Global Decomposition Numbers for Cellular Algebras

Sebastian Oehms, Universität Stuttgart ^{1}

1 Introduction

We introduce a concept of global decomposition numbers for cellular algebras A over a principal ideal domain R. Such algebras have been defined by Graham and Lehrer in [GL]. In the category of R-finitely generated A-modules, you always have a set of standard resp. costandard modules, the former of which Graham and Lehrer called cell representations and which we will denote by △(λ) resp. ▽(λ) (for the indicated connections with quasihereditary algebras we refer to [KX]). Here λ runs through a poset Λ which comes along with the definition of A.

Now, if there is an R-algebra structure on a field K, i.e. a ring homomorphism from R to K, it is shown in [GL] that there is a unique maximal A_{K} := (K ⊗_{R} A)-submodule in △_{K}(λ) := K ⊗_{R} △(λ) (not necessarily proper) and that the corresponding nonzero simple quotients L_{K}(λ) are absolutely irreducible and give all the irreducibles A_{K}-modules. In this situation you can define decomposition numbers d_{K}^{λμ} as the integer coefficients of △_{K}(λ) in the Grothendieck group of A_{K} with respect to the basis element L_{K}(μ).

In this paper we are going to define analogues of L_{K}(μ) and d_{K}^{λμ} with respect to the algebra A over the original ground ring R, such that you can get the above described simple modules and decomposition numbers by specialising. As examples for A we will treat the group algebras of the symmetric groups S_{3} and S_{4} over the ring ℤ. In this case, the standard modules △(λ) are known as Specht modules and the index set Λ consists of partitions of n. It turns out that the global decomposition numbers contain informations on the module category of ℤS_{4}, which vanishes in all specialisations, i.e. which can not be reconstructed from the knowledge of ordinary decomposition numbers for all characteristics.

This work is purely conceptual, the proofs being almost trivial. But anyway, it provides a new view and calculus for decomposition numbers. Since more information survives in this calculus, there might be a chance to understand things better.

2 General Concepts

Let R be an arbitrary unital commutative ring and A a unital R-algebra, i.e. a unital ring A together with a ring homomorphism from R into the center of A sending 1 to 1. We denote by mod_{R} the category of finitely generated R-modules and by mod_{A} the category of right A-modules which are finitely generated as R-modules. Assume that the isomorphism classes of mod_{R} and mod_{A} form sets 𝕊_{R} and 𝕊_{A}. The monoidal structure of the category mod_{R} – coming along with the tensorproduct over R – induces the structure of a commutative monoid on 𝕊_{R}. Thus the multiplication of two elements u, v ∈ 𝕊_{R} given by representatives U, V ∈ mod_{R} is defined as the isomorphism class of U ⊗_{R} V. The unit element 1 is the isomorphism class of R. and in addition you have an element 0 corresponding to the zero module. In a similar manner there is an action of the monoid 𝕊_{R} from the left on the set 𝕊_{A}, since the tensor product U ⊗_{R} M of an R-module U with a right A-module gives again a right A-module.

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

Now let ˜ \underline{R} := ℤ𝕊_{R} be the monoid algebra on 𝕊_{R} over the integers and ˜ \underline{A} the free abelian group on 𝕊_{A}. The above action of 𝕊_{R} on 𝕊_{A} induces a ˜ \underline{R}-module structure on ˜ \underline{A}. Let I_{R} be the abelian subgroup of ˜ \underline{R} generated be the expressions U′ −U +U′′ for all U, U′, U′′ ∈ 𝕊_{R}, such that there is a short split exact sequence (from now on we will not deitinguish between elements in 𝕊_{R} resp. 𝕊_{A} and representatives for these elements in mod_{R} resp. mod_{A}).

0 → U′ → U → U′′ → 0.

Since tensoring split exact sequences yields again split exact sequences, I_{R} is obviously an ideal in ˜ \underline{R}. We denote the residue class ring by

\underline{R} := ˜ \underline{R}/I_{R}.

Similarily, in ˜ \underline{A} we consider the abelian subgroup I_{A} generated by all expressions M′ − M + M′′, such that there is a short exact sequence

0 → M′ → M → M′′ → 0

of right A-modules which is split as a sequence of R-modules. It is obviously an ˜ \underline{R}- submodule of ˜ \underline{A} and clearly one has

I_{R}I_{A} ⊆ I_{A}.

Therefore the quotient

\underline{A} := ˜ \underline{A}/I_{A}

can be considered as an \underline{R}-module. Before proceeding let us give some examples. For this purpose denote by 𝕐_{R} ⊆ 𝕊_{R} the set of isomorphism classes corresponding to the indecomposable R-modules and by 𝕏_{R} ⊆ 𝕊_{R} the set corresponding to cyclic R-modules, i.e. modules isomorphic to R/I for some ideal I in R. Note that 𝕏_{R} is a submonoid of 𝕊_{R} and both contain 1 and 0. Therefore, the ℤ-linear span of 𝕏_{R} in \underline{R} is a subring.

Now if R is noetherian, then the ring \underline{R} is generated by the residue classes of 𝕐_{R}. If in addition the Krull-Schmidt theorem holds in mod_{R}, that is, if decomposition into indecomposables is unique, then \underline{R} is free abelian on 𝕐_{R}\\{0\}.

Let us look in more detail to the case where R is a principal ideal domain. Here you have 𝕐_{R} ⊆ 𝕏_{R} and the former is a submonoid as well. More precisely, for each pair (p, l) of a prime p ∈ R and a positive integer l ∈ ℕ you have an element in 𝕐_{R} whose representative is R/(p^{l}) and which we denote by [p^{l}]. From the main theorem on modules over principal ideal domains it follows that for different pairs (p, l) ≠ (q, k) you get different elements [p^{l}] ≠ [q^{k}] and that all elements of 𝕐_{R} – except 1 and 0 – are of this form. The multiplication in 𝕐_{R} for elements other than 1 and 0 is given by the rule

[p^{l}][q^{k}] = \{ \begin{matrix}0 & if p ≠ q \\ [p^{min(l,k)}] & if p = q \end{matrix} .  (1)

Now \underline{R} is canonically isomorphic to the integral monoid algebra of 𝕐_{R} factored by the one dimensional span of the element 0 ∈ 𝕐_{R}. In the case R is a principal ideal domain we will always identify \underline{R} in this way and use the notation [p^{l}] for basis elements as above. In the same manner we write [n] for the residue class of the isomorphism type of R/(n) if n ∈ R is arbitrary. Thus [n] = [p_{1}^{l_{1}}] + [p_{2}^{l_{2}}] + … + [p_{k}^{l_{k}}] if n = p_{1}^{l_{1}}p_{2}^{l_{2}} … p_{k}^{l_{k}} is the primedecomposition of n in R. In the case where R is a field, \underline{R} is canonically isomorphic to ℤ and \underline{A} is the Grothendieck group G_{0}(A) of A considered as a ℤ-module.

