Orthogonality and 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

Orthogonality and formulae for Kostka–Foulkes polynomials

Let $𝕂=\mathbb{Z}\left[t\right]\text{.}$ If $f=\sum _{\mu \in P}{f}_{\mu }{x}^{\mu }\in 𝕂\left[P\right]$ let

$$\begin{array}{cc}\bar{f}=\sum _{\mu \in P}{f}_{\mu }{x}^{-\mu },\text{and}{\left[f\right]}_1=f_0=\text{(coefficient of 1 in}f\text{).}& \text{(3.1)}\end{array}$$

Define a symmetric bilinear form

$$\begin{array}{cc}{⟨,⟩}_t:𝕂\left[P\right]\times 𝕂\left[P\right]⟶𝕂\text{by}{⟨f,g⟩}_t=\frac{1}{\mid W\mid }{\left[f\bar{g}\prod _{\alpha \in R}\frac{1-x^{\alpha }}{1-t{x}^{\alpha }}\right]}_1\text{.}& \text{(3.2)}\end{array}$$

"Specializing" $t$ at the values 0 and 1 gives inner products

$${⟨,⟩}_0:𝕂\left[P\right]\times 𝕂\left[P\right]⟶𝕂\text{and}{⟨,⟩}_1:𝕂\left[P\right]\times 𝕂\left[P\right]⟶𝕂$$

with

$$\begin{array}{cc}{⟨f,g⟩}_0=\frac{1}{\mid W\mid }{\left[f\bar{g}\prod _{\alpha \in R}(1-x^{\alpha })\right]}_1\text{and}{⟨f,g⟩}_1=\frac{1}{\mid W\mid }{\left[f\bar{g}\right]}_1\text{.}& \text{(3.3)}\end{array}$$

Let $\lambda$ and $\mu \in P^{+}\text{.}$ Then

$${⟨m_{\lambda },m_{\mu }⟩}_1=\frac{1}{\mid W_{\lambda }\mid }{\delta }_{\lambda \mu },{⟨s_{\lambda },s_{\mu }⟩}_0={\delta }_{\lambda \mu },\text{and}{⟨P_{\lambda },P_{\mu }⟩}_t=\frac{1}{W_{\lambda }\left(t\right)}{\delta }_{\lambda \mu }\text{.}$$

