Centralizer Coalgebras, FRT-Construction and Symplectic

Monoids

Sebastian Oehms^{1}, Mathematisches Institut B, Universität Stuttgart, D70550 Stuttgart, Germany

1 Introduction

L. Faddeev, N. Reshetikhin and L. Takhtadjian [RTF] introduced a construction to obtain quantum deformations of coordinate rings of classical groups. General considerations about this so called FRT-construction can be found for instance in [Ma],[Ta], [Ha], [Su] and also in many textbooks on quantum groups. Our approach differs from former ones in the following three aspects:

First, we focus attention to the graded matric bialgebra which arises in the first step of the construction. This means that we rather look at quantizations of appropriate closed monoids instead of classical groups. Especially we look at the homogenous summands of these graded bialgebras. These are coalgebras which can be defined in a dual way to centralizer algebras of subsets in an endomorphism ring. We therefore call them centralizer coalgebras and investigate their relationship to the corresponding centralizer algebras.

Further, we work over arbitrary noetherian integral domains as base rings. This makes sense since all known examples are already well defined over rings of integral Laurent polynomials, i.e. ℤ[X, X^{−1}] (in the indeterminate X). We will see that there are tremendous differences to the theory over fields, especially concerning the comparison of centralizer algebras and coalgebras. For instance the centralizer coalgebra may have R-torsion. We present the following criterion for R-projectivity: This property holds if and only if the centralizer algebra is stable under base changes. Furthermore, we will see that the latter property is always valid for centralizer coalgebras.

Finally, the FRT-construction in the ordinary form depends on exactly one endomorphism which usually is a quantum Yang-Baxter operator in the applications. Here we give another version of the FRT-construction which can be applied to sets of endomorphisms. This generalization is necessary to describe the coordinate rings of classical symplectic and orthogonal monoids by use of an FRT-type construction. We demonstrate this in the symplectic case, giving some improvements of results by S. Doty [Dt]. Furthermore, applying our results on centralizer coalgebras we obtain an integral form for the symplectic Schur algebra defined by S. Donkin in [Do2] without any use of the hyperalgebra or Kostant ℤ-form. As an additional incredient we need a symplectic version of the straightening formula for bideterminants the proof of which covers all of the last two sections.

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

2 Centralizer Coalgebras

Let R be a noetherian integral domain, V a free R-module with a fixed basis {v_{1}, …, v_{n}}. Let e_{i}^{j} denote the corresponding basis of matrix units for E := End_{R}(V). The algebra structure on E induces a coalgebra structure on the dual R-module E^{∗} := Hom_{R}(E, R). On the dual basis elements the comultiplication Δ and the counit ϵ are given by

Δ(e_{i}^{∗}^{j}) = ∑_{k=1}^{n} e_{i}^{∗}^{k} ⊗ e_{k}^{∗}^{j}, ϵ(e_{i}^{∗}^{j}) = δ_{ij}.

We will be engaged with epimorphic coalgebra images of E^{∗} since they correspond to subalgebras of E. Consider the isomorphism ϑ_{tr} : E → E^{∗} of R-modules induced by the nondegenerate bilinear form corresponding to the matrix trace map tr : E → R, i.e. ϑ_{tr}(μ)(ν) := tr(μν). Note that ϑ_{tr} maps e_{i}^{j} onto e_{j}^{∗}^{i}. Let A ⊆ E be an arbitrary subset and L(A) the R-linear span in E of all commutators [ν, μ] = νμ − μν where ν runs through A and μ runs through E. We set K(A) := ϑ_{tr}(L(A)) ⊆ E^{∗}. If A consists of just one element ν we use abbreviations L(ν) and K(ν) for L(A) and K(A).

Lemma 2.1 K(A) is a coideal in E^{∗} for each subset A ⊆ E.

Sketch of Proof: Since the sum of coideals is a coideal again we only need to consider the case A = {ν}. Using an explicit expression for ν in terms of the basiselements e_{i}^{j} one calculates for arbitrary numbers k, l ∈ {1, …, n}:

Δ(ϑ_{tr}([ν, e_{k}^{l}])) = ∑_{m}(e_{l}^{∗}^{m} ⊗ ϑ_{tr}([ν, e_{k}^{m}]) + ϑ_{tr}([ν, e_{m}^{∗}^{l}]) ⊗ e_{m}^{∗}^{k},

which just means

Δ(K(ν)) ⊆ K(ν) ⊗ E^{∗} + E^{∗} ⊗ K(ν).

In a simialar way one shows ϵ(K(ν)) = 0. A more detailed proof can be found in section 1.3 of [Oe]. □

If C is an arbitrary coalgebra such that V is a C-comodule with structure map τ_{V} : V → V ⊗ C we denote the set of C-comodule endomorphisms by

End_{C}(V) := {μ ∈ E| (μ ⊗ id_{C}) ∘ τ_{V} = τ_{V} ∘ μ}.

Clearly this is a subalgebra of E. The proof of the next lemma is similar to the preceeding one.

Lemma 2.2 Let C be a coalgebra together with an epimorphism π : E^{∗} → C. Then

μ ∈ End_{C}(V) ⟺ K(μ) ⊆ ker(π).

Corollary 2.3 Let M be a coideal in E^{∗} and μ, ν ∈ E. Then K(μ) ⊆ M and K(ν) ⊆ M implies K(μν) ⊆ M and K(νμ) ⊆ M.

We now define the centralizer coalgebra of the subset A ⊆ E as

M(A) := E^{∗}/K(A).

According to lemma 2.2 we have

A ⊆ End_{M(A)}(V).

Furthermore M(A) is the largest epimorphic image of E^{∗} with this property. By the corollary M(A) does not change if A is substituted by its algebraic span. Before going deeper into the analysis of relationships between M(A) and the centralizer algebra

C(A) := End_{A}(V) = {μ ∈ E| [μ, ν] = 0, for all ν ∈ A}

we give a presentation of M(A) by generators and relations, which is convenient for practical use.

The residue classes of the basis elements e_{i}^{∗}^{j} with respect to any coideal in E^{∗} will always be denoted by x_{ij} where i, j ∈ \underline{n} := \{1, …, n\}. If μ = ∑_{i,j=1}^{n} a_{ij}e_{i}^{j} ∈ E is arbitrary we write

μx_{ij} := ∑_{k=1}^{n} a_{ik}x_{kj} and x_{ij}μ := ∑_{k=1}^{n} x_{ik}a_{kj}.  (1)

Now, if N is a subset of E and A its algebraic span then M(A) is defined by the generators x_{ij} and the relations

μx_{ij} = x_{ij}μ for all μ ∈ N, i, j ∈ \underline{n}.  (2)

As consequences one has relations of the same form where μ runs through all of A.

3 Comparison Theorems

Remember the definition of the complement

U^{⊥} := {f ∈ W^{∗} = Hom_{R}(W, R)| f(u) = 0 ∀u ∈ U}

of a submodule U in an R-module W and the definition of the evaluation map

Ev_{W} : W → W^{∗}^{∗}, given by Ev_{W} (x)(y) := y(x) x ∈ W, y ∈ W^{∗}

In W^{∗}^{∗}^{∗} we have the following commutativity rule

Ev_{W^{∗}}(U^{⊥}) = Ev_{W} (U)^{⊥}.  (3)

Turning to our special situation we first note that tr(b[x, a]) = tr(x[a, b]) for all x, a, b ∈ E. Since the bilinear form induced by tr is nondegenerate, it follows:

ϑ_{tr}([x, a])(b) = tr(b[x, a]) = 0 for all x ∈ E ⟺ [a, b] = 0.  (4)

The following fundamental lemma of this section is easy to prove now.

Lemma 3.1 We have K(A)^{⊥} = Ev_{E}(C(A)) and K(A)^{⊥}^{⊥} = Ev_{E^{∗}}(C(A)^{⊥})

Proof: According to the definition Ev_{E}(b) ∈ K(A)^{⊥} if and only if Ev_{E}(b)(ϑ_{tr}([x, a])) = ϑ_{tr}([x, a])(b) = 0 for all x ∈ E and a ∈ A. Applying (4) this is the case if and only if [a, b] = 0 for all a ∈ A, thus if and only if b ∈ C(A). The second equation follows from the first by use of equation (3). □

Remember that the dual module C^{∗} = Hom_{R}(C, R) of a coalgebra C always posseses the structure of an algebra by use of the convolution product

μν := (μ ⊗ ν) ∘ Δ where μ, ν ∈ C^{∗}.