We call a right A-module M ∈ mod_{A} reducible by R-sums iff M contains an A-submodule N which is a direct summand of R-modules in M. If M is not reducible by R sums we call it irreducible by R sums. The concept is clearly invariant under isomorphisms of A-modules. Let 𝕀_{A} be the subset of 𝕊_{A} consisting of isomorphism classes of by R sums irreducible A-modules. If R is noetherian it is easily seen that \underline{A} is generated by 𝕀_{A} as a ℤ-module.

3 Base Changes

Let S be another commutative ring which might be considered as an R-algebra via some ring homomorphism φ : R → S. In this situation there is a functor

φ := S ⊗_{R} − : mod_{R} → mod_{S}

leading to a map 𝕊_{φ} : 𝕊_{R} → 𝕊_{S} of monoids, given by 𝕊_{φ}(U) = V iff V ≅ S ⊗_{R} U. In the second step this induces a ring homomorphism ˜ \underline{φ} : ˜ \underline{R} → ˜ \underline{S} carrying the ideal I_{R} into the ideal I_{S}, since tensoring short split exact sequences yields again short split exact sequences. Thus we end up with a ring homomorphism

\underline{φ} : \underline{R} → \underline{S}.

In the case where S is a field, this is a ring homomorphism to the integers given on a residue class [M] in \underline{R} for some M ∈ 𝕊_{R} by

\underline{φ}([M]) = dim_{S}(S ⊗_{R} M).

For a principal ideal domain you therefore have \underline{φ}(1) = 1 and

\underline{φ}([p^{l}]) = \{ \begin{matrix}0 & if (p) ≠ ker(R → S) \\ 1 & if (p) = ker(R → S) \end{matrix} .  (2)

Turning to the algebra A we write A_{S} := S ⊗_{R} A for the base extended algebra. As above, there is a functor

ψ := S ⊗_{R} − : mod_{A} → mod_{A_{S}}

leading to a map 𝕊_{ψ} : 𝕊_{A} → 𝕊_{A_{S}}. This induces a ℤ-module homomorphism ˜ \underline{ψ} : ˜ \underline{A} → ˜ \underline{A_{S}} and since ˜ \underline{ψ}(I_{A}) ⊆ I_{A_{S}} finally a ℤ-module homomorphism

\underline{ψ} : \underline{A} → \underline{A_{S}}.

Concerning the action of 𝕊_{R} on 𝕊_{A} one obviously has 𝕊_{ψ}(UM) = 𝕊_{φ}(U)𝕊_{ψ}(M) for all U ∈ 𝕊_{R} and M ∈ 𝕊_{A} which leads to

\underline{ψ}(ra) = \underline{φ}(r)\underline{ψ}(a) for all r ∈ \underline{R}, a ∈ \underline{A}

If we interpret \underline{S} as an \underline{R}-algebra via \underline{φ}, we can restate this as

Proposition 3.1 There is an \underline{S}-module homomorphism

\underline{ψ}_{S} : \underline{S} ⊗\underline{R} \underline{A} → \underline{A_{S}}

given by \underline{ψ}_{S}(s ⊗ a) := s\underline{ψ}(a) for all s ∈ \underline{S} and a ∈ \underline{A}.

Now, on the other hand there are functors

φ′ : mod_{S} → mod_{R}, ψ′ : mod_{A_{S}} → mod_{A}

carrying an S-module resp. A_{S}-module M to the same abelian group, but inflating the action of S resp. A_{S} to an action of R resp. A on M. Going again through the above procedure, one obtains maps of abelian groups

\underline{φ′} : \underline{S} → \underline{R}, \underline{ψ′} : \underline{A_{S}} → \underline{A}.

If the map R → S is surjective, then you have φ ∘ φ′(U) ≅ U and ψ ∘ ψ′(M) ≅ M for all S-modules U and A_{S}-modules M. Therefore in this case, one obtains

\underline{φ} ∘ \underline{φ′} = id\underline{S} and \underline{ψ} ∘ \underline{ψ′} = id\underline{A_{S}}.

In the case where R → S is surjektiv, there corresponds an element in 𝕏_{R} to S, i.e. the isomorphism class of the R-module S. This leads to a corresponding element [S] in \underline{R}. Denote the left multiplication by [S] in \underline{A} by l_{S}.

Proposition 3.2 Suppose that the R-algebra structure map R → S on S is surjective. Then \underline{ψ′} gives an embedding of \underline{A_{S}} into \underline{A} as a direct summand, and you have

\underline{ψ′} ∘ \underline{ψ} = l_{S}.

Furthermore, the \underline{S}-module homomorphism \underline{ψ}_{S} of Proposition 3.1 is an isomorphism with inverse given by \underline{ψ}_{S}^{−1}(a) := 1\underline{S} ⊗ \underline{ψ′}(a).

Proof: The first two statements are clear from definitions and what has been said before (note that [S] is an idempotent and therefore l_{S} a projection map of \underline{R}-modules). For the third one, it is enough to show that κ(a) := 1\underline{S} ⊗ \underline{ψ′}(a) defines an inverse as a map of ℤ-modules. Now \underline{ψ}_{S} ∘ κ = id is directly clear from the above righthand formula, whereas κ∘\underline{ψ}_{S} = id holds by surjectivity of \underline{φ} following from the lefthand formula. □

The set 𝕐_{R} ∩ 𝕏_{R} contains the set of fields F which are epimorphic images of R and which we denote by 𝔽_{R}. These are precisely the elements of 𝕏_{R} corresponding to simple Rmodules F = R/I, one for each maximal ideal I of R. For any F ∈ 𝔽_{R} the ℤ-module \underline{A_{F}} is the Grothendieck group of A_{F} as mentioned before. Therefore the proposition shows, that all such Grothendieck groups are embedded in \underline{A} as direct summands and that a projection on them is given by multiplication with the idempotent [F] ∈ \underline{R}. We set

I^{0} := \{a ∈ \underline{A}| [F]a = 0 ∀F ∈ 𝔽_{R}\}.

