Formulae for Kostka–Foulkes polynomials

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

Last update: 27 September 2012

Formulae for Kostka–Foulkes polynomials

For $\lambda \in P$ let $s_{\lambda }$ denote the Weyl character, as defined in (2.6). The Kostka-Foulkes polynomials, or $q$-weight multiplicities, $K_{\lambda \mu }\left(t\right),$ $\lambda ,\mu \in P^{+},$ are defined by the change of basis formula

$$\begin{array}{cc}{s}_{\lambda }=\sum _{\mu \in P^{+}}{K}_{\lambda \mu }\left(t\right)P_{\mu }(x;t),& \text{(3.12)}\end{array}$$

where the Macdonald spherical functions $P_{\mu }(x;t)$ are as in (2.9).

Fro each $\alpha \in R^{+}$ define the raising operator $R_{\alpha }:P\to P$ by

$$\begin{array}{cc}{R}_{\alpha }\lambda =\lambda +\alpha ,\text{and define}\left(R_{{\beta }_1}\dots R_{{\beta }_l}\right)s_{\lambda }=s_{R_{{\beta }_1}\dots R_{{\beta }_l}\lambda },& \text{(3.13)}\end{array}$$

for any sequence ${\beta }_1,\dots ,{\beta }_l$ of positive roots. Using the straightening law for Weyl characters (2.7),

$$s_{\mu }={(-1)}^{\ell \left(w\right)}{s}_{w\circ \mu },\text{where}w\circ \mu =w(\mu +\rho )-\rho ,$$

any $s_{\mu }$ is equal to 0 or to $\pm s_{\lambda }$ with $\lambda \in P^{+}\text{.}$ Composing the action of raising operators on Weyl characters should be avoided. For example, if ${\alpha }_i$ is a simple root then (since $⟨\rho ,{\alpha }_i^{\vee }⟩=1\text{)}$ $s_{-{\alpha }_i}=-s_{s_i\circ \left(-{\alpha }_i\right)}=-s_{s_i(\rho -{\alpha }_i)-\rho }=-s_{-{\alpha }_i}$ giving that $s_{-{\alpha }_i}=0$ and so

$$R_{{\alpha }_i}\left(R_{{\alpha }_i}{s}_{-2{\alpha }_i}\right)=R_{{\alpha }_i}{s}_{-{\alpha }_i}=R_{{\alpha }_i}·0=0,\text{whereas}\left(R_{{\alpha }_i}{R}_{{\alpha }_i}\right)s_{-2{\alpha }_i}=s_0=1\text{.}$$

Let $Q^{+}$ be the set of nonnegative integral linear combinations of positive roots. Define the q-analogue of the partition function $F(\gamma ;t),$ $\gamma \in P,$ by

$$\begin{array}{cc}\prod _{\alpha \in R^{+}}\frac{1}{1-t{x}^{\alpha }}=\sum _{\gamma \in Q^{+}}F(\gamma ;t)x^{\gamma },\text{and}F(\gamma ;t)=0,\text{if}\gamma \notin Q^{+}\text{.}& \text{(3.14)}\end{array}$$

Let $\lambda ,\mu \in P^{+}\text{.}$ Let $t_{\mu }$ be the translation in $\mu$ as defined in (1.10) and let $n_{\mu }$ be the longest element of the double coset $W{t}_{\mu }W\text{.}$ Let $W_{\mu }\left(t\right)$ be as in (2.8), $P_{\mu }(x;t)$ as in (2.9) and let ${⟨,⟩}_t$ be the inner product defined in (3.2). For $y,w\in \stackrel{\sim }{W}$ let $P_{yw}\in \mathbb{Z}\left[t^{\pm \frac{1}{2}}\right]$ denote the Kazhdan-Lusztig polynomial defined in (1.26)-(1.27) and let ${\rho }^{\vee }=\frac{1}{2}\sum _{\alpha \in R^{+}}{\alpha }^{\vee }\text{.}$

1. $K_{\lambda ,\mu }\left(t\right)=W_{\mu }\left(t\right){⟨s_{\lambda },P_{\mu }(x;t)⟩}_t\text{.}$
2. $K_{\lambda ,\mu }\left(t\right)=\text{coefficient of}{s}_{\lambda }\text{in}\left({\displaystyle \prod _{\alpha \in R^{+}}}\frac{1}{1-t{R}_{\alpha }}\right)s_{\mu }\text{.}$
3. $K_{\lambda ,\mu }\left(t\right)={\displaystyle \sum _{w\in W}}{(-1)}^{\ell \left(w\right)}F(w(\lambda +\rho )-(\mu +\rho );t)\text{.}$
4. $K_{\lambda ,\mu }\left(t\right)=t^{⟨\lambda -\mu ,{\rho }^{\vee }⟩}{P}_{x,n_{\lambda }}\left(t^{-1}\right),\text{for any}x\in W{t}_{\mu }W\text{.}$