Here, we have identified μ⊗ν with its image under the natural homomorphism C^{∗}⊗C^{∗} → (C ⊗ C)^{∗}. Note that this construction is functorial. In the special case C = E^{∗} we obtain an algebra structure on E^{∗}^{∗}. Furthermore it is easy to show that the evaluation map Ev_{E} : E → E^{∗}^{∗} is an isomorphism of algebras.

Lemma 3.2 Let C be a coalgebra together with an epimorphism π : E^{∗} → C. Set K := ker(π). Then ρ := Ev_{E}^{−1} ∘ π^{∗} : C^{∗} → E is a monomorphism of R-algebras and im(ρ) = Ev_{E}^{−1}(K^{⊥}).

Proof: By functoriality the dual map π^{∗} : C^{∗} → E^{∗}^{∗} is an algebra homomorphism. Since Hom_{R}(−, R) is exact on the right, π^{∗} is injective. One easily shows im(π^{∗}) = K^{⊥}. This completes the proof, since Ev_{E} is an algebra isomorphism as mentioned above. □

Theorem 3.3 (First Comparison Theorem) The centralizer algebras C(A) and the dual of the centralizer coalgebra M(A)^{∗} are isomorphic to each other. An isomorphism is given by ρ.

Proof: This follows immediately from lemmas 3.1 and 3.2. □

Now, let us compare the dual of C(A) with M(A). To this aim we consider the dual map J^{∗} : E^{∗} → C(A)^{∗} of the inclusion J : C(A) ↪ E. Because of Ev_{E^{∗}}(U) ⊆ U^{⊥}^{⊥} for arbitrary submodules U and according to Lemma 3.1 we have

K(A) ⊆ Ev_{E^{∗}}^{−1}(K(A)^{⊥}^{⊥}) = Ev_{E^{∗}}^{−1}(Ev_{E^{∗}}(C(A)^{⊥})) = C(A)^{⊥} = ker(J^{∗}).

Therefore J^{∗} factors to an R-module homomorphism

θ : M(A) → C(A)^{∗}.

Lemma 3.4 The kernel of θ is precisely the torsion submodule of M(A)

Proof: From general results of commutative algebra (cf. [Oe] Anhang A 1.2) it follows that the torsion submodule of M(A) coincides with Ev_{E^{∗}}^{−1}(K(A)^{⊥}^{⊥})/K(A). By the above calculations this is just the kernel C(A)^{⊥}/K(A) of θ. □

This immediately implies Corollary 3.5 (Criterion of torsion freeness) The following statements are equivalent:

(a) M(A) is torsion free

(b) K(A) = C(A)^{⊥}

(c) θ is injective.

Remark 3.6 The map θ is surjective if and only if the extension group Ext_{R}^{1}(E/C(A), R) is trivial, in particular if C(A) is a direct summand in E.

At the beginning of this section we have constructed an algebra structure on the dual module of a coalgebra. Conversely we should obtain a coalgebra structure on the dual of an algebra A. Whereas this is not possible in general, it can be done under certain restrictions to the R-module structure of the algebra A. To be more precise, the natural R-homomorphism from A^{∗} ⊗ A^{∗} into (A ⊗ A)^{∗} must be an isomorphism. This is the case if A is projective and finitely generated. Therefore, under these circumstances the construction is always possible and functorial, that is: duals of algebra maps become coalgebra maps. Thus we obtain

Theorem 3.7 (Second Comparison Theorem) Suppose the centralizer algebra C(A) of A is a direct summand in E as an R-module. Then C(A)^{∗} can be turned into a coalgebra. The map θ : M(A) → C(A)^{∗} is an epimorphism of coalgebras, whose kernel is just the torsion submodule of M(A).

4 Change of Base Rings

Let S be another noetherian integral domain together with a homomorphism R → S of rings. We will consider S as an R-algebra via this homomorphism. In this situation there is a functor from the category of R-modules into the category of S-modules given by

W^{S} := S ⊗_{R} W and α^{S} := id_{S} ⊗ α : U^{S} → W^{S}  (5)

to each pair of R-modules W and U and an R-homomorphism α : U → W. We will study the behaviour of the construction of centralizer coalgebras under these functors. It will turn out that the centralizer coalgebra behaves better than the centralizer algebra. To start with let

E_{S} := End_{S}(V^{S}), ζ_{S} : E^{S} → E_{S}

where ζ_{S} is given by ζ_{S}(s ⊗ e)(t ⊗ v) := st ⊗ e(v) for all s, t ∈ S, e ∈ End_{R}(V), v ∈ V. Let J_{A} : A ↪ E be the natural embedding of the subalgebra A ⊆ E and let

A_{S} := im(ζ_{S} ∘ J_{A}^{S})

denote the image of A^{S} in E_{S}. Further there are natural homomorphisms

η_{S} : C(A)^{S} → C(A_{S}) = End_{A_{S}}(V^{S}) ⊆ E_{S}

on generators given in a similar way to ζ_{S}. Note, that both, ζ_{S} and η_{S} are homomorphisms of algebras connected by the equation

J_{C(A_{S})} ∘ η_{S} = ζ_{S} ∘ J_{C(A)}^{S}.

Thus η_{S} is injective if and only if J_{C(A)}^{S} is injective (J stands for the corresponding embeddings). This may fail if C(A) is not a pure R-submodule in E. Further η_{S} may fail to be surjective (see the example following theorem 4.3). C(A) is called stable under base change if η_{S} is an isomorphism of for all choices for S. Now, in analogy to η_{S} we are going to consider natural homomorphisms

μ_{S} : M(A)^{S} → M(A_{S}) = (E_{S})^{∗}/K(A_{S})

We will show that they are isomorphisms independent of the choices for A, R and S. To this claim we consider

χ_{S} := ζ_{S}^{∗}^{−1} ∘ ψ_{E} : E^{∗}^{S} → (E_{S})^{∗}

where ψ_{E} : E^{∗}^{S} → E^{S}^{∗} is the natural homomorphism given by ψ_{E}(s ⊗ f)(t ⊗ e) := stf(e) on genorators with s, t ∈ S, f ∈ E^{∗} e ∈ W. It is easy to check that χ_{S} is a homomorphism of coalgebras if the coalgebra structure on E^{∗}^{S} is defined in a canonical way (for details see [Oe] section 1.5). Further, the verification of the commutativity rule

χ_{S} ∘ ϑ_{tr}^{S} = ϑ_{tr}_{S} ∘ ζ_{S}  (6)

is straightforward, as well. Here ϑ_{tr}_{S} : E_{S} → (E_{S})^{∗} is the isomorphism induced by the matrix trace map tr_{S} : E_{S} → S. Setting

L(A_{S}) :=< [ν, μ]|ν ∈ A_{S}, μ ∈ E_{S} >_{S−mod},

we obtain

im(ζ_{S} ∘ J_{L(A)}^{S}) = L(A_{S}),

since ζ_{S} is an isomorphism of S-algebras. By definition we have K(A)_{S} = ϑ_{tr}_{S}(L(A)_{S}) and im(J_{K(A)}^{S}) = im(ϑ_{tr}^{S} ∘ J_{L(A)}^{S}). Using (6) this yields

im(χ_{S} ∘ J_{K(A)}^{S}) = K(A_{S}).  (7)

Here again, we have used the symbol J to indicate embeddings of R-submodules. Note that in particular K := im(J_{K(A)}^{S}) is a coideal in E^{∗}^{S} since χ_{S} is an isomorphism of coalgebras and therefore M(A)^{S} ≅ E^{∗}^{S}/K is a coalgebra. Finally, we are able to define the natural homomorphism μ_{S} as the factorization of χ_{S} which exists by (7). We immediately obtain

Theorem 4.1 (Change of Base Rings) For any noetherian integral domain S which is an R-algebra and any R-subalgebra A of E there is a natural homomorphism

μ_{S} : S ⊗_{R} M(A) → M(S ⊗_{R} A)

which is an isomorphism of S-coalgebras.

This means that M(A) is stable under base changes for all choices of A and R. If the R-algebra S is a field, it follows from theorems 3.3 and 4.1 that

dim_{S}(C(A_{S})) = dim_{S}(M(A_{S})^{∗}) = dim_{S}((M(A)^{S})^{∗}) = dim_{S}(M(A)^{S})  (8)

Now, for a noetherian integral domain R it is known from commutative algebra that an R-module W is projective if and only if the dimension of W^{S} is independent of the field S. Thus we obtain

Corollary 4.2 Let F be the field of fractions of R. Then M(A) is projective if and only if dim_{S}(C(A_{S})) = dim_{F} (C(A_{F})) holds for each field S.