As an intersection of the kernels of the \underline{R}-module homomorphisms l_{F} this is obviously a submodule. To see that for an R-algebra S the map \underline{ψ} takes I^{0} to the corresponding submodule I_{S}^{0} of \underline{A_{S}}, one uses the fact that the R-algebra structure map ρ : R → S induces an injective map τ : 𝔽_{S} → 𝔽_{R} on a representative F = S/I given by τ(F) := [R/ρ^{−1}(I)]. It is easily seen that the equation [τ(F)]a = [F]\underline{ψ}(a) holds for all F ∈ 𝔽_{S} and a ∈ \underline{A}. Thus, setting

\underline{A}^{0} := \underline{A}/I^{0}

it follows that \underline{ψ} factors to a map \underline{ψ}^{0} : \underline{A}^{0} → \underline{A_{S}}^{0}. In the case where S is a field you have 𝔽_{S} = \{1\} and therefore I_{S}^{0} = \{0\}. This means that \underline{A}^{0} still maps to the Grothendieck group G_{0}(A_{S}) for any field S that is an R-algebra.

4 Residual Series

As we have seen, factoring out the submodule I^{0} we don’t cut off more infomation as is contained in the collection of all Grothendieck groups for fields. However, we can save additional informations, that is, factoring out some proper submodule of I^{0} instead of I^{0} if it remains possible to define global decomposition numbers in the corresponding quotient of \underline{A}. We now give the definition of such a one.

For each isomorphism class C ∈ 𝕏_{R} of cyclic R-modules there is a well defined map res_{C} : 𝕊_{R} → 𝕊_{R} to each U ∈ 𝕊_{R} given by the isomorphism class of the kernel of the map

U → (R/I) ⊗_{R} U, u ↦ 1 ⊗ u,

where I is the ideal in R such that R/I is of isomorphism type C. Thus res_{C}(U) is just the isomorphism class of the R-modul IU. It is easily seen that res_{C} maps short split exact sequences to short split exact sequences, such that it induces an additive map on \underline{R} which we denote by the same symbol. Now, if we take for U a right A-module M, then of course res_{C}(M) is again a right A-module. Therefore one obtains a similar additive map res_{C} : \underline{A} → \underline{A} for which we will use the same symbol as well and call it the C-residuum map. We write res_{C}^{i} for iterating res_{C} i times and define

N^{i} := \{a ∈ \underline{A}| res_{F}^{i} (a) ∈ I^{0} ∀F ∈ 𝔽_{R}\}.

These are of course abelian subgroups of \underline{A} as the intersection of preimages of I^{0} under the additive maps res_{F}^{i}. Later on, we will see that they in fact are \underline{R}-submodules in the case where R is a principle ideal domain. But in general this is not true.

We now define inductively a descending chain I^{0} ⊇ I^{1} ⊇ … of \underline{R}-submodules. The definition of I^{0} has been given. For i > 1 let I^{i} be the submodule of I^{i−1} generated by I^{i−1} ∩ N^{i}. Set

I^{∞} := ⋂_{i=0}^{∞} I^{i} and \underline{A}^{i} := \underline{A}/I^{i}, \underline{A}^{∞} := \underline{A}/I^{∞}.

Let us illustrate this in the case of a principle ideal domain. Then F ∈ 𝔽_{R} is of the form F = R/(p) for a prime p, that is, [F] = [p] in the above introduced notation. We write res_{p} := res_{F} for short. Since \underline{R} is generated by 𝕐_{R} as a ℤ-module we only need to calculate for prime powers q^{l}

res_{p}([q^{l}]) = \{ \begin{matrix}[q^{l}] & if p ≠ q \\ [q^{l−1}] & if p = q and l > 1 \\ 0 & if p = q and l = 1 \end{matrix} .  (3)

The p-residuum of 1 and 0 is obviously 1 resp. 0. This, together with (1) shows that res_{p} is multiplicative, i.e. res_{p}(ra) = res_{p}(r)res_{p}(a) for r, a ∈ \underline{R} resp. a ∈ \underline{A}. Therefore, the N^{i} are submodules in \underline{A} as mentioned above, and it follows I^{i} = I^{i−1} ∩ N^{i}. This means that an element a ∈ \underline{A} is zero in \underline{A}^{∞}, iff for all i ∈ ℕ_{0} and all primes p, q ∈ R you have [q]res_{p}^{i}(a) = 0. Here res_{p}^{0} has to be defined as the identity map.

From (1) and (3) it is easy to see that [q]res_{p}(a) = [q]a if q ≠ p. Therefore, [p]res_{p}^{i}(a) = 0 implies [q]res_{p}^{i}(a) = 0 for all primes q if a is assumed to lie in I^{0}. This shows

Lemma 4.1 Let R be a principal ideal domain. Two elements a, b of \underline{A} coincide in \underline{A}^{∞} iff they have identical residual series for all primes p. The latter one is for x ∈ \underline{A} defined by

Res_{p}(x) := ([p]res_{p}^{0}(x), [p]res_{p}^{1}(x), [p]res_{p}^{2}(x), …)

A similar statement holds with respect to \underline{A}^{i} and the first i entries of both series.

It should be remarked that res_{F} is not multiplicative in general. As an example, take the ring R := ℤ[x, x^{−1}] of Laurent polynomials in the indeterminant x which is important in the context of Hecke algebras, q-Schur algebras, Birman-Murakami-Wenzl algebras, etc. Let F := R/(p, x − 1) ≅ GF (p) be the Galois field to the prime p ∈ ℤ with R-module structure given by the natural projection. In the same manner, U := R/(x − 1) ≅ ℤ and V := R/(p) ≅ GF (p)[x, x^{−1}] are considered as R-modules. Then you have res_{F} (U) ≅ U, res_{F} (V) ≅ V, and since U ⊗_{R} V ≅ F it follows

res_{F} ([U])res_{F} ([V]) = [F] ≠ 0 = res_{F} ([U][V]).

