Intertwinning spaces associated with q-analogues of the Young symmetrizers in the Hecke algebra

Publishedin”Parameterspaces”BanachCenterPub.36PolishAcad.ofSciencesWarszawa 1996.

Gérard Duchamp^{1} and Sungsoon Kim^{2}

Résumé. — Soit H, l’algèbre de Hecke du groupe symétrique dont une base est (T_{w})_{w∈𝔖_{n}}. À chaque sous ensemble S ⊂ [1,n − 1], on associe deux idempotents _{S} et ∇_{S} qui sont les q-déformations des symétriseur et antisymétriseur du sous groupe de Young 𝔖_{I_{S}} engendré par les transpositions simples {(i,i + 1)}_{i∈S}. Nous donnons ici des bases explicites pour le sous espace d’entrelacement _{S_{1}}H∇_{S_{2}}, indexées par les doubles classes 𝔖_{I_{S1}} \𝔖_{n}/𝔖_{I_{S2}} . Nous calculons également des bases et les caractères des idéaux

ℑ(I,J) = _{S_{1}}H∇_{S_{2}}H.

Abstract. — Let H be the Hecke algebra of the symmetric group with basis (T_{w})_{w∈𝔖_{n}}. To each subset S ⊂ [1,n−1], we associate two idempotents _{S} and ∇_{S} which are q-deformations of the symmetrizer and antisymmetrizer relative to the Young subgroup 𝔖_{I_{S}} generated by the simple transpositions {(i,i+1)}_{i∈S}. We give here explicit bases for the intertwinning space _{S_{1}}H∇_{S_{2}}, indexed by the double classes 𝔖_{I_{S1}} \𝔖_{n}/ 𝔖_{I_{S2}} . We also compute bases and characters of the right ideals

ℑ(I,J) = _{S_{1}}H∇_{S_{2}}H.

1 L.I.R., Université de Rouen, Place E. Blondel, B.P. 118, 76134 Mont-Saint-Aignan Cedex, France 2 L.I.T.P., Université Paris 7, 2 place Jussieu, 75251 Paris Cedex 05, France

Introduction

Aquestionofimportance,inrepresentationtheoryofgroups,istheconstructionofprimitive idempotents in a group algebra. For the symmetric group, after the early construction by A. Young (see [Ru], [JK] for details), several complete sets of primitive idempotents (i.e. generating irreducible ideals) have been proposed ([Ju], [Mo], [Mu1] and [DJ], [Gy], [Mu2] for the Hecke algebra).

The original idea of Young (see e.g. [Fu], [JK]) for computing a primitive idempotent in the algebra of the symmetric group consists in starting from a tableau without repetition

5 7 2 4 1 3 6

and derive two idempotents, the row symmetrizer P (i.e. the sum of the permutations that preserve each row, in the picture 𝔖_{{1,3,6}} ×𝔖_{{2,4}} ×𝔖_{{5,7}}) and the column antisymmetrizer N (i.e. the alternating sum of the permutations that preserve each column, in the picture 𝔖_{{1,2,5}} × 𝔖_{{3,4,7}} × 𝔖_{{6}}). Now, P N is (up to a scalar) an idempotent and the associated representation is irreducible, of index the shape of the tableau.

The idempotence of P N is due to two facts, the first being that P.ℂ[𝔖_{n}].N is one dimensional and the second being that there is a natural isomorphism

P.ℂ[𝔖_{n}].N ≅ Hom_{𝔖_{n}}(N.𝔖_{n},P.𝔖_{n})

Let H be the Hecke algebra of the symmetric group with basis (T_{w})_{w∈𝔖_{n}}. The preceding construction does not give in general idempotents in the this algebra (even in the simplest case of a tableau of shape (1,2)). However, for a single row (or column), we get the following q-analogues of Young symmetrizers and antisymmetrizers

= ∑_{w∈𝔖_{n}} T_{w} and ∇ = ∑_{w∈𝔖_{n}}(−q)^{l_{max}−l(w)}T_{w}

with l_{max} = \frac{n(n−1)}{2}.

We can get also, with the same formulas, analogs of the Young row (resp. column) symmetrizers providing that the filling of the tableau is done with consecutive numbers in the rows (resp. columns). This explains why Young’s construction cannot be adapted trivially. The elements so obtained, denoted _{ω_{I}} and ∇_{ω_{I}} are particular cases of the _{μ} and ∇_{μ}, μ ∈ 𝔖_{n} constructed elsewhere using solutions of the Yang-Baxter equation [DKLLST].

