{1} = (i_{1}, …, i_{l−1}), i^{2} = (j′, j), i^{3} = (i_{l+1}, …, i_{r}).

To k ∈ \underline{n} we set i(k) := i^{1} + (k, k′) + i^{3} and calculate

β_{l}(v_{i}) = v_{is_{l}} + (y − 1)v_{i} − (y − 1) ∑_{k>j} q^{ρ_{k}−ρ_{j}}ϵ_{k}v_{i(k)}.

Since (y − 1) ∑_{k=1}^{n} q^{ρ_{k}−ρ_{j}}ϵ_{k}ϵ_{j}v_{i(k)} = (y − 1)γ_{l}(v_{i}) we obtain the equation

β_{l}(v_{i}) = v_{is_{l}} + (y − 1)(id_{V^{⊗r}} − γ_{l})(v_{i}) + (y − 1) ∑_{k≤j} q^{ρ_{k}−ρ_{j}}v_{i(k)}.

But i(j) = is_{l} and

f(i(k)) = f(i^{1}) + f((k, k′)) + f(i^{3}) < f(i^{1}) + f((j′, j)) + f(i^{3}) = f(i(j′)) = f(i)

for all k < j by (9), yielding the asserted congruence modulo W_{i}. If i_{l} < i_{l+1} the interesting case is i′_{l+1} = i_{l} =: j ≤ m. Here the assertion immediately follows from the calculation

β_{l}(v_{i}) = v_{is_{l}} − (y − 1) ∑_{k>j′} q^{ρ_{k}+ρ_{j}}v_{i(k)},

because f(i(k)) < f(i) for all k > j′. □

Remark 8.9 Since γ_{l}(V^{⊗r}) ⊆ J_{r} for 1 ≤ l < r the lemma implies that \overline{W}_{i} + J_{r} is invariant under A_{r}. But, W_{i} = ∑_{f(j)<f(i)} \overline{W}_{j} and thus, W_{i} + J_{r} must be invariant, as well.

Corollary 8.10 Let j ∈ I(n, r) and l ∈ \underline{r}. Then to each k ∈ I(n, r) satisfying f(k) < f(j) there is a_{jk}(s_{l}) in R (not neccessarily nonzero) such that the following equations hold in K for all i ∈ I(n, r):

T_{q}^{λ}(i : j) ≀ β_{l}^{−1} = h_{j_{l+1}j_{l}}T_{q}^{λ}(i : js_{l}) + ∑_{f(k)<f(j)} a_{jk}(s_{l})T_{q}^{λ}(i : k) if j_{l} > j_{l+1},

T_{q}^{λ}(i : j) ≀ β_{l} = yh_{j_{l+1}j_{l}}T_{q}^{λ}(i : js_{l}) + ∑_{f(k)<f(j)} a_{jk}(s_{l})T_{q}^{λ}(i : k) if j_{l} ≤ j_{l+1}.

Proof: By definition of bideterminants, this follows immediately from the lemma. □

Corollary 8.11 Let j ∈ I(n, r) and w ∈ S_{λ′}. Then there is an invertible element a_{j}(w) ∈ R and to each k ∈ I(n, r) satisfying f(k) < f(j) another element a_{jk}(w) in R such that the following equations hold in K for all i ∈ I(n, r):

T_{q}^{λ}(i : j) = a_{j}(w)T_{q}^{λ}(i : jw) + ∑_{f(k)<f(j)} a_{jk}(w)T_{q}^{λ}(i : k).

Proof: We use induction on the length of w. If this is zero there is nothing to prove. If not, we write w = w′s_{l} where w′, s_{l} ∈ S_{λ′} and l(w′) = l(w) − 1. By induction hypothesis we have

T_{q}^{λ}(i : j) = a_{j}(w′)T_{q}^{λ}(i : jw′) + ∑_{f(k)<f(j)} a_{jk}(w′)T_{q}^{λ}(i : k).

But by Lemma 8.4 we have T_{q}^{λ}(i : jw′) = −T_{q}^{λ}(i : jw′) ≀ β_{l}^{−1} as well as T_{q}^{λ}(i : jw′) = −T_{q}^{λ}(i : jw′) ≀ β_{l}. Thus, the assertion follows from the preceding corollary and the fact that f(j) = f(jw′). □

Setting h_{i}(w) := ∏ h_{i_{w(k)}i_{w(j)}}, where the product runs over all pairs 1 ≤ j < k ≤ r such that w(j) > w(k), the more explicit formula a_{j}(w) = (−1)^{l(w)}y^{t}h_{j}(w) for the invertible coefficients is easy to verify but will not be needed any more. Here, t is the number of factors h_{ij} in h_{j}(w) such that i ≥ j. Now we turn to a more special situation for the multi-index i.

Lemma 8.12 Let i ∈ I(n, r) satisfy i_{1} ≤ i_{2} ≤ … ≤ i_{r} and let w ∈ S_{r} be arbitrary. Then the following congruence relation holds in V^{⊗r} modulo the R-submodule W′_{i} = W_{i} + J_{r}:

β(w^{−1})(v_{i}) ≡ y^{l(w)}h_{i}(w)v_{iw}.

Proof: We use induction on r, the case r = 1 being trivial. For r > 1 we embed S_{r−1} as the parabolic subgroup of S_{r} generated by s_{1}, …, s_{r−2} which fix r. If w ∈ S_{r−1} there is nothing to prove by the induction hypothesis. Otherwise, we write w = w′s where w′ ∈ S_{r−1} and s := s_{r−1}s_{r−2} … s_{j+1}s_{j} for an appropriate j < r. We may assume l(w) = l(w′) + r − j. By the induction hypothesis and remark 8.9 we calculate

β(w^{−1})(v_{i}) ≡ β(s^{−1})(y^{l(w′)}h_{i}(w′)v_{iw′}) = y^{l(w′)}h_{i}(w′)β_{j}β_{j+1} … β_{r−1}(v_{iw′}) ≡
y^{l(w′)}y^{r−j}h_{i}(w′)h_{i_{w′(r)}i_{w′(r−1)}}h_{i_{w′(r)}i_{w′(r−2)}} … h_{i_{w′(r)}i_{w′(j)}}v_{iw′s} = y^{l(w)}h_{i}(w)v_{iw}.

Here, Lemma 8.8 has to be applied r − j times and we have used the assumption on i implying i_{w′(j)}, i_{w′(j+1)}, …, i_{w′(r−1)} ≤ i_{r} = i_{w′(r)}. □

Corollary 8.13 Let j ∈ I(n, r) satisfy j_{l} ≤ j_{l+1} ≤ … ≤ j_{k−1} ≤ j_{k} for some 1 ≤ l < k ≤ r and w ∈ S_{r} satisfy w(i) = i for 1 ≤ i ≤ l and k < i ≤ r. Then, to each k ∈ I(n, r) satisfying f(k) < f(j) there is an element a′_{jk}(w) in R such that the following equations hold in K for all i ∈ I(n, r):

T_{q}^{λ}(i : j) ≀ β(w) = y^{l(w)}h_{j}(w^{−1})T_{q}^{λ}(i : jw^{−1}) + ∑_{f(k)<f(j)} a′_{jk}(w)T_{q}^{λ}(i : k).

Proof: As for the proof of Corollary 8.10, this follows easily from the preceeding lemma together with Lemma 8.5 by the definition of bideterminants. □

9 The Weak Straightening Algorithm

We are now able to give the proof of the following weak form of the straightening algorithm.

Proposition 9.1 (Weak Quantum Symplectic Straightening Algorithm) Let λ ∈ Λ^{+}(r) be a partition of r and j ∈ I(n, r)\I_{λ}. Then to each k ∈ I(n, r) satisfying k ⊲ j there is an element a_{jk} ∈ R such that in K the following congruence relation holds for all i ∈ I(n, r):

T_{q}^{λ}(i : j) ≡ ∑_{k⊲j}a_{jk}T_{q}^{λ}(i : k) mod K(> λ).

Proof: We divide into the following two cases

1. j is not λ-column standard.

2. j is λ-column standard but not λ-row standard.

Case 1:

By assumption there are two consecutive indices j_{l} and j_{l+1} in j = (j_{1}, …, j_{r}) such that j_{l} ⪰ j_{l+1} and s_{l} = (l, l + 1) ∈ S_{λ′}. If j_{l} = j_{l+1}, we have T_{q}^{λ}(i : j) = 0 by Corollary 8.2, implying our assertion. In the case j_{l+1} ≻ j_{l} we apply Corollary 8.11:

T_{q}^{λ}(i : j) = a_{j}(s_{l})T_{q}^{λ}(i : js_{l}) + ∑ a_{jk}(s_{l})T_{q}^{λ}(i : k).

The multi-indices k in the sum satisfy f(k) < f(j) and consequently k ⊲ j. Finally, since f(j) = f(js_{l}) and js_{l} occurs before j in the lexicographic order on I(n, r) we have js_{l} ⊲ j as well.

Case 2:

In principle we follow the lines of the proof of [Ma, 2.5.7], using our tools developed in the previous section to substitute corresponding classical results. However, two additional difficulties lead to some complications. Firstly, the lack of commutativity inside A_{R,q}^{s}(n) forces us to work with a fixed basic tableau. But, the change of basic tableaux in the above mentioned proof can be compensated for by Lemma 8.7. Secondly, our tools from the previous section rather prefer the usual order < on \underline{n} instead of ≺, which is the right one for the combinatoricis of tableaux.