5 Application to Cellular Algebras

We now restrict A to a special type of R-algebra which have been described using a couple of axioms by Graham and Lehrer ([GL]) and which are called cellular algebras. Instead of listing these axioms we rather explain the properties of A and mod_{A} which will be needed in the sequel. For more details we refer to [GL].

1. There is a finite poset Λ and for each λ ∈ Λ a pair △(λ), ▽(λ) ∈ mod_{A} of Amodules called the standard and the costandard module of type λ. Both are free and isomorphic to each other as R-modules and there is an A-module homomorphism

f_{λ} : △(λ) → ▽(λ).

2. For each commutative ring S being an R-algebra, A_{S} := S ⊗_{R} A is again a cellular algebra and the functor ψ := S ⊗_{R} − carries △(λ), ▽(λ), f_{λ} to the corresponding objects and morphisms △_{S}(λ), ▽_{S}(λ), f_{S,λ} with respect to A_{S}.

3. For each field K being an R-algebra, the image L_{K}(λ) of f_{K,λ} is an irreducible A_{K}- module as long as f_{K,λ} is not the zero map. All irreducible A_{K}-modules occur in this way. Furthermore, if λ ≠ μ and f_{K,λ} ≠ 0 ≠ f_{K,μ}, then the (nesessarily absolutely) irreducible modules L_{K}(λ) and L_{K}(μ) are non isomorphic.

4. For a field K and μ ∈ Λ such that f_{K,μ} ≠ 0, let d_{K}^{λμ} be the multiplictiy of L_{K}(μ) as a composition factor of △_{K}(λ). Then d_{K}^{μμ} = 1 and d_{K}^{λμ} ≠ 0 implies λ ≤ μ.

Let us compare these statements with corresponding definitions and results in [GL]. We assume that A is finitely generated as an R-module. This implies the finiteness of Λ. Now, as we consider right modules, our △(λ) corresponds to W (λ)^{∗} in [GL]. The costandard modules are defined as their duals

▽(λ) := Hom_{R}(△(λ), R)

where the action of A is given by (la)(d) := l(da^{∗}) for all l ∈ Hom_{R}(△(λ), R), a ∈ A and d ∈ △(λ). Here ∗ denotes the anti involution on A which exists by one of the axioms. Now, on W (λ) there is a symmetric bilinear form ϕ_{λ} with ϕ_{λ}(a^{∗}x, y) = ϕ_{λ}(x, ay). Since W (λ) coincides with W (λ)^{∗} as an R-module we have a bilinear form on △(λ) with ϕ_{λ}(xa, y) = ϕ_{λ}(x, ya^{∗}) (since the operation on W (λ)^{∗} is just the left action on W (λ) pulled to the right via ∗). This in turn leads to the A-module homomorphism f_{λ} being defined by f_{λ}(x)(y) := ϕ_{λ}(x, y).

The compatibility with the functor ψ is obvious (compare (1.8) in [GL]). The statements concerning the L_{K}(λ) follow from (3.2) and (3.4) of [GL]. For this purpose, note that the radical of ϕ_{λ} is nothing but the kernel of f_{λ} where again the action of A is pulled from left to right via ∗. Finally, the statement on the decomposition numbers d_{K}^{λμ} follows from (3.6) of [GL].

If K is a field and all f_{K,λ} are isomorphisms, then according to Theorem (3.8) of [GL] A_{K} is semisimple. For the field Q of fractions on an integral domain R this is by flatness the case iff f_{λ} is injective for all λ ∈ Λ. If this is the case call A generic semisimple. Even for such an A it is possible that f_{K,λ} = 0 for some field K. For a prime p we will throughout denote by F := [p] = R/(p) the corresponding residue class field for short. We set

Λ_{p} := {λ ∈ Λ| f_{F,λ} ≠ 0}, Λ_{p}^{−} := Λ\Λ_{p}, P_{λ} := {p ∈ R| p prim, λ ∈ Λ_{p}^{−}}.

According to (3.10) in [GL] for a field K, the algebra A_{K} is quasi-hereditary if f_{K,λ} ≠ 0 for all λ ∈ Λ. Let us call A integrally quasi-hereditary if Λ_{p}^{−} = ∅ for all primes p ∈ R.

We now define a global version of L_{K}(λ) in \underline{A}. Throughout, we will not distinguish between right A-modules and their residue classes in \underline{A}, \underline{A}^{i} or \underline{A}^{∞} any more. Let T(λ) be the cokernel of f_{λ}. We set

L(λ) := ▽(λ) − T(λ) ∈ \underline{A}.

Note that L(λ) is not the residue class of some A-module in general. But if K is a field and an R-algebra, then the map \underline{ψ} of chapter 3 carries this element to the residue class of L_{K}(λ) in the Grothendieck group G_{0}(A_{K}).

Let L be the \underline{R}-linear span of all L(λ) in \underline{A}^{∞}. We wish to show that for a generic semisimple cellular algebra A over a pid you always have L = \underline{A}^{∞}. From now on, let us assume that R is a pid. We call an element r ∈ \underline{R} positive if in the unique expression with respect to the ℤ-basis 𝕐_{R}\\{0\} of \underline{R} all integer coefficients are nonnegative. Write \underline{R}^{+} for the submonoid of all these elements and L^{+} for the \underline{R}^{+} span of \{L(λ)| λ ∈ Λ\} in L. Note that x ∈ \underline{R}^{+} is zero, iff [p]x = 0 for all primes p, whereas x ∈ \underline{R} is zero, iff the residual series Res_{p}(x) := ([p]res_{p}^{i}(x), [p]res_{p}^{1}(x), …) in \underline{R} is zero for all primes p.

Lemma 5.1 Let A be cellular and M a right A-module. If M is an R-torsion module, then you have M ∈ L^{+} as elements of \underline{A}^{∞}.

Proof: Since the R-annihilator of x ∈ M is contained in the R-annihilator of xa for all a ∈ A, it follows that the R-module decomposition of M into primary components is in fact a decomposition of A-modules. Thus we may assume that M is primary as an R-module. Let p be the corresponding prime. Now for all primes q ≠ p the residual series Res_{q}(M) is constant zero. Let us consider Res_{p}(M). Recall that F := R/(p) and λ ∈ Λ_{p} means f_{F,λ} ≠ 0, that is L_{F} (λ) = [p]L(λ) ≠ 0. Since these simple A_{F} -modules form a basis of the Grothendieck group of A_{F}, there are unique nonnegative integers k_{λ}^{i} such that

[p]res_{p}^{i}(M) = ∑_{λ∈Λ_{p}} k_{λ}^{i}[p]L(λ).

