Symmetric and alternating functions and their q-analogues

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

Last update: 26 September 2012

Symmetric and alternating functions and their q-analogues

Let 10 and ε0 be the elements of the finite Hecke algebra H which are determined by

102=10 and Ti10= q10, for all1in, ε02=ε0 and Tiε0= (-q-1) ε0, for all1in.

In terms of the basis {TwwW} of H these elements have the explicit formulae

10= 1W0(q2) wW q(w)Tw, andε0= 1W0(q-2) wW (-q)-(w) Tw, (2.1)

where W0(t)= wW t(w). (To define these elements one should adjoin the element W0(q2)-1 to 𝕂 or to H.) The elements 10 and ε0 are q-analogues of the elements in the group algebra of W given by

1=1W wWwand ε=1W wW (-1)(w)w, (2.2)

and the vector spaces 10H10 and ε0H10 are q-analogues of the vector spaces (more precisely, free 𝕂=[q,q-1]-modules) of symmetric functions and alternating functions,

𝕂[P]W = { f𝕂[P] wf=ffor allwW } =1𝕂[P], 𝒜 = { f𝕂[P] wf=(-1)(w) ffor allwW } =ε𝕂[P], (2.3)

respectively, where the action of W on 𝕂[P] is as defined in 1.29.

For μP let the orbit Wμ and the stabilizer Wμ of μ be defined by

Wμ= {wμwW} andWμ= {wWwμ=μ} .

Then define

mμ= γWμ xγ= W Wμ 1xμ, aμ=wW (-1)(w) wxμ=Wε xμ, Mμ=10 xμ10, Aμ=ε0xμ 10. (2.4)

Theorem 2.2 below shows that the elements in (2.4) which are indexed by elements of P+ and P++ form bases (over 𝕂) of 𝕂[P]W, 𝒜, 10H10, and ε0H10. This will be a consequence of the following straightening rules. The straightening law for the Mμ given in the following Proposition is a generalization of [Mac1995, III §2 Ex. 2].

For γP let mγ, aγ, Mγ, and Aγ be as defined in (2.4). Let αi be a simple root and let μP be such that d=μ,αi0. Then

msiμ=mμ, asiμ=-aμ, andAsiμ= -Aμ.

Letting t=q-2, Mμ=Msiμ if d=0, and if d>0 then

Msiμ = tMμ + ( j=1 d/2-1 (t2-1) tj-1 Mμ-jαi ) + { (t-1)td/2-1 Mμ-(d/2)αi , ifdis even, 0 , ifdis odd.

Proof.

Proposition 2.1 implies that, for all μP and wW,

mwμ=mμ, awμ= (-1)(w) aμ,and Awμ (-1)(w) Aμ. (2.5)

Let 𝕂=[q,q-1]. As free 𝕂-modules

𝕂[P]W has basis {mλλP+}, 10H10 has basis {MλλP+}, 𝒜 has basis {aμμP++}, ε0H10 has basis {AμμP++}.

Proof.

For λP define the Schur function, or Weyl character, by

sλ=aλ+ρaρ, whereρ=12 αR+α. (2.6)

The straightening law for aμ in (2.5) implies the following straightening law for the Schur functions. If μP and wW then, by (2.5) and the definition of sμ,

(-1)(w) sμ= (-1)(w) aμ+ρ aρ = aw(μ+ρ)-ρ+ρ aρ =swμ, (2.7)

where wμ=w(μ+ρ)-ρ.

The dot action of the Weyl group W on 𝔥* which is appearing here, wμ=t-ρw tρμ=(tρ-1) wtρμ, is ordinary action of W on 𝔥* except with the "center" shirted to -ρ. For the root system of type C2, in Example 1.1, the picture is

Hα1+α2 Hα1 Hα2 Hα1+2α2 C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C ρ s1ρ s2ρ Hα1+α2 Hα1 Hα2 Hα1+2α2 C 0 s10 s20 -ρ the orbitWρ the orbitW0

The following proposition shows that the Weyl characters sλ are elements of 𝕂[P]W. The equaility in part (a) is the Weyl denominator formula, a generalization of the factorization of the Vandermonde determinant det(xin-j) =1i,jn (xi-xj). In the remainder of this section we shall abuse language and use the term "vector space" in place of "free 𝕂= [q,q-1] module".

Let P+, P++, 𝕂[P]W and 𝒜 be as in (1.7) and (2.4) and let ρ be as in (1.8).

  1. If f𝒜 then f is divisible by aρ and aρ=xρ αR+ (1-x-α)
  2. The set {sλλP+} is a basis of 𝕂[P]W.
  3. The maps P+P++ λλ+ρ and Φ:𝕂[P]W 𝒜 faρf sλaλ+ρ are a bijection and a vector space isomorphism, respectively.

Proof.

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).

page history