		Proof.
	(a) This follows from the third equality in Proposition 3.1 and the definition of $K_{\lambda \mu }\left(t\right)\text{.}$ (b) Since $$\begin{array}{ccc}{P}_{\mu }(x;t)W_{\mu }\left(t\right)\prod _{\alpha \in R}\frac{1}{1-t{x}^{\alpha }}& =& \sum _{w\in W}w\left(x^{\mu }\prod _{\alpha \in R^{+}}\frac{1-t{x}^{-\alpha }}{1-x^{-\alpha }}\right)\prod _{\alpha \in R}\frac{1}{1-t{x}^{\alpha }}\\ & =& \sum _{w\in W}w\left(x^{\mu +\rho }\frac{1}{x^{\rho }\prod _{\alpha \in R^{+}}(1-x^{-\alpha })(1-t{x}^{\alpha })}\right)\\ & =& \frac{1}{a_{\rho }}\sum _{w\in W}{(-1)}^{\ell \left(w\right)}w\left(\prod _{\alpha \in R^{+}}\left(\frac{1}{1-t{x}^{\alpha }}\right)x^{\mu +\rho }\right),\end{array}$$ it follows that $$\begin{array}{ccc}{K}_{\lambda \mu }\left(t\right)& =& \left(\text{coefficient of}{P}_{\mu }(x;t)\text{in}{s}_{\lambda }\right)={⟨s_{\lambda },W_{\mu }\left(t\right)P_{\mu }(x;t)⟩}_t\\ & =& {⟨s_{\lambda },W_{\mu }\left(t\right)P_{\mu }(x;t)\prod _{\alpha \in R}\frac{1}{1-t{x}^{\alpha }}⟩}_0\\ & =& \text{coefficient of}{s}_{\lambda }\text{in}\frac{1}{a_{\rho }}\sum _{w\in W}{(-1)}^{\ell \left(w\right)}w\left(\prod _{\alpha \in R^{+}}\left(\frac{1}{1-t{x}^{\alpha }}\right)x^{\mu +\rho }\right)\\ & =& \text{coefficient of}{s}_{\lambda }\text{in}\left(\prod _{\alpha \in R^{+}}\frac{1}{1-t{R}_{\alpha }}\right)s_{\mu }\text{.}\end{array}$$ (c) $$\begin{array}{ccc}{K}_{\lambda \mu }\left(t\right)& =& \text{coefficient of}{s}_{\lambda }\text{in}\frac{1}{a_{\rho }}\sum _{w\in W}{(-1)}^{\ell \left(w\right)}w\left(\prod _{\alpha \in R^{+}}\left(\frac{1}{1-t{x}^{\alpha }}\right)x^{\mu +\rho }\right)\\ & =& \text{coefficient of}{a}_{\lambda +\rho }\text{in}\sum _{w\in W}{(-1)}^{\ell \left(w\right)}w\left(\left(\sum _{\gamma \in Q^{+}}F(\gamma ;t)x^{\gamma }\right)x^{\mu +\rho }\right)\\ & =& \text{coefficient of}{x}^{\lambda +\rho }\text{in}\sum _{w\in W}{(-1)}^{\ell \left(w\right)}w\left(\sum _{\gamma \in Q^{+}}F(\gamma ;t)x^{\gamma +\mu +\rho }\right)\\ & =& \sum _{w\in W}{(-1)}^{\ell \left(w\right)}F(w(\lambda +\rho )-(\mu +\rho );t),\end{array}$$ since $w^{-1}(\gamma +(\mu +\rho ))=\lambda +\rho$ implies $\gamma =w(\lambda +\rho )-(\mu +\rho )\text{.}$ (d) Let $\lambda \in P^{+}\text{.}$ By Theorem 2.9 and Lemma 2.7 $$\begin{array}{ccc}\sum _{x\le n_{\lambda }}{q}^{-(\ell \left(n_{\lambda }\right)-\ell \left(x\right))}{P}_{x,n_{\lambda }}\left(q^2\right)T_x& =& C_{n_{\lambda }}^{\prime }=q^{-\ell \left(w_0\right)}{W}_0\left(q^2\right)s_{\lambda }{1}_0\\ & =& q^{-\ell \left(w_0\right)}{W}_0\left(q^2\right)\sum _{\mu \le \lambda }{K}_{\lambda \mu }\left(q^{-2}\right)P_{\mu }(x;q^{-2})1_0\\ & =& q^{-\ell \left(w_0\right)}{W}_0\left(q^2\right)\sum _{\mu \le \lambda }{K}_{\lambda \mu }\left(q^{-2}\right)\frac{W_0\left(q^{-2}\right)}{W_{\mu }\left(q^{-2}\right)}{M}_{\mu }\\ & =& \sum _{\mu \le \lambda }{K}_{\lambda \mu }\left(q^{-2}\right)\sum _{x\in W{t}_{\mu }W}{q}^{\ell \left(x\right)-\ell \left(n_{\mu }\right)}{T}_x\text{.}\end{array}$$ Hence, for $\mu \le \lambda$ and $x\in W{t}_{\mu }W,$ $$K_{\lambda \mu }\left(q^{-2}\right)=q^{\ell \left(n_{\mu }\right)-\ell \left(n_{\lambda }\right)}{P}_{x,n_{\lambda }}\left(q^2\right)\text{.}$$ By (2.15) and (2.16), $$\ell \left(n_{\mu }\right)-\ell \left(n_{\lambda }\right)=\ell \left(t_{\mu }\right)+\ell \left(w_0\right)-(\ell \left(t_{\lambda }\right)+\ell \left(w_0\right))=2⟨\mu ,{\rho }^{\vee }⟩-2⟨\lambda ,{\rho }^{\vee }⟩,$$ and the result follows on replacing $q^{-2}$ by $t\text{.}$ $\square$