To start, let l ∈ \underline{r} be the smallest index such that j_{l} is larger than its right hand neighbour j_{l′} in the λ-tableaux of j. Assume that the entry j_{l} lies in the s-th column j_{λ}^{s} and that j_{l′} lies in the s + 1-th column j_{λ}^{s+1}, where 1 ≤ s < λ_{1}. Clearly, l′ = l + λ′_{s}. Let t be the index of the row containing both entries. We picture this by

T_{j}^{λ} = … \begin{matrix}j_{λ}^{s} & j_{λ}^{s+1} \\ ⋮ & ⋮ \\ j_{l−1} & j_{l′−1} & t − 1 \\ j_{l} & j_{l′} & t \\ j_{l+1} & j_{l′+1} & t + 1 \\ ⋮ & ⋮ \end{matrix} ….

By assumption we have … ≺ j_{l′−1} ≺ j_{l′} ≺ j_{l} ≺ j_{l+1} ≺ . . .. Now, we refine the dual partition λ′ of λ to a composition η ∈ Λ(p + 2, r), where p := λ_{1} is the number of columns of the diagram of λ. More precisely, we split the s-th and (s + 1)-th column in front of and below the t-th row, explicitly

η_{i} := \{ \begin{matrix}λ′_{i} & i < s \\ t − 1 & i = s \\ λ′_{s} − t + 1 & i = s + 1 \\ t & i = s + 2 \\ λ′_{s+1} − t & i = s + 3 \\ λ′_{i−2} & i > s + 3 \end{matrix} , μ_{i} := \{ \begin{matrix}η_{i} & i ≤ s \\ η_{s+1} + η_{s+2} & i = s + 1 \\ η_{i+1} & i > s + 2 \end{matrix} .

Obviously, this η is the coarsest refinement of the partition λ′ and the composition μ ∈ Λ(p + 1, r) defined above. Let us split the multi-index j according to η as follows:

j_{η}^{s} = (j_{h}, …, j_{l−1}), j_{η}^{s+1} = (j_{l}, …, j_{h+k−1}),
j_{η}^{s+2} = (j_{h+k}, …, j_{l′}), j_{η}^{s+3} = (j_{l′+1}, …, j_{h+k+k′−1}).

Here, h := l − t + 1 = λ′_{1} + … + λ′_{s−1} + 1 is the index of the first entry of the s-th column and k := λ′_{s} (resp. k′ := λ′_{s+1}) are the lengths of both columns in question. We have

j_{λ}^{s} = j_{η}^{s} + j_{η}^{s+1}, j_{λ}^{s+1} = j_{η}^{s+2} + j_{η}^{s+3} and set j_{μ}^{s+1} := j_{η}^{s+1} + j_{η}^{s+2}.

In order to apply Laplace Duality 8.6 to the pair (λ, μ) of compositions we have to choose coset representatives of S_{η} = S_{λ′} ∩ S_{μ} in S_{λ′} and S_{μ} carefully. From the theory of parabolic subgroups of reflection groups it is known that each right coset of S_{η} in S_{μ} ≅ S_{μ_{1}} ×…×S_{μ_{p+1}} contains a unique element of minimal length called the distinguished right coset representative, in fact one looks for coset representatives of S_{η_{s+1}} × S_{η_{s+2}} in S_{μ_{s+1}}. We choose these for our set X. The property l(wu) = l(w) + l(u) for u ∈ X and w ∈ S_{η} follows from that theory as well. Similarily, one finds a set Y of distinguished left coset representatives of S_{η} in S_{λ′} satisfying l(vw) = l(v) + l(w) for w ∈ S_{η} and v ∈ Y.

We will not apply Laplace Duality to the original index pair i, j. This is the trick to handle the transition from the order < to ≺, mentioned above. Instead of j we consider rather j′ := jw where w ∈ S_{μ_{s+1}} ⊆ S_{r} is chosen in such a way that j′_{l} < j′_{l+1} < … < j′_{l′−1} < j′_{l′} and j′_{i} = j_{i} for 1 ≤ i < l and l′ < i ≤ r (the embedding of S_{μ_{s+1}} is understood according to the composition μ). This w exists uniquely since j_{μ}^{s+1} = (j_{l}, …, j_{l′}) contains exactly μ_{s+1} = λ′_{s} + 1 elements by the assumption j_{h+k} ≺ j_{h−k+1} ≺ … ≺ j_{l′} ≺ j_{l} ≺ … ≺ j_{h+k−1} on j. Now, by Laplace-Duality we obtain

∑_{u∈X}(−y)^{−l(u)}T_{q}^{λ}(i : j′) ≀ β(u) = ∑_{v∈Y}(−y)^{−l(v)}β(v) ≀ t_{q}^{μ}(i : j′).  (10)

With help of Lemma 8.7 the right hand side of this equation can be written as a linear combination of bideterminants T_{q}^{μ′}(k : l). Thus the right hand side is seen to lie in K(> λ) as soon we have shown that μ′ > λ. But, this follows since the longest column being removed from the diagram of λ to obtain the diagram of μ′ has length λ′_{s}, whereas a column of length μ_{s+1} = λ′_{s} + 1 has to be added to the diagram of μ′. On the left hand side of (10) we may apply Corollary 8.13 by construction of the multi-index j′:

(−y)^{−l(u)}T_{q}^{λ}(i : j′) ≀ β(u) = sign(u)h_{j′}(u^{−1})T_{q}^{λ}(i : j′u^{−1}) + ∑(−y)^{−l(u)}a′_{j′k}(u)T_{q}^{λ}(i : k), the sum running over all k satisfying f(k) < f(j′) = f(j). Now, for all u ∈ X we have ũ := uw^{−1} ∈ S_{μ_{s+1}} since w is contained in S_{μ_{s+1}}. Furthermore, there is a unique coset representative u_{0} ∈ X satisfying S_{η}u_{0} = S_{η}w and this is the only one for which the corresponding ũ lies in S_{η}. Therefore, in the case u ≠ u_{0} there is an e such that l ≤ e < h + k and h + k ≤ ũ^{−1}(e) ≤ l′. Choose such an e for each u ∈ X. In doing so, we are assigning a transposition û := (l, e) to each u which is contained in S_{η}. In the case of u_{0} we set û_{0} := ũ_{0} ∈ S_{η}. Applying Corollary 8.11 to û one calculates

T_{q}^{λ}(i : j′u^{−1}) = T_{q}^{λ}(i : jũ^{−1}) = a_{jũ^{−1}}(û)T_{q}^{λ}(i : jũ^{−1}û) + ∑ a_{(jũ^{−1})k}(û)T_{q}^{λ}(i : k),

where the sum is running over all k satisfying f(k) < f(jũ^{−1}) = f(j), again. For these k we set

ā_{jk} := ∑_{u∈X}(−y)^{−l(u)}a′_{j′k}(u) + sign(u)h_{j′}(u^{−1})a_{(jũ^{−1})k}(û),

whereas in the case f(k) = f(j) we write

ā_{jk} := \{ \begin{matrix}sign(u)h_{j′}(u^{−1})a_{jũ^{−1}}(û) & if there exist u ∈ X, k = jũ^{−1}û \\ 0 & otherwise \end{matrix} .

Observe, that ā_{jj} occurs in this definition for u = u_{0}. We assert that for u ≠ u_{0} the multi-index k := jũ^{−1}û occurs before j in the lexicographic order with respect to ≺. For by construction of ũ and û we have

k_{l} = j_{ũ^{−1}û(l)} = j_{ũ^{−1}(t)} ∈ {j_{h+k}, j_{h+k+1}, …, j_{l′}}

and consequently k_{l} ≺ j_{l}. But this implies k ≺ j, since k_{i} = j_{i} for i < l. Thus we obtain

−ā_{jj}T_{q}^{λ}(i : j) ≡ ∑_{k⊲j} ā_{jk}T_{q}^{λ}(i : k) mod K(> λ).

Since the coefficient −ā_{jj} is invertible, the asserted congruence relation holds as well. □

It should be remarked, that the proof works with any other order on \underline{n} instead of ≺, as well. The proof of the strong part of the algorithm (Proposition 7.2) can be given right now in the (initial) case r = 2 and we are going to do this not only because it is very instructive, but also because we will need a basis of A_{R,q}^{s}(n, 2) in order to proceed to the general case.

If r = 2 there are exactly two partitions in Λ^{+}(m, 2) for m ≥ 2, namely 2ω_{1} and ω_{2}. In the first case we have I_{2ω_{1}} = I_{2ω_{1}}^{mys}, that is, the weak and the strong form of the straightening algorithm coincide. Turning to ω_{2} there is exactly one element in I_{ω_{2}}\I_{ω_{2}}^{mys}, namely j = (m, m′). By (7) we obtain in K = A_{R,q}^{sh}(n, 2)

T_{q}^{ω_{2}}(i : (m, m′)) = −q^{m} ∑_{i=1}^{m−1} q^{−i}T_{q}^{ω_{2}}(i : (i, i′))

yielding Proposition 7.2 in the case r = 2 since (i, i′) ⊲ (m, m′) for all i < m.

Remark 9.2 If we had used the notion of symplectic standard tableau instead of the reverse version we would have to consider (1′, 1) instead of (m, m′) in the last step above. This would force us to work with a reverse version of the order choosen on ℕ _{0}^{m} (as in [O2, section 7]). But this would cause some trouble concerning Lemma 8.8. One way out could be a manipulation of the Yang-Baxter operator β conjugating it by the twofold tensorproduct of the appropriate permutation on \underline{n}. Thus, one has to decide between working with the familiar version of β or following the familiar notion of tableaux.