For all l ∈ ℕ there is an A_{F} -epimorphism

[p]res_{p}^{l−1}(M) = p^{l−1}M/p^{l}M → p^{l}M/p^{l+1}M = [p]res_{p}^{l}(M)

given by multiplication by p. This shows that k_{λ}^{i}−k_{λ}^{i+1} is nonnegative. There is a smallest number j, such that res_{p}^{l}(M) = 0 for all l ≥ j, since M is a torsion module. We set

N := ∑_{λ∈Λ_{p}}(∑_{i=1}^{j}(k_{λ}^{i−1} − k_{λ}^{i})[p^{i}])L(λ).

This is obviously contained in L^{+}. By Lemma 4.1 it remains to show that Res_{q}(M) = Res_{q}(N) for all primes q. For q ≠ p both series are identically zero. For q = p by multiplicativity of res_{p}^{i}, it is enough to show

[p]res_{p}^{i}(∑_{i=1}^{j}(k_{λ}^{i−1} − k_{λ}^{i})[p^{i}]) = k_{λ}^{i}[p]

for all i. This can be calculated with help of (3). □

Corollary 5.2 For a generic semisimple cellular algebra A all costandard modules ▽(λ) are contained in L^{+} as elements of \underline{A}^{∞}.

Proof: This is immediate from the definition of L(λ) and the Lemma 5.1 since under the above assumption T(λ) is an R-torsion module. □

Lemma 5.3 Let Q be the field of fractions on R, M and N two full A-lattices in the same A_{Q}-module V. Then in \underline{A}^{∞} the identity M = N holds.

Proof: By Lemma 4.1 we have to show that the residual series of M and N coincide for all primes p. The epimorphism from [p]res_{p}^{i}(M) to [p]res_{p}^{i+1}(M) considered in the proof of Lemma 5.1 and being induced by multiplication by p must be an isomorphism in the case of a free module. Therefore, the residual series of M and N are constant. Thus the proof is finished as soon as we have shown [p]M = [p]N for all primes p. Since the sum M + N in V again is a full lattice,we can reduce to the case M ⊆ N. Now let R_{p} be the localisation of R at the prime ideal (p) and M′ := R_{p}M, N′ := R_{p}N the corresponding R_{p} lattices in V. Multiplication by [p] – the residue class in \underline{R_{p}} of the field F = R_{P} /(p) – is defined in \underline{A_{R_{p}}} as well and leads to the same elements [p]M′ = [p]M resp. [p]N′ = [p]N in the Grothendieck group G_{0}(A_{F}) (see Propositions 3.1 and 3.2). Finishing the proof now is standard (cf. [CR], 16.16). □

Applying Lemma 5.3 to the image of f_{λ} and ▽(λ) as lattices in ▽_{Q}(λ), one immediately gets

Corollary 5.4 For a generic semisimple cellular algebra A corresponding standard and costandard modules △(λ) and ▽(λ) coincide in \underline{A}^{∞} for all λ ∈ Λ.

Theorem 5.5 Let A be a generic semisimple cellular algebra. Then each right A-module M ∈ mod_{A} is contained in L^{+} as an element of \underline{A}^{∞}. In particular \underline{A}^{∞} = L.

Proof: By Lemma 5.1 we can reduce to the case where M is free as an R-module, since an arbitrary M can be written M = M_{f} + M_{t} with M_{f} free and M_{t} an R-torsion module (even in \underline{A}). Let Q be the field of fractions on R. Since A is generic semisimple, M_{Q} := Q⊗_{R}M decomposes into a direct sum of simple modules L_{Q}(λ) = △_{Q}(λ) = ▽_{Q}(λ). Let N ∈ mod_{A} be the direct sum of the corresponding ▽(λ). Both M and N are full lattices in M_{Q} so that it follows M = N in \underline{A}^{∞} by Lemma 5.3. Since N ∈ L^{+} by Corollary 5.2 the proof is finished. □

Throughout, let us now assume that A is generic semisimple. Note that in this case the set P_{λ} defined above is finite for all λ. Set z_{λ} := 1 − ∑_{p∈P_{λ}}[p^{l_{p}}] ∈ \underline{R} where l_{p} is the largest number such that [p^{l_{p}}]L(λ) = 0.

Proposition 5.6 There are uniquely determined elements d^{λμ} ∈ \underline{R} assigned to each pair λ, μ ∈ Λ which satisfy the following conditions:

(a) d^{λμ} ∈ \underline{R}^{+} for all λ ≠ μ and d^{λλ} = z_{λ}

(b) [p]d^{λμ} = 0 for all μ ∈ Λ_{p}^{−} (c) ▽(λ) = ∑_{μ∈Λ} d^{λμ}L(μ) holds in \underline{A}^{∞}.

Proof: To show existence, there are elements g^{λμ} ∈ \underline{R}^{+} such that T (λ) = ∑_{μ∈Λ} g^{λμ}L(μ) by Lemma 5.1. Furthermore, the proof of this lemma shows that [p]g^{λμ} = 0 for all μ ∈ Λ_{p}^{−}. If λ ∈ Λ_{p} we multiply the equation ▽(λ) = L(λ)+T (λ) by [p]. Because [p]▽(λ) = ▽_{F} (λ) and [p]L(λ) = L_{F} (λ), we get [p](g^{λλ}+1) = [p]d_{F}^{λλ} = [p] thus [p]g^{λλ} = 0. Because g^{λλ} ∈ \underline{R}^{+}, this in turn implies g^{λλ} = 0 for all λ. Setting d^{λλ} := z_{λ} and d^{λμ} := g^{λμ} for λ ≠ μ we are finished since ▽(λ) = L(λ) + T (λ) and obviously L(λ) = z_{λ}L(λ).