With the notation of Remark 3.3(b), it follows from Theorem 3.4(a) and $s_0=P_0(x;t)$ that

$$\begin{array}{cc}\begin{array}{ccc}{K}_{\lambda ,0}\left(t\right)& =& W_0\left(t\right){⟨s_{\lambda },P_0(x;t)⟩}_t=W_0\left(t\right){⟨s_{\lambda },s_0⟩}_t\\ & =& \sum _{k\ge 0}\text{dim}\left({\text{Hom}}_{𝔤}(L\left(\lambda \right),{\mathscr{H}}^k)t^k\right)\text{.}\end{array}& \text{(3.15)}\end{array}$$

This is an important formula for the Kostka-Foulkes polynomial in the case that $\mu =0\text{.}$

Let $J\subset \{{\alpha }_1,\dots ,{\alpha }_n\}$ be a subset of the set of simple roots and let

$$\begin{array}{cc}{𝔥}_J^{*}=ℝ\text{-span}\{{\alpha }_j\in J\},R_J=R\cap {𝔥}_J^{*},R_J^{+}=R^{+}\cap {𝔥}_J^{*},& \text{(3.16)}\end{array}$$ $$\begin{array}{cc}{W}_J=⟨s_j\mid {\alpha }_j\in J⟩,\text{and}{P}_J^{+}=P^{+}\cap {𝔥}_J^{*},& \text{(3.17)}\end{array}$$

so that $R_J$ is a parabolic subsystem of the root system $R,$ $R_J^{+}$ is the set of positive roots of $R_J,$ $W_J$ is the Weyl group of $R_J,$ and $P_J^{+}$ is the set of dominant integral weights for $R_J\text{.}$ Let ${𝔥}_J^{\perp }$ be the orthogonal complement to ${𝔥}_J^{*}$ with respect to the inner product $⟨,⟩$ so that

$$\begin{array}{cc}{𝔥}^{*}={𝔥}_J^{*}\oplus {𝔥}_J^{\perp },\text{and write}\mu ={\mu }_J+{\mu }_J^{\perp },& \text{(3.18)}\end{array}$$

to denote the decomposition of an element $\mu \in {𝔥}^{*}$ as a sum of ${\mu }_J\in {𝔥}_J^{*}$ and ${\mu }_J^{\perp }\in {𝔥}_J^{\perp }\text{.}$

Let $J$ be a subset of the set of simple roots $\{{\alpha }_1,\dots ,{\alpha }_n\}$ and use notations as in (3.16-3.18). Then, for $\lambda ,\mu \in P^{+},$

$$K_{\lambda \mu }\left(t\right)=\text{coefficient of}{s}_{\lambda }\text{in}\left(\prod _{\alpha \in (R^{+}\backslash R_J^{+})}\frac{1}{1-t{R}_{\alpha }}\right)\sum _{{\lambda }_J\in P_J^{+}}{K}_{{\lambda }_J{\mu }_J}\left(t\right)s_{{\lambda }_J+{\mu }_J^{\perp }},$$

where $K_{{\lambda }_J{\mu }_J}\left(t\right)$ are Kostka-Foulkes polynomials for the root system $R_J\text{.}$

