Induction and restriction

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 3 March 2013

Induction and restriction

Restriction to $H_n\text{.}$ Let $H_n$ be the subalgebra of ${\stackrel{\sim }{H}}_n$ generated by $T_1,\dots ,T_{n-1}\text{.}$ The elements $T_w,w\in S_n,$ form a basis of $H_n\text{.}$ Since $q$ is not a root of unity the subalgebra $H_n$ of ${\stackrel{\sim }{H}}_n$ is semisimple. The irreducible representations of $H_n$ are indexed by the partitions $\nu ⊢n$ and these representations are $q-analogues$ of the irreducible representations of the symmetric group $S_n\text{.}$ The following result describes the decomposition of the restriction to $H_n$ of the irreducible $\stackrel{\sim }{H}\text{-module}$ ${\stackrel{\sim }{H}}^{(c,\lambda /\mu )}\text{.}$

Theorem 6.1. Let ${\stackrel{\sim }{H}}^{(c,\lambda /\mu )}$ be the irreducible representation of the affine Hecke algebra ${\stackrel{\sim }{H}}_n$ which is defined in Theorem 4.1. Then

$${\stackrel{\sim }{H}}^{(c,\lambda /\mu )}{\downarrow }_{H_n}^{{\stackrel{\sim }{H}}_n}=\sum _{\nu ⊢n}{c}_{\mu \nu }^{\lambda }{H}^{\nu },$$

where $c_{\mu \nu }^{\lambda }$ is the classical Littlewood-Richardson coefficient and $H^{\nu }$ is the irreducible $H_n\text{-module}$ indexed by the partition $\nu \text{.}$

		Proof.
	If $\beta =({\beta }_1,\dots ,{\beta }_{\ell })$ is a composition of $n$ let ${\gamma }_{\beta }={\gamma }_{{\beta }_1}\times \dots \times {\gamma }_{{\beta }_{\ell }}\in S_{{\beta }_1}\times \dots \times S_{{\beta }_{\ell }}$ where ${\gamma }_1=(1,2,\dots ,r)\in S_r$ (in cycle notation). Let ${\chi }^{(c,\lambda /\mu )}\left(T_{{\gamma }_{\beta }}\right)$ be the trace of the action of the element $T_{\gamma \beta }\in H_n$ on the ${\stackrel{\sim }{H}}_n\text{-module}$ ${\stackrel{\sim }{H}}^{(c,\lambda /\mu )}\text{.}$ With notations as in Theorem 4.1 $${\chi }^{(c,\lambda /\mu )}\left(T_{{\gamma }_{\beta }}\right)=\sum _{Q\in {ℱ}^{\lambda /\mu }}{T}_{{\gamma }_{\beta }}{v}_Q{\mid }_{v_Q},$$ and one can copy (without change) the proof of Theorem 2.20 in [HRa1996] and obtain $${\chi }^{(c,\lambda /\mu )}\left(T_{{\gamma }_{\beta }}\right)=\sum _{\mu \subseteq {\lambda }^{\left(1\right)}\subseteq \dots \subseteq {\lambda }^{\left(\ell \right)}=\lambda }\Delta ({\lambda }^{\left(1\right)}/\mu )\Delta ({\lambda }^{\left(2\right)}/{\lambda }^{\left(1\right)})\dots \Delta ({\lambda }^{\left(\ell \right)}/{\lambda }^{(\ell -1)}),$$ where the sum is over all sequences $\mu \subseteq {\lambda }^{\left(1\right)}\subseteq \dots \subseteq {\lambda }^{\left(\ell \right)}=\lambda$ such that $\mid {\lambda }^{\left(i\right)}/{\lambda }^{(i-1)}\mid ={\beta }_i$ and the factor $\Delta ({\lambda }^{\left(i\right)}/{\lambda }^{(i-1)})$ is given by $$\Delta (\lambda /\mu )=\{\begin{array}{cc}{(q-q^{-1})}^{cc-1}\prod _{bs\in CC}{q}^{c\left(bs\right)-1}{(-q^{-1})}^{r\left(bs\right)-1},& \text{if}\hspace{0.17em}\lambda /\mu \hspace{0.17em}\text{is a border strip,}\\ 0,& \text{otherwise.}\end{array}$$ In the formula for $\Delta (\lambda /\mu ):$ a border strip is a skew shape with at most one box in each diagonal, $CC$ is the set of connected components of $\lambda /\mu ,$ $cc$ is the number of connected components of $\lambda /\mu ,$ $r\left(bs\right)$ is the number of rows of $bs,$ and $c\left(bs\right)$ is the number of columns of $bs\text{.}$ Let $s_{\lambda }$ denote the Schur function (see [Mac1995]) and define $$q_r=\sum _{m=1}^r{(-q^{-1})}^{r-m}{q}^{m-1}{s}_{\left(m{1}^{r-m}\right)}\text{.}$$ By Proposition 6.11(a) in [HRa1995], $$q_r{s}_{\mu }=\sum _{\lambda }\Delta (\lambda /\mu )s_{\lambda }\text{.}$$ Letting $q_{\beta }=q_{{\beta }_1}\dots q_{{\beta }_{\ell }}$ one can inductively apply this formula to obtain $$q_{\beta }{s}_{\mu }=\sum _{\lambda }{\chi }^{(c,\lambda /\mu )}\left(T_{{\gamma }_{\beta }}\right)s_{\lambda }\text{.}$$ Thus, with notations as in [Mac1995] Chapt. I, $$\begin{array}{ccc}{\chi }^{(c,\lambda /\mu )}\left(T_{{\gamma }_{\beta }}\right)& =& ⟨q_{\beta }{s}_{\mu },s_{\lambda }⟩\\ & =& ⟨q_{\beta },s_{\lambda /\mu }⟩,& \text{by [Mac1995] I (5.1),}\\ & =& \sum _{\nu }{c}_{\mu \nu }^{\lambda }⟨q_{\beta },s_{\nu }⟩,& \text{by [Mac1995] I (5.3),}\\ & =& \sum _{\nu }{c}_{\mu \nu }^{\lambda }{\chi }^{\nu }\left(T_{{\gamma }_{\beta }}\right),& \text{[Ram1991-2] Theorem 4.14,}\end{array}$$ where ${\chi }^{\nu }\left(T_{{\gamma }_{\beta }}\right)$ denotes the irreducible character of $H_n$ evaluated at the element $T_{{\gamma }_{\beta }}\text{.}$ The result follows since, by [Ram1991-2] Theorem 5.1, the characters of $H_n$ are determined by their values on the elements $T_{{\gamma }_{\beta }}\text{.}$ $\square$