A complete set of primitive idempotents for the generic Hecke algebra have been defined (see [DJ], [DKLLST], [Gy]) with q-analogues of Young symmetrizers associated with partitions.

The idempotence of these elements is based (as for the symmetric group) on the combinatorial fact that, for I and J conjugate, the spaces _{ω_{I}} H∇_{ω_{J}} and ∇_{ω_{J}} H _{ω_{I}} are both one dimensional.

The preceding property still holds when I and J are no longer partitions but compositions associated to conjugate partitions. (This fact is characteristic free and independent of the choice of q.) Then, when the Hecke algebra is semi-simple, we can derive primitive idempotents of the form _{ω_{I}} h′∇_{ω_{J}} h′′ (see Corollary 1.3. below).

Our main concern here is the combinatorial structure of the spaces _{ω_{I}} H∇_{ω_{J}} and of right ideals

ℑ(I,J) = _{w_{I}} H∇_{w_{J}} H

i.e. the sum of the ideals _{w_{I}} h∇_{w_{J}} H for h ∈ H (I, J are arbitrary compositions). A motivation also to the study of these more general spaces is the exploration of alternative computations for elements of Kazdhan-Lusztig basis as it can be easily checked that the elements ∇_{ω_{I}} and _{ω_{I}} belong to invariant bases.

The paper is organized as follows. In a first section, we give the classical isomorphism

e.H.f ≅ Hom_{H}(f.H,e.H).

Dimensions and bases of the intertwinning space _{ω_{I}} .H.∇_{ω_{J}} are then given. These bases are indexed by well-known combinatorial objects, namely matrices with entries in {0,1} and prescribed row (and column) sums.

The second part is devoted to the study of the structure of ℑ(I,J) (isotypic decomposition, bases and characters).

Aknowledgments

The authors are grateful to the referee for many useful remarks and corrections.

I. Intertwinning spaces.

I.1. Intertwinning two right direct summand ideals.

Let H be a k-algebra (k is a ring) and U,V two right H-modules. In classical representation theory [CR] Hom_{H}(U,V) is called the intertwinning space between U and V. In case U,V are principal (right) ideals of H generated by idempotents (i.e. factors of decompositions H = I ⊕ J where I and J are ideals) the space Hom_{H}(U,V) has a concrete realization within H.

PROPOSITION I.1. — Let e,f be two idempotents of H. Then the mapping

ψ : eHf ⟶ Hom_{H}(fH,eH)

defined by

ψ(ehf) : fx ⟶ ehfx

is an isomorphism

Proof. — It is obvious that ψ(ehf) ∈ Hom_{H}(fH,eH). Now, for θ ∈ Hom_{H}(fH,eH) we have θ(f) ∈ eH but, θ(f) = θ(f^{2}) = θ(f)f ∈ eHf. The reader may then easily verify that

ϕ : Hom_{H}(fH,eH) ⟶ eHf

defined by ϕ(θ) = θ(f) is the inverse isomorphism.

I.2. Background for symmetrizers in the Hecke algebra.

Let us begin with some standard facts about combinatorics of partitions.

Let I = (i_{1},i_{1}, ⋯,i_{k}) be a vector with (strictly) positive integral entries. This object is called a composition of |I| = ∑_{1≤r≤k} i_{r} whereas the entries (i_{r}) are called the parts of I. The set

CS(I) = {i_{1},i_{1} + i_{2}, ⋯,|I|} = { ∑_{1≤r≤s} i_{r}}_{1≤s≤k}

is the set of cumulated sums of I. A decreasing composition J is called a partition of |J| and we write J ⊢ |J|. Partitions can be represented by plane diagrams of boxes, Ferrers’ diagrams. Such a diagram can be transposed (i.e. transformed by the orthogonal symmetry with axis x = y), the result being the diagram of the conjugate partition I . For example

Partition 322  331, conjugate of 322

Now, to each composition, corresponds a unique partition I^{0} obtained by reordering the parts and we will say that two compositions I, J are conjugate iff I^{0} = J^{0} (see [An]).

We will also need some well known results about the Hecke algebra of the symmetric group (which is a deformation of the algebra of the group).

