## Formulae for Kostka–Foulkes polynomials

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

$sλ=∑μ∈P+ Kλμ(t) Pμ(x;t), (3.12)$

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

Fro each $\alpha \in {R}^{+}$ define the raising operator ${R}_{\alpha }:\phantom{\rule{0.2em}{0ex}}P\to P$ by

$Rαλ=λ+α, and define ( Rβ1… Rβl ) sλ= s Rβ1… Rβlλ , (3.13)$

for any sequence ${\beta }_{1},\dots ,{\beta }_{l}$ of positive roots. Using the straightening law for Weyl characters (2.7),

$sμ=(-1)ℓ(w) sw∘μ,where w∘μ=w(μ+ρ)-ρ,$

any ${s}_{\mu }$ is equal to 0 or to $±{s}_{\lambda }$ 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 $⟨\rho ,{\alpha }_{i}^{\vee }⟩=1\text{)}$ ${s}_{-{\alpha }_{i}}=-{s}_{{s}_{i}\circ \left(-{\alpha }_{i}\right)}=-{s}_{{s}_{i}\left(\rho -{\alpha }_{i}\right)-\rho }=-{s}_{-{\alpha }_{i}}$ giving that ${s}_{-{\alpha }_{i}}=0$ and so

$Rαi (Rαis-2αi) =Rαis-αi= Rαi·0=0, whereas (RαiRαi) s-2αi=s0=1.$

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

$∏α∈R+ 11-txα= ∑γ∈Q+F (γ;t)xγ, andF(γ;t) =0,ifγ∉ Q+. (3.14)$

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), ${P}_{\mu }\left(x;t\right)$ as in (2.9) and let ${⟨,⟩}_{t}$ be the inner product defined in (3.2). For $y,w\in \stackrel{\sim }{W}$ let ${P}_{yw}\in ℤ\left[{t}^{±\frac{1}{2}}\right]$ 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. ${K}_{\lambda ,\mu }\left(t\right)={W}_{\mu }\left(t\right){⟨{s}_{\lambda },{P}_{\mu }\left(x;t\right)⟩}_{t}\text{.}$
2. ${K}_{\lambda ,\mu }\left(t\right)=\text{coefficient of}\phantom{\rule{0.2em}{0ex}}{s}_{\lambda }\phantom{\rule{0.2em}{0ex}}\text{in}\phantom{\rule{0.2em}{0ex}}\left(\prod _{\alpha \in {R}^{+}}\frac{1}{1-t{R}_{\alpha }}\right){s}_{\mu }\text{.}$
3. ${K}_{\lambda ,\mu }\left(t\right)=\sum _{w\in W}{\left(-1\right)}^{\ell \left(w\right)}F\left(w\left(\lambda +\rho \right)-\left(\mu +\rho \right);t\right)\text{.}$
4. ${K}_{\lambda ,\mu }\left(t\right)={t}^{⟨\lambda -\mu ,{\rho }^{\vee }⟩}{P}_{x,{n}_{\lambda }}\left({t}^{-1}\right),\phantom{\rule{0.2em}{0ex}}\text{for any}\phantom{\rule{0.2em}{0ex}}x\in W{t}_{\mu }W\text{.}$

