Formulae for Kostka–Foulkes polynomials

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 27 September 2012

Formulae for Kostka–Foulkes polynomials

For λP let sλ denote the Weyl character, as defined in (2.6). The Kostka-Foulkes polynomials, or q-weight multiplicities, Kλμ(t), λ,μP+, are defined by the change of basis formula

sλ=μP+ Kλμ(t) Pμ(x;t), (3.12)

where the Macdonald spherical functions Pμ(x;t) are as in (2.9).

Fro each αR+ define the raising operator Rα:PP by

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

for any sequence β1,,βl of positive roots. Using the straightening law for Weyl characters (2.7),

sμ=(-1)(w) swμ,where wμ=w(μ+ρ)-ρ,

any sμ is equal to 0 or to ±sλ with λP+. Composing the action of raising operators on Weyl characters should be avoided. For example, if αi is a simple root then (since ρ,αi =1) s-αi=- ssi(-αi) =-ssi(ρ-αi)-ρ =-s-αi giving that s-αi=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(γ;t), γP, by

αR+ 11-txα= γQ+F (γ;t)xγ, andF(γ;t) =0,ifγ Q+. (3.14)

Let λ,μP+. Let tμ be the translation in μ as defined in (1.10) and let nμ be the longest element of the double coset WtμW. Let Wμ(t) be as in (2.8), Pμ(x;t) as in (2.9) and let ,t be the inner product defined in (3.2). For y,wW let Pyw[t±12] denote the Kazhdan-Lusztig polynomial defined in (1.26)-(1.27) and let ρ=12 αR+ α.

  1. Kλ,μ(t)= Wμ(t) sλ,Pμ (x;t) t .
  2. Kλ,μ(t)= coefficient ofsλ in ( αR+ 11-tRα ) sμ.
  3. Kλ,μ(t)= wW (-1)(w) F ( w(λ+ρ)- (μ+ρ);t ) .
  4. Kλ,μ(t)= tλ-μ,ρ Px,nλ (t-1), for anyxWtμW.


(a) This follows from the third equality in Proposition 3.1 and the definition of Kλμ(t).

(b) Since

Pμ(x;t) Wμ(t) αR 11-txα = wWw ( xμ αR+ 1-tx-α 1-x-α ) αR 11-txα = wWw ( xμ+ρ 1 xρ αR+ (1-x-α) (1-txα) ) = 1aρ wW (-1)(w)w ( αR+ ( 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) αR 11-txα 0 = coefficient ofsλin 1aρ wW (-1)(w)w ( αR+ ( 11-txα ) xμ+ρ ) = coefficient ofsλin ( αR+ 11-tRα ) sμ.


Kλμ(t) = coefficient ofsλin 1aρ wW (-1)(w)w ( αR+ ( 11-txα ) xμ+ρ ) = coefficient ofaλ+ρin wW (-1)(w)w ( ( γQ+ F(γ;t) xγ ) xμ+ρ ) = coefficient ofxλ+ρin wW (-1)(w)w ( γQ+ F(γ;t) xγ+μ+ρ ) = wW (-1)(w)F ( w(λ+ρ)- (μ+ρ);t ) ,

since w-1 (γ+(μ+ρ)) =λ+ρ implies γ=w(λ+ρ) -(μ+ρ).

(d) Let λP+. By Theorem 2.9 and Lemma 2.7

xnλ q - ( (nλ)- (x) ) Px,nλ (q2)Tx = Cnλ= q-(w0) W0(q2)sλ10 = q-(w0) W0(q2) μλ Kλμ (q-2) Pμ(x;q-2) 10 = q-(w0) W0(q2) μλ Kλμ (q-2) W0(q-2) Wμ(q-2) Mμ = μλ Kλμ (q-2) xWtμW q(x)-(nμ) Tx.

Hence, for μλ and xWtμW,

Kλμ(q-2) =q(nμ)-(nλ) Px,nλ(q2).

By (2.15) and (2.16),

(nμ)- (nλ)= (tμ)+ (w0)- ((tλ)+(w0)) =2μ,ρ -2λ,ρ,

and the result follows on replacing q-2 by t.

With the notation of Remark 3.3(b), it follows from Theorem 3.4(a) and s0=P0(x;t) that

Kλ,0(t) = W0(t) sλ,P0(x;t)t =W0(t) sλ,s0t = k0 dim ( Hom𝔤 ( L(λ), k ) tk ) . (3.15)

This is an important formula for the Kostka-Foulkes polynomial in the case that μ=0.

Let J{α1,,αn} be a subset of the set of simple roots and let

𝔥J*=-span {αjJ}, RJ=R𝔥J*, RJ+=R+𝔥J*, (3.16) WJ= sjαjJ ,andPJ+= P+𝔥J*, (3.17)

so that RJ is a parabolic subsystem of the root system R, RJ+ is the set of positive roots of RJ, WJ is the Weyl group of RJ, and PJ+ is the set of dominant integral weights for RJ. Let 𝔥J be the orthogonal complement to 𝔥J* with respect to the inner product , so that

𝔥*=𝔥J* 𝔥J, and writeμ=μJ+ μJ, (3.18)

to denote the decomposition of an element μ𝔥* as a sum of μJ𝔥J* and μJ𝔥J.

Let J be a subset of the set of simple roots {α1,,αn} and use notations as in (3.16-3.18). Then, for λ,μP+,

Kλμ(t)= coefficient ofsλ in ( α(R+\RJ+) 11-tRα ) λJPJ+ KλJμJ(t) sλJ+μJ,

where KλJμJ(t) are Kostka-Foulkes polynomials for the root system RJ.


By the third equation in the proof of Theorem 3.4(b), Kλμ(t) is the coefficient of aλ+ρ in

wW (-1)(w)w ( αR+ (11-txα) xμ+ρ ) = 1WJ wW (-1)(w)w vWJ (-1)(v)v ( ( α(R+\RJ+) 11-txα ) ( αRJ+ 11-txα ) xμJ+ρJ xμJ+ρJ ) = 1WJ wW (-1)(w)w ( ( α(R+\RJ+) 11-txα ) xμJ+ρJ vWJ (-1)(v)v ( αRJ+ 11-txα ) xμJ+ρJ ) = 1WJ wW (-1)(w)w ( ( α(R+\RJ+) 11-txα ) xμJ+ρJ λJPJ+ KλJμJ (t) aαJ+ρJ ) ,

where the last equality follows from Theorem 3.4(b) applied to the root system RJ. Expanding aλJ+ρJ gives that Kλμ(t) is the coefficient of aλ+ρ in

1WJ λJPJ+ KλJμJ (t)wW (-1)(w)w ( ( α(R+\RJ+) 11-txα ) xμJ+ρJ vWJ (-1)(v)v xλJ+ρJ ) = 1WJ λJPJ+ KλJμJ (t)wW (-1)(w)w vWJ (-1)(v)v ( ( α(R+\RJ+) 11-txα ) xμJ+ρJ+λJ+ρJ ) = λJPJ+ KλJμJ (t)wW (-1)(w)w ( ( α(R+\RJ+) 11-txα ) xλJ+μJ+ρ )

from which the desired formula follows by dividing by aρ and converting to raising operators (as in the proof of Theorem 3.4(b) above).


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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