Theorem 4.3 (Criterion of Projectivity) The following statements are equivalent:

(a) The centralizer coalgebra M(A) is projectiv.

(b) The centralizer algebra C(A) = End_{A}(V) is stable under base change.

(c) M(A) is torsion free and C(A) a direct summand in E.

Proof: First assume (a). Then the sequence

0 → K(A) → E^{∗} → M(A) → 0

is split and consequently the same is true for

0 → M(A)^{∗} → E^{∗}^{∗} → E^{∗}^{∗}/K(A)^{⊥} → 0.

Since Ev_{E} induces an isomorphism between E^{∗}^{∗}/K(A)^{⊥} and E/C(A) according to lemma 3.1 it follows that E/C(A) is projective, as well. Thus C(A) is a direct summand in E proving (c).

Part (a) follows from (c) by theorem 3.7, since the dual of a projective module is projective again. To verify (b) we therefore may assume both (a) and (c). Since C(A) is a direct summand J_{C(A)}^{S} is injective for all R-algebras S. Consequentely all η_{S} are injective (see above). To show surjectivity note that the image im(ζ_{S} ∘ J_{C(A)}^{S}) = im(J_{C(A_{S})} ∘ η_{S}) of C(A)^{S} in E_{S} must be a direct summand therein, since ζ_{S} is an isomorphism and im(J_{C(A)}^{S}) a direct summand in E^{S}. Therefore, to show that this submodule of E_{S} coincides with C(A_{S}), it is enough to verify that both have the same rank (the dimension of the Gtensored module over the field G of fractions on S). But these ranks must indeed be the same as can be seen from the following calculations

dim_{G}(C(A)^{G}) = dim_{F} (C(A)^{F}) = dim_{F} (C(A_{F})) = dim_{G}(C(A_{G})).

where the left-hand-side equation holds by projectivity of C(A), the right-hand-side one by corollary 4.2 and the one in the middle since η_{F} is an isomorphism by flatness of the field F of fractions on R. This establishes (b).

Now assume (b). This implies that the map J_{C(A)}^{S} induced by the embedding J_{C(A)} is injective for all S. By commutative algebra arguments one concludes that C(A) is a direct summand in the R-free module E, in particular it is projective. Now, let S be a field. Since η_{S} is an isomorphism we have

dim_{S}(C(A)^{S}) = dim_{S}(C(A_{S})).

The left-hand-side is independent of S by projectivity of C(A). Thus by corollary 4.2 M(A) is projective yielding (a). □

Example: Let R = ℤ and V = ℤ^{4}. Further let

a := ( \begin{matrix}0 & 2 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 2 \end{matrix} ) ∈ E = End_{ℤ}(ℤ^{4})

and A :=< a > be the subalgebra in E generated by a. Each field can be considered as a ℤ-algebra. For a field of characteristic different from 2 the minimum polynomial of a^{S} is t^{2} − 2t, but in characteristic 2 it is t^{2}. This means that the algebra A_{S} is two dimensional for each field S. It follows that A is a direct summand in E. But nevertheless M(A) is not projective (free), because in the case of a field of characteristic different from 2 a^{S} is diagonalisable and one calculates dim_{S}(C(A_{S})) = 2^{2} + 2^{2} = 8, while in the case of characteristic 2 we have dim_{S}(C(A_{S})) = 9. Therefore M(A) can’t be projective in view of corollary 4.2.

5 FRT-Construction

The r fold tensor product of V is denoted by V^{⊗r}. Since it is a free R-module, as well, we may apply all results of former sections to a situation where V is substituted by V^{⊗r}, E by E_{r} := End_{R}(V^{⊗r}) and E^{∗} by E_{r}^{∗} := Hom_{R}(E_{r}, R). There are natural isomorphisms between E_{r} and E^{⊗r} and between E_{r}^{∗} and E^{∗}^{⊗r}. We will omit them in our notation and consider them as identity maps. Set V^{⊗0} := E_{0} := R.

Suppose we are given a family A = (A_{r})_{r∈ℕ_{0}} of R-algebras such that A_{r} ⊆ E_{r}. According to lemma 2.1 there is an associated family of coideals K_{r} in E_{r}^{∗}.

K_{r} := K(A_{r}).  (9)

By the above identification we consider T (E^{∗}) := ⨁_{r∈ℕ_{0}} E_{r}^{∗} as the tensor algebra on the Rmodule E^{∗}. There is a unique coalgebra structure on this tensor algebra, extending the one from E^{∗} and turning T (E^{∗}) into a bialgebra. Furthermore the E_{r}^{∗} are subcoalgebras dual to the algebras E_{r}, i.e. the coalgebra structure on E_{r}^{∗} is the one considered above. Epimorphic images of this bialgebra are called matric bialgebras (cf. [Ta]). Let us investigate under what circumstances the coideal

I := ⊕_{r∈ℕ_{0}}K_{r}  (10)

becomes an ideal in T (E^{∗}) and thus consequently is a biideal. To this claim we consider inclusion maps s_{r}, t_{r} : E_{r} → E_{r+1} given by

s_{r}(μ) = μ ⊗ id_{V} , t_{r}(μ) = id_{V} ⊗ μ, μ ∈ E_{r}

and focus attention to a special situation, which is general enough for our applications. We start with an arbitrary subset N ⊆ E_{2} and define a family A inductively beginning with A_{0} := R, A_{1} := R · id_{V} and A_{2} as the algebraic span of N in E_{2}, than continuing by the formula

A_{r} := ⟨s_{r−1}(A_{r−1}) + t_{r−1}(A_{r−1})⟩_{Alg} for r > 2.

We call this a by N induced family of subalgebras of E_{r}.

Proposition 5.1 Let A be the family induced by a subset N ⊆ E_{2}. Then the coideal I defined in (10) is a homogenous biideal in T (E^{∗}) generated by the coideal K_{2}.

Proof: First we show that I is a homogenous ideal. To this claim take a ∈ K_{r} and b ∈ E_{u}^{∗}. We have to show a ⊗ b ∈ K_{r+u} and b ⊗ a ∈ K_{r+u}. The choice of the element a can be reduced to a generator a = ϑ_{tr}([μ, ν]) of the R-module K_{r} with μ ∈ E_{r} and ν ∈ A_{r}. Letting ν̂ := s_{r+u−1} ∘ s_{r+u−2} ∘ … ∘ s_{r}(ν) = ν ⊗ id_{V^{⊗u}} ∈ A_{r+u} and μ̂ := μ ⊗ b̄, where b̄ ∈ E_{u} is the preimage of b under ϑ_{tr} one obtains

a ⊗ b = ϑ_{tr}([μ, ν]) ⊗ b = ϑ_{tr}([μ̂, ν̂]) ∈ K_{r+u}.

Similary b ⊗ a ∈ K_{r+u} and the first statement is established.

For the second part denote by J the homogenous ideal in T (E^{∗}) generated by K_{2}. Since I is a homogenous ideal containing K_{2} we get J ⊆ I. For the reverse inclusion we show K_{r} ⊆ J by induction on r. Clearly K_{0} = K_{1} = (0) and K_{2} are contained in J. Now suppose r > 2. Letting J_{r} := J ∩E_{r}^{∗} one obtains J_{r} = J_{r−1} ⊗E^{∗} +E^{∗} ⊗J_{r−1}. By induction hypothesis we see K_{r−1} = J_{r−1}. Therefore

J_{r} = K_{r−1} ⊗ E^{∗} + E^{∗} ⊗ K_{r−1} = K(M)

where M := s_{r−1}(A_{r−1}) + t_{r−1}(A_{r−1}) ⊆ E_{r}. But since A_{r} is generated by M as an algebra we finally see from corollary 2.3 K_{r} = K(A_{r}) = K(M) = J_{r}. □

According to the proposition we may assign a graded matric bialgebra to each subset N ⊆ E_{2} by

M(N) := T (E^{∗})/I  (11)

whose homogenous summand are the centralizer coalgebras M(A_{r}). This bialgebra will be called the FRT-construction corresponding to the subset N. The reader familiar with the ordinary FRT-construction will recognize the latter one as the special case where N consists of just one element β (use the description (12) below). In this case we write M(β) := M(N). Usually in the applications this β is a quantum Yang-Baxter operator leading to a representation of the Artin braid groups on the modules V^{⊗r}. In this situaton the algebras A_{r} are just the images of the corresponding group algebras (over R) under this representation.

