4 research outputs found
A combinatorial formula for graded multiplicities in excellent filtrations
A filtration of a representation whose successive quotients are isomorphic to
Demazure modules is called an excellent filtration. In this paper we study
graded multiplicities in excellent filtrations of fusion products for the
current algebra . We give a combinatorial formula for the
polynomials encoding these multiplicities in terms of two dimensional lattice
paths