		Proof.
	By the third equation in the proof of Theorem 3.4(b), $K_{\lambda \mu }\left(t\right)$ is the coefficient of $a_{\lambda +\rho }$ in $$\begin{array}{cc}\multicolumn{2}{c}{\sum _{w\in W}{(-1)}^{\ell \left(w\right)}w\left(\prod _{\alpha \in R^{+}}\left(\frac{1}{1-t{x}^{\alpha }}\right)x^{\mu +\rho }\right)}\\ =& \frac{1}{\mid W_J\mid }\sum _{w\in W}{(-1)}^{\ell \left(w\right)}w\sum _{v\in W_J}{(-1)}^{\ell \left(v\right)}v\left(\left(\prod _{\alpha \in (R^{+}\backslash R_J^{+})}\frac{1}{1-t{x}^{\alpha }}\right)\left(\prod _{\alpha \in R_J^{+}}\frac{1}{1-t{x}^{\alpha }}\right)x^{{\mu }_J+{\rho }_J}{x}^{{\mu }_J^{\perp }+{\rho }_J^{\perp }}\right)\\ =& \frac{1}{\mid W_J\mid }\sum _{w\in W}{(-1)}^{\ell \left(w\right)}w\left(\left(\prod _{\alpha \in (R^{+}\backslash R_J^{+})}\frac{1}{1-t{x}^{\alpha }}\right)x^{{\mu }_J^{\perp }+{\rho }_J^{\perp }}\sum _{v\in W_J}{(-1)}^{\ell \left(v\right)}v\left(\prod _{\alpha \in R_J^{+}}\frac{1}{1-t{x}^{\alpha }}\right)x^{{\mu }_J+{\rho }_J}\right)\\ =& \frac{1}{\mid W_J\mid }\sum _{w\in W}{(-1)}^{\ell \left(w\right)}w\left(\left(\prod _{\alpha \in (R^{+}\backslash R_J^{+})}\frac{1}{1-t{x}^{\alpha }}\right)x^{{\mu }_J+{\rho }_J}\sum _{{\lambda }_J\in P_J^{+}}{K}_{{\lambda }_J{\mu }_J}\left(t\right)a_{{\alpha }_J+{\rho }_J}\right),\end{array}$$ where the last equality follows from Theorem 3.4(b) applied to the root system $R_J\text{.}$ Expanding $a_{{\lambda }_J+{\rho }_J}$ gives that $K_{\lambda \mu }\left(t\right)$ is the coefficient of $a_{\lambda +\rho }$ in $$\begin{array}{cc}\multicolumn{2}{c}{\frac{1}{\mid W_J\mid }\sum _{{\lambda }_J\in P_J^{+}}{K}_{{\lambda }_J{\mu }_J}\left(t\right)\sum _{w\in W}{(-1)}^{\ell \left(w\right)}w\left(\left(\prod _{\alpha \in (R^{+}\backslash R_J^{+})}\frac{1}{1-t{x}^{\alpha }}\right)x^{{\mu }_J^{\perp }+{\rho }_J^{\perp }}\sum _{v\in W_J}{(-1)}^{\ell \left(v\right)}v{x}^{{\lambda }_J+{\rho }_J}\right)}\\ =& \frac{1}{\mid W_J\mid }\sum _{{\lambda }_J\in P_J^{+}}{K}_{{\lambda }_J{\mu }_J}\left(t\right)\sum _{w\in W}{(-1)}^{\ell \left(w\right)}w\sum _{v\in W_J}{(-1)}^{\ell \left(v\right)}v\left(\left(\prod _{\alpha \in (R^{+}\backslash R_J^{+})}\frac{1}{1-t{x}^{\alpha }}\right)x^{{\mu }_J^{\perp }+{\rho }_J^{\perp }+{\lambda }_J+{\rho }_J}\right)\\ =& \sum _{{\lambda }_J\in P_J^{+}}{K}_{{\lambda }_J{\mu }_J}\left(t\right)\sum _{w\in W}{(-1)}^{\ell \left(w\right)}w\left(\left(\prod _{\alpha \in (R^{+}\backslash R_J^{+})}\frac{1}{1-t{x}^{\alpha }}\right)x^{{\lambda }_J+{\mu }_J^{\perp }+\rho }\right)\end{array}$$ from which the desired formula follows by dividing by $a_{\rho }$ and converting to raising operators (as in the proof of Theorem 3.4(b) above). $\square$

Acknowledgements

The research of A. Ram was partially supported by the National Science Foundation (DMS-0097977), the National Security Agency (MDA904-01-1-0032) and by EPSRC Grant GR K99015 at the Newton Institute for Mathematical Sciences. The research of K. Nelsen was partially supported by the National Science Foundation (DMS-0097977 and a VIGRE grant) and the National Security Agency (MDA904-01-1-0032).

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