10 Quantum Symplectic Exterior Algebra

We are going to prepare the proof of Proposition 7.2 for general r. Since we need a q-analogue of [O2, Lemma 8.1], we have to investigate the quantum symplectic exterior algebra. We start with its definition which can be found in many textbooks on quantum groups (for instance [CP, chapter 7]). It is defined as the quotient of the tensor algebra T (V) = ⨁_{r∈ℕ_{0}} V^{⊗r} by a certain ideal. We denote it by ⋀_{R,q}(n) and write the symbol ∧ for multiplication in this algebra. Setting

c_{i} := q^{i}v_{i′} ∧ v_{i}, and d_{i} := −q^{−i}v_{i} ∧ v_{i′}

for i ∈ \underline{m}, the relations holding in ⋀_{R,q}(n) can be written as

v_{k} ∧ v_{l} = −q^{−1}v_{l} ∧ v_{k},  (11)
v_{i} ∧ v_{i′} = −q^{−2}v_{i′} ∧ v_{i} − (y − 1)q^{−i} ∑_{j=i+1}^{m} c_{j}  (12)

v_{i′} ∧ v_{i} = −q^{2}v_{i} ∧ v_{i′} + (y − 1)q^{i} ∑_{j=i+1}^{m} d_{j}  (13)
v_{k} ∧ v_{k} = 0  (14)

where k > l, k ≠ l′ and i ∈ \underline{m} is assumed. Note that the third relation is a consequence of the former two, and clearly the remaining three types of relations imply that the set

B := \{v_{I}| I ⊆ \underline{n}\}

is a set of R -linear generators of ⋀_{R,q}(n) where we have set

v_{I} := v_{i_{1}} ∧ v_{i_{2}} ∧ … ∧ v_{i_{r}}, if I := {i_{1}, …, i_{r}} and i_{1} < i_{2} < … < i_{r}.

In contrast to [O2, section 7] we take the usual order < on \underline{n} here for technical reasons. A subset I ⊆ \underline{n} ordered in that way will be called an ordered subset in the sequel. Using the diamond lemma of ring theory one shows that B in fact is an R-basis of ⋀_{R,q}(n) (cf. [Ha, p. 157], [O1, Satz 3.8.3, B.1.1]). Furthermore, ⋀_{R,q}(n) is a graded algebra since the relations are homogeneous of degree two. A basis for the r-th homogeneous summand ⋀_{R,q}(n, r) is given by the subset B_{r} of B corresponding to the set P (n, r) of subsets I ⊆ \underline{n} having cardinality |I| = r.

Before proceeding, let us briefly argue why ⋀_{R,q}(n) possesses the structure of an A_{R,q}^{s}(n)- comodule algebra (for details we refer to [O1, Satz 3.8.2]). First, the reader should convince himself that we may reduce to the case R = Z := ℤ [q, q^{−1}], the ring of integral Laurent polynomials in the indeterminate q (use [O2, Theorem 4.1]). In this case it is easy to see that the defining equations for ⋀_{R,q}(n) precisely span the kernel of the endomorphism β − q^{2}id_{V^{⊗2}} (consider the eigenspace decomposition of β in the case of the field of fractions 𝕂 := ℚ (q) of Z). Therefore, it is enough to show that this kernel is a A_{R,q}^{s}(n)-subcomodule of V^{⊗2}, since this property is transmitted to the spanned ideal. Now, if C := End_{A_{2}}(V^{⊗2}) is the centralizer algebra of the algebra A_{2} spanned by β and γ, this kernel clearly is a C-submodule of V^{⊗2}. Since C is the dual algebra to the coalgebra A_{R,q}^{s}(n, 2) ([O2, Theorem 3.3]), the reader might think that this completes the argument. The following counterexample shows that some caution is needed at this point: Consider the ℤ -module on abstract generators x_{ij}:

M := ℤ x_{11} ⊕ ℤ x_{22} ⊕ (ℤ /2ℤ)x_{21} ⊕ (ℤ /2ℤ)x_{12}.

It can be considered as a ℤ -coalgebra with comultiplication and augmentation beeing defined in the usual way on the generators. In fact it is the centralizer coalgebra (as

defined in [O2, section 2]) of the endomorphism μ on ℤ^{2} given by the matrix ( \begin{matrix}1 & 0 \\ 0 & 3 \end{matrix} ).

The centralizer algebra of μ clearly is given by

C := End_{μ}(ℤ^{2}) = \{( \begin{matrix}k & 0 \\ 0 & l \end{matrix} ) | k, l ∈ ℤ \} .

Thus, the span of the element v_{1} = (1, 0) is a C-submodule of ℤ^{2}. On the other hand τ_{V} (v_{1}) = v_{1} ⊗ x_{11} + v_{2} ⊗ x_{21} is not contained in v_{1} ⊗ M, since x_{21} ≠ 0 and therefore this span is not an M-subcomodule of ℤ^{2}.

However, it is easy to see that such things cannot happen if the coalgebra is free as an R-module ([O1, Lemma 1.5.4]), which was not the case in the counterexample. To finish our argument that ⋀_{R,q}(n) is an A_{R,q}^{s}(n)-comodule algebra, we have to show that the coalgebra A_{Z,q}^{s}(n, 2) is a free Z-module. But we already have seen at the end of the previous section that the straightening formula is valid in the case r = 2 and consequently B_{2} is a set of generators for A_{R,q}^{s}(n, 2). Now, over the field of fractions 𝕂 one calculates |B_{2}| = (\binom{n}{2} + n)^{2} + (\binom{n}{2} − 1)^{2} + 1^{2} = dim_{𝕂}(C) = dim_{𝕂}(A_{𝕂 ,q}^{s}(n, 2)) as the square sum of eigenspace dimensions of the endomorphism β. Since A_{𝕂 ,q}^{s}(n, 2) ≅ 𝕂 ⊗_{Z} A_{Z,q}^{s}(n, 2) by [O2, Theorem 4.1], the set B_{2} must be linearly independent as well. Thus our argument that ⋀_{R,q}(n) is an A_{R,q}^{s}(n)-comodule algebra is finished.

One might guess that the coefficient functions of ⋀_{R,q}(n, r) corresponding to the basis B_{r} consist of bideterminants as in the classical case. In fact this is true and we are going to prove this below. Denoting the structure map by τ_{∧} : ⋀_{R,q}(n) → ⋀_{R,q}(n) ⊗ A_{R,q}^{s}(n) (we will use the same symbol in the case of A_{R,q}^{sh}(n) later on) the explicit formula is

τ_{∧}(v_{J}) = ∑_{I∈P (n,r)} v_{I} ⊗ T_{q}^{ω_{r}}(i : j)  (15)

where i = (i_{1}, …, i_{r}) and j = (j_{1}, …, j_{r}) are the multi-indices corresponding to the ordered subsets I := {i_{1}, …, i_{r}} and J = {j_{1}, …, j_{r}}, respectively. Let us first treat the ingredients needed in the proof of that formula.

Lemma 10.1 Let π_{r} : V^{⊗r} → ⋀_{R,q}(n, r) be the natural projection. Then the endomorphism κ_{λ} := ∑_{w∈S_{λ}}(−y)^{−l(w)}β(w) factors through π_{r}, i.e. there is a homomorphism of R-modules ν_{r} : ⋀_{R,q}(n, r) → V^{⊗r} such that κ_{r} = ν_{r} ∘ π_{r}.

Proof: Again, we may assume R = Z = ℤ [q, q^{−1}]. In this case we know that the defining ideal of ⋀_{R,q}(n) is generated by the kernel of (id_{V^{⊗2}} − y^{−1}β). Thus, the assertion immediately follows from Lemma 8.1. □

Let I_{ω_{r}}^{<} := {i ∈ I(n, r)| i_{1} < i_{2} < … < i_{r}} be the set of multi-indices corresponding to the ordered subsets I ∈ P (n, r).

Lemma 10.2 Let F_{r} be the R-linear span of {v_{j}| j ∈ I(n, r)\I_{ω_{r}}^{<}} in V^{⊗r}. Then for all w ∈ S_{r}\{id} and i ∈ I_{ω_{r}}^{<} it follows that β(w)(v_{i}) ∈ F_{r}.

Proof: We use induction on r. The case r = 2 directly follows from the formulas

β(v_{(k,l)}) = qv_{(l,k)}  (16)
β(v_{(i,i′)}) = v_{(i′,i)} + (y − 1) ∑_{j=1}^{i−1} q^{j−i}v_{(j′,j)}  (17)

which are valid for k < l, k ≠ l′ and i ≤ m. If r > 2, we embed S_{r−1} as the subgroup of S_{r} that fixes the letter r (cf. proof of Lemma 8.12). If w ∈ S_{r−1}, there is nothing to prove since F_{r−1} ⊗ V ⊆ F_{r}. Otherwise, we may write β(w) = β(w′)β_{r−1}β_{r−2} … β_{l} where w′ ∈ S_{r−1} and l ≤ r − 1.

First consider the case where i′_{l} is not contained in {i_{l+1}, …, i_{r}}. Applying β_{r−1}β_{r−2} … β_{l} to v_{i} we only have to use (16) but not (17). Consequently, we have β_{r−1}β_{r−2} … β_{l}(v_{i}) = q^{r−l+1}v_{i_{1}} … v̂_{i_{l}} … v_{i_{r}}v_{i_{l}}. Here, v̂_{i_{l}} denotes the omission of v_{i_{l}}. This element obviously lies in F_{r}, proving the assertion in the case w′ = id. If w′ is not the identity map we have β(w′)(q^{r−l+1}v_{i_{1}} … v̂_{i_{l}} … v_{i_{r}}v_{i_{l}}) ∈ F_{r−1} ⊗ v_{i_{l}} ⊆ F_{r} by the induction hypothesis since (i_{1}, …, î_{l}, …, i_{r}) ∈ I_{ω_{r−1}}^{<}.