		Proof.
	Letting $W\lambda$ denote the $W$-orbit of $\lambda ,$ the first equality follows from $$\mid W_{\lambda }\mid {⟨m_{\lambda },m_{\mu }⟩}_1=\frac{\mid W_{\lambda }\mid }{\mid W\mid }\sum _{\gamma \in W\lambda ,\nu \in W\mu }{\left[x^{\gamma }{x}^{-\nu }\right]}_1={\delta }_{\lambda \mu }\frac{\mid W_{\lambda }\mid }{\mid W\mid }\sum _{\gamma \in W\lambda }1={\delta }_{\lambda \mu }\text{.}$$ If $\lambda ,\mu \in P^{+},$ $$\begin{array}{ccc}{⟨s_{\lambda },s_{\mu }⟩}_0& =& \frac{1}{\mid W\mid }{\left[\overline{a_{\rho }{s}_{\lambda }}{a}_{\rho }{s}_{\mu }\right]}_1=\frac{1}{\mid W\mid }{\left[\overline{a_{\lambda +\rho }}{a}_{\mu +\rho }\right]}_1\\ & =& \frac{1}{\mid W\mid }\sum _{v,w\in W}{(-1)}^{\ell \left(v\right)}{(-1)}^{\ell \left(w\right)}{\left[x^{-v(\lambda +\rho )}{x}^{w(\mu +\rho )}\right]}_1\\ & =& {\delta }_{\lambda \mu }\frac{1}{\mid W\mid }\sum _{v\in W}{(-1)}^{\ell \left(v\right)}{(-1)}^{\ell \left(v\right)}={\delta }_{\lambda \mu },\end{array}$$ giving the second statement. By Lemma 2.5(b) the matrix $K^{-1}$ given by the values ${\left(K^{-1}\right)}_{\lambda \mu }$ in the equation $$P_{\lambda }(x;t)=\sum _{\mu }{\left(K^{-1}\right)}_{\lambda \mu }{s}_{\mu },$$ has entries in $\mathbb{Z}\left[t\right]$ and is upper triangular with 1's on the diagonal, that is, ${\left(K^{-1}\right)}_{\lambda \lambda }=1$ and ${\left(K^{-1}\right)}_{\lambda \mu }=0$ unless $\mu \le \lambda \text{.}$ Since $P_{\lambda }(x;1)=m_{\lambda }$ the matrix $k^{-1}$ describing the change of basis $$m_{\lambda }=\sum _{\mu }{\left(k^{-1}\right)}_{\lambda \mu }{s}_{\mu },$$ is the specialization of $K^{-1}$ at $t=1$ and so $k^{-1}$ has entries in $\mathbb{Z}$ and is upper triangular with 1's on the diagonal. Then the matrix $A=K^{-1}{k}^{-1}$ giving the change of basis $$\begin{array}{cc}{P}_{\lambda }(x;t)=\sum _{\nu \le \lambda }{A}_{\lambda \nu }{m}_{\mu },& \text{(3.4)}\end{array}$$ has $A_{\lambda \mu }\in \mathbb{Z}\left[t\right],$ $A_{\lambda \lambda }=1,$ and $A_{\lambda \mu }=0$ unless $\mu \le \lambda \text{.}$ Let $Q^{+}$ be the set of nonnegative integral linear combinations of positive roots. Then $$\begin{array}{ccc}{P}_{\mu }(x;t)W_{\mu }\left(t\right)\left(\prod _{\alpha \in R}\frac{1-x^{\alpha }}{1-t{x}^{\alpha }}\right)& =& \sum _{w\in W}w\left(x^{\mu }\prod _{\alpha \in R^{+}}\frac{1-x^{\alpha }}{1-t{x}^{\alpha }}\right)\\ & =& \sum _{w\in W}w\left(x^{\mu }\prod _{\alpha \in R^{+}}(1+\sum _{r>0}{t}^{r-1}(t-1)x^{r\alpha })\right)\\ & =& \sum _{w\in W}w\left(\sum _{\nu \in Q^{+}}{c}_{\nu }{x}^{\mu +\nu }\right)\\ & =& \sum _{\nu \in Q^{+}}{c}_{\nu }\left(\sum _{w\in W}w{x}^{\mu +\nu }\right),\end{array}$$ where $c_{\nu }\in \mathbb{Z}\left[t\right]$ and $c_0=1\text{.}$ Hence $$\begin{array}{cc}{P}_{\mu }(x;t)W_{\mu }\left(t\right)\prod _{\alpha \in R}\frac{1-x^{\alpha }}{1-t{x}^{\alpha }}=\mid W_{\mu }\mid m_{\mu }+\sum _{\gamma >\mu }{B}_{\mu \gamma }{m}_{\gamma }=\sum _{\gamma \ge \mu }{B}_{\mu \gamma }{m}_{\gamma },& \text{(3.5)}\end{array}$$ with $B_{\mu \gamma }\in \mathbb{Z}\left[t\right]$ and $B_{\mu \mu }=\mid W_{\mu }\mid \text{.}$ Assume that $\lambda \le \mu$ if $\lambda$ and $\mu$ are comparable. Then, by using (3.4) and (3.5), $$\begin{array}{ccc}{⟨P_{\lambda },P_{\mu }⟩}_t& =& \frac{1}{W_{\mu }\left(t\right)}{⟨P_{\lambda },P_{\mu }{W}_{\mu }\left(t\right)\prod _{\alpha \in R}\frac{1-x^{\alpha }}{1-t{x}^{\alpha }}⟩}_1\\ & =& \frac{1}{W_{\mu }\left(t\right)}{⟨\sum _{\nu \le \lambda }{A}_{\lambda \nu }{m}_{\nu },\sum _{\gamma \ge \mu }{B}_{\mu \gamma }{m}_{\gamma }⟩}_1\text{.}\end{array}$$ Since $A_{\lambda \lambda }=1$ and $B_{\mu \mu }=\mid W_{\mu }\mid$ the result follows from ${⟨m_{\lambda },m_{\mu }⟩}_1={\mid W_{\lambda }\mid }^{-1}{\delta }_{\lambda \mu }\text{.}$ $\square$

The following theorem shows that the spherical functions $P_{\lambda }(x;t)$ are uniquely determined by the triangularity in (3.4) and the orthogonality in the third equality of Proposition 3.1.

Let $𝕂=\mathbb{Z}\left[t\right]\text{.}$ The spherical functions $P_{\lambda }(x;t)$ are the unique elements of $𝕂{\left[P\right]}^W$ such that

1. $P_{\lambda }=m_{\lambda }+{\displaystyle \sum _{\mu <\lambda }}{A}_{\lambda \mu }{m}_{\mu },$
2. ${⟨P_{\lambda },P_{\mu }⟩}_t=0\text{if}\lambda \ne \mu \text{.}$

		Proof.
	Assume that the $P_{\mu }$ are determined for $\mu <\lambda \text{.}$ Then the condition in (a) can be rewritten as $$P_{\lambda }=m_{\lambda }+\sum _{\mu <\lambda }{C}_{\lambda \mu }{P}_{\mu },$$ for some constants $C_{\lambda \mu }\text{.}$ Take the inner product on each side with $P_{\nu },$ $\nu <\lambda ,$ and use property (b) to get the system of equations $$0={⟨m_{\lambda },P_{\nu }⟩}_t+\sum _{\mu <\lambda }{C}_{\lambda \mu }{⟨P_{\mu },P_{\nu }⟩}_t={⟨m_{\lambda },P_{\nu }⟩}_t+C_{\lambda \nu }{⟨P_{\nu },P_{\nu }⟩}_t\text{.}$$ Hence $$C_{\lambda \nu }=\frac{-{⟨m_{\lambda },P_{\nu }⟩}_t}{{⟨P_{\nu },P_{\nu }⟩}_t},\text{for each}\nu <\lambda ,$$ and this determines $P_{\lambda }\text{.}$ $\square$

