aa r X i v : . [ m a t h . R T ] D ec The Garnir Relations for Weyl Groups
Sait Halıcıo˘glu
Department of MathematicsAnkara University06100 Tando˘gan AnkaraTurkey
Abstract.
The so-called Garnir relations are very important in the representation theory of the symmetricgroups. In this paper, we give a possible extension of Garnir relations for Weyl groups in general.
There are well known constructions of the irreducible representations and of the irreduciblemodules, called Specht modules, for the symmetric groups S n which are based on combinatorialconcepts connected with Young tableaux and tabloids (see, e.g. [4]). In [5] Morris described apossible extension of this work to Weyl groups in general. An alternative and improved approachwas described by the present author and Morris [1]. Later, Halıcıo˘glu [2] develop the theoryfurther and show how a K -basis for Specht modules can be constructed in terms of standardtableaux and tabloids. The so-called Garnir relations are very important in the representationtheory of the symmetric groups. The main aim of this paper is to give a possible extension ofGarnir relations for Weyl groups in general . We now state some results on Weyl groups which are required later . Any unexplained notationmay be found in J E Humphreys [3], James and Kerber [4], Halıcıo˘glu and Morris [1], Halıcıo˘glu[2].Let V be l -dimensional Euclidean space over the real field R equipped with a positive definiteinner product ( , ) for α ∈ V , α = 0 , let τ α be the ref lection in the hyperplane orthogonal to1 , that is , τ α is the linear transformation on V defined by τ α ( v ) = v − α , v )( α , α ) α for all v ∈ V . Let Φ be a root system in V and π a simple system in Φ with correspondingpositive system Φ + and negative system Φ − . Then , the W eyl group of Φ is the finite reflectiongroup W = W (Φ), which is generated by the τ α , α ∈ Φ. We now give some of the basic facts presented in [1].Let Ψ be a subsystem of Φ with simple system J ⊂ Φ + and Dynkin diagram ∆ and Ψ = r [ i =1 Ψ i , where Ψ i are the indecomposable components of Ψ, then let J i be a simple system in Ψ i ( i = 1 , , ..., r ) and J = r [ i =1 J i . Let Ψ ⊥ be the largest subsystem in Φ orthogonal to Ψ and let J ⊥ ⊂ Φ + be the simple system of Ψ ⊥ .Let Ψ ′ be a subsystem of Φ which is contained in Φ \ Ψ, with simple system J ′ ⊂ Φ + andDynkin diagram ∆ ′ , Ψ ′ = s [ i =1 Ψ ′ i , where Ψ ′ i are the indecomposable components of Ψ ′ ; let J ′ i be a simple system in Ψ ′ i ( i = 1 , , ..., s ) and J = s [ i =1 J ′ i . Let Ψ ′⊥ be the largest subsystemin Φ orthogonal to Ψ ′ and let J ′⊥ ⊂ Φ + be the simple system of Ψ ′⊥ . Let ¯ J stand for theordered set { J , J , ..., J r ; J ′ , J ′ , ..., J ′ s } , where in addition the elements in each J i and J ′ i arealso ordered. Let T ∆ = { w ¯ J | w ∈ W} . The pair { J, J ′ } is called a usef ul system in Φif W ( J ) ∩ W ( J ′ ) = < e > and W ( J ⊥ ) ∩ W ( J ′⊥ ) = < e > . The elements of T ∆ are called∆ − tableaux , the J and J ′ are called the rows and the columns of { J, J ′ } respectively. Two∆-tableaux ¯ J and ¯ K are row − equivalent , written ¯ J ∼ ¯ K , if there exists w ∈ W ( J ) suchthat ¯ K = w ¯ J . The equivalence class which contains the ∆-tableau ¯ J is denoted by { ¯ J } and iscalled a ∆ − tabloid . Let τ ∆ be set of all ∆-tabloids. Then τ ∆ = {{ d ¯ J } | d ∈ D Ψ } , where D Ψ = { w ∈ W | w ( j ) ∈ Φ + f or all j ∈ J } is a distinguished set of coset representatives of W (Ψ) in W . The group W acts on τ ∆ as σ { wJ } = { σwJ } for all σ ∈ W . Let K be arbitraryfield, let M ∆ be the K -space whose basis elements are the ∆-tabloids. Extend the action of W on τ ∆ linearly on M ∆ , then M ∆ becomes a K W -module. Let κ J ′ = X σ ∈ W ( J ′ ) s ( σ ) σ and e J,J ′ = κ J ′ { ¯ J } where s ( σ ) = ( − l ( σ ) is the sign function and l ( σ ) is the length of σ . Then e J,J ′ is called thegeneralized ∆ − polytabloid associated with ¯ J . Let S J,J ′ be the subspace of M ∆ generated by e wJ,wJ ′ where w ∈ W . Then S J,J ′ is called a generalized Specht module . A useful system2 J, J ′ } in Φ is called a good system if d Ψ ∩ Ψ ′ = ∅ for d ∈ D Ψ then { dJ } appears withnon-zero coefficient in e J,J ′ . If { J, J ′ } is a good system , then it was also proved in [1] that S J,J ′ is irreducible. The following are proved in [2].A good system { J, J ′ } is called a very good system in Φ if d ≤ d ′ for all d ∈ D Ψ ∩ D Ψ ′ , d ′ ∈ D Ψ such that d ′ = dσρ , where ρ ∈ W ( J ) , σ ∈ W ( J ′ ).If { J, J ′ } is a very good system in Φ , then { e dJ,dJ ′ | d ∈ D Ψ ∩ D Ψ ′ } is linearly independent overK.A very good system { J, J ′ } is called a perf ect system in Φ if { e dJ,dJ ′ | d ∈ D Ψ ∩ D Ψ ′ } is a basisfor S J,J ′ . The proof given by M . H . Peel [6] can be modified to prove that if U and V are subgroupsof W and Y = U V , then( X σ ∈ U s ( σ ) σ )( X σ ∈ V s ( σ ) σ ) = | U ∩ V | ( X σ ∈ Y s ( σ ) σ ) Let Φ be a root system and let { J, J ′ } be a very good system in Φ. Let wJ be a given ∆-tableau,where w ∈ W . We want to find elements of M ∆ which annihilate the given ∆-polytabloid e wJ,wJ ′ .By ( 2.1 ) if w = dρ , where d ∈ D Ψ ′ and ρ ∈ W ( J ′ ) we have e wJ,wJ ′ = s ( ρ ) e dJ,dJ ′ . Hence wemay assume that w ∈ D Ψ ′ . Lemma 3.1
Let { J, J ′ } be a very good system in Φ and let Ψ ∗ be a subsystem of Φ withsimple system J ∗ . Let d ∈ D Ψ ′ . Suppose there exists a and onto f unctionw → w ′ of the set W ( J ∗ ) W ( dJ ′ ) with the f ollowing property :(3 . f or each w ∈ W ( J ∗ ) W ( dJ ′ ) , there exists an element ρ w ∈ W ( dJ ) such thatρ w = e , s ( ρ w ) = − , w ′ = wρ w ; f urther ( w ′ ) ′ = w. T hen ( X σ ∈ W ( J ∗ ) s ( σ ) σ ) e dJ,dJ ′ = 0 . Proof