An application where N must consist of two elements will be given in the next section. Here the A_{r} are images of the Brauer centralizer algebras under Brauer’s representations corresponding to the symplectic groups. The bialgebra M(N) will turn out to be the coordinate ring of a certain symplectic monoid. It is a remarkable fact, that the second operator is only needed in the classical situation, whereas in the quantum case it lies in the algebraic span of the quantum Yang-Baxter operator (see [Oe], Bemerkung 2.5.1, 2.5.4). Thus, the ordinary FRT-construction behaves singularly (in a certain sense) when specializing the deformation parameter to 1, i.e. in the classical limit.

We close this section giving a more convenient description of M(N). We denote by I(n, r) the set of maps from from \underline{r} := \{1, …, r\} to \underline{n} := \{1, …, n\} and call the elements i ∈ I(n, r) multi-indices writing i = (i_{1}, …, i_{r}), where i_{j} ∈ \underline{n} for j ∈ \underline{r}. The residue classes of the multiplicative generators e_{i}^{∗}^{j} + I of M(N) where i, j ∈ \underline{n} will be denoted by x_{ij}. For pairs of multi-indices i, j ∈ I(n, r) we introduce the abbreviation

x_{ij} := x_{i_{1}j_{1}}x_{i_{2}j_{2}} … x_{i_{r}j_{r}}.

Using the notation introduced in (1) we obtain a presentation of M(N) by generators and relations given as follows

M(N) = ⟨x_{ij}, i, j ∈ \underline{n}| μx_{ij} = x_{ij}μ, i, j ∈ I(n, 2), μ ∈ N⟩ .  (12)

The verification of this formula follows from the second statement of proposition 5.1 together with (2).

6 Example: Symplectic Monoids

Let n = 2m be even. We will apply the FRT-construction to two endomorphisms β, γ ∈ E_{2} = E ⊗ E. In order to define them we have to introduce some notation. First consider the involution i′ := n − i + 1 on \underline{n}, that is

(1′, 2′, …, n′) = (n, n − 1, …, 1).

Further set ϵ_{i} := 1 if i ≤ m and ϵ_{i} := −1 if i > m and define

β := ∑_{i,j∈\underline{n}} e_{i}^{j} ⊗ e_{j}^{i}

γ := ∑_{i,j∈\underline{n}} ϵ_{i}ϵ_{j}e_{i}^{j′} ⊗ e_{i′}^{j}.

The first endomorphism is just the flip operator on V ⊗ V, whereas the second is an integer multiple of a projection map whose kernel is just the kernel of the linear form on V ⊗ V corresponding to the canonical skew bilinear form on V (see below) and whose image is the one dimensional span of a skew bivector. Our object of interest will be the FRT-construction

A_{R}^{s}(n) := M({β, γ}).

According to the preceeding section it is a graded matric bialgebra whose homogenous summands

A_{R}^{s}(n, r) := M(A_{r})

are the centralizer coalgebras of algebras A_{r} which are generated by endomorphisms

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

for i = 1, …, r − 1. Using the notation of [We] the Brauer centralizer algebra D_{r}(x) (x an element in R) is generated by symbols g_{i} and e_{i} for i = 1, …, r − 1 and the assignment g_{i} ↦ β_{i} and e_{i} ↦ γ_{i} defines a representation of D_{r}(−n) on V^{⊗r}. Thus A_{r} is just the image of D_{r}(−n) under this representation, in H. Wenzl’s notation from [We]: A_{r} = B_{r}(Sp(n)). R. Brauer showed in [Br] that this is just the centralizer algebra of the symplectic group Sp_{ℂ}(n) acting on V^{⊗r} if R = ℂ. One of our aims is to generalize this to the case of an arbitrary algebraically closed field K instead of ℂ.

Another problem, connected with the former, is to show that the centralizer algebra C(A_{r}) of A_{r} which will turn out to be the symplectic Schur algebra S_{0}(n, r) defined by S. Donkin in [Do2] is stable under base changes. In view of theorem 4.3 this is equivalent to the projectivity of the coalgebra A_{R}^{s}(n, r) as an R-module. For this purpose we are going to construct a basis for the latter one. The procedure follows [Oe] where the more general quantum case is treated. But it will become more transparent in the much simpler classical case. First note that there is an epimorphism of graded bialgebras from

A_{R}(n) := R[x_{11}, x_{12}, …, x_{nn}]

to A_{R}^{s}(n) leaving the symbols x_{ij} fixed (we use x_{ij} as symbols for residue classes of e_{i}^{∗}^{j} in all cases of matric bialgebras). For A_{R}(n) is just the FRT-construction M(β) where the relations coming from the endomorphism γ are omitted. This is because x_{i_{2}j_{1}}x_{i_{1}j_{2}} = βx_{ij} = x_{ij}β = x_{i_{1}j_{2}}x_{i_{2}j_{1}} just give the ordinary commutativity relations. The kernel of this bialgebra epimorphism is the ideal in A_{R}(n) which is generated by the polynomials γx_{ij} = x_{ij}γ where i, j ∈ I(n, 2). To write down these polynomials explicitly let us fix some notation:

f_{ij} := ∑_{k=1}^{n} ϵ_{k}x_{ik}x_{jk′}, f̄_{ij} := ∑_{k=1}^{n} ϵ_{k}x_{ki}x_{k′j} ∈ A_{R}(n, 2).

Setting i = (i, j) and j = (k, l) we obtain

γx_{ij} = \{ \begin{matrix}ϵ_{j}f̄_{kl} & i = j′ \\ 0 & i ≠ j′ \end{matrix} and x_{ij}γ = \{ \begin{matrix}ϵ_{l}f_{ij} & k = l′ \\ 0 & k ≠ l′ \end{matrix}

Therefore we have

A_{R}^{s}(n) = A_{R}(n)/F  (13)

where F is the ideal in A_{R}(n) generated by the set

F := {f_{ij}, f̄_{ij}, f_{ll′} − f̄_{kk′}| 1 ≤ i < j ≤ n, i ≠ j′, 1 ≤ l ≤ k ≤ m}.  (14)

If R = K is an algebraically closed field we can interpret this in terms of algebraic geometry, i.e. we can look at the vanishing set of F in the monoid M_{K}(n) of n × nmatrices. It is easy to see that this is just the closed submonoid

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

in M_{K}(n) called the symplectic monoid by S. Doty [Dt] and which has been considered by D.J. Grigor’ev [Gg] first. 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 function d : SpM_{K}(n) → K is called the coefficient of dilation. It is neccesarily a regular function on SpM_{K}(n) and already well defined in A_{K}^{s}(n), explicitly:

d = ϵ_{k}f_{kk′} = ϵ_{k}f̄_{kk′} ∈ A_{R}^{s}(n, 2).  (15)

Note that this is independent of k ∈ \underline{n} by the relations in A_{K}^{s}(n). Furthermore d is a group-like element of this bialgebra (cf. [Oe] 2.1.1). The set GL_{K}(n) ∩ SpM_{K}(n) of invertible elements in SpM_{K}(n) is precisely the group GSp_{K}(n) of symplectic similitudes. S. Doty showed in [Dt] that SpM_{K}(n) in fact coincides with the Zariski-closure of GSp_{K}(n) in M_{K}(n). We will obtain this as an easy consequence of the results presented below.

To write down a basis for A_{R}^{s}(n, r) we need some combinatorics. The set of partitions of r is denoted by Λ^{+}(r). It contains subsets Λ^{+}(l, r) which consist of partitions having not more than l parts. We write partitions as l-tuples λ := (λ_{1}, λ_{2}, …, λ_{l}) of nonnegative integers λ_{i} in descending order λ_{1} ≥ λ_{2} ≥ … ≥ λ_{l} ≥ 0 such that λ_{1} + … + λ_{l} = r. 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). The column lengths of the diagram lead to another partition λ′ ∈ Λ^{+}(λ_{1}, r) called the dual of the partition λ, i.e. λ′_{i} := |{j| λ_{j} ≥ i}|. Let S_{r} denote the symmetric group on r symbols and S_{λ} the standard Young subgroup of S_{r} corresponding to the partition λ. This is the subgroup fixing the subsets \{1, …, λ_{1}\}, \{λ_{1} + 1, …, λ_{1} + λ_{2}\}, … of \underline{r}. In the above example λ = (3, 2, 2, 1) the standard Young subgroup of S_{8} corresponding to the dual partition λ′ fixes {1, 2, 3, 4}, {5, 6, 7} and {8}.