We next consider the case i′_{l} ∈ {i_{l+1}, …, i_{r}}. This forces i_{l} ≤ m because i_{l} < i′_{l}. Let i′_{l} = i_{k}. As above, we have β_{k−2}β_{k−3} … β_{l}(v_{i}) = q^{k−l−2}v_{i_{1}} … v̂_{i_{l}} … v_{i_{k−1}}v_{i_{l}}v_{i_{k}}v_{i_{k+1}} … v_{i_{r}}. Applying β_{k−1} to this expression, we have to use (17) for the first time. But for each basis element v_{j} occurring as a summand in the resulting expression we have j_{k} ≤ i_{l} ≤ m. Similar things happen concerning the remaining β_{k}, …, β_{r−1}. Thus, for each v_{j} occurring as a summand in β_{r−1}β_{r−2} … β_{l}(v_{i}), it follows j_{r} ≤ i_{l} ≤ m. On the other hand, for each such summand there must exist an h < r where j_{h} > m. This is because j must contain a pair \{i, i′\} for some i ∈ \underline{m}, since this was the case for the multi-index i we started with. Consequently, we obtain j ∈ F_{r} in this case too. □

Let the coefficient matrices of the R-module homomorphisms π_{r}, κ_{r} and ν_{r} (from Lemma 10.1) be given by

π_{r}(v_{j}) = ∑_{I∈P (n,r)} π_{Ij}v_{I}, κ_{r}(v_{j}) = ∑_{i∈I(n,r)} κ_{ij}v_{i} and ν_{r}(v_{J}) = ∑_{i∈I(n,r)} ν_{iJ}v_{i}.

Now, if j ∈ I_{ω_{r}}^{<} corresponds to the ordered set J ∈ P (n, r) we have π_{r}(v_{j}) = v_{J} yielding ν_{iJ} = κ_{ij} by Lemma 10.1. From Lemma 10.2 it follows κ_{r}(v_{j}) ≡ v_{j} modulo F_{r}. Thus, for a pair i, j ∈ I_{ω_{r}}^{<} of multi-indices corresponding to ordered sets I, J ∈ P (n, r), we obtain ν_{iJ} = κ_{ij} = δ_{IJ} (Kronecker symbol). Finally, from κ_{r} = ν_{r} ∘ π_{r} we see for all i ∈ I_{ω_{r}}^{<} and j ∈ I(n, r)

κ_{ij} = ∑_{K∈P (n,r)} ν_{iK}π_{Kj} = π_{Ij}.

We are now ready to give the proof of equation (15). We calculate

τ_{∧}(v_{J}) = ∑_{k∈I(n,r)} v_{k_{1}} ∧ … ∧ v_{k_{r}} ⊗ x_{kj} =

∑_{k∈I(n,r)} ∑_{I∈P (n,r)} π_{Ik}v_{I} ⊗ x_{kj} = ∑_{I∈P (n,r)} v_{I} ⊗ ∑_{k∈I(n,r)} κ_{ik}x_{kj}.

But, this is exactly what we wanted by the definition T_{q}^{ω_{r}}(i : j) = κ_{r} ≀x_{ij} of bideterminants (using notation (1)).

The formula we just have proved has some useful consequences concerning the comultiplication and augmentation of A := A_{R,q}^{s}(n). These are valid for any pair i, j ∈ I_{ω_{r}}^{<} of multi-indices corresponding to ordered sets I, J ∈ P (n, r) and follow directly with the help of the comodule axioms (τ_{∧} ⊗ id_{A}) ∘ τ_{∧} = (id_{∧} ⊗ Δ) ∘ τ_{∧} and (id_{∧} ⊗ ϵ) ∘ τ_{∧} = id_{∧}:

Δ(T_{q}^{ω_{r}}(i : j)) = ∑_{k∈I_{ωr}^{<}} T_{q}^{ω_{r}}(i : k) ⊗ T_{q}^{ω_{r}}(k : j),  (18)

ϵ(T_{q}^{ω_{r}}(i : j)) = δ_{ij}.  (19)

Another useful consequence is the following corollary:

Corollary 10.3 Let a_{j} ∈ R be such that ∑_{j∈I(n,r)} a_{j}v_{j_{1}} ∧ v_{j_{2}} ∧ … ∧ v_{j_{r}} = 0. Then for all i ∈ I(n, r) we have

∑_{j∈I(n,r)} a_{j}T_{q}^{ω_{r}}(i : j) = ∑_{j∈I(n,r)} a_{j}T_{q}^{ω_{r}}(j : i) = 0.

Proof: By Lemma 10.1 and the assumption we have

κ_{r}( ∑_{j∈I(n,r)} a_{j}v_{j}) = ∑_{j∈I(n,r)} a_{j}κ_{r}(v_{j}) = 0.

Consequently, for all k ∈ I(n, r) we obtain ∑_{j∈I(n,r)} a_{j}κ_{kj} = 0 and therefore

∑_{j∈I(n,r)} a_{j}T_{q}^{ω_{r}}(i : j) = ∑_{k,j∈I(n,r)} a_{j}x_{ik}κ_{kj} = 0.

The equation with exchanged indices is deduced by an application of the involution ^{∗}. □

11 Proof of Proposition 7.2

First, we have to state the q-analogue of [O2, Lemma 8.1], one of the principal ingredients in the proof of the symplectic straightening algorithm in the classical case. In order to define the quantum analogue to the ideal N considered there we have to look in more detail at the elements c_{i} and d_{i} defined in the previous section. Setting y = q^{2} as before, let us write down the fundamental relations between them,

y^{i}d_{i} = yc_{i} + (y − 1) ∑_{j=i+1}^{m} c_{j} and y^{−i}c_{i} = y^{−1}d_{i} + (y^{−1} − 1) ∑_{j=i+1}^{m} d_{j}.  (20)

These imply c_{i} ∧ d_{i} = d_{i} ∧ c_{i} = 0 according to (14) and furthermore,

c_{i}^{2} := c_{i} ∧ c_{i} = (y^{−1} − 1) ∑_{j=i+1}^{m} c_{i} ∧ c_{j} and d_{i}^{2} = (y − 1) ∑_{j=i+1}^{m} d_{i} ∧ d_{j}.  (21)

This stands in remarkable contrast to the classical and even quantum linear case where such expressions vanish. On the other hand, the elements c_{i} and d_{i} commute with each other, exactly as in the classical case. Consequently, the elements d_{K} := d_{k_{1}} ∧d_{k_{2}} ∧…∧d_{k_{a}} are defined independently on the order of the elements of the subset K := {k_{1}, …, k_{a}} ⊆ \underline{m}. Again, we write P (m, a) for the collection of all subsets K of \underline{m} whose cardinality is a. Set

D_{a} := ∑_{K∈P (m,a)} d_{K}

and let N be the ideal in ⋀_{R,q}(n) generated by the elements D_{1}, D_{2}, …, D_{m}. We call an ordered subset I ∈ P (n, r) reverse symplectic if the multi-index iw obtained from I by ordering its elements according to ≺ (obtained from i by a suitable permutation w ∈ S_{r} such that i_{(w(1)} ≺ i_{w(2)} ≺ … ≺ i_{w(r)}) is ω_{r}-reverse symplectic standard.

Proposition 11.1 Let I ∈ P (n, r) be non reverse symplectic. Then, to each J ∈ P (n, r) such that the inequality f(j) < f(i) holds for the corresponding multi-indices i and j, there exists a_{IJ} ∈ R such that in ⋀_{R,q}(n) the following congruence relation holds:

v_{I} ≡ ∑_{J∈P (n,r), f(j)<f(i)} a_{IJ}v_{J} mod N.

Proposition 11.2 The semi bialgebra A_{R,q}^{sh}(n) acts trivially on the elements D_{a}, that is τ_{∧}(D_{a}) = 0.

We postpone the very technical proofs of both propositions to separate sections below.

Let us prove Proposition 7.2 in the case λ = ω_{r}, first. Take j ∈ I(n, r)\I_{ω_{r}}^{mys}. Using the weak part of the straightening algorithm 9.1, we may assume j ∈ I_{ω_{r}}\I_{ω_{r}}^{mys}. This means j_{1} ≺ j_{2} ≺ … ≺ j_{r}. In order to apply our lemmas we have to change orders from ≺ to <. Let w ∈ S_{r} be such that j_{w(1)} < j_{w(2)} < … < j_{w(r)}, that is, jw is a multiindex corresponding to a non reverse symplectic ordered set J ∈ P (n, r) in the sense of Proposition 11.1. Application of this proposition to v_{J} yields

X := v_{J} − ∑_{K∈P (n,r), f(k)<f(j)} a_{jk}v_{K} ∈ N

since f(j) = f(jw). According to Proposition 11.2, the element τ_{∧}(X) must be zero. Applying (15), we obtain the following equation holding in ⋀_{R,q}(n, r) ⊗ K:

∑_{I∈P (n,r)} v_{I} ⊗ (T^{ω_{r}}(i : jw) − ∑_{K∈P (n,r), k⊲j}a_{jk}T^{ω_{r}}(i : k)) = 0.

Since {v_{I}| I ∈ P (n, r)} is a basis of ⋀_{R,q}(n, r), each individual summand in the summation over P (n, r) must be zero. Together with Corollary 8.11, this gives the desired result in the case of multi-indices i corresponding to ordered subsets I ∈ P (n, r), that is i ∈ I_{ω_{r}}^{<}. The case for general i can be deduced from this using

T_{q}^{ω_{r}}(i : j) = ∑_{K∈P (n,r)} ν_{iK}T_{q}^{ω_{r}}(k : j),

which follows from the formula κ_{r} = ν_{r} ∘ π_{r} we met in the previous section.