Let k be a ring, q ∈ k, n ∈ ℕ^{∗}. The Hecke algebra H_{n,q}^{k} is the k-associative algebra with unit presented with generators (T_{i})_{1≤i<n} and relations

T_{i}T_{j} = T_{j}T_{i} |i − j| ≥ 2
(T_{i} − 1)(T_{i} + q) = 0 1 ≤ i < n
T_{i}T_{i+1}T_{i} = T_{i+1}T_{i}T_{i+1} 1 ≤ i < n − 1.

As a consequence of Murphy and Diper-James papers [Mu12][DJ] it turns out that, if q.[n]!_{q} is invertible in k ([r]_{q} := 1+q +⋯ q^{r−1} and [n]!_{q} = [1]_{q}.[2]_{q}. · · · [n]_{q}), H_{n,q}^{k} is a direct product of full matrix algebras.

H_{n,q}^{k} ≅ ∏_{λ⊢n} M(n_{λ},k)  (I.1)

n_{λ} being the dimension of the irreducible representation of shape λ of the symmetric group (the numbers n_{λ} can be computed directly by the ”hook formula” [Fu]).

If k is a field (I.1) implies that H_{n,q}^{k} is semi-simple and the decomposition (I.1) is unique. Each component M(n_{λ},k) is called a block of H_{n,q}^{k}, its neutral element is a central element that we will denote e_{λ} and call the central idempotent of shape λ. This idempotent splits into indecomposables ones that are primitive idempotents of shape λ. We here briefly indicate how to construct an exhaustive (this means at least one for each shape) family of them.

Let I = (i_{1},i_{2}, ⋯ i_{k}) be a composition of n. To I we associate the partition

[1,n] = I_{1} ⊎ I_{2} ⊎ ⋯ ⊎ I_{k}

where I_{s} is the interval [1 + i_{1} + ⋯ + i_{s−1}, i_{1} + i_{2} + ⋯ i_{s}].

The subgroup

𝔖_{I} = {σ ∈ 𝔖_{n} | (∀s ≤ k)(σ(I_{s}) = I_{s})}

is also the subgroup generated by the elementary transpositions σ_{i} = (i,i+1) with i ∉ CS(I). This subgroup is called the Young subgroup associated with I. 𝔖_{I} is naturally isomorphic (by restrictions) to

𝔖_{i_{1}} × 𝔖_{i_{2}} ⋯ × 𝔖_{i_{k}}  (I.2)

and, in particular, its cardinality is I! = i_{1}!i_{2}! ⋯ i_{k}!. There is a unique longest element ω_{I} in 𝔖_{I} which acts as the reversal in every interval I_{s}. More precisely, with 0 ≤ h ≤ i_{s} − 1

ω_{I}(1 + i_{1} + ⋯ i_{s−1} + h) = i_{1} + i_{2} ⋯ + i_{s} − h

Define elements

I = ∑_{μ∈𝔖_{I}} T_{μ} ; ∇_{I} = ∑_{μ∈𝔖_{I}} (−q)^{l(ω_{I})−l(μ)}T_{μ} = ∑_{μ∈𝔖_{I}} (−q)^{l(ω_{I}μ)}T_{μ}

In the spirit of Fulton [Fu], they will be called the symmetrizer (resp. antisymmetrizer) relative to 𝔖_{I}. These elements act as scalars on the elementary generators of 𝔖_{I} and therefore define idempotents (see [DKLLST]).

PROPOSITION I.2. — For i ∉ CS(I) we have

i) T_{i} _{I} = _{I}T_{i} = q _{I} ii) T_{i}∇_{I} = ∇_{I}T_{i} = −∇_{I} iii) If q,I! ∈ k^{∗} then \frac{1}{q.I!} _{I} and \frac{1}{I!}∇_{I} are idempotents. iv) If I,J are conjugate, the spaces _{I}.H.∇_{J} and ∇_{J}.H. _{I} are linearly independant over

k and one-dimensional.