To each partition λ ∈ Λ^{+}(r) and a pair of multi-indices i, j ∈ I(n, r) one defines a bideterminant T^{λ}(i : j) ∈ A_{R}(n, r) by

T^{λ}(i : j) := ∑_{w∈S_{λ′}} sign(w)x_{i(jw)} = ∑_{w∈S_{λ′}} sign(w)x_{(iw)j.}

where iw := (i_{w(1)}, i_{w(2)}, …, i_{w(r)}). These are products of minor determinants, one factor for each column, the size of which correspond to the length of the column. By (13) they can be interpreted as elements of A_{R}^{s}(n, r), as well. We wish to write down a basis of the latter R-module consisting of such bideterminants. Since they are too large in number one needs a criterion to single out the right ones. This can be done using λ-tableaux. These are constructed from the diagram of λ by inserting the components of a multi-index 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} .

We put a new order ≪ on the set \underline{n}, namely 1′ ≪ 1 ≪ 2′ ≪ 2 ≪ … ≪ m′ ≪ m. 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 in T_{i}^{λ} 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-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 i rows. The corresponding subset of I_{λ} will be denoted by I_{λ}^{sym}.

The notion of symplectic standard tableaux traces back to R.C. King [Ki] and it has appeard in a lot of work concerning symplectic groups and their representation theory (for details see [Do3]).

It is well known from invariant theory (cf. [Mr], section 2.5) that the collection of all bideterminants T^{λ}(i : j) where λ runs through Λ^{+}(n, r) and i, j run through I_{λ} form a basis of A_{R}(n, r). Similarily we will prove in the next section

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

B_{r} := \{d^{l}T^{λ}(i : j)| 0 ≤ l ≤ \frac{r}{2}, λ ∈ Λ^{+}(m, r − 2l), i, j ∈ I_{λ}^{sym}\}.

Before proving this let us have a look at some consequences. The first one generalizes theorem 9.5 (a) of [Dt] avoiding the restriction to characteristic zero. Furthermore, it contains corollary 5.5 (f) of that paper for algebraically closed fields.

Corollary 6.2 Let K be an algebraically close field. Then A_{K}^{s}(n) coincides with the coordinate ring of the Zariski-closure \overline{GSp_{K}(n)} of GSp_{K}(n) in M_{K}(n). In particular SpM_{K}(n) is identical to \overline{GSp_{K}(n)}. Therefore, a complete set of generators of the vanishing ideal of \overline{GSp_{K}(n)} in A_{K}(n) is given by the set F (defined in equation 14).

Proof: Let A_{0}(n) be the coordinate ring of \overline{GSp_{K}(n)} and A_{0}(n, r) its r-th homogenous summand. In [Do2] the symplectic Schur algebra S_{0}(n, r) is defined as the dual algebra to the coalgebra A_{0}(n, r). The dimension of the latter one is given by Weyl’s character formula and therefore independent of the field K (cf. [Do2] p. 77). On the other hand there is an epimorphism of graded bialgebras from A_{K}^{s}(n) to A_{0}(n) since \overline{GSp_{K}(n)} is closed in SpM_{K}(n) and the latter one has been defined as the vanishing set of the ideal F by which A_{K}^{s}(n) is defined. But by our basis theorem 6.1 the dimension of A_{K}^{s}(n, r) is independent of the field K, as well. Thus, the proof can be finished looking at the case K = ℂ and using Doty’s theorem 9.5 (a) or alternately by a direct calculation of |B_{r}| = dim_{ℂ}((A_{0}(n, r))) (see proposition 7.1 below). □

By theorem 4.1 we have isomorphisms

K ⊗_{ℤ} A_{ℤ}^{s}(n, r) ≅ A_{K}^{s}(n, r), K ⊗_{ℤ} A_{ℤ}^{s}(n) ≅ A_{K}^{s}(n).

Since A_{K}^{s}(n) has been recognized to be the coordinate ring of SpM_{K}(n) = \overline{GSp_{K}(n)} we may interpret the spectrum of the ring A_{ℤ}^{s}(n) as an integral monoid scheme SpM_{ℤ}(n). Accordingly, an integral form for the symplectic Schur algebra can be obtained as the dual algebra

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

of its homogenous summands. By theorem 4.3 and 6.1 this is stable under base change, that is, tensoring by a field K gives the symplectic Schur algebra S_{K}^{s}(n, r) = S_{0}(n, r) = Hom_{K}(A_{K}^{s}(n, r), K) defined over that field. An integral form for symplectic Schur algebras exists, as well, by S. Donkin’s work on generalized Schur algebras (see [Do2]). But his approach is quite different using the theory of Lie algebras in particular the Kostant ℤform. In both cases the notion of symplectic Schur algebras can be extended to more general integral domains R instead of ℤ leading to identical concepts. In our case this is S_{R}^{s}(n, r) := Hom_{R}(A_{R}^{s}(n, r), R). By theorem 3.3 we conclude

Corollary 6.3 Over any noetherian integral domain R the symplectic Schur algebra is isomorphic to the centralizer algebra of the Brauer algebra D_{r}(−n).

For a field of characteristic zero this has been proved by S. Doty, too ([Dt] corollary 9.3. (c)). It should be remarked, that the basis dual to B_{r} together with the anti-involution defined by matrix transposition give a cell datum for the symplectic Schur algebra in the sense of J. Graham and G. Lehrer (cf. [Oe], 4.2.5). Thus, its representation theory can be developed easily to the extent of the treatment of cellular algebras in [GL].

7 Proof of theorem 6.1

Let us first reduce to showing that B_{r} is a set of generators for the R-module A_{R}^{s}(n, r). This can be done by the following proposition where notations from the proof of corollary 6.2 are used.

Proposition 7.1 |B_{r}| = dim_{ℂ}(A_{0}(n, r)).

Proof: We use [Do2] p. 74 ff. First the reader may check that our definition of A_{0}(n) and A_{0}(n, r) is identical to the one given there. According to [Do2] and 2.2c in [Do1] we have

dim_{ℂ}(A_{0}(n, r)) = ∑_{λ∈π_{0}(n,r)} dim_{ℂ}(Y_{0}(λ))^{2}  (16)

where Y_{0}(λ) := Ind_{B_{0}}^{GSp_{ℂ}(n)}(ℂ_{λ}) is the irreducible GSp_{ℂ}(n) module induced from the linear character ℂ_{λ} of the Borel subgroup B_{0} (notations taken from [Do2]). Here λ runs through the set π_{0}(n, r) of dominant weights corresponding to the irreducibles occuring in V^{⊗r}. If T_{0} denotes the maximal torus of GSp_{ℂ}(n) we may consider the weights λ as the group-like elements in its coordinate ring. More precisely λ ∈ π_{0}(n, r) is of the form

λ = x_{11}^{μ_{1}}x_{22}^{μ_{2}} … x_{mm}^{μ_{m}}d^{l}

as can be seen from the argumentation in [Do2]. Here, μ := (μ_{1}, …, μ_{m}) ∈ Λ^{+}(m, r − 2l) is a partition of r − 2l in not more than m parts and 0 ≤ l ≤ \frac{r}{2} an integer. Restricting to the symplectic group Sp_{ℂ}(n) we have to set the coefficient of dilation d equal to 1. Thus the restriction of λ to the maximal torus of Sp_{ℂ}(n) is just the dominant weight

λ̄ = x_{11}^{μ_{1}}x_{22}^{μ_{2}} … x_{mm}^{μ_{m}}

for the symplectic group itself. Furthermore, it is easy to show that restricting the GSp_{ℂ}(n)-module structure of Y_{0}(λ) to the symplectic group gives the module Ȳ (λ̄) := IndSp_{ℂ}(n) ; B̄(ℂλ̄) induced from the linear character ℂλ̄ of the Borel subgroup B̄ = B_{0}∩Sp_{ℂ}(n) of Sp_{ℂ}(n) (for details see [Oe], 3.3.3). But the dimension of the latter one is known to be the cardinality of I_{μ}^{sym} (see [Do3] theorem 2.3 b for instance). Thus, we obtain

dim_{ℂ}(A_{0}(n, r)) = ∑_{0≤l≤ \frac{r}{2}} ∑_{μ∈Λ^{+}(m,r−2l)} |I_{μ}^{sym}|^{2} = |B_{r}|.