Next we conssider the general case of λ. Here, we can proceed exactly as in the classical case. Again, we may assume j ∈ I_{λ}\I_{λ}^{mys} by the weak part of the straightening algorithm. Let λ′ = (μ_{1}, …, μ_{p}) be the dual partition (p = λ_{1}). We spilt j into p multi-indices j^{l} ∈ I(n, μ_{l}). where for each l ∈ \underline{p} the entries of j^{l} are taken from the l-th column of T_{j}^{λ}. The same thing can be done with i. Since j is not λ-reverse symplectic standard but standard there must be a column s such that j^{s} is not ω_{μ_{s}}-reverse symplectic standard. Applying the result to the known case of T_{q}^{ω_{μs}}(i^{s} : j^{s}), we obtain

T_{q}^{λ}(i : j) = T_{q}^{ω_{μ_{1}}}(i^{1} : j^{1})T_{q}^{ω_{μ_{2}}}(i^{2} : j^{2}) ⋯ T_{q}^{ω_{μs}}(i^{s} : j^{s}) ⋯ T_{q}^{ω_{μp}}(i^{p} : j^{p})

≡ ∑ a_{j^{s}k^{s}}T_{q}^{ω_{μ_{1}}}(i^{1} : j^{1}) ⋯ T_{q}^{ω_{μs}}(i^{s} : k^{s}) ⋯ T_{q}^{ω_{μp}}(i^{p} : j^{p}) = ∑ a_{jk}T_{q}^{λ}(i : k).

Therein, k^{s} ∈ I(n, μ_{s}) satisfies k^{s} ⊲ j^{s}, k ∈ I(n, r) is constructed from j by replacing the entries of j^{s} by that of k^{s} and a_{jk} is the same as a_{j^{s}k^{s}} for the corresponding k^{s}. The product formula for bideterminants applied above is valid by our choice of basic λ-tableaux inserting the numbers 1, …, r column by column top down (otherwise the non-commutativity of A_{R,q}^{s}(n) would cause some trouble). From (9) we see k ⊲ j and the proof of 7.2 is completed.

12 Proof of Proposition 11.1

For convenience, in in the following sections we abbreviate y = q^{2} and denote multiplication in ⋀_{R,q}(n) by juxtaposition instead of ∧. Further, to sets K, L ⊆ \underline{m} we define integers

v(K, L) := |{(k, l) ∈ K × L| k > l}|.

Lemma 12.1 Let a ∈ \underline{m}. If \underline{m} = L ∪ M is a partition of \underline{m} into disjoint subsets L and M then to each K ∈ P (m, a) there is an integer s(K, L) such that

D_{a} = ∑_{K∈P (m,a)} y^{s(K,L)}c_{K∩L}d_{K∩M}.

If K ⊆ M, the integer s(K, L) equals v(K, L).

Sketch of Proof: In order to prove this, one has to use a more general statement in which the set \underline{m} is substituted by \{l, l + 1, …, m\} for some l ∈ \underline{m}. After this, the results can be proved straightforwardly using induction on m − l and the formulas (20) and (21). For details we refer to [O1, Lemma 3.9.1 and B.2]. □

To a set I ∈ P (n, r) we associate the followng subsets of \underline{m}:

I^{−} := I ∩ \underline{m}, I^{+} := \{i ∈ \underline{m}| i′ ∈ I\} and I^{0} := I^{−} ∩ I^{+}.

Lemma 12.2 Let I ∈ P (n, r) be such that I^{0} = ∅ and J ⊆ \underline{m}. Set s. = |J| and T := \underline{m}\(I^{−} ∪ I^{+}). If J ⊆ T and a ∈ \underline{m} is such that s < a then we have

v_{I}d_{J}D_{a−s} = v_{I} ∑_{S∈P (m,a),J⊆S⊆T} y^{v(S\J,I^{+}∪J)}d_{S}.

Proof: We apply Lemma 12.1 to L := I^{+} ∪ J and M := \underline{m}\L. Since J ⊆ T there is an invertible element b ∈ R such that v_{I}d_{J} = bd_{J}v_{I}. Now, we see that v_{I}d_{J}c_{K∩L}d_{K∩M} vanishes for each K ∈ P (m, a − s) that is not contained in T \J. For, if K contains an element of I^{+}, the basis element v_{I} will be annihilated multiplying by c_{K∩L} whereas, in the case K ∩ I^{+} ≠ ∅ multiplication by d_{K∩M} results in zero. Here, we have used the commutativity between c_{K∩L} and d_{K∩M}. Finally, if K ∩ J ≠ ∅, we have d_{J}c_{K∩L} = 0. A set K ⊆ T \J is obviously contained in M. By the second part of the lemma we have s(K, L) = v(K, L) = v(K, I^{+} ∪ J). Thus, setting S = J ∪ K the assertion follows. □

Lemma 12.3 Let M ⊆ K ⊆ \underline{m} be fixed. Then

∑_{L⊆K\M}(−1)^{|L|}y^{v(K,L)} = \{ \begin{matrix}1 & K = M \\ 0 & K ≠ M \end{matrix}

Proof: Clearly the sum is 1 if K = M. For K ≠ M we show by induction on n := |K| > 0 that the sum is zero. Starting with the case n = 1 we have M = ∅ and ∑_{L⊆K}(−1)^{|L|}y^{v(K,L)} = y^{v(K,∅)} − y^{v(K,K)} = 1 − 1 = 0. For the induction step, let x ∈ K be minimal and set K̂ := K\x. If x ∉ M we calculate

∑_{L⊆K\M}(−1)^{|L|}y^{v(K,L)} = ∑_{L⊆K̂\M}(−1)^{|L|}y^{v(K,L)} + ∑_{L⊆K̂\M}(−1)^{|L|+1}y^{v(K,L∪{x})}.

Since x is minimal, v(K, L) = v(K̂, L) and v(K, L ∪ {x}) = v(K̂, L) + n − 1 if L ⊆ K̂\M. Now, we may apply the induction hypothesis to K̂ which results in

∑_{L⊆K\M}(−1)^{|L|}y^{v(K,L)} = (1 − y^{n−1}) ∑_{L⊆K̂\M}(−1)^{|L|}y^{v(K̂,L)} = 0.

In the case x ∈ M, we write M̂:= M\{x}. Similarly, we calculate

∑_{L⊆K\M}(−1)^{|L|}y^{v(K,L)} = ∑_{L⊆K̂\M̂}(−1)^{|L|}y^{v(K̂,L)},

where the right hand side is zero by the induction hypothesis. □

Lemma 12.4 Let I ∈ P (n, r) with r > m and set a = |I^{0}|, Î := I\{i, i′| i ∈ I^{0}} and T := \underline{m}\(I^{+} ∪ I^{−}). Then there is an invertible a_{I} ∈ R such that the following equation holds:

vÎ ∑_{J⊆T}(−1)^{|J|}y^{v(J,Î^{+}∪J)}d_{J}D_{a−|J|} = a_{I}v_{I}.

Proof: Let T̂ = T ∪ I^{0} and observe that Î^{+} = I^{+}\I^{0}. Because r > m, we must have a ≥ 1. On the other hand 2a + |Î| = r > m implies a > m − |Î| − |I^{0}| = |T | ≥ |J|. Using Lemma 12.2 we calculate

vÎ ∑_{J⊆T}(−1)^{|J|}y^{v(J,Î^{+}∪J)}d_{J}D_{a−|J|} = vÎ ∑_{J⊆T}(−1)^{|J|}y^{v(J,Î^{+}∪J)} ∑_{S∈P (m,a),J⊂S⊆T̂} y^{v(S\J,Î^{+}∪J)}d_{S} =

vÎ ∑_{J⊆S⊆T̂,J∩I^{0}=∅}(−1)^{|J|}y^{v(S,J)+v(S,Î^{+})}d_{S} = vÎ ∑_{S⊆T̂} y^{v(S,Î^{+})}d_{S} ∑_{J⊆S\I^{0}}(−1)^{|J|}y^{v(S,J)}.

But by Lemma 12.3, the last term equals y^{v(I^{0},Î^{+})}vÎd_{I^{0}}. Using relation (11) of the exterior algebra, this can be transformed into v_{I} up to some invertible mutiple a_{I}. □

We are now able to prove Proposition 11.1 by induction on the Lie rank m. In the case where m = 1 both sets of P (2, 1) = {{1}, {2}} are reverse symplectic. In P (2, 2) there is just one set. namely I = {1, 2}, for which we have v_{I} = −qd_{1} = −qD_{1} ∈ N_{2}. Thus there is nothing to prove here.

For the induction step we embed ⋀_{R,q}(n − 2) into ⋀_{R,q}(n) sending v_{i} to v_{i+1}. It is easy to check that this indeed leads to an embedding of algebras. Using the induction hypothesis we may treat the case where I ⊆ \underline{n}\\{1, n\} without much effort. Some caution is needed only concerning the difference between the two ideals N of ⋀_{R,q}(n − 2) and ⋀_{R,q}(n) respectively. For a single element, this difference can be expressed as a sum of basis elements v_{L} with 1, n ∈ L, which all are smaller than v_{I} in our order. If I ∩ {1, n} ≠ ∅ we may apply the induction hypothesis to the set Î:= I\{1, n} in case Î is non reverse symplectic too:

vÎ ≡ ∑_{Ĵ⊆\underline{n}\\{1,n\}, |Î|=|Ĵ|, f(ĵ)<f(î)} aÎ ; ĴvĴ + ∑_{\{1,n\}⊆L⊆\underline{n}} a_{ÎL}v_{L} mod N.

Again, the terms a_{ÎL}v_{L} compensate for the difference between the two ideals N. Multiplying this equation by v_{1} from the left (respectively by v_{n} from the right, respectively by both from both sides) yields the assertion because the elements v_{L} vanish, and f(ĵ) < f(î) implies f(j) < f(i), where j is the multi-index attached to the set J = Ĵ∪ (I\Î).

It remains to prove the assertion in the case where Î is reverse symplectic. Here we need the preparations of this section. By the reverse symplectic condition we have ∑_{i=j}^{m} λ_{i} ≤ m−j+1 for all j > 1, where (λ_{1}, …, λ_{m}) = f(i). Because I itself is non reverse symplectic we must have r = |I| = ∑_{i=1}^{m} λ_{i} > m. According to Lemma 12.4 we conclude v_{I} ∈ N. But this implies the assertion of Proposition 11.1 in the remaining case, too.

