Induction and restriction

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 3 March 2013

Induction and restriction

Restriction to Hn. Let Hn be the subalgebra of Hn generated by T1,,Tn-1. The elements Tw,wSn, form a basis of Hn. Since q is not a root of unity the subalgebra Hn of Hn is semisimple. The irreducible representations of Hn are indexed by the partitions νn and these representations are q-analogues of the irreducible representations of the symmetric group Sn. The following result describes the decomposition of the restriction to Hn of the irreducible H-module H(c,λ/μ).

Theorem 6.1. Let H(c,λ/μ) be the irreducible representation of the affine Hecke algebra Hn which is defined in Theorem 4.1. Then

H(c,λ/μ) HnHn= νn cμνλHν,

where cμνλ is the classical Littlewood-Richardson coefficient and Hν is the irreducible Hn-module indexed by the partition ν.


If β=(β1,,β) is a composition of n let γβ=γβ1× ×γβ Sβ1×× Sβ where γ1=(1,2,,r) Sr (in cycle notation). Let χ(c,λ/μ)(Tγβ) be the trace of the action of the element TγβHn on the Hn-module H(c,λ/μ). With notations as in Theorem 4.1

χ(c,λ/μ) (Tγβ)= Qλ/μ TγβvQ vQ,

and one can copy (without change) the proof of Theorem 2.20 in [HRa1996] and obtain

χ(c,λ/μ) (Tγβ)= μλ(1)λ()=λ Δ(λ(1)/μ) Δ(λ(2)/λ(1)) Δ(λ()/λ(-1)),

where the sum is over all sequences μλ(1)λ()=λ such that λ(i)/λ(i-1) =βi and the factor Δ(λ(i)/λ(i-1)) is given by

Δ(λ/μ)= { (q-q-1)cc-1 bsCC qc(bs)-1 (-q-1)r(bs)-1, ifλ/μis a border strip, 0, otherwise.

In the formula for Δ(λ/μ): a border strip is a skew shape with at most one box in each diagonal, CC is the set of connected components of λ/μ, cc is the number of connected components of λ/μ, r(bs) is the number of rows of bs, and c(bs) is the number of columns of bs.

Let sλ denote the Schur function (see [Mac1995]) and define

qr=m=1r (-q-1)r-m qm-1 s(m1r-m).

By Proposition 6.11(a) in [HRa1995],

qrsμ=λ Δ(λ/μ)sλ.

Letting qβ=qβ1qβ one can inductively apply this formula to obtain

qβsμ=λ χ(c,λ/μ) (Tγβ)sλ.

Thus, with notations as in [Mac1995] Chapt. I,

χ(c,λ/μ) (Tγβ) = qβsμ,sλ = qβ,sλ/μ, by [Mac1995] I (5.1), = νcμνλ qβ,sν, by [Mac1995] I (5.3), = νcμνλ χν (Tγβ), [Ram1991-2] Theorem 4.14,

where χν(Tγβ) denotes the irreducible character of Hn evaluated at the element Tγβ. The result follows since, by [Ram1991-2] Theorem 5.1, the characters of Hn are determined by their values on the elements Tγβ.

Classically, the Littlewood-Richardson coefficients describe

  1. The decomposition of the tensor product of two irreducible polynomial representations of GLn(), and
  2. The decomposition of an irreducible representation of Sk×S when it is induced to Sk+.

Theorem 6.2 gives an exciting new way of interpreting these coefficients. They describe

  1. The decomposition of an irreducible representation Hn when it is restricted to the subalgebra Hn.

Induction from “Young subalgebras”. Let k and be such that k+=n. Let

Hk = the subalgebra ofHn generated byTi,1 ik-1andxi ,1ik, H = the subalgebra ofHn generated byTi,k+1 in-1,and xi,k+1in.

In this way Hk×H is naturally a subalgebra of Hn.

Let (a,θ) be a placed skew shape with k boxes and let (b,ϕ) be a placed skew shape with boxes. Number the boxes of θ with 1,,k (as in Section 2, along diagonals from southwest to northeast) and number the boxes of ϕ with k+1,,n, in order to match the imbeddings of Hk and H in Hn. Let H(a,θ) and H(b,ϕ) be the corresponding representations of Hk and H as defined in Theorem 4.1.

Let θ*vϕ (resp. θ*hϕ) be the skew shape obtained by placing θ and ϕ adjacent to each other in such a way that box(k+1) of ϕ is immediately above (resp. to the left of) boxk of θ. Let ab be the content function given by

(ab)(boxi)= { a(boxi), if1ik, b(boxi), ifk+1i,

Theorem 6.2. With notations as above,

Ind HkH Hn ( H(a,θ) H(b,ϕ) ) = H ( ab, θ*vϕ ) + H ( ab, θ,*hϕ )

in the Grothendieck ring of finite dimensional representations of Hn.


Let Sn/(Sk×S) be the set of minimal length coset representatives of the cosets of Sk×S in Sn. The module M= Ind HkH Hn ( H(a,θ) H(b,ϕ) ) has basis

Tw(vLvQ) wherewSn/ (Sk×S), LθandQ ϕ.

By repeatedly applying the relations (1.4) we obtain

xi (Tw(vLvQ)) =Twxw-1(i) (vLvQ)+ u<wbuTu (vLvQ),

for some constants bu. From this we can see that the action of xi on M is an upper triangular matrix with eigenvalues q2c(P(i)) where P runs over the standard tableaux of shapes θ*vϕ and θ*hϕ. It follows that M has

Sn Card(θ) Card(ϕ) Sk×S

distinct weights. Since this number is exactly the dimension of M, it follows that every generalized weight space of M is one dimensional and thus that M is calibrated. By Theorem 4.1, all irreducible calibrated representations are of the form H(c,λ/μ) for some placed skew shape (c,λ/μ) and by Lemma 2.2 this placed skew shape is completely determined by any one of the weights of the module H(c,λ/μ). Thus, our analysis of the weights of M implies that both H(ab,θ*vϕ) and H(ab,θ*hϕ) are composition factors of M. The result follows since

dim(H(ab,θ*vϕ)) + dim(H(ab,θ*hϕ)) =dim(M).

A ribbon is a skew shape which has at most one box in each diagonal.

Corollary 6.3. Let c be the content function given by c(boxi)=i-1, for 1in. Let t=(t1,,tn) =(1,q2,q2(n-1)) and let vt be the one dimensional module for [X]= [ x1±1 ,, xn±1 ] given by

xivt=tivt, for all1in.

In the Grothendieck ring of finite dimensional Hn-modules

Ind [X] Hn (vt)= λ/μ H(c,λ/μ) ,

where the sum is over all connected ribbons λ/μ with n boxes.


Since [X]= H1 H1 H1 Hn this result can be obtained by repeatedly applying Theorem 6.2.

Theorem 6.2 and Corollary 6.3 are Hn-module realizations of the Schur function identities in [Mac1995] I §5 Ex 21 (a),(b). The module Ind [X] Hn (vt) is a principal series module for Hn (see [Kat1981]). The identity in Corollary 6.3 describes the composition series of this principal series module. Using the methods of [Ram1998] and [Ram1998-2, (1.2) Ex. 2] one can obtain a generalization of this identity (and thus of [Mac1995] I §5 Ex 21 (b)) which holds for affine Hecke algebras of arbitrary Lie type.

Notes and References

This is an excerpt of the preprint entitled Skew shape representations are irreducible authored by Arun Ram in 1998.

Research supported in part by National Science Foundation grant DMS-9622985, and a Postdoctoral Fellowship at Mathematical Sciences Research Institute.

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