Proof. — i), ii) For every i ∉ CS(I) let R_{i} (resp. D_{i}) be the set of μ ∈ 𝔖_{I} s.t. l(σ_{i}μ) = l(μ)+1 (resp. l(σ_{i}μ) = l(μ) − 1) it is straightforward that D_{i} = σ_{i}.R_{i} then

I = ( ∑_{μ∈R_{i}} T_{μ} + ∑_{μ∈D_{i}} T_{μ}) = (1 + T_{i})( ∑_{μ∈R_{i}} T_{μ})

the claim follows from the fact that T_{i}(T_{i} + 1) = q(T_{i} + 1) the other identities of i) and ii) have a similar proof.

iii) Is an immediate consequence of i) and ii). iv) if I,J are conjugate, it is easy to verify that (see also below Remark I.3) there is only

one ν ∈ 𝔖_{n} such that

i ∉ CS(I), j ∉ CS(J) ⟹ l(σ_{i}ν) = l(ν) + 1, l(νσ_{j}) = l(ν) + 1

It follows from i) and ii) that _{I}.T_{ν}.∇_{J} is a generator of _{I}.H.∇_{J}, now, from general theory of Coxeter groups, one has l(ω_{I}.ν.ω_{J}) = l(ω_{I}) + l(ν) + l(ω_{J}) = m and then

I.T_{ν}.∇_{J} = T_{ω_{I}νω_{J}} + ∑_{l(μ)<m} R_{μ}T_{μ}

this proves that _{I}.T_{ν}.∇_{J} is torsion free and then the claim.

From general theory of semi-simple algebras, Proposition I.2 iv) above implies that

COROLLARY I.3. — Suppose that k is a field and H_{n,q}^{k} is semi-simple. Let I,J be two conjugate compositions. Then there exists elements μ′,μ′′ ∈ 𝔖_{n} such that _{I}T_{μ′}∇_{J}T_{μ′′} is a primitive idempotent of shape I.

In the previous case, elements _{I}h′ and ∇_{J}h′′ can be defined by means of a solution of the Yang-Baxter equation (see Remark I.3. ii) below). We must say now a word of this construction.

Recall that the action of 𝔖_{n} on the right on the words of lenth n, is defined by

(a_{1}a_{2} ⋯ a_{n}).σ := a_{σ(1)}a_{σ(2)} ⋯ a_{σ(n)}

Let then σ_{i_{1}}σ_{i_{2}} ⋯ σ_{i_{p}} be a reduced decomposition of the permutation μ. The fact that this decomposition is reduced implies that the sequence

\{ \begin{matrix}w_{0} = 12 ⋯ n \\ w_{k} = w_{k−1}.σ_{i_{k}} \end{matrix}

is such that, at each step, w_{k} = uijv and w_{k+1} = ujiv with i < j. Then supposing that [r]_{q} ∈ k^{∗} for r < n and setting s_{k} := \frac{[j−i−1]_{q}}{[j−i]_{q}}. The products

(∇_{i_{1}} + s_{1})(∇_{i_{2}} + s_{2}) ⋯ (∇_{i_{p}} + s_{p})
( _{i_{1}} − s_{1})( _{i_{2}} − s_{2}) ⋯ ( _{i_{p}} − s_{p})

do not depend on the choice of the reduced decomposition of μ, but only of μ (see [DKLLST] for complete proof). We will denote the first ∇_{μ} and the second _{μ}. The link with the symmetrizers become transparent with the following formulas [DKLLST]

ω_{I} = ∑_{h∈𝔖_{I}} T_{h} ∇_{ω_{I}} = ∑_{h∈𝔖_{I}} (−q)^{l(ω_{I}h)}T_{h}

Remark I.2. — It is not hard to see that the leading term of _{μ} (resp. ∇_{μ}) is T_{μ}. In other words, we have

μ = T_{μ} + ∑_{l(ν)<l(μ)} R_{ν}(q)T_{ν} ∇_{μ} = T_{μ} + ∑_{l(ν)<l(μ)} S_{ν}(q)T_{ν}

with R_{ν},S_{ν} ∈ k(q).

From now on, we will denote _{ω_{I}} (resp. ∇_{ω_{I}}) instead of _{I} (resp. ∇_{I}).

I.3. Bases for _{ω_{I}} .H.∇_{ω_{J}}.