13 Proof of Proposition 11.2

In order to prove the proposition we have to consider generalizations of the elements D_{1}, …, D_{m}, which are defined for any l ≤ m by

D_{a,l} := ∑_{K∈P (l,a)} d_{K}.

For a positive integer k, define the y^{−1}-integer {k}_{y^{−1}} := 1 + y^{−1} + y^{−2} + … + y^{−k+1} ∈ R.

Lemma 13.1 Let a ∈ \underline{m}. Then we have

D_{1,l}D_{a,l} ≡ {a + 1}_{y^{−1}}D_{a+1,l}

modulo the ideal spanned by D_{1}.

Proof: Using the above introduced notations we may write the right hand side of (21) as d_{l}^{2} = (y − 1) ∑_{i=l+1}^{m} d_{i}. Since ∑_{i=l+1}^{m} d_{i} + D_{1,l} = D_{1}, we deduce d_{l}^{2} ≡ (y^{−1} − 1)d_{l}D_{1,l} modulo D_{1}.

We proceed by induction on l. If l = 1, both sides are zero if a > 1. In the case a = 1 we have to show that d_{1}^{2} ≡ 0. But by (21) yd_{1}^{2} = (y − 1)d_{1}D_{1} ≡ 0, implying d_{1}^{2} ≡ 0 since y is invertible.

For the induction step we write D_{a,l} = d_{l}D_{a−1,l−1} + D_{a,l−1} and obtain

D_{1,l}D_{a,l} = d_{l}^{2}D_{a−1,l−1} + d_{l}D_{a,l−1} + D_{1,l−1}(d_{l}D_{a−1,l−1} + D_{a,l−1}) ≡
((y^{−1} − 1){a}_{y^{−1}} + 1 + {a}_{y^{−1}})d_{l}D_{a,l−1} + {a + 1}_{y}D_{a+1,l−1}.

But (y^{−1} − 1){a}_{y^{−1}} + 1 + {a}_{y^{−1}} = {a + 1}_{y^{−1}}. Thus the lemma follows. □

It should be remarked that a similar formula holds even with = instead of ≡ if in the definition of D_{a,l} the sets K are taken as subsets of {l, …, m} instead of {1, …, l}. In that formula y^{−1} has to be replaced by y ([O1, Lemma 3.9.1]).

We introduce some new conventions. To an ordered subset J = \{j_{1}, j_{2}, …, j_{a}\} ⊆ \underline{m} we define corresponding multi-indices by

j′j := (j_{1}, j′_{1}, j_{2}, j′_{2}, …, j_{a}, j′_{a}), jj′ := (j_{1}, j_{2}, …, j_{a}, j′_{a}, j′_{a−1}, …, j′_{1}).

Furthemore, we write

u_{J} := − ∑_{j∈J} j.

From the definition of d_{J}, we have d_{J} = q^{u_{J}} v_{j′j}. By the relations of the exterior algebra there is another integer a_{J} such that v_{j′j} = q^{a_{J}} v_{jj′}. By (15) and Corollary 10.3 we calculate

τ_{∧}(d_{J}) = ∑_{I∈P (n,2a)} q^{u_{J} +a_{J}} v_{I} ⊗ T_{q}^{ω_{2a}}(i : jj′) = ∑_{I∈P (n,2a)} q^{u_{J}} v_{I} ⊗ T_{q}^{ω_{2a}}(i : j′j).

Setting

G_{i,l,a} = ∑_{J∈P (l,a)} q^{u_{J}} T_{q}^{ω_{2a}}(i : j′j),

we may write τ_{∧}(D_{a}) = ∑_{I∈P (n,2a)} v_{I} ⊗ G_{i,m,a}. Therefore the proposition which we have to prove in this section holds if and only if

G_{i,m,a} = 0, for all i ∈ I(n, r) and a ∈ \underline{m},

since the v_{I} form a free basis of the comodule. We will prove this equation with the help of the Laplace-Extension which is a special case of Laplace-Duality (Proposition 8.6) applied to the partitions

λ_{t} := (2a − t + 1, \underbrace{1, 1, …, 1}_{(t − 1)-times}) ∈ Λ^{+}(t, 2a).

Caution: this symbol should not confused with the t-th component of a partition λ. A bideterminant T_{q}^{λ_{t}}(i : j) is the product of a t × t minor determinant with a monomial

T_{q}^{λ_{t}}(i : j) = T_{q}^{ω_{t}}((i_{1}, …, i_{t}) : (j_{1}, …, j_{t}))x_{i_{t+1}j_{t+1}}x_{i_{t+2}j_{t+2}} … x_{i_{2a}j_{2a}}.

Let L_{t} denote the set of distinguished left coset representatives of S_{λ_{t−1}} in S_{λ_{t}}. Using basic transpositions s_{i} this set can be written down explicitly

L_{t} = {id_{,}s_{t−1}, s_{t−2}s_{t−1}, …, s_{1}s_{2} … s_{t−1}}.

Setting

μ_{t} := ∑_{w∈L_{t}}(−y)^{−l(w)}β(w)

the quantum symplectic (left) Laplace-Extention deduced from Proposition 8.6 reads

μ_{t} ≀ T_{q}^{λ_{t−1}}(i : j) = T_{q}^{λ_{t}}(i : j).

There is a very useful recursive calculation rule for the endomorphisms μ_{t}:

−y^{−1}μ_{t}β_{t} = μ_{t+1} − id_{.}  (22)

Before we state the fundamental lemma of this section we remind the reader of the addition of multi-indices, for example j′j + (kk′) = (j_{1}, j′_{1}, …, j_{a}, j′_{a}, k, k′).

Lemma 13.2 Let l, a ∈ \underline{m} and i ∈ I(n, 2a). Then

G_{i,l,a} = ∑_{J∈P (l,a−1),k∈\underline{l}} q^{u_{J} −k}T_{q}^{λ_{2a−1}}(i : j′j + (kk′)) ≀ (id − y^{−a}β_{2a−1}).

Proof: First we treat the case where a = 1. Here we have G_{i,l,1} = ∑_{j=1}^{l} q^{−1}T_{q}^{ω_{2}}(i : (jj′)) by definition. Since P (l, 0) = ∅ the summation on the right hand side is over k ∈ \underline{l} too. Furthermore,

T_{q}^{λ_{1}}(i : (kk′)) ≀ (id − y^{−1}β_{1}) = x_{i_{1}k}x_{i_{2}k′} ≀ (id − y^{−1}β_{1}) = T_{q}^{ω_{2}}(i : (kk′)).

Thus both sides of the equation are identical. For the general case we use induction on l. In the case l = 1 we necessarily have a = 1 and there is nothing to prove. In order to prove the induction step we may assume a > 1 and l > 1. We divide the summation on the right hand side into three subsums:

(A) l ∈ J  (B) l ∉ J, k = l  (C) l ∉ J, k < l.

Using (6) we calculate for the subsum (A)

∑_{J∈P (l−1,a−2),k∈\underline{l}} q^{u_{J} −k−l}T_{q}^{λ_{2a−1}}(i : j′j + (ll′kk′)) ≀ (id − y^{−a}β_{2a−1}) =

∑_{J∈P (l−1,a−2),k∈\underline{l−1}} q^{u_{J} −k−l}T_{q}^{λ_{2a−1}}(i : j′j + (ll′kk′)) − y^{−a}q^{u_{J} +k−3l}T_{q}^{λ_{2a−1}}(i : j′j + (ll′k′k))

+q^{−2l} ∑_{J∈P (l−1,a−2)} q^{u_{J}} T_{q}^{λ_{2a−1}}(i : j′j + (ll′ll′)) − y^{−a}q^{u_{J}} T_{q}^{λ_{2a−1}}(i : j′j + (ll′l′l)).

The bideterminant T_{q}^{λ_{2a−1}}(i : j′j + (ll′l′l)) vanishes by Corollary 8.2. Unfortunately the other seprated bideterminant is not zero in general. By Corollary 10.3 we deduce from v_{ll′} = −v_{l′l} + (y^{−1} − 1) ∑_{k=1}^{l−1} q^{l−k}v_{kk′} the equation

T_{q}^{λ_{2a−1}}(i : j′j + (ll′ll′)) = (y^{−1} − 1)q^{l} ∑_{k=1}^{l−1} q^{−k}T_{q}^{λ_{2a−1}}(i : j′j + (kk′l′l)).