 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 $Pμ(x;t) Wμ(t) ∏α∈R 11-txα = ∑w∈Ww ( xμ ∏α∈R+ 1-tx-α 1-x-α ) ∏α∈R 11-txα = ∑w∈Ww ( xμ+ρ 1 xρ ∏α∈R+ (1-x-α) (1-txα) ) = 1aρ ∑w∈W (-1)ℓ(w)w ( ∏α∈R+ ( 11-txα ) xμ+ρ ) ,$ it follows that $Kλμ(t) = ( coefficient ofPμ (x;t)in sλ ) = ⟨ sλ,Wμ(t) Pμ(x;t) ⟩ t = ⟨ sλ,Wμ(t) Pμ(x;t) ∏α∈R 11-txα ⟩ 0 = coefficient ofsλin 1aρ ∑w∈W (-1)ℓ(w)w ( ∏α∈R+ ( 11-txα ) xμ+ρ ) = coefficient ofsλin ( ∏α∈R+ 11-tRα ) sμ.$ (c) $Kλμ(t) = coefficient ofsλin 1aρ ∑w∈W (-1)ℓ(w)w ( ∏α∈R+ ( 11-txα ) xμ+ρ ) = coefficient ofaλ+ρin ∑w∈W (-1)ℓ(w)w ( ( ∑γ∈Q+ F(γ;t) xγ ) xμ+ρ ) = coefficient ofxλ+ρin ∑w∈W (-1)ℓ(w)w ( ∑γ∈Q+ F(γ;t) xγ+μ+ρ ) = ∑w∈W (-1)ℓ(w)F ( w(λ+ρ)- (μ+ρ);t ) ,$ since ${w}^{-1}\left(\gamma +\left(\mu +\rho \right)\right)=\lambda +\rho$ implies $\gamma =w\left(\lambda +\rho \right)-\left(\mu +\rho \right)\text{.}$ (d) Let $\lambda \in {P}^{+}\text{.}$ By Theorem 2.9 and Lemma 2.7 $∑x≤nλ q - ( ℓ(nλ)- ℓ(x) ) Px,nλ (q2)Tx = Cnλ′= q-ℓ(w0) W0(q2)sλ10 = q-ℓ(w0) W0(q2) ∑μ≤λ Kλμ (q-2) Pμ(x;q-2) 10 = q-ℓ(w0) W0(q2) ∑μ≤λ Kλμ (q-2) W0(q-2) Wμ(q-2) Mμ = ∑μ≤λ Kλμ (q-2) ∑x∈WtμW qℓ(x)-ℓ(nμ) Tx.$ Hence, for $\mu \le \lambda$ and $x\in W{t}_{\mu }W,$ $Kλμ(q-2) =qℓ(nμ)-ℓ(nλ) Px,nλ(q2).$ By (2.15) and (2.16), $ℓ(nμ)- ℓ(nλ)= ℓ(tμ)+ ℓ(w0)- (ℓ(tλ)+ℓ(w0)) =2⟨μ,ρ∨⟩ -2⟨λ,ρ∨⟩,$ 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 ${s}_{0}={P}_{0}\left(x;t\right)$ that

$Kλ,0(t) = W0(t) ⟨sλ,P0(x;t)⟩t =W0(t) ⟨sλ,s0⟩t = ∑k≥0 dim ( Hom𝔤 ( L(λ), ℋk ) tk ) . (3.15)$

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

Let $J\subset \left\{{\alpha }_{1},\dots ,{\alpha }_{n}\right\}$ be a subset of the set of simple roots and let

$𝔥J*=ℝ-span {αj∈J}, RJ=R∩𝔥J*, RJ+=R+∩𝔥J*, (3.16)$ $WJ= ⟨sj∣αj∈J⟩ ,andPJ+= P+∩𝔥J*, (3.17)$

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

$𝔥*=𝔥J*⊕ 𝔥J⊥, and writeμ=μJ+ μJ⊥, (3.18)$

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 $\left\{{\alpha }_{1},\dots ,{\alpha }_{n}\right\}$ and use notations as in (3.16-3.18). Then, for $\lambda ,\mu \in {P}^{+},$

$Kλμ(t)= coefficient ofsλ in ( ∏α∈(R+\RJ+) 11-tRα ) ∑λJ∈PJ+ KλJμJ(t) sλJ+μJ⊥,$

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 $∑w∈W (-1)ℓ(w)w ( ∏α∈R+ (11-txα) xμ+ρ ) = 1∣WJ∣ ∑w∈W (-1)ℓ(w)w ∑v∈WJ (-1)ℓ(v)v ( ( ∏α∈(R+\RJ+) 11-txα ) ( ∏α∈RJ+ 11-txα ) xμJ+ρJ xμJ⊥+ρJ⊥ ) = 1∣WJ∣ ∑w∈W (-1)ℓ(w)w ( ( ∏α∈(R+\RJ+) 11-txα ) xμJ⊥+ρJ⊥ ∑v∈WJ (-1)ℓ(v)v ( ∏α∈RJ+ 11-txα ) xμJ+ρJ ) = 1∣WJ∣ ∑w∈W (-1)ℓ(w)w ( ( ∏α∈(R+\RJ+) 11-txα ) xμJ+ρJ ∑λJ∈PJ+ KλJμJ (t) aαJ+ρJ ) ,$ 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 $1∣WJ∣ ∑λJ∈PJ+ KλJμJ (t)∑w∈W (-1)ℓ(w)w ( ( ∏α∈(R+\RJ+) 11-txα ) xμJ⊥+ρJ⊥ ∑v∈WJ (-1)ℓ(v)v xλJ+ρJ ) = 1∣WJ∣ ∑λJ∈PJ+ KλJμJ (t)∑w∈W (-1)ℓ(w)w ∑v∈WJ (-1)ℓ(v)v ( ( ∏α∈(R+\RJ+) 11-txα ) xμJ⊥+ρJ⊥+λJ+ρJ ) = ∑λJ∈PJ+ KλJμJ (t)∑w∈W (-1)ℓ(w)w ( ( ∏α∈(R+\RJ+) 11-txα ) xλJ+μJ⊥+ρ )$ 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).