For uniqueness, take (d^{λμ})′ to be a second set of such elements. Since d^{λλ} = (d^{λλ})′ by definition and for λ ≠ μ both elemens are in \underline{R}^{+}, it is enough to show [p]d^{λμ} = [p](d^{λμ})′ for all primes p. In the case μ ∈ Λ_{p}^{−} this follows from condition (b). Using (b) and multiplying (c) by [p] we get from Proposition 3.2 [p]d^{λμ} = [p]d_{F}^{λμ} = [p](d^{λμ})′ for all μ ∈ Λ_{p} where d_{F}^{λμ} are the usual decomposition numbers for A_{F}. To this claim, note that △_{F} (λ) = [p] △ (λ) = [p] ▽ (λ) = ▽_{F} (λ) in G_{0}(A_{F}) by Corollary 5.4 and Proposition 3.2. This completes the proof. □

For any field K you have \underline{φ}(d^{λμ}) = d_{K}^{λμ} where \underline{φ} : \underline{R} → \underline{K} = ℤ is the map induced by the base change to K (see chapter 3). Therefore, we call these elements the global decomposition numbers of A.

Proposition 5.7 If λ ≠ μ then d^{λμ} ≠ 0 implies λ < μ.

Proof: Since 0 ≠ d^{λμ} ∈ \underline{R}^{+} there must be a prime p ∈ R such that [p]d^{λμ} ≠ 0. This implies d_{F}^{λμ} ≠ 0 for the residue class field F = R/(p). The corresponding known result for fields then implies the statement of the proposition. □

For any total order on Λ refining the given partial order, the matrix D := (d^{λμ})_{λ,μ∈Λ} is therefore upper triangular with respect to this total order. Setting e^{λλ} := 1 and e^{λμ} := d^{λμ} for λ ≠ μ one gets a unitriangular (and so invertible) matrix E := (e^{λμ})_{λ,μ∈Λ}. Note that condition (c) of proposition 5.6 holds for the e^{λμ}, too. We call D the global decomposition matrix and E the regular decomposition matrix. The fact that E is invertible implies that \underline{A}^{∞} is generated as an \underline{R} module by the costandard modules ▽(λ), too.

Let us determine the relations among the L(λ) resp. the ▽(λ). Denote by X_{λ} the canonical basis elements of the free \underline{R}-module \underline{R}^{Λ}. We define two epimorphisms σ, τ : \underline{R}^{Λ} → \underline{A}^{∞} by σ(X_{λ}) := L(λ) and τ(X_{λ}) := ▽(λ). If ϵ is the automorphism of \underline{R}^{Λ} given by the regular decomposition matrix E, i.e. ϵ(X_{λ}) := ∑_{μ∈Λ} e^{λμ}X_{μ}, we clearly have σ ∘ϵ = τ. Let K_{τ} and K_{σ} be the kernels of τ and σ respectively. We only need to determine K_{τ} since K_{σ} = ϵ(K_{τ}).

Let p ∈ R be a prime such that Λ_{p}^{−} ≠ ∅. Since for all μ ∈ Λ_{p} we have d^{μμ} = 1 the costandard modules {▽_{F} (μ)| μ ∈ Λ_{p}} form a basis of the Grothendieck group G_{0}(A_{F}). Therefore there are unique integers a_{λμ,p} for all λ ∈ Λ_{p}^{−} and μ ∈ Λ_{p} such that

[p] ▽ (λ) = ∑_{μ∈Λ_{p}} a_{λμ,p}[p] ▽ (μ).

For each pair (λ, p) with λ ∈ Λ_{p}^{−} we set

u_{λ,p} := X_{λ} − ∑_{μ∈Λ_{p}} a_{λμ,p}X_{μ},

and let K′_{τ} be the ℤ-linear span of all elements [p^{l}]u_{λ,p} for such pairs (λ, p) and positive integers l ∈ ℕ.

Proposition 5.8 We have K_{τ} = K′_{τ}.

Proof: Since the residual series of the costandard modules are constant we have for a prime q

[q]res_{q}^{i}(τ([p^{l}]u_{λ,p})) = [q]res_{q}^{i}([p^{l}])(▽(λ) − ∑_{μ∈Λ_{p}} a_{λμ,p} ▽ (μ)).

Now, [q]res_{q}^{i}([p^{l}]) = 0 whenever q ≠ p by the formulas (1) and (3). Multiplying the bracket term by [p] gives zero by construction of the numbers a_{λμ,p}. Therefore Res_{q}^{i}(τ([p^{l}]u_{λ,p})) = 0 for all primes q, thus K′_{τ} ⊆ K_{τ}.

Now let x := ∑_{λ∈Λ} b_{λ}X_{λ} be in K_{τ} with b_{λ} ∈ \underline{R}. We first claim that there exists i ∈ ℕ_{0} such that [p]res_{p}^{j}(b_{λ}) = 0 for all j ≥ i, λ ∈ Λ and primes p ∈ R. For if there is λ_{0} ∈ Λ and a prime p ∈ R such that for all i ∈ ℕ_{0} there exists j ≥ i giving [p]res_{p}^{j}(b_{λ_{0}}) ≠ 0, it follows that the coefficient of 1 in the unique presentation of b_{λ_{0}} with respect to the ℤ-basis 𝕐_{R}\\{0\} of \underline{R} is nonzero. Now, if Q is the field of fractions on R and \underline{φ} : \underline{R} → \underline{Q} = ℤ the corresponding map of chapter 3 we must have \underline{φ}(b_{λ_{0}}) ≠ 0. But this implies

\underline{ψ}(τ(x)) = ∑_{λ∈Λ} \underline{φ}(b_{λ}) ▽_{Q} (λ) ≠ 0

since all costandard modules form a basis of G_{0}(A_{Q}). This contradict τ(x) = 0 in \underline{A}^{∞}.

We now proceed by induction on such a number i. For i = 0 we must have x = 0 since then the residual series of all b_{λ} are zero for all primes. Assume i > 0. There are uniquely determined numbers h_{λ,p} ∈ ℤ such that [p]h_{λ,p} = [p]res_{p}^{i−1}(b_{λ}). Set

y := ∑_{λ∈Λ} ∑_{p∈P_{λ}} h_{λ,p}[p^{i}]u_{λ,p} ∈ K′_{τ}.

Then z := x − y ∈ K_{τ}. Write z = ∑_{λ∈Λ} c_{λ}X_{λ}. By construction of y we have [p]res_{p}^{j}(c_{λ}) = [p]res_{p}^{j}(b_{λ}) = 0 for all j ≥ i and all primes p. We claim [p]res_{p}^{i−1}(c_{λ}) = 0 for all primes p and λ ∈ Λ. If this is shown the induction hypothesis applies to z giving x = y + z ∈ K′_{τ} as required.