In this section, we construct a basis of _{ω_{I}} H∇_{ω_{J}} with elements of the type _{ω_{I}} T_{σ}∇_{ω_{J}}. This basis is closely related to the minimal representatives of the double classes 𝔖_{I}\𝔖_{n}/𝔖_{J} (see [Ke]). Let us state their properties

PROPOSITION I.5. — Let I,J be two compositions of n, and W (I,J) be the set of σ ∈ 𝔖_{n} such that

l(ω_{I}σω_{J}) = l(ω_{I}) + l(σ) + l(ω_{J})  (I.3)

then

i) W (I,J) is the set of elements of minimal length in each double class 𝔖_{I}.μ.𝔖_{J} and, hence, W (I,J) is a set of representatives for 𝔖_{I}\𝔖_{n}/𝔖_{J}.

ii) ( _{ω_{I}} T_{σ}∇_{ω_{J}})_{σ∈W (I,J)} is a basis of the k-module _{ω_{I}} H∇_{ω_{J}}.

Proof. — i) Is classical.

ii) The family ( _{ω_{I}} T_{μ}∇_{ω_{J}})_{μ∈W (I,J)} is linearly independant over k as the dominant term of _{ω_{I}} T_{μ}∇_{ω_{J}} is T_{ω_{I}μω_{J}}. Now, let M be the linear span of the elements ( _{ω_{I}} T_{μ}∇_{ω_{J}})_{μ∈W (I,J)}, it is sufficient to prove that every _{ω_{I}} T_{ν}∇_{ω_{J}} belongs to M for ν ∈ 𝔖_{n}. Suppose not then, it would exist α ∈ 𝔖_{n} of minimal length with _{ω_{I}} T_{α}∇_{ω_{J}} ∉ M. Now, α ∉ W (I,J) and then either α = σ_{i}α_{1} for some i ∉ CS(I) or α = α_{2}σ_{j} for some j ∉ CS(J) (or both). Then, for example in the first case,

ω_{I} T_{σ_{i}α_{1}}∇_{ω_{J}} = _{ω_{I}} T_{σ_{i}}T_{α_{1}}∇_{ω_{J}} = q _{ω_{I}} T_{α_{1}}∇_{ω_{J}}

a contradiction. The other case is similar.

Remarks I.3. — i) Let I = (i_{1},i_{2}, ⋯ i_{r}), J = (j_{1},j_{2}, ⋯ j_{s}). Then the set W (I,J) is in one-to-one correspondence with the set of r × s-matrices with entries in {0,1} and column (resp. row) sums given by I (resp. J). We first need a description of ω_{I}.W (I,J), the elements of which have a simple characterization in terms of subwords on consecutive letters and increasing conditions. Namely an element μ belongs to ω_{I}.W (I,J) iff μ:

a) Has the decreasing subwords of ω_{I}.

b) Is increasing on each interval defined by J.

We now describe the correspondence on an example (for practical reasons, as the parts involved will be small, we will write I = 1311 instead of I = (1,3,1,1))

For example, with I = 1311, J = 231, the permutations μ are (written as words) 143562,451362,461352. The matrices to be found have 3 rows and 4 columns. Index the columns by the subwords (condition (a) above), here (1, 432, 5, 6) and the rows by the intervals of places where the permutation has to be increasing (condition (b) above), here 12,345,6. For the first permutation we have

subwords 1 432 5 6 ↓ places
• • - -  12
- • • •  345
- •  - -  6
P ermutation 143562

And, for the two remaining elements of ω_{1311}.W (1311,231)

\begin{matrix}1 & 432 & 5 & 6 \\ - & • & • & - & 12 \\ • & • & - & • & 345 \\ - & • & - & - & 6 \\ P ermutation 451362 \end{matrix}  \begin{matrix}1 & 432 & 5 & 6 \\ - & • & - & • & 12 \\ • & • & • & - & 345 \\ - & • & - & - & 6 \\ P ermutation 461352 \end{matrix}

ii) For ν ∈ W (I,J), the leading term of _{ω_{I}} T_{ν}∇_{ω_{J}} is T_{ω_{I}νω_{J}} because lengths add (see Proposition I.6.). So any family of the form

( _{ω_{I}} B_{ν}∇_{ω_{J}})_{ν∈W (I,J)}

such that the leading term of B_{ν} is T_{ν}, is also a basis of _{ω_{I}} H∇_{ω_{J}}, triangular w.r.t.