□

Observe that by theorem 4.1 the proof of 6.1 can be reduced to the case R = ℤ, since the definition of bideterminants over R and ℤ respectively commutes with the isomorphism μ_{R} when R is considered as a ℤ-algebra. Now, suppose we have shown that B_{r} generates A_{ℤ}^{s}(n, r) ⊆ A_{ℂ}^{s}(n, r) as a ℤ-module. Then the image of B_{r} in A_{0}(n, r) under the epimorphism considered in the proof of 6.2 is a set of generators, too. By the above proposition it must be a basis of A_{0}(n, r). Consequentely, there can’t be any relations among the elements of B_{r}, especially none with integer coefficients, giving the desired result.

The proof that B_{r} is indeed a set of generators will follow from a symplectic version of the famous straightening formula. For convenience of the reader we will first state the algorithm leading to the classical straightening formula. To do so, we put an order on the set Λ^{+}(r) of partitions of r writing λ < μ if the dual λ′ occurs before the dual μ′ in the lexicographical 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 abbreviate A := A_{R}(n, r) and define A(> λ) resp. A(≥ λ) to be the R-linear span in A of all bideterminants T^{μ}(i : j) such that μ > λ resp. μ ≥ λ. For λ = ω_{r} we set A(> ω_{r}) := 0. Clearly A = A(≥ α_{r}).

Proposition 7.2 (Classical 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 = jw ^{2} for some w ∈ S_{r} and k ≪ j there is an element a_{jk} ∈ R such that in A the following congruence relation holds for all i ∈ I(n, r):

T^{λ}(i : j) ≡ ∑_{k≪j} a_{jk}T^{λ}(i : k) mod A(> λ).

Here, the order on I(n, r) is the lexicographical one according to the given order on \underline{n}, in our case ≪. A proof of the proposition can be found for example in [Mr], 2.5.7.

Let us now state the symplectic analogue. First consider the algebra

A_{R}^{sh}(n) := A_{R}^{s}(n)/ ⟨d⟩

2this is missing in the statement of 2.5.7 in [Mr] but can be seen directly from the proof given there

where ⟨d⟩ is the ideal in A_{R}^{s}(n) generated by the coefficient of dilation. It is graded since d is a homogenous element but not a bialgebra because an augmentation map is missing. In fact, it turns out that (in the case where R = K is an algebraically closed field) it is the coordinate ring of the semigroup SpH_{K}(n) := SpM_{K}(n)\GSp_{K}(n) of noninvertible elements in the symplectic monoid (see remark 7.5). Let us abbreviate its submodule of homogenous elements of degree r by A′ := A_{R}^{sh}(n, r) and define A′(> λ) and A′(≥ λ) in the same manner as above. Further, 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_{m}, b_{m−1}, …, b_{1}) appears before (a_{m}, a_{m−1}, …, a_{1}) in the lexicographical order. Next, we obtain an order ⊲ on ℕ_{0}^{m} × I(n, r) in a lexicographical way, as well:

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

Finally, this gives a 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 7.3 (Symplectic Straightening Algorithm) Let λ ∈ Λ^{+}(r) be a partition of r and j ∈ I(n, r)\I_{λ}^{sym}. Then to each k ∈ I(n, r) satisfying k⊲j there is an element a_{jk} ∈ R such that in A′ the following congruence relation holds for all i ∈ I(n, r):

T^{λ}(i : j) ≡ ∑_{k⊲j}a_{jk}T^{λ}(i : k) mod A′(> λ).

Before proving this, let us deduce that B_{r} is a set of generators for A_{R}^{s}(n, r). First note that multiplication by the coefficient of dilation d leads to an exact sequence for r > 1

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

Therefore, by induction on r we can reduce to showing that

{T^{λ}(i : j)| λ ∈ Λ^{+}(m, r), i, j ∈ I_{λ}^{sym}}

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

{T^{λ}(i : j)| i, j ∈ I_{λ}^{sym}}

is a set of generators of A′(≥ λ)/A′(> λ) for each partition λ. This can be deduced from the straightening algorithm 7.3: Since I(n, r) is a finite set, the elemination of multiindices j not being λ-symplectic standard in an expression T^{λ}(i : j) must terminate. This gives the straightening formula concerning the right-hand-side argument of T^{λ}(i : j):

Corollary 7.4 (Symplectic Straightening Formula) Let λ ∈ Λ^{+}(r) be a partition of r and j ∈ I(n, r). Then, to each k ∈ I_{λ}^{sym} there is an element a_{jk} ∈ R, such that in A′ we have for all i ∈ I(n, r):

T^{λ}(i : j) ≡ ∑_{k∈I_{λ}^{sym}} a_{jk}T^{λ}(i : k) mod A′(> λ).

Now, there is an algebra automorphism θ on A_{R}(n) induced by matrix transposition and given by θ(x_{ij}) = x_{ji} on generators. Of course it is an anti-automorphism of coalgebras. It can be readily seen that θ(F) = F and θ(d) = d. Therefore, it factors to an automorphism of A_{R}^{sh}(n) which will be denoted by the same symbol. From the definition of bideterminants we see θ(T^{λ}(i : j)) = T^{λ}(j : i).

Applying θ to the congruence relation in 7.4 we see that a non λ-symplectic standard entry i on the left-hand-side entry of a bideterminant can be eleminated, too, not affecting the right-hand-side entry. Thus, it must be possible to write T^{λ}(i : j) as a sum of bideterminants T^{λ}(k : l) modulo A′(> λ) where k, l ∈ I_{λ}^{sym}. Therefore, the proof of theorem 6.1 is finished as soon as proposition 7.3 is established.

Remark 7.5 A symplectic straightening formula similar to 7.4 can be obtained as a special case of a theorem by de Concini ([Co], theorem 2.4) where m = 2r (in the notation taken from there). The algebra denoted A in that paper becomes the coordinate ring of the semigroup denoted SpH_{K}(n) above. But, note that this result is not strong enough for our purpose since we don’t know wether A = A_{K}^{sh}(n) or not. On the other hand, the latter identity follows from theorem 6.1 in a similar way to corollary 6.2 using theorem 3.6 of [Co].

Also, the symplectic straightening formula is related to the treatment of symplectic Schurmodules in [Do3] and [Ia], section 6, as can be seen from the proof of lemma 8.1 below. Concerning the latter paper it should be noted that the algebra A_{K}^{sp}(\overline{n}) defined there as the coordinate ring of the symplectic group itself is only filtered by ∑_{t=0}^{r} A_{K}^{sp}(\overline{n}, t) but not graded by A_{K}^{sp}(\overline{n}, r). In fact, it can be deduced from the above remark that A_{K}^{sh}(n) is the corresponding graded algebra, that is A_{K}^{sp}(\overline{n}, r) = A_{K}^{sh}(n, r) as K-vector spaces.

8 Proof of proposition 7.3

First, some considerations about the exterior algebra

⋀_{R}(n) := T (V)/ ⟨v_{i} ⊗ v_{j} + v_{j} ⊗ v_{i}, v_{k} ⊗ v_{k}; i, j, k ∈ \underline{n}⟩

are needed. It is easy to see and well known that this graded R-algebra is a comodule algebra for the bialgebra A_{R}(n), i.e. it is an A_{R}(n)-comodule such that multiplication is a morphism of comodules. In fact, the homogenous summands ⋀_{R}(n, r) of ⋀_{R}(n) are comodules for the coalgebras A_{R}(n, r). Since A_{R}^{s}(n) is an epimorphic image of A_{R}(n) the exterior algebra is a comodule algebra for the latter one, as well.