Keep p fixed and let λ ∈ Λ_{p}^{−}. The coefficient of X_{λ} in y is of the form h_{λ,p}[p^{i}] + ∑_{q≠p}[q^{i}]v_{q} with some v_{q} ∈ \underline{R} being nonzero only for a finite number of primes q. Since [p]res_{p}^{i−1}([q^{i}]v_{q}) = [p][q^{i}]res_{p}^{i−1}(v_{q}) = 0, it follows [p]res_{p}^{i−1}(c_{λ}) = [p]res_{p}^{i−1}(b_{λ}) − [p]h_{λ,p} = 0 by definition of h_{λ,p}. Making use of τ(z) = 0 we calculate

[p]res_{p}^{i−1}(τ(z)) = ∑_{λ∈λ}[p]res_{p}^{i−1}(c_{λ})res_{p}^{i−1}(▽(λ)) = ∑_{λ∈Λ_{p}}[p]res_{p}^{i−1}(c_{λ})[p] ▽ (λ) = 0,

where in addition we used the fact that the residual series of ▽(λ) is constant. Since {▽_{F} (λ)| λ ∈ Λ_{p}} is a basis of G_{0}(A_{F}) the coefficients [p]res_{p}^{i−1}(c_{λ}) must be zero for λ ∈ Λ_{p} as well. □

Clearly K′_{τ} = 0 iff A is integrally quasi-hereditary. Thus we have proved

Theorem 5.9 The \underline{R}-module \underline{A}^{∞} is free iff A is integrally quasi-hereditary. If this is the case \{L(λ)| λ ∈ Λ\} and \{▽(λ)| λ ∈ Λ\} are bases of \underline{A}^{∞}. Furthermore, the global and regular decomposition matrices coincide and the decomposition numbers are uniquely determined by condition (3) of Proposition 5.6.

Examples of integrally quasi-hereditary generic semisimple cellular algebras are given by Schur algebras and generalizations of them.

6 Examples

As examples we will treat the ℤ-group algebras A := ℤS_{n} of symmetric groups. Here the standard modules △(λ) are known to be the Specht modules ([GL], 5.7, see [JK], chapter 7 for details on Specht-modules). The reader not familiar with cellular algebras may check imediately that the conditions stated in the begining of section 5 can be realised using this modules. They are labelled by the set Λ = Λ(n)^{+} of partitions of n and are defined as submodules of permutation modules M^{λ} on the cosets of parabolic subgroups corresponding to the partition λ ∈ Λ. As first example, let us look at λ := λ_{n−1} := (n − 1, 1). Then M^{λ} is the permutation representation on the cosets of the copy of S_{n−1} in S_{n} fixing the last element n. As a ℤ-module this is free of rank n with basis e_{1}, …, e_{n}, say. The operation of S_{n} from the right is given by e_{i}π := e_{π^{−1}(i)}. The Specht module △(λ) is the free ℤ-submodule with basis b_{i} := e_{i+1} − e_{i} for i = 1, …, n − 1.

The symmetric bilinear form ϕ_{λ} is given by restriction of the canonical bilinear form on M^{λ} (given by the unit matrix) to △(λ). Thus the Gram matrix of ϕ_{λ} is nothing but the Cartan matrix corresponding to the Dynkin diagram of type A_{n−1}. This in turn is just the coefficient matrix of f_{λ} : △(λ) → ▽(λ) = Hom_{ℤ}(△(λ), ℤ) with respect to the basis b_{i} on △(λ) and the corresponding dual basis b_{i}^{∗} on ▽(λ). The image of f_{λ} is consequently the span of the columns of this matrix and is easily calculated to be the span of nb_{1}^{∗} and b_{i}^{∗} − ib_{1}^{∗} for i = 2, …, n − 1. Therefore T (λ) – the cokernel of f_{λ} – is generated by b_{1}^{∗} and is obviously isomorphic to ℤ/nℤ as a ℤ-module.

Let us show that the operation of ℤS_{n} on T (λ) is trivial. We only need to check this for the long cycle σ := (12 … n) or its inverse and τ := (12). On △(λ) the operation is given by b_{1}σ = − ∑_{i=1}^{n−1} b_{i} and b_{i}σ = b_{i−1} for i > 1 in the case of σ whereas we have b_{1}τ = −b_{1}, b_{2}τ = b_{1} + b_{2} and b_{i}τ = b_{i} for all i > 2. Transposing the corresponding coefficient matrices we get b_{1}^{∗}σ^{−1} = b_{2}^{∗} − b_{1}^{∗} = b_{1}^{∗}τ. Since b_{2}^{∗} − b_{1}^{∗} is congruent to b_{1}^{∗} modulo the image of f_{λ} we are done.

The Specht module of the partition λ_{n} := (n) is just the one-dimensional trivial representation with bilinear form ϕ_{λ_{n}} being given by the trivial Gram matrix. Therefore △(λ_{n}) = ▽(λ_{n}) = L(λ_{n}) holds in \underline{A}. Moreover, we have shown the following equation in \underline{A}:

▽(λ_{n−1}) = L(λ_{n−1}) + [n]L(λ_{n}).

In the case of the symmetric group S_{3} there are three partitions λ_{1} := (1^{3}) = (1, 1, 1), λ_{2} := (2, 1) and λ_{3} := (3). The corresponding Specht modules are free ℤ-modules of rank 1, 2 and 1, respectively. The cases λ_{2} and λ_{3} have been treated in general above. The Specht module △(λ_{1}) is the sign representation and f_{λ}(b_{1}) = 3!b_{1}^{∗}. Therefore we have T (λ_{1}) = [3!] ▽ (λ_{1}) = ([2] + [3]) ▽ (λ_{1}). From the ordinary decomposition matrices of S_{3} for the primes 2 and 3 we get [2] ▽ (λ_{1}) = [2]L(λ_{3}) and [3] ▽ (λ_{1}) = [3]L(λ_{3}) leading to the global decomposition matrix of S_{3}:

Global Decomposition Matrix of S_{3}

L(λ) λ_{1} λ_{2} λ_{3}
▽(λ)  (1^{3}) (2, 1) (3)
λ_{1} (1^{3}) 1 − [6] [3]  [2]
λ_{2} (2, 1) 1 [3]
λ_{3} (3)  1

Note that the corresponding equations according to condition (c) of Proposition 5.6 even hold in \underline{A}. This will not be true in the next example S_{4} as well as the fact that the global decomposition matrix can entirely be constructed from the knowledge of the ordinary decomposition matrices for the primes 2 and 3. On the other hand, the ordinary decomposition numbers are detected from the above by multiplying the whole table by [2] resp.