( _{ω_{I}} T_{ν}∇_{ω_{J}})_{ν∈W (I,J)}

in particular, this is the case with _{ω_{I}ν}∇_{ω_{J}} as one has the factorization _{ω_{I}ν} = _{ω_{I}} B_{ν} for some B_{ν} with leading term T_{ν} (see [DKLLST] for details).

iii) In general, we deduce from Propositions I.1 and I.5, that

|W (I,J)| = ∑_{λ⊢n} m′_{λ}.m′′_{λ}

where m′_{λ} (resp. m′′_{λ}) is the number of irreducible components of shape λ in _{ω_{I}} .H (resp. ∇_{ω_{J}} H). In particular, if I,J are conjugate compositions then |W (I,J)| = 1 and _{ω_{I}} H∇_{ω_{J}} is one dimensional. In the semi-simple case this allows to construct primitive idempotents for all possible shapes (see [DKLLST]).

iv) Notice that it can happen that W (I,J) = ∅. This is the case, for example, with I = J = [n], n ≥ 2.

I.4. Characters.

When k is of characteristic 0 and q = 1 one can compute the characteristics of _{ω_{I}} H and ∇_{ω_{J}} H considered as right 𝔖_{n}-modules. Here, we have

char( _{ω_{I}} H) = S^{I} = S_{i_{1}}S_{i_{2}} ⋯ S_{i_{k}} char(∇_{ω_{J}} H) = Λ^{J} = Λ_{i_{1}}Λ_{i_{2}} ⋯ Λ_{i_{k}}

If I,J are conjugate, these symmetric functions have only one common irreducible component S_{λ}, λ being the diagram with row weights given by I and column weights given by J (in fact, we could write λ = I) [L&S-1,2]. We recover the fact that _{ω_{I}} H∇_{ω_{J}} is one-dimensional in this case.

I.5. Permutation of parts.

The symmetric group 𝔖_{k} acts on compositions of n into k parts by

I^{σ} = (i_{1},i_{2} ⋯ i_{k})^{σ} = (i_{σ(1)},i_{σ(2)}, ⋯ i_{σ(k)})

Here we show that _{ω_{I}} (resp. ∇_{ω_{I}}) and _{ω_{I^{σ}}} (resp. ∇_{ω_{I^{σ}}}) are conjugate.

PROPOSITION I.7. — Let I be a composition of n belonging to ℕ^{k}. For every σ ∈ 𝔖_{k} there exists τ(σ) ∈ 𝔖_{n} such that

ω_{I^{σ}} = T_{τ(σ)} _{ω_{I}} T_{τ(σ)}^{−1} ∇_{ω_{I^{σ}}} = T_{τ(σ)}∇_{ω_{I}} T_{τ(σ)}^{−1}

Proof. — This is a direct consequence of the fact that the Young subgroups 𝔖_{ω_{I^{σ}}} and 𝔖_{ω_{I}} are conjugate.

COROLLARY I.8. — ℑ(I^{σ_{1}},J^{σ_{2}}) = T_{τ(σ_{1})}ℑ(I,J)

This explains how the results for general I,J can be computed from the case when I,J are partitions.

II. Structure of ℑ(I,J).

II.1. Isotypic decomposition of ℑ(I,J).

From now, we suppose that k is a field and q.[n!]_{q} ≠ 0 (this implies that H_{n,q} is semisimple). The following result is classical and can be derived from the thesis of Hoefsmith [Ho] or the orthogonal family of idempotents of Jucys-Murphy [Ju][Mu2].

PROPOSITION II.1. — There is a family of orthogonal central idempotents (e_{λ})_{λ⊢n}. Hence

H_{n,q} = ⊕_{λ⊢n}e_{λ}.H_{n,q}

each summand being a matrix algebra

e_{λ}.H_{n,q} ≅ M(n_{λ},k)

where n_{λ} is the dimension of the representation of shape λ.

We derive the following decomposition of ℑ(I,J)

THEOREM II.2. — ℑ(I,J) = ⊕_{{λ⊢n \atop (S_{λ},Λ^{J})≠0}} _{ω_{I}} H.e′_{λ}.H where e′_{λ} is any minimal idempotent of

shape λ. In particular, there exists h′,h′′ ∈ H such that e′_{λ} = _{ω_{λ}}h′∇_{ω_{λ}}h̃′′.

