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Macdonald Polynomials and Demazure Characters

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Introduction

We will here discuss the connection between nonsymmetric Macdonald polynomials and the characters of Demazure modules for ^sl(n) as given in [3]. We assume a familiarity with affine (untwisted) Lie algebras, specifically ^sl(n), but will give all necessary facts about Demazure modules and nonsymmetric Macdonald polynomials here.

Nonsymmetric Macdonald Polynomials

Recall that nonsymmetric Macdonald polynomials Eλ(z1,zn,q,t) are indexed by compositions λNn and that they form a basis of C(q,t)[z1,,zn]. Henceforth we specialize to t=0, and write

Eλ=Eλ(z1,,zn,q,0)

.

We can generate these polynomials recursively via the endomorphisms Φ,H0,H1,,Hn1 acting on the space Z[q,q1][z1,,zn] (note that when we specialize to t=0 we drop from the space C(q,t)[z1,,zn] to Z[q,q1][z1,,zn]). Φ,H1,,Hn1, are defined such that

¯Hi=sizi+11sizizi+11in1

Φf(z1,,zn)=znf1(q1zn,z1,,zn1)

There is an ˉH0 too but we will not discuss it. The recursive rules tell us that after setting E(0n)=1, then

qλ1ΦE(λ1,,λn)=E(λ2,,λn,λ1+1)

qλ1λn+1¯H0Eλ=E(λn1,λ2,,λn1,λ1+1)if λ1>λn1

and otherwise qλ1λn+1¯H0Eλ=Eλ. Finally,

¯HiEλ=Esiλif λi<λi+1

and otherwise ¯HiEλ=Eλ. These operators should be reminiscent of the action of the Weyl group of ^sl(n) on compositions.

As an example suppose that for n=3 we want to generate E(1,2,1). Then we could apply the composition ¯H2Φ4 to E(0,0,0) to get

Φ(E(0,0,0))=E(0,0,1)=z3,

Φ(E(0,0,1))=E(0,1,1)=z2z3,

Φ(E(0,1,1))=E(1,1,1)=z1z2z3,

Φ(E(1,1,1))=E(1,1,2)=z1z2z23,

¯H2(E(1,1,2))=E(1,2,1)=z1z22z3+z1z2z23

Demazure Modules

In this section we let g be a Kac-Moody algebra associated with Cartan datum (h,Π,Π,P,P). We closely follow chapter 2 of [1]. Recall that a g-module V is a weight module if it admits a weight space decomposition:

V=μhVμ

where

Vμ={vV|hv=μ(h)vfor allhh}

A vector vVμ is called a weight vector of weight μ if eiv=0 for all iI, v is called a maximal weight vector. The dimension dimVμ is called the weight multiplicity of μ. When dimVμ< for all μ, the

character of V is defined to be

chV=μdimVμeμ

where eμ are formal basis elements of the group algebra F[h] with multiplication eλeμ=eλ+μ. We call a g-module V a highest weight module of highest weight λh if there exists a nonzero vector vλV such that

eivλ=0for all iI,

hvλ=λ(h)vfor all hh,

V=U(g)vλ( or Uvλ=V),

where we here use the decomposition U(g)UU0U+ of the universal enveloping algebra of g. An element Λh is a dominant integral weight if Λ belongs to the set,

P+={ΛP|λ(hi)Z0for all iI}

The irreducible highest weight g-modules V(Λ) where Λ is a dominant integral weight have the special property that the Chevalley generaters ei and fi are locally nilpotent on V(Λ). This allows us to construct a well-defined automorphism

τi=(expfi)(exp(ei))(expfi)

where the action of τi on weight spaces is given by

τiVλ=Vsiλfor all iI,λwt(V)

here si denotes the generator of the Weyl group associated with g with index i.

If we still assume that Λ is a dominant integral weight, V=V(Λ) the unique irreducible highest weight ^sl(n)-module with highest weight Λ, then the weight space Vw(Λ) of weight w(Λ) generates a U+(^sl(n)-module, Ew(Λ) which is called a Demazure module. Note that Demazure modules are finite dimensional, and also that they form a filtration on V(Λ) which is compatible with the Bruhat order on W:

wwEw(Λ)Ew(Λ)

We can also define Demazure operators that act on the group ring of the weight lattice P:

Δi=1eαisi1eαi

where si is the simple reflection in the Weyl group with respect to simple root αi. To wW with reduced decomposition w=si1si2sij we can then associate the Demazure operator

Δw=Δi1Δi2Δij

,

There is a nice connection between characters and Demazure operators given by the formula [2]:

χ(Ew(Λ))=Δw(eΛ).

The Connection

We let Λ0,Λ1,,Λn1 be the n-fundamental weights of ^sl(n). Recall that these Λi are defined such that (Λi,αj)=δij. Finally,

δ=n1i=0αi.

For the connection between Eλ and characters of Demazure modules we want to relate the action of ¯Hi and Φ to operators on P. More specifically, we would like a commutative diagram

Demazure Diagrams II.jpg

We can get this by defining π:Z[q,q1][z1,,zn]P on generators by

π(zi)=eΛi1Λi,π(zn)=eΛn1Λ0,π(q)=eδ.

(note that this definition is slightly different to that found in the paper). We get a similar commutative diagram for Φ:

Demazure Diagrams.jpg

The main result of [3] is then that through the homomorphism π, we can identify

qu(λ)+u(ηλ)Eλwithχ(Ew(Λi))

where u(λ) and ηλ (this is a partition) depend only on λ and i=|λ|modn and where w is an specific affine Weyl group element defined such that w acts on η|λ| to give λ.

References

  1. J. Hong and S.J. Kang. Introduction to quantum groups and crystal bases, volume 42 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002.
  2. S. Kumar, Demazure character formula in arbitrary Kac-Moody setting, Invent. Math. 89 (1987), 395-423.
  3. Y. Sanderson, On the connection between Macdonald polynomials and Demazure characters, J. Algebraic Combin. 11 (2000), no.3, 269-275.

Macdonald Polynomials and Demazure Characters is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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