[3] and then setting [2] = 1 resp. [3] = 1.

Before proceeding to the next example let us write down all the relations between the L(λ) with the help of Proposition 5.8. We have to consider two pairs (p, λ) such that λ ∈ Λ_{p}^{−}. These are (2, λ_{1}) and (3, λ_{1}). We calculate u_{λ_{1},2} = X_{λ_{1}} −X_{λ_{3}} and u_{λ_{1},3} = X_{λ_{1}} −X_{λ_{2}} +X_{λ_{3}}. This gives ϵ(u_{λ_{1},2}) = X_{λ_{1}} + [3]X_{λ_{2}} + ([2] − 1)X_{λ_{3}} and ϵ(u_{λ_{1},3}) = X_{λ_{1}} + ([3] − 1)X_{λ_{2}} + ([2] − [3] + 1)X_{λ_{3}} leading to the following complete set of relations for i, j ∈ ℕ and k, l ∈ ℤ:

(k[2^{i}] + l[3^{j}])L(λ_{1}) = l([3^{j}] − [3])L(λ_{2}) + (k([2^{i}] − [2]) + l([3] − [3^{j}]))L(λ_{3}).

Turning to S_{4} we have 5 partitions λ_{1} := (1^{4}), λ_{2} := (2, 1^{2}), λ_{3} := (2, 2), λ_{4} := (3, 1) and λ_{5} := (4). Let us first picture the global decomposition matrix and then comment it:

Global Decomposition Matrix of S_{4}

L(λ) λ_{1} λ_{2} λ_{3} λ_{4} λ_{5}
▽(λ)  (1^{4}) (2, 1^{2}) (2, 2) (3, 1) (4)
λ_{1} (1^{4}) 1 − [24]  [3]  [8]
λ_{2} (2, 1^{2})  1 − [2]  [8] [2]
λ_{3} (2, 2)  1 − [2] [2] [3]
λ_{4} (3, 1)  1 [4]
λ_{5}  (4)  1

First note that the entries [8] and [4] cannot be reconstructed from the knowledge of the ordinary decomposition matrices of S_{4}. As in the preceeding example the first row follows easily from T (λ_{1}) = [4!] ▽ (λ_{1}) and the knowledge of [2] ▽ (λ_{1}) = [2]L(λ_{5}) and [3] ▽ (λ_{1}) = [3]L(λ_{3}). Note that [8] ▽ (λ_{1}) = [8]L(λ_{5}) does not hold in \underline{A} but surely in \underline{A}^{∞}. In contrast, the equations corresponding to the rows 3, 4, 5 hold in \underline{A} the last two of which have been treated in the beginning of this chapter. We leave the calculation of T (λ_{3}) as an exercise and turn to the most interesting case T (λ_{2}).

Here the Specht module is a free ℤ-module of rank three with basis b_{1}, b_{2}, b_{3}, say. The long cycle σ := (1234) and τ := (12) operate from right by b_{1}σ = b_{1}−b_{2}+b_{3}, b_{2}σ = b_{1}, b_{3}σ = b_{2} and b_{1}τ = −b_{1}, b_{2}τ = b_{3}, b_{2}τ = b_{3}. The Gram matrix of the bilinear form is given by

2 ( \begin{matrix}3 & 1 & −1 \\ 1 & 3 & 1 \\ −1 & 1 & 3 \end{matrix} )

and therefore the image of f_{λ} is spanned by 8b_{1}^{∗}, 8b_{2}^{∗} and 2(b_{1}^{∗} − b_{2}^{∗} + b_{3}^{∗}). Clearly, T (λ_{2}) is isomorphic to [8] + [8] + [2] as a ℤ-module. But anyway, as an A-module it is irreducible by ℤ-sums as defined in chapter 2. Therefore it cannot be splitted in \underline{A} anymore. But we can calculate in \underline{A}^{∞} as in the proof of Lemma 5.1. From the ordinary decomposition matrix for the prime 2, we get [2]T (λ_{2}) = [2](L(λ_{4})+L(λ_{5})) and by dimension arguments the 2-residual series must be

Res_{2}(T (λ_{2})) = ([2](L(λ_{4}) + L(λ_{5})), [2]L(λ_{4}), [2]L(λ_{4}), 0, …)

which completes the verification. In order to determine the relations among the L(λ) one calculates

ϵ(u_{λ_{1},2}) = X_{λ_{1}} + [3]X_{λ_{3}} + ([8] − 1)X_{λ_{5}}
ϵ(u_{λ_{1},3}) = X_{λ_{1}} + ([3] − 1)X_{λ_{3}} − [2]X_{λ_{4}} + ([8] − [3] + 1)X_{λ_{5}}
ϵ(u_{λ_{2},2}) = X_{λ_{2}} + ([8] − 1)X_{λ_{4}} + ([2] − [4])X_{λ_{5}}
ϵ(u_{λ_{3},2}) = X_{λ_{3}} + ([2] − 1)X_{λ_{4}} + ([3] − [4] + 1)X_{λ_{5}}.

As in the case of S_{3} a complete set of relations can be obtained from these formulas with help of Proposition 5.8.

Problems:

• Does the global decomposition matrix depend on the cell datum of the algebra A, i.e. on the choice of △(λ) and f_{λ} etc. or is it independed up to permutations on rows and columns?

• Is there an expansion of this theory to the groundring R := ℤ[q, q^{−1}] if you take additional structure on the algebra A into acount, for example informations on the open subset in Spec(R) corresponding to specializations for A being semisimple.

References

[CR] Curtis, C.W., Reiner, I.: Methods of Representation Theory, Vol. I, Wiley. 1981.

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

[JK] James, J., Kerber, A.: The Representation Theory of the Symmetric Groups. Encycl. of Math. Appl. Vol. 16, Addison Wesley, 1981.

[KX] König, S., Xi, C.: On the structure of cellular algebras. preprint.