As usual we write v_{i} ∧ v_{j} for the residue class of v_{i} ⊗ v_{j} in ⋀_{R}(n) and denote arbitrary multiplications by ∧, too. For a subset I := \{i_{1}, …, i_{r}\} ⊆ \underline{n} ordered i_{1} ≪ i_{2} ≪ … ≪ i_{r} by the given order ≪ on \underline{n} (called an ordered subset in the sequel) we use the abbreviation v_{I} := v_{i_{1}} ∧ v_{i_{2}} ∧ … ∧ v_{i_{r}}. The elements v_{I} give a basis of ⋀_{R}(n) if I ranges over all subsets of \underline{n}. A basis for ⋀_{R}(n, r) is obtained when I ranges over all subsets of cardinality r, the collection of which will be denoted by P (n, r). The comodule structure of ⋀_{R}(n, r) can be described in a simple way using bideterminants for the partition ω_{r} := (1^{r}) = (1, 1, …, 1). These are just the usual r × r-minor determinants. Denoting the structure map by τ_{∧} : ⋀_{R}(n) → ⋀_{R}(n) ⊗ A_{R}(n) (we will use the same symbol in the case A_{R}^{s}(n) later on) we explicitly have

τ_{∧}(v_{J}) = ∑_{I∈P (n,r)} v_{I} ⊗ T^{ω_{r}}(i : j)  (17)

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. Define d_{k} := v_{k′} ∧ v_{k} and d_{K} := d_{k_{1}} ∧ d_{k_{2}} ∧ … ∧ d_{k_{a}} for a subset K := \{k_{1}, …, k_{a}\} ⊆ \underline{m} of cardinality a. Again, we write P (m, a) for the collection of all such subsets K. Note that the d_{k} are in the center of the exterior algebra and in particular commute with each other. Thus d_{K} is defined independent of the order of the elements of K. Set

D_{a} := ∑_{K∈P (m,a)} d_{K}

and let N be the ideal in ⋀_{R}(n) generated by the elements D_{1}, D_{2}, …, D_{m}. One crucial point in the proof of proposition 7.3 is to show that the elements D_{a} are invariant under the bialgebra A_{R}^{s}(n). But first, let us establish a prototype straightening algorithm inside the graded algebra ⋀_{R}^{s}(n) with respect to the set I_{ω_{r}}^{sym}. We call an ordered subset I ∈ P (n, r) symplectic if the corresponding multi-index i is ω_{r}-symplectic standard.

Lemma 8.1 Let I ∈ P (n, r) be non symplectic. Then, to each J ∈ P (n, r) such that the inequallity f(j) < f(i) holds for corresponding multi-indices i and j there is a_{IJ} ∈ R, such that in ⋀_{R}(n) the following congruence relation holds:

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

Proof: We follow [Do3]. Clearly, we may reduce to the case R = ℤ since ⋀_{R}(n) ≅ R ⊗_{ℤ}⋀_{ℤ}(n) and the canonical isomorphism therein respects the basis elements v_{I} and the ideal N. If K ⊆ \underline{m} then in ⋀_{R}(n) we calculate as in [Do3], 2.2.

(∑_{k∈K} d_{k})^{a} = a! ∑_{L∈P (m,a), L⊆K} d_{L}  (18)

In the case K = \underline{m} this implies D_{1}^{a} = a!D_{a}. Set x := ∑_{k∈K} d_{k} and y := D_{1} − x. It follows from (18) that the equation

x^{a} = (−1)^{a}y^{a} + ∑_{b=1}^{a} \binom{a}{b}(−y)^{a−b}b!D_{b}

is divisible by a!. Setting X_{a}^{K} := ∑_{L∈P (m,a), L⊆K} d_{L} and M := \underline{m}\K this yields

X_{a}^{K} = (−1)^{a}X_{a}^{M} + ∑_{b=1}^{a}(−1)^{a−b}X_{a−b}^{M}D_{b}

since ⋀_{ℤ}(n) is a free ℤ-module. We obtain X_{a}^{K} ≡ (−1)^{a}X_{a}^{M} modulo the ideal N and in the special case K ∈ P (m, a)

d_{K} ≡ (−1)^{a} ∑_{L∈P (m,a), L∩K=∅} d_{L} mod N.  (19)

The proof can be finished now in a similar way to the proof of the Symplectic Carter-Lusztig Lemma in [Do3]: Since I is not symplectic there is a number s ∈ \underline{r} such that I = {i_{1}, …, i_{r}} contains an element i_{s} satisfying i_{s} < s if i_{s} ≤ m or i′_{s} < s if i_{s} > m. We assume s to be as small as possible with this property. Let k, l ∈ \underline{m} be the unique numbers with {k, k′} = {i_{s}, i′_{s}} and {l, l′} = {i_{s−1}, i′_{s−1}}. By minimality of s we have l ≥ s − 1. On the other hand, i_{s−1} ≪ i_{s} implies l ≤ k. This gives s − 1 ≤ l ≤ k < s, that is k = l and s = k + 1.

Now let I_{s} := {i_{1}, …, i_{s}} be the ordered subset of the first s entries of I and K be the set of all p ∈ \underline{m} such that both p and p′ are contained in I_{s}. Setting K′ := \{p′| p ∈ K\} and H := I\(K ∪ K′) we obtain v_{I} = v_{H}d_{K} since the elements d_{p} are in the center of ⋀_{ℤ}(n). Therefore, by equation (19) it remains to show that for all subsets L ∈ P (m, a) which do not intersect K we have f(j) < f(i) where j is the multi-index corresponding to the ordered set J := H ∪ L ∪ L′. Since v_{J} = v_{H}d_{L} = 0 if L or L′ := {p′| p ∈ L} contains an element of H we further may assume H ∩L = H ∩L′ = ∅. Now, suppose L ⊆ \underline{k}. Let G be the set of all g ∈ \underline{k}\K such that g or g′ lies in I_{s}. Since I_{s}\(K ∪ K′) ⊆ H the intersection of L and G is empty, as well. This gives a contradiction

k + 1 = s = |I_{s}| = 2|K| + |G| = |K| + |L| + |G| = |K ∪ L ∪ G| ≤ k

for K, L and G are disjoint subsets of \underline{k} by assumption on L. Thus, the largest element t of L must be greater than k, whereas all elements of K are smaller or equal to k. If f(i) = (b_{1}, …, b_{m}) and f(j) = (a_{1}, …, a_{m}) it follows that a_{t} = b_{t} + 2 > b_{t} and a_{l} = b_{l} for t < l ≤ m. By definition of our order on ℕ_{0}^{m} this means f(j) < f(i) completing the proof. □

As mentioned before, the crucial point in the proof of proposition 7.3 is contained in the following

Proposition 8.2 The elements D_{a} are invariant under the bialgebra A_{R}^{s}(n). The corresponding group-like elements in A_{R}^{s}(n) are the a-th powers of the coefficient of dilation, more precisely:

τ_{∧}(D_{a}) = D_{a} ⊗ d^{a} ∈ ⋀_{R}(n, 2a) ⊗ A_{R}^{s}(n, 2a)

Proof: In the case a = 1 this easily follows from (15) using the relations given by the set F defined in (14). If we could divide by a!, we would be able to finish the proof right now using D_{1}^{a} = a!D_{a} and the fact that multiplication is a morphism of A_{R}^{s}(n)- comodules. But, as this is not possible in general (note, that we don’t know if A_{ℤ}^{s}(n) is a free ℤ-module, yet) we have to proceed in another way. We set r = 2a and

G_{i} := ∑_{L∈P (m,a)} T^{ω_{r}}(i : j(L))

where j(L) is the multi-index corresponding to the ordered subset L ∪ L′ = {l, l′| l ∈ L} of \underline{n}. If i is the multi-index corresponding to the ordered subset I ∈ P (n, r) we also write G_{I} = G_{i}. By (17) we have

τ_{∧}(D_{a}) = ∑_{I∈P (n,r)} v_{I} ⊗ G_{I}.

Therefore, it remains to show that G_{I} = d^{a} if there is a K ∈ P (m, a) such that I = K ∪K′ and G_{I} = 0 otherwise.

If i = (i_{1}, …, i_{r}) is a multi-index and l ≤ a we set i^{l} := (i_{2l−1}, i_{2l}) ∈ I(n, 2). Since the result is clear in the case a = 1 we know

G_{i^{l}} = \{ \begin{matrix}0 & i′_{2l−1} ≠ i_{2l} \\ d & i′_{2l−1} = i_{2l} ≤ m \\ −d & i′_{2l−1} = i_{2l} > m \end{matrix} .

Therefore, the element G′_{i} := ∏_{l=1}^{a} G_{i^{l}} is zero or ±d^{a} depending on i. Let us investigate this in more detail. If σ ∈ S_{r} denotes the involution defined by σ(2l−1) = 2l, σ(2l) = 2l−1 for l ∈ \underline{a} and W the centralizer of σ in S_{r}, then clearly G′_{i} = 0 if and only if G′_{iw} = 0 for all w ∈ W. The group W is isomorphic to the Weyl group of type C_{a}. It is a semi-direct product of the normal subgroup W^{1} consisting of all π ∈ W which permute neighboured pairs together with W^{0}, the subgroup generated by the transpositions (2l−1, 2l) for l ∈ \underline{a}. W^{1} is isomorphic to S_{a}, whereas the group W^{0} can be identified with (ℤ/2ℤ)^{a}. Choose a set H of left coset representatives for W in S_{r} the element representing W itself being id\underline{r}.