Since β_{2a−1}β_{2a−2}(v_{j′j+(k′kll′)}) = q^{2}v_{j′j+(k′llk)} and β_{2a−1}^{−1}β_{2a−2}^{−1}(v_{j′j+(kk′ll′)}) = q^{−2}v_{j′j+(kllk′)} we may again deduce from Corollary 10.3 that

T_{q}^{λ_{2a−1}}(i : j′j + (ll′kk′)) = T_{q}^{λ_{2a−1}}(i : j′j + (kk′ll′)) ≀ β_{2a−1}^{−1}β_{2a−2}^{−1}
T_{q}^{λ_{2a−1}}(i : j′j + (ll′k′k)) = T_{q}^{λ_{2a−1}}(i : j′j + (k′kll′)) ≀ β_{2a−1}β_{2a−2}

Here, in addition, we have used the equations v_{(ll′k′)} = q^{2}v_{(k′ll′)} and v_{(ll′k)} = q^{−2}v_{(kll′)} which are valid inside the exterior algebra. Modulo γ the congruence relation β^{−1} ≡ (y^{−1}β + (y^{−1} − 1)id) holds. Therefore, modulo the ideal G_{2a} of A_{2a} the congruence

β_{2a−1}^{−1}β_{2a−2}^{−1} ≡ y^{−2}β_{2a−1}β_{2a−2} + y^{−1}(y^{−1} − 1)(β_{2a−1} + β_{2a−2}) + (y^{−1} − 1)^{2}id

is valid which implies

T_{q}^{λ_{2a−1}}(i : j′j + (kk′ll′)) ≀ β_{2a−1}^{−1}β_{2a−2}^{−1} = y^{−2}T_{q}^{λ_{2a−1}}(i : j′j + (kk′ll′)) ≀ β_{2a−1}β_{2a−2}
+y^{−1}(y^{−1} − 1)T_{q}^{λ_{2a−1}}(i : j′j + (kk′ll′)) ≀ β_{2a−1} − (y^{−1} − 1)T_{q}^{λ_{2a−1}}(i : j′j + (kk′ll′)).

by Lemma 8.3. Here we have also used th fact that T_{q}^{λ_{2a−1}}(i : j′j + (kk′ll′)) ≀ β_{2a−2} = −T_{q}^{λ_{2a−1}}(i : j′j + (kk′ll′)) by Lemma 8.4. So far, our calculations lead to the following expression for the subsum (A):

∑_{J∈P (l−1,a−2),k∈\underline{l−1}} q^{u_{J} −k−l−4}(T_{q}^{λ_{2a−1}}(i : j′j + (kk′ll′)) ≀ β_{2a−1}β_{2a−2}

−y^{−(a−1)+k−(l−1)}T_{q}^{λ_{2a−1}}(i : j′j + (k′kll′)) ≀ β_{2a−1}β_{2a−2}
+(1 − y)T_{q}^{λ_{2a−1}}(i : j′j + (kk′ll′)) ≀ β_{2a−1}).

Now we apply Laplace-Extension twice to the first and second term and once to the third term. In case of the first term this gives

T_{q}^{λ_{2a−1}}(i : j′j + (kk′ll′)) = μ_{2a−1}μ_{2a−2} ≀ T_{q}^{λ_{2a−3}}(i : j′j + (kk′ll′)).

Note that we may commute the β_{2a_{1}}β_{2a−2} from the right of T_{q}^{λ_{2a−3}}(i : j′j + (kk′ll′)) to the left by (3), since a bideterminant corresponding to the partition λ_{2a−3} is a monomial on the part where β_{2a−1} and β_{2a−2} are operating on. A similar fact is true concerning the third term of the sum with respect to λ_{2a−2} and β_{2a−1}. Furthermore, note that for fixed J and arbitrary k ∈ I(n, r) we have

∑_{k=1}^{l−1} −y^{−(a−1)+k−(l−1)}T_{q}^{λ_{2a−1}}(k : j′j + (k′kll′)) = ∑_{k=1}^{l−1} −y^{−(a−1)}T_{q}^{λ_{2a−1}}(k : j′j + (kk′ll′)) ≀ β_{2a−3}

by (6). Substituting these equations into the above expression for the subsum (A) gives

∑_{J∈P (l−1,a−2),k∈\underline{l−1}} q^{u_{J} −k−l−4}(μ_{2a−1}μ_{2a−2}β_{2a−1}β_{2a−2}≀T_{q}^{λ_{2a−3}}(i : j′j+(kk′ll′))≀(id−y^{−(a−1)}β_{2a−3})

+(1 − y)μ_{2a−1}β_{2a−1} ≀ T_{q}^{λ_{2a−2}}(i : j′j + (kk′ll′))).

To the first term of that sum we can apply the induction hypothesis. To this claim note that the symbol ≀ on the right of the bideterminant stands for a sum over the bideterminants left multi-index. In order to apply the induction hypothesis this summation has to be commuted with the summation under the ∑-symbol. In a similar way we can apply Lemma 13.1 together with Corollary 10.3 to the second term of the sum above. This results in

∑_{J∈P (l−1,a−1)} q^{u_{J} −l}(y^{−2}μ_{2a−1}μ_{2a−2}β_{2a−1}β_{2a−2} ≀ T_{q}^{λ_{2a−2}}(i : j′j + (ll′))

+y^{−1}(y^{−1} − 1){a − 1}_{y^{−1}}μ_{2a−1}β_{2a−1} ≀ T_{q}^{λ_{2a−2}}(i : j′j + (ll′))).

Since μ_{2a−2} and β_{2a−1} commute, (22) yields

y^{−2}μ_{2a−1}μ_{2a−2}β_{2a−1}β_{2a−2} = (−y^{−1}μ_{2a−1}β_{2a−1})(−y^{−1}μ_{2a−2}β_{2a−2}) = (μ_{2a} − id)(μ_{2a−1} − id).

Similarily one calculates

y^{−1}(y^{−1} − 1){a − 1}_{y^{−1}}μ_{2a−1}β_{2a−1} = (1 − y^{−(a−1)})(μ_{2a} − id).

Finally we obtain the following expression for the subsum (A):

∑_{J∈P (l,a),l∈J} q^{u_{J}} (μ_{2a}μ_{2a−1} − μ_{2a−1} − y^{−(a−1)}(μ_{2a} − id)) ≀ T_{q}^{λ_{2a−2}}(i : j′j).

The calculation of subsum (B) only needs one application of Laplace-Extension together with the commutation of β_{2a−1} from the right of the bideterminant to the left:

∑_{J∈P (l,a),l∈J} q^{u_{J}} T_{q}^{λ_{2a−1}}(i : j′j) ≀ (id − y^{−a}β_{2a−1})

= ∑_{J∈P (l,a),l∈J} q^{u_{J}} (μ_{2a−1} + y^{−(a−1)}(μ_{2a} − id)) ≀ T_{q}^{λ_{2a−2}}(i : j′j).

Thus subsum (A) and (B) together equal

∑_{J∈P (l,a),l∈J} q^{u_{J}} μ_{2a}μ_{2a−1} ≀ T_{q}^{λ_{2a−2}}(i : j′j) = ∑_{J∈P (l,a),l∈J} q^{u_{J}} T_{q}^{λ_{2a}}(i : j′j)

To the subsum (C) the induction hypothesis can be applied directly and we see that it equals

G_{i,l−1,a} = ∑_{J∈P (l−1,a)} q^{u_{J}} T_{q}^{λ_{2a}}(i : j′j).

Thus, it follows that all three subsums add up to G_{i,l,a}. □

Now we are able to prove Proposition 11.2. As we have seen above, we have to show G_{i,m,a} = 0 for a = 1, …, m and i ∈ I(n, 2a). From (7) we already know G_{i,m,1} = 0 for all i ∈ I(n, 2). We will deduce the general case with help of Lemma 13.2. Let a > 1 and i ∈ I(n, 2a) be arbitrary. We apply Laplace-Extension to the formula of the lemma:

G_{i,m,a} = ∑_{J∈P (m,a−1),k∈\underline{l}} q^{u_{J} −k}μ_{2a−1} ≀ T_{q}^{λ_{2a−2}}(i : j′j + (kk′)) ≀ (id − y^{−a}β_{2a−1}).

As in the proof of the lemma we may commute (id − y^{−a}β_{2a−1}) to the other side of the bideterminant. Let (μ_{ih})_{i,h∈I(n,2a)} be the coefficient matrix of the endomorphism μ_{2a−1}(id − y^{−a}β_{2a−1}) with respect to the canonical basis. We denote the multi-index consisting of the first 2a − 2 indices of h by h̄ := (h_{1}, …, h_{2a−2}) and obtain

G_{i,m,a} = ∑_{h∈I(n,2a)} μ_{ih} ∑_{J∈P (m,a−1)} q^{u_{J}} T_{q}^{ω2a−2}(h̄ : j′j) ∑_{k=1}^{m} q^{−k}x_{h_{2a−1}k}x_{h_{2a}k′}

= ∑_{h∈I(n,2a)} μ_{ih}G_{h̄,m,a−1} ∑_{k=1}^{m} q^{−k}x_{h_{2a−1}k}x_{h_{2a}k′} = 0,

since G_{h̄,m,a−1} = 0 for all h ∈ I(n, 2a) by the induction hypothesis.

14 Finishing the proof of Theorem 6.1

Let us briefly recall what we have done so far. With respect to the proof that B_{r} is a basis, we have reduced the fact that it is a set of generators in section 7 to the verification of Proposition 7.1, which we just have completed. Further we already know from section 6 that the axiom (C2^{∗}) of a cellular coalgebra is valid. It remains to show axiom (C3^{∗}) and the fact that B_{r} is linearly independent.

Let us start with the latter task. It is clearly enough to consider the case where R = Z = ℤ [q, q^{−1}] Any relation among elements of B_{r} with coefficients from Z is a relation with coefficients from the field of fractions 𝕂 on Z, too. Thus, we only have to show |B_{r}| = dim_{𝕂} A_{𝕂 ,Q}^{s}(n, r). Now, A_{𝕂 ,Q}^{s}(n, r) is the centralizer coalgebra of the Birman-Mu-rakami-Wenzl algebra acting on V^{⊗r}. Consequently, by the comparison theorems from [O2] the dimension in question is the same as the dimension of the centralizer algebra of that action on V^{⊗r}. But this dimension can be deduced from well-known results from the theory of quantum groups, for which we refer to [CP].

With regard to this claim note that, although our β differs from the corresponding Quantum Yang Baxter operator in [CP, chapter 7 (p. 237)], by a multiple of q^{−1}, the centralizer algebra must be the same. By [CP, Theorem 10.2.5], this algebra is just the image of the quantized universal enveloping algebra (QUE) corresponding to the Dynkin diagram C_{m} under its action on V^{⊗r}. Now, by [CP, Proposition 10.1.13 and Theorem 10.1.14], the tensor space V^{⊗r} decomposes into irreducibles as a QUE-module because q ∈ 𝕂 is transcendental over ℚ. These irreducibles are indexed by the highest weights of the symplectic group and have corresponding dimensions. The weights occurring are the same as in the case of the symplectic group and correspond precisely to the elements of the set Λ from the definition of B_{r} (cf. [O2, 7.1]). It follows from work of R.C. King that these dimensions are just |M(λ)| ([Ki],cf. [Do2]). Consequently, we obtain the required identity:

dim_{𝕂} A_{𝕂 ,Q}^{s}(n, r) = ∑_{λ∈Λ} |M(λ)|^{2} = |B_{r}|.