By ( 2.3 ) we have( X σ ∈W ( J ∗ ) s ( σ ) σ ) e dJ,dJ ′ = ( X σ ∈ W ( J ∗ ) s ( σ ) σ )( X σ ∈ W ( dJ ′ ) s ( σ ) σ ) { dJ } = | W ( J ∗ ) ∩ W ( dJ ′ ) | ( X σ ∈W ( J ∗ ) W ( dJ ′ ) s ( σ ) σ ) { dJ } w ′ ) ′ = w , and thus the function is an involution. If w ′ = wρ w = w then ρ w = e and s ( ρ w ) = 1 . This is a contradiction. Then the function is an involution without fixed points.Suppose that w , w ∈ W ( J ∗ ) W ( dJ ′ ) and w = w . If w = w ′ then w ′ = w (the functionis an involution). Hence w ′ = w = w ρ w and w ′ = w = w ρ w . If w = w ρ w ρ w then ρ w = ρ w and so w = w . This is a contradiction. But \ i =1 { w i , w ′ i } = ∅ and W ( J ∗ ) W ( dJ ′ )is a disjoint union of 2-elements sets { w , w ′ } . Hence X σ ∈W ( J ∗ ) W ( dJ ′ ) s ( σ ) σ ) { dJ } is a sum ofterms of the form ( s ( σ ) σ + s ( σ ′ ) σ ′ ) { dJ } .By ( 3.1 ) we have( s ( σ ) σ + s ( σ ′ ) σ ′ ) { dJ } = ( s ( σ ) σ + s ( σ ) s ( ρ σ ) σρ σ ) { dJ } = ( s ( σ ) σ − s ( σ ) σρ σ ) { dJ } = ( s ( σ ) σ ) { dJ } − ( s ( σ ) σρ σ ) { dJ } = ( s ( σ ) σ ) { dJ } − ( s ( σ ) σ ) { dJ } = 0Then( X σ ∈ W ( J ∗ ) s ( σ ) σ ) e dJ,dJ ′ = 0 . If σ ∈ W ( dJ ′ ) then σe dJ,dJ ′ = s ( σ ) e dJ,dJ ′ and so s ( σ ) σe dJ,dJ ′ = e dJ,dJ ′ . Now let H = { σ ∈ W ( J ∗ ) | s ( σ ) σe dJ,dJ ′ = e dJ,dJ ′ } . Then H = W ( J ∗ ) ∩ W ( dJ ′ ) and H is a subgroup of W ( J ∗ ). Now let C be a set of left coset representatives of H in W ( J ∗ ) such that C contains theidentity element . Theorem 3.2
Let { J, J ′ } be a very good system in Φ and d ∈ D Ψ ′ . T hene dJ,dJ ′ = − ( X σ ∈ C \{ e } s ( σ ) σ ) e dJ,dJ ′ Proof
It follows directly from Lemma 3.1 by cancelling the factor | H | . Definition 3.3
Let σ , σ , ..., σ k be coset representatives for H in W ( J ∗ ) and let G dJ,dJ ′ = k X j =1 s ( σ j ) σ j G dJ,dJ ′ is called Garnir element . 4 xample 3.4
Let Φ = G with simple system π = { α , α } = { ǫ − ǫ , − ǫ + ǫ + ǫ } Let a α + a α be denoted by a a and τ α , τ α be denoted by τ , τ respectively.Let Ψ = A + ˜A be the subsystem of G with simple system J = { , } and Dynkindiagram ∆. Let Ψ ′ = A be another subsystem of G with simple system J ′ = { } . Then the∆-tabloids are: { J } = { ,
32; 11 } , { τ J } = { ,
31; 10 } , { τ τ J } = { , − } Let d = τ ∈ D Ψ ′ . Then { dJ } = {− ,
32; 21 } . In this case W ( dJ ) = { e, τ , τ τ τ τ τ , τ τ τ τ τ τ } and W ( dJ ′ ) = { e, τ τ τ τ τ } .If Ψ ∗ = A is the subsystem of G with simple system J = { , } , then W ( J ∗ ) = { e, τ , τ τ τ τ τ , τ τ τ , τ τ τ τ , τ τ τ τ } . Since W ( J ∗ ) W ( dJ ′ ) = W ( J ∗ ) and w ′ = wτ and further ( w ′ ) ′ = w for all w ∈ W ( J ∗ ) W ( dJ ′ ), then Lemma 3.1 conditions are satisfied.Then H = W ( J ∗ ) ∩ W ( dJ ′ ) = W ( dJ ′ ) , C = { e, τ , τ τ τ } and the Garnir element G dJ,dJ ′ = e − τ − τ τ τ . By Theorem 3.2 we have e dJ,dJ ′ = τ e dJ,dJ ′ + ( τ τ τ ) e dJ,dJ ′ = e J,J ′ − e τ J,τ J ′ References [1] S . Halıcıo˘glu and A . O . Morris ,
Specht M odules f or W eyl Groups , Contributionsto Algebra and Geometry, Vol.34 (1993), No.2, 257–276.[2] S. Halıcıo˘glu,
A Basis f or Specht M odules f or W eyl Groups , Turkish J. Math.,18(3):311-326, 1994.[3] J . E . Humphreys,
Introduction to Lie algebras and representation theory , GraduateTexts in Mathematics, Volume 9 ( Springer-Verlag , Berlin , 1972 )[4] G . D . James , A . Kerber ,
T he Representation T heory of the Symmetric Group ,Addison-Wesley Publishing Company ( London , 1981 ) .[5] A . O . Morris,