Classically, the Littlewood-Richardson coefficients describe

1. The decomposition of the tensor product of two irreducible polynomial representations of ${GL}_n\left(ℂ\right),$ and
2. The decomposition of an irreducible representation of $S_k\times S_{\ell }$ when it is induced to $S_{k+\ell }\text{.}$

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

3. The decomposition of an irreducible representation ${\stackrel{\sim }{H}}_n$ when it is restricted to the subalgebra $H_n\text{.}$

Induction from “Young subalgebras”. Let $k$ and $\ell$ be such that $k+\ell =n\text{.}$ Let

$$\begin{array}{ccc}{\stackrel{\sim }{H}}_k& =& \text{the subalgebra of}\hspace{0.17em}{\stackrel{\sim }{H}}_n\hspace{0.17em}\text{generated by}\hspace{0.17em}{T}_i,1\le i\le k-1\hspace{0.17em}\text{and}\hspace{0.17em}{x}_i,1\le i\le k,\\ {\stackrel{\sim }{H}}_{\ell }& =& \text{the subalgebra of}\hspace{0.17em}{\stackrel{\sim }{H}}_n\hspace{0.17em}\text{generated by}\hspace{0.17em}{T}_i,k+1\le i\le n-1,\hspace{0.17em}\text{and}\hspace{0.17em}{x}_i,k+1\le i\le n\text{.}\end{array}$$

In this way ${\stackrel{\sim }{H}}_k\times {\stackrel{\sim }{H}}_{\ell }$ is naturally a subalgebra of ${\stackrel{\sim }{H}}_n\text{.}$

Let $(a,\theta )$ be a placed skew shape with $k$ boxes and let $(b,\varphi )$ be a placed skew shape with $\ell$ boxes. Number the boxes of $\theta$ with $1,\dots ,k$ (as in Section 2, along diagonals from southwest to northeast) and number the boxes of $\varphi$ with $k+1,\dots ,n,$ in order to match the imbeddings of ${\stackrel{\sim }{H}}_k$ and ${\stackrel{\sim }{H}}_{\ell }$ in ${\stackrel{\sim }{H}}_n\text{.}$ Let ${\stackrel{\sim }{H}}^{(a,\theta )}$ and ${\stackrel{\sim }{H}}^{(b,\varphi )}$ be the corresponding representations of ${\stackrel{\sim }{H}}_k$ and ${\stackrel{\sim }{H}}_{\ell }$ as defined in Theorem 4.1.

Let $\theta {*}_v\varphi$ (resp. $\theta {*}_h\varphi \text{)}$ be the skew shape obtained by placing $\theta$ and $\varphi$ adjacent to each other in such a way that ${\text{box}}_{(k+1)}$ of $\varphi$ is immediately above (resp. to the left of) ${\text{box}}_k$ of $\theta \text{.}$ Let $a\otimes b$ be the content function given by

$$(a\otimes b)\left({\text{box}}_i\right)=\{\begin{array}{cc}a\left({\text{box}}_i\right),& \text{if}\hspace{0.17em}1\le i\le k,\\ b\left({\text{box}}_i\right),& \text{if}\hspace{0.17em}k+1\le i\le \ell ,\end{array}$$

