Last update: 3 March 2013

**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 $\nu \u22a2n$
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

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 $${\chi}^{(c,\lambda /\mu )}\left({T}_{{\gamma}_{\beta}}\right)=\sum _{Q\in {\mathcal{F}}^{\lambda /\mu}}{T}_{{\gamma}_{\beta}}{v}_{Q}{\mid}_{{v}_{Q}},$$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, $$\begin{array}{ccc}{\chi}^{(c,\lambda /\mu )}\left({T}_{{\gamma}_{\beta}}\right)& =& \u27e8{q}_{\beta}{s}_{\mu},{s}_{\lambda}\u27e9\\ & =& \u27e8{q}_{\beta},{s}_{\lambda /\mu}\u27e9,& \text{by [Mac1995] I (5.1),}\\ & =& \sum _{\nu}{c}_{\mu \nu}^{\lambda}\u27e8{q}_{\beta},{s}_{\nu}\u27e9,& \text{by [Mac1995] I (5.3),}\\ & =& \sum _{\nu}{c}_{\mu \nu}^{\lambda}{\chi}^{\nu}\left({T}_{{\gamma}_{\beta}}\right),& \text{[Ram1991-2] Theorem 4.14,}\end{array}$$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

- The decomposition of the tensor product of two irreducible polynomial representations of ${GL}_{n}\left(\u2102\right),$ and
- 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

- 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

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,

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 $${T}_{w}({v}_{L}\otimes {v}_{Q})\phantom{\rule{2em}{0ex}}\text{where}\hspace{0.17em}w\in {S}_{n}/({S}_{k}\times {S}_{\ell}),L\in {\mathcal{F}}^{\theta}\hspace{0.17em}\text{and}\hspace{0.17em}Q\in {\mathcal{F}}^{\varphi}\text{.}$$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 ${b}_{u}\in \u2102\text{.}$ 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 $$\frac{\mid {S}_{n}\mid \text{Card}\left({\mathcal{F}}^{\theta}\right)\text{Card}\left({\mathcal{F}}^{\varphi}\right)}{\mid {S}_{k}\times {S}_{\ell}\mid}$$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 $\u2102{v}_{t}$ be the one dimensional module for
$\u2102\left[X\right]=\u2102[{x}_{1}^{\pm 1},\dots ,{x}_{n}^{\pm 1}]$
given by

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

$${\text{Ind}}_{\u2102\left[X\right]}^{{\stackrel{\sim}{H}}_{n}}\left(\u2102{v}_{t}\right)=\sum _{\lambda /\mu}{\stackrel{\sim}{H}}^{(c,\lambda /\mu )},$$where the sum is over all connected ribbons $\lambda /\mu $ with $n$ boxes.

Proof. | |

Since $\u2102\left[X\right]={\stackrel{\sim}{H}}_{1}\otimes {\stackrel{\sim}{H}}_{1}\otimes \dots \otimes {\stackrel{\sim}{H}}_{1}\subseteq {\stackrel{\sim}{H}}_{n}$ 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 ${\text{Ind}}_{\u2102\left[X\right]}^{{\stackrel{\sim}{H}}_{n}}\left(\u2102{v}_{t}\right)$ 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.

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.