Proof. — Of course, we have

ℑ(I,J) = ⊕_{λ⊢n}e_{λ}.ℑ(I,J) = ⊕_{{λ⊢n \atop (S_{λ},Λ_{J})≠0}} _{ω_{I}} .H.e_{λ}.∇_{ω_{J}} .H

as, if (S_{λ},Λ^{J}) = 0, then H.e_{λ}.∇_{ω_{J}} .H = {0} and H.e_{λ}.ℑ(I,J) = _{ω_{I}} .H.e_{λ}.∇_{ω_{J}} .H = {0} (indeed, this is the case also if (S_{λ},S^{J}) = 0). Otherwise, there exists a minimal idempotent e′_{λ} of shape λ such that ∇_{ω_{J}} .H ⊃ e′_{λ}.H and, hence

H.e_{λ}.H ⊃ H.e_{λ}∇_{ω_{J}} .H ⊃ H.e′_{λ}.H = H.e_{λ}.H

this proves the claim.

Remark II.1. — Let 𝔍 be an ideal of H then, with the previous notations, we have

𝔍e_{λ} = 𝔍e′_{λ}.H

this ideal is the sum of the components of ”type λ” in 𝔍. We will denote this ideal 𝔍_{λ}. One has

dim(𝔍_{λ}) = m_{λ}.n_{λ}

where m_{λ} is the multiplicity of the representation of type λ and n_{λ} its dimension.

Here, we have

e_{λ}.ℑ(I,J) = _{ω_{I}} .H.e′_{λ}.H = ( _{ω_{I}} .H)_{λ}

II.2. Character and basis of ℑ(I,J).

Let, as above, m_{λ} be the multiplicity of e′_{λ}.H in ℑ(I,J). From Remark II.1. we deduce that, if (S_{λ},Λ_{J}) ≠ 0,

m_{λ} = dim(Hom_{H}( _{ω_{I}} .H,e′_{λ}.H)) = dim( _{ω_{I}} .H.e′_{λ})

this allows us to construct bases of ℑ(I,J).

PROPOSITION II.3. — Let ( _{ω_{I}} .h_{i}e′_{λ})_{i∈I_{λ}} be a basis in each non-vanishing _{ω_{I}} .H.e′_{λ} and (e′_{λ}.h′_{j})_{j∈J_{λ}} a basis of e′_{λ}.H. Then

( _{ω_{I}} h_{i}e′_{λ}h′_{j})_{(i,j)∈I_{λ}×J_{λ}}

is a basis of ℑ_{λ}(I,J).

Proof. — Let ( _{ω_{I}} .h_{i}e′_{λ})_{i∈I_{λ}} (resp. (e′_{λ}.h′_{j})_{j∈J_{λ}}) be a basis of _{ω_{I}} .H.e′_{λ} (resp. e′_{λ}.H). It is obvious that ( _{ω_{I}} h_{i}e′_{λ}h′_{j})_{(i,j)∈I_{λ}×J_{λ}} generates in ℑ_{λ}(I,J). But remark II.1. states that dim(ℑ_{λ}(I,J)) = card(I_{λ} × J_{λ}) and, then, this family is a basis of ℑ_{λ}(I,J).

Remark II.2. — Now, since

ℑ(I,J) = ⊕_{λ⊢n}ℑ_{λ}(I,J)

the sum being taken over all λ such that (S_{λ},S^{I})(S_{λ},Λ^{J}) ≠ 0 (otherwise ℑ_{λ}(I,J) = {0}) we get a basis of ℑ(I,J) by union of bases of Proposition II.3.

Example. — With I = 211, J = 22, we have S^{211} = S_{4} + 2S_{13} + S_{22} + S_{112} and Λ^{22} = S_{22} + S_{112} + S_{1111} then ℑ_{λ}(I,J) ≠ 0 for λ = 211 and λ = 22. We get a basis of ℑ_{211}: 2341∇_{3214} and a basis of ℑ_{22}: _{2413}∇_{2143}. Then the desired basis of ℑ(211,22) is

{ _{2341}∇_{3214}, _{2413}∇_{2143}}

III. Conclusion

Bases and characters of the ideal ℑ(I,J) have been described. In general, bases for an isotypic component are indexed by two indices. For the second range, we have a nice description for a basis of e′_{λ}.H in case e′_{λ} is choosen to be of the form _{ω_{λ}}h′∇_{ω_{λ}}h̃′′ [DKLLST][DJ].

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