Theorem 6.2. With notations as above,

$${\text{Ind}}_{{\stackrel{\sim }{H}}_k\otimes {\stackrel{\sim }{H}}_{\ell }}^{{\stackrel{\sim }{H}}_n}({\stackrel{\sim }{H}}^{(a,\theta )}\otimes {\stackrel{\sim }{H}}^{(b,\varphi )})={\stackrel{\sim }{H}}^{(a\otimes b,\theta {*}_v\varphi )}+{\stackrel{\sim }{H}}^{(a\otimes b,\theta ,{*}_h\varphi )}$$

in the Grothendieck ring of finite dimensional representations of ${\stackrel{\sim }{H}}_n\text{.}$

		Proof.
	Let $S_n/(S_k\times S_{\ell })$ be the set of minimal length coset representatives of the cosets of $S_k\times S_{\ell }$ in $S_n\text{.}$ The module $M={\text{Ind}}_{{\stackrel{\sim }{H}}_k\otimes {\stackrel{\sim }{H}}_{\ell }}^{{\stackrel{\sim }{H}}_n}({\stackrel{\sim }{H}}^{(a,\theta )}\otimes {\stackrel{\sim }{H}}^{(b,\varphi )})$ has basis $$T_w(v_L\otimes v_Q)\text{where}\hspace{0.17em}w\in S_n/(S_k\times S_{\ell }),L\in {ℱ}^{\theta }\hspace{0.17em}\text{and}\hspace{0.17em}Q\in {ℱ}^{\varphi }\text{.}$$ By repeatedly applying the relations (1.4) we obtain $$x_i\left(T_w(v_L\otimes v_Q)\right)=T_w{x}_{w^{-1}\left(i\right)}(v_L\otimes v_Q)+\sum _{u<w}{b}_u{T}_u(v_L\otimes v_Q),$$ for some constants $b_u\in ℂ\text{.}$ From this we can see that the action of $x_i$ on $M$ is an upper triangular matrix with eigenvalues $q^{2c\left(P\left(i\right)\right)}$ where $P$ runs over the standard tableaux of shapes $\theta {*}_v\varphi$ and $\theta {*}_h\varphi \text{.}$ It follows that $M$ has $$\frac{\mid S_n\mid \text{Card}\left({ℱ}^{\theta }\right)\text{Card}\left({ℱ}^{\varphi }\right)}{\mid S_k\times S_{\ell }\mid }$$ 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 ${\stackrel{\sim }{H}}^{(c,\lambda /\mu )}$ for some placed skew shape $(c,\lambda /\mu )$ and by Lemma 2.2 this placed skew shape is completely determined by any one of the weights of the module ${\stackrel{\sim }{H}}^{(c,\lambda /\mu )}\text{.}$ Thus, our analysis of the weights of $M$ implies that both ${\stackrel{\sim }{H}}^{(a\otimes b,\theta {*}_v\varphi )}$ and ${\stackrel{\sim }{H}}^{(a\otimes b,\theta {*}_h\varphi )}$ are composition factors of $M\text{.}$ The result follows since $$\text{dim}\left({\stackrel{\sim }{H}}^{(a\otimes b,\theta {*}_v\varphi )}\right)+\text{dim}\left({\stackrel{\sim }{H}}^{(a\otimes b,\theta {*}_h\varphi )}\right)=\text{dim}\left(M\right)\text{.}$$ $\square$

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 $c\left({\text{box}}_i\right)=i-1,$ for $1\le i\le n\text{.}$ Let $t=(t_1,\dots ,t_n)=(1,q^2,q^{2(n-1)})$ and let $ℂv_t$ be the one dimensional module for $ℂ\left[X\right]=ℂ[x_1^{\pm 1},\dots ,x_n^{\pm 1}]$ given by

$$x_i{v}_t=t_i{v}_t,\text{for all}\hspace{0.17em}1\le i\le n\text{.}$$

In the Grothendieck ring of finite dimensional ${\stackrel{\sim }{H}}_n\text{-modules}$

$${\text{Ind}}_{ℂ\left[X\right]}^{{\stackrel{\sim }{H}}_n}\left(ℂv_t\right)=\sum _{\lambda /\mu }{\stackrel{\sim }{H}}^{(c,\lambda /\mu )},$$

where the sum is over all connected ribbons $\lambda /\mu$ with $n$ boxes.

		Proof.
	Since $ℂ\left[X\right]={\stackrel{\sim }{H}}_1\otimes {\stackrel{\sim }{H}}_1\otimes \dots \otimes {\stackrel{\sim }{H}}_1\subseteq {\stackrel{\sim }{H}}_n$ this result can be obtained by repeatedly applying Theorem 6.2. $\square$

Theorem 6.2 and Corollary 6.3 are ${\stackrel{\sim }{H}}_n\text{-module}$ realizations of the Schur function identities in [Mac1995] I §5 Ex 21 (a),(b). The module ${\text{Ind}}_{ℂ\left[X\right]}^{{\stackrel{\sim }{H}}_n}\left(ℂv_t\right)$ is a principal series module for ${\stackrel{\sim }{H}}_n$ (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.

page history