Now, if i corresponds to an ordered set I = K ∪ K′ for some K ∈ P (a, n) the inequality G′_{iπ} ≠ 0 holds for a permutation π ∈ S_{r} if and only if π ∈ W. Thus, for h ∈ H we have G′_{ih} = 0 if h ≠ id and G′_{i} = d^{a}. If there is no K such that I = K ∪ K′ one clearly has G′_{ih} = 0 for all h ∈ H. Therefore, the proof is finished as soon as we have shown

G_{i} = ∑_{h∈H} sign(h)G′_{ih}.  (20)

To this claim let μ := (a, a) ∈ Λ^{+}(2, r) be the partition of r whose diagram consists of two rows of length a. To a multi-index l = (l_{1}, l_{2}, …, l_{a}) ∈ I(m, a) another multi-index j(l) := (l′_{1}, l_{1}, l′_{2}, l_{2}, …, l′_{a}, l_{a}) ∈ I(n, r) can be associated. Using this notation and reading T^{μ}(i : j(l)) as a product of a 2 × 2-determinants we obtain the formula

G′_{i} = ∑_{l∈I(m,a)} T^{μ}(i : j(l)) ∈ A_{R}^{s}(n, r).

On the other hand, using Laplace Duality (see for example [Mr] 2.5.1) we calculate

T^{ω_{r}}(i : j) = ∑_{h∈H, π∈W^{1}} sign(h)T^{μ}(ihπ : j) = ∑_{h∈H} sign(h) ∑_{π∈W^{1}} T^{μ}(ih : jπ).  (21)

Therein, note that W^{0} is precisely the column stabilizer of the basic tableaux T_{b}^{μ}, whereas the column stabilizer of T_{b}^{ω_{r}} is all of S_{r} (here b := (1′, 1, 2′, 2, …, a′, a)). Also, note that HW^{1} is a set of left coset representatives of W^{0} and that all permutations of W^{1} are even. Furthermore, in the right-hand-side equation we have used the commutativity between the 2 × 2-determinant factors of T^{μ}(ihπ : j) = T^{μ}(ih : jπ^{−1}). Now, the proof of (20) can be reduced to the verification of

∑_{L∈P (m,r)} ∑_{π∈W^{1}} ∑_{h∈H} sign(h)T^{μ}(ih : j(L)π) = ∑_{l∈I(m,a)} ∑_{h∈H} sign(h)T^{μ}(ih : j(l))  (22)

To this claim we associate to a multi-index l ∈ I(m, a) its contents |l| = λ = (λ_{1}, …, λ_{m}) which is defined by λ_{i} := |{1 ≤ t ≤ a| l_{t} = i}|. It is a composition of a into m parts, that is an m-tuple of non negative integers λ_{i} summing up to a. These compositions count the set of S_{a}-orbits in I(m, a). Denoting the set of all such compositions by Λ(m, a) we can write down the right-hand term in (22) as a sum of subsums ∑_{λ∈Λ(m,r)} Σ_{λ} each of which is given by

Σ_{λ} := ∑_{|l|=λ} ∑_{h∈H} sign(h)T^{μ}(ih : j(l))

Now, the subsum Σ_{ω_{a}} (for ω_{a} = (1^{a})) is just the left-hand-side in (22). Therefore, it remains to show that all other subsums are zero. To this claim we denote the cardinality of the standard Young subgroup S_{λ} of S_{a} corresponding to the composition λ ∈ Λ(m, a) by k_{λ} := |S_{λ}|. If k = (k_{1}, …, k_{a}) ∈ I(m, a) is the unique multi-index with contents λ and k_{1} ≤ k_{2} ≤ … ≤ k_{r} (the initial index corresponding to λ) then S_{λ} is just the stabilizer of k in S_{a}. Identifying S_{a} with W^{1} it is the stabilizer of j(k) ∈ I(n, r) in W^{1}. Applying (21) again we obtain

T^{ω_{r}}(i : j(k)) = ∑_{h∈H} sign(h) ∑_{π∈W^{1}} T^{μ}(ih : j(k)π) =

k_{λ} ∑_{h∈H} sign(h) ∑_{|l|=λ} T^{μ}(ih : j(l)) = k_{λ}Σ_{λ}.

Now, T^{ω_{r}}(i : j(k)) must be zero since |k| ≠ ω_{a} implies that k contains at least one number twice. We obtain k_{λ}Σ_{λ} = 0 and because this equation is already valid in the free ℤ-module A_{ℤ}(n, r) we conclude Σ_{λ} = 0 for all λ ≠ ω_{a} completing the proof. □

Let us prove proposition 7.3 in the case λ = ω_{r} first. Take j ∈ I(n, r)\I_{ω_{r}}^{sym}. Using the classical straighening algorithm 7.2 we may assume j ∈ I_{ω_{r}}\I_{ω_{r}}^{sym} (observe that k = jw implies f(k) = f(j)). This means, that j is a multi-index corresponding to a non symplectic ordered set J ∈ P (n, r) in the sense of lemma 8.1. Application of the latter one yields

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

According to proposition 8.2 τ_{∧}(X) must be contained in ⋀_{R}(n)⊗ < d > where < d > denotes the ideal in A_{R}^{s}(n) generated by d. Applying (17) we obtain the following equation in ⋀_{R}(n, r) ⊗ A_{R}^{sh}(n, r):

∑_{I∈P (n,r)} v_{I} ⊗ (T^{ω_{r}}(i : j) − ∑_{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}(n, r) each individual summand in the summation over P (n, r) must be zero. 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 general case for i can be deduced from this, easily (see [Oe], 3.11.4).

Now, lets turn to the general case of λ. Again, we may assume j ∈ I_{λ}\I_{λ}^{sym} by the classical 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-symplectic standard but standard there must be a column s such that j^{s} is not ω_{μ_{s}}-symplectic standard. Applying the result to the known case of T^{ω_{μs}}(i^{s} : j^{s}) we obtain

T^{λ}(i : j) = T^{ω_{μ_{1}}}(i^{1} : j^{1})T^{ω_{μ_{2}}}(i^{2} : j^{2}) … T^{ω_{μs}}(i^{s} : j^{s}) … T^{ω_{μp}}(i^{p} : j^{p})

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

Therein, k^{s} ∈ I(n, μ_{s}) satisfies k^{s} ⊲ j^{s}, k ∈ I(n, r) is constructed from j 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}. One easily checks k ⊲ j and the proof of 7.3 is completed.

References

[Br] Brauer, R.: On Algebras which are connected with the semisimple continuous Groups. Annals of Math. Vol. 38, No. 4 (1937), 857-871.

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

[Do1] Donkin, S.: On Schur Algebras and Related Algebras I, Journal of Algebra 104 (1986), 310-328.

[Do2] 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.

[Do3] 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.

[Gg] Grigor’ev, D. J.: An Analogue of the Bruhat Decomposition for the Closure of the Cone of a Chevalley Group of the Classical Series. Soviet Math. Dokl., Vol. 23 (1981), No. 2.

[Ha] Hayashi, T.: Quantum Deformation of Classical Groups. Publ. RIMS, Kyoto Univ. 28 (1992), 57-81.

[Ia] Iano-Fletcher, M.: Polynomial Representations of Symplectic Groups. Thesis, University of Warwick, 1990.

[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.

[Ma] Manin, Y.I.: Quantum Groups and Non-Commutative Geometry. CRM, Univesitè de Montrèal, 1988.

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

[Oe] Oehms, S.: Symplektische q-Schur-Algebren, Thesis, University of Stuttgart.

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

[Su] Sudbery, A.: Matrix-Element Bialgebras Determined by Quadratic Coordinate Algebras. J. of Algebra, Vol 158, (1993), 375-399.

[Ta] Takeuchi, M.: Matric Bialgebras and Quantum Groups. Israel J. of Math., Vol. 72, Nos. 1-2, (1990), 232-251.

[We] Wenzl, H.: On the structure of Brauer’s centalizer algebras. Annals of Math., 128 (1988), 173-193.