We now verify axiom (C3^{∗}). We abbreviate K := A_{R,q}^{s}(n, r). Let D\underline{λ} ; i,j ∈ B_{r} where \underline{λ} = (λ, l) ∈ Λ and i, j ∈ M(λ). As d_{q}^{l} is grouplike and Δ a homomorphism of algebras we calculate using (18) that

Δ(D\underline{λ} ; i,j) = (d_{q}^{l} ⊗ d_{q}^{l})Δ(T_{q}^{λ}(i : j)) = ∑_{h∈I_{λ}^{<}} D\underline{λ} ; i,h ⊗ D\underline{λ} ; h,j.

Here, as in section 10, I_{λ}^{<} is the set of multi-indices that are λ-column-standard with respect to the usual order < on \underline{n} (see section 7). Now, according to the straightening formula 7.1 (after application of ^{∗}) to each h ∈ I_{λ}^{<} and k ∈ M(λ) there is an element a_{hk} ∈ R (unique by the linear independence of B_{r}) such that

D\underline{λ} ; h,j ≡ ∑_{k∈M(λ)} a_{hk}D\underline{λ} ; k,j mod K(> \underline{λ}).

We set

h(k, i) := ∑_{h∈I_{λ}^{<}} D\underline{λ} ; i,ha_{hk} ∈ K(≥ \underline{λ})

and obtain

Δ(D\underline{λ} ; i,j) ≡ ∑_{k∈M(λ)} h(k, i) ⊗ D\underline{λ} ; k,j mod K(≥ \underline{λ}) ⊗ K(> \underline{λ}).

This completes the verification of axiom (C3^{∗}) and hence the proof of Theorem 6.1.

15 Quasi-Heredity of the Symplectic q-Schur Algebra

In [GL] Graham and Lehrer have presented a nice criterion for quasi-heridity of a cellular algebra which we will now verify in our case. This will prove Theorem 6.4.

To this aim we have to investigate the bilinear form ϕ\underline{λ} on the standard modules W (\underline{λ}). We must show that they are not zero ([GL, 3.10]. Let us first calculate the Gram-matrix of ϕ\underline{λ} with respect to the basis \{C\underline{λ} ; i| i ∈ M(\underline{λ})\} of W (\underline{λ}). We abbreviate its entries by ϕ_{ij} := ϕ\underline{λ}(C\underline{λ} ; i, C\underline{λ} ; j) ∈ R. According to the definition in [GL, 2.3], these numbers satisfy

C\underline{λ} ; i,kC\underline{λ} ; l,j ≡ ϕ_{kl}C\underline{λ} ; i,j mod S_{R,q}^{s}(n, r)(< \underline{λ}).

Such a congruence relation is valid with ϕ_{kl} being independent of i and j by the axioms of a cellular algebra (see [GL, 1.7]). Dualizing this congruence we obtain the following counterpart in the cellular coalgebra K = A_{R,q}^{s}(n, r):

Δ(D\underline{λ} ; i,j) ≡ ∑_{k,l∈M(\underline{λ})} ϕ_{kl}D\underline{λ} ; i,k ⊗ D\underline{λ} ; l,j

modulo K(≥ \underline{λ}) ⊗ K(> \underline{λ}) + K(> \underline{λ}) ⊗ K(≥ \underline{λ}). According to the calculations for the verification of axiom (C3^{∗}) in the previous section we see using the notations from there that

ϕ_{kl} = ∑_{h∈I_{λ}^{<}} a_{hk}a_{hl}.  (23)

The bilinear form ϕ\underline{λ} is different from zero if this is the case for a single entry ϕ_{kl}. We calculate ϕ_{kk} where k is the λ-tableau T_{k}^{λ} = T with constant rows T (i, j) := m + i for all 1 ≤ i ≤ m and 1 ≤ j ≤ λ_{j}. Obviously T is a standard tableau with respect to both orders on \underline{n}, namely < as well as ≺. Furthermore the reverse symplectic condition holds because T (i, j) = (m − i + 1)′. Consequently we have k ∈ M(\underline{λ}) ∩ I_{λ}^{<}. Note that k does not contain any pair of associated indices (i, i′). This implies that each multi-index h for which a_{hk} ≠ 0 must have the same content as k, as can be seen from the proof of the straightening algorithm. The content η := |k| of k is given by

η_{i} = \{ \begin{matrix}0 & i ≤ m \\ λ_{i−m} & i > m. \end{matrix}

Note, that k is the only element in I_{λ}^{<} having this content. This means a_{hk} = 0 if h ≠ k and we conclude

ϕ_{kk} = ∑_{h∈I_{λ}^{<}} a_{hk}^{2} = a_{kk}^{2} = 1.

By [GL, Remark 3.10], this finishes the proof of Theorem 6.4.

16 Outlook

Dualizing the coalgebra map A_{R,q}^{s}(n, r−2) →^{·d_{q}} A_{R,q}^{s}(n, r) from the sequence (8), one obtains an epimorphism of algebras from S_{q}^{s}(n, r) to S_{q}^{s}(n, r−2). On a basis element C\underline{λ} ; i,j it is given by subtracting 1 from l in \underline{λ} = (λ, l) and keeping i, j fixed. Its kernel is the linear span of those basis elements which occur in the case l = 0. This forces a recursive structure on the representation theory of these algebras in a similar way as is known for the Birman-Murakami-Wenzl algebras (see [BW]). In addition these epimorphisms can be used to define an inverse limit of the symplectic q-Schur algebras in a similar way as has been worked out for the type A q-Schur algebra in [GR, section 6.4]. It seems to be plausible that accordingly the quantized universal enveloping algebra embeds into this inverse limit.

Concerning analogues to the orthogonal case, note that Lemma 10.1 will not work here. Maybe, a way out is to consider coefficient functions of the symmetric algebra, i.e. the elements

∑_{w∈S_{λ}} y^{−l(w)}β(w) ≀ x_{ij} = ∑_{w∈S_{λ}} y^{−l(w)}x_{ij} ≀ β(w)

instead of bideterminants, which are coefficient functions of the exterior algebra.

References

[BW] Birman, J., Wenzl, H,: Braids, Link Polynomials and a new Algebra. Tansactions of the Amer. Math. Soc., Vol. 313, No. 1 (1989), 249-273.

[CP] Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press. 1994.

[Co] De Concini C.: Symplectic Standard Tableaux. Advances in Mathematics 34 (1979), 1-27.

[DD] Dipper, R., Donkin, S.: Quantum GL_{n}. Proc. London Math. Soc. 63 (1991), 165-211.

[DJ] Dipper, R., James, G.: The q-Schur Algebra. Proc. London Math. Soc. (3) 59 (1989), 23-50.

[DJ2] Dipper, R., James, G.: q-tensor space and q-Weyl modules, Trans. A.M.S. 327 (1991), 251-282.

[DJM] Dipper, R., James, G., Mathas, A.: Cyclotomic q-Schur Algebras. Math. Zeitschrift 229 (1998), 385-416.

[Do1] Donkin, S.: Good Filtrations of Rational Modules for Reductive Groups. Arcata Conf. on Repr. of Finite Groups. Proceedings of Symp. in Pure Math., Vol. 47 (1987), 69-80.

[Do2] Donkin, S.: Representations of symplectic groups and the symplectic tableaux of R.C. King. Linear and Multilinear Algebra, Vol. 29 (1991), 113-124.

[Dt] Doty, S.: Polynomial Representations, Algebraic Monoids, and Schur Algebras of Classical Type. J. of Pure and Applied Algebra, 123 (1998), 165-199.

[GL] Graham, J.J., Lehrer, G.I.: Cellular Algebras. Invent. Math. 123 (1996), 1-34.

[Gr] Green, J.A.: Combinatorics and the Schur algebra. J. of Pure and Appl. Alg. 88 (1993), 89-106.

[GR] Green, R.M.: q-Schur algebras and quantized enveloping algebras. Thesis. University of Warwick, 1995.

[HH] Hashimoto, M., Hayashi, T.: Quantum Multilinear Algebra. Tohoku Math. J., 44 (1992), 471-521.

[Ha] Hayashi, T.: Quantum Groups and Quantum Determinants. J. of Algebra 152 (1992), 146-165.

[Ki] King, R.C.: Weight multiplicity for classical groups., Group Theoretical Methods in Physics (fourth International Colloquium, Nijmegen 1975), Lecture Notes in Physics 50, Springer 1975.

[KX] König, S., Xi, C.: On the structure of cellular algebras. Algebras and Modules II, Proceedings of ICRA VIII (Geiranger), CMS Conference Proceedings.

[Ma] Martin, S.: Schur Algebras and Representation Theory. Cambridge University Press, 1993.

[O1] Oehms, S.: Symplektische q-Schur-Algebren, Thesis, University of Stuttgart. Shaker Verlag Aachen, 1997.

[O2] Oehms, S.: Centralizer Coalgebras, FRT-Construction and Symplectic Monoids. J. of Algebra 244 (2001), 19-44.

[RTF] Reshetikhin, N. Y., Takhtajan, L. A., Faddeev, L. D.: Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990), 193-225.

[Tk] Takhtajan, L.A.: Lectures on Quantum Groups. In: Introduction to Quantum Group and Integrable Massive Models of Quantum Field Theory (Hrsg.: M.-L. Ge, B.-H. Zhao. World Scientific, 1990.

[We] Wenzl, H.: Quantum Groups and Subfactors of Type B, C and D. Commun. Math. Phys. 133 (1990), 383-432.