(a) The inner product ${⟨,⟩}_t$ arises naturally in the context of $p$-adic groups. Let $S^1=\{z\in ℂ\mid \mid z\mid =1\}$ and view the $x^{\lambda },$ $\lambda \in P,$ as characters of

$$\begin{array}{cc}T=\text{Hom}(P,S^1)\text{via}\begin{array}{cccc}{x}^{\lambda }:& T& ⟶& C^{*}\\ & s& ⟼& s\left(\lambda \right)\text{.}\end{array}& \text{(3.6)}\end{array}$$

Let $ds$ be the Haar measure on $T$ normalized so that

$$\begin{array}{cc}⟨x^{\lambda },x^{\mu }⟩={\int }_T{x}^{\lambda }\left(s\right)\overline{x^{\mu }\left(s\right)}ds={\delta }_{\lambda \mu }\text{.}& \text{(3.7)}\end{array}$$

Letting ${\mathbb{Q}}_p$ be the field of $p$-adic number, Macdonald [Mac1971, (5.1.2)] showed that the Plancherel measure for the $p$-adic Chevalley group $G\left({\mathbb{Q}}_p\right)$ corresponding to the root system $R$ is given by

$$\begin{array}{cc}d\mu \left(s\right)=\frac{W_0\left(p^{-1}\right)}{\mid W\mid }\prod _{\alpha \in R}\frac{1-x^{\alpha }\left(s\right)}{1-p^{-1}{x}^{\alpha }\left(s\right)}\text{.}& \text{(3.8)}\end{array}$$

The corresponding inner product is

$$W_0\left(p^{-1}\right){⟨f,g⟩}_{p^{-1}}={\int }_{T}f\left(s\right)\overline{g\left(s\right)}d\mu \left(s\right),\text{for}f,g\in C\left(T\right),$$

where $C\left(T\right)$ is the vector space of continuous functions on $T\text{.}$

(b) The inner product ${⟨,⟩}_t$ arises naturally in another representation theoretic context. The complex semisimple Lie algebra $𝔤$ corresponding to the root system $R$ acts on $S\left({𝔤}^{*}\right),$ the ring of polynomials on $𝔤,$ by the (co-)adjoint action. As graded $𝔤$-modules the characters of $S\left({𝔤}^{*}\right)$ and the subring of invariants $S{\left({𝔤}^{*}\right)}^{𝔤}$ are

$$\begin{array}{cc}\begin{array}{ccc}\text{grch}\left(S\left({𝔤}^{*}\right)\right)& =& \left(\prod _{i=1}^r\frac{1}{1-t}\right)\left(\prod _{\alpha \in R}\frac{1}{1-t{x}^{\alpha }}\right)\text{and}\\ \text{grch}\left(S{\left({𝔤}^{*}\right)}^{𝔤}\right)& =& \prod _{i=1}^r\frac{1}{1-t^{d_i}}=\frac{1}{W_0\left(t\right)}\prod _{i=1}^r\frac{1}{1-t},\end{array}& \text{(3.9)}\end{array}$$

where $r$ is the rank of $𝔤$ and $d_1,\dots ,d_r$ are the degrees of the Weyl group $W\text{.}$ Let $\mathscr{H}$ denote the vector space of harmonic polynomials. An important theorem of Kostant [Kos1963, Theorem 0.2] gives

$$\begin{array}{cc}S\left({𝔤}^{*}\right)\cong S{\left({𝔤}^{*}\right)}^{𝔤}\otimes \mathscr{H},\text{and thus,}\text{grch}\left(\mathscr{H}\right)=W_0\left(t\right)\prod _{\alpha \in R}\frac{1}{1-t{x}^{\alpha }}\text{.}& \text{(3.10)}\end{array}$$

If $L\left(\lambda \right)$ denotes the finite dimensional irreducible $𝔤$-module of highest weight $\lambda \in P^{+}$ then $L\left(\lambda \right)$ has character $s_{\lambda }$ and using the notation of (3.2),

$$\begin{array}{cc}\sum _{k\ge 0}\text{dim}\left({\text{Hom}}_{𝔤}(L\left(\lambda \right),L\left(\mu \right)\otimes {\mathscr{H}}^k)t^k\right)& \text{(3.11)}\end{array}\begin{array}{ccc}& =& {⟨s_{\lambda },s_{\mu }{W}_0\left(t\right)\prod _{\alpha \in R}\frac{1}{1-t{x}^{\alpha }}⟩}_0\\ & =& W_0\left(t\right){\left[s_{\lambda }\overline{s_{\mu }}\prod _{\alpha \in R}\frac{1-x^{\alpha }}{1-t{x}^{\alpha }}\right]}_1=W_0\left(t\right){⟨s_{\lambda },s_{\mu }⟩}_t,\end{array}$$

where ${\mathscr{H}}^k$ is the vector space of degree $k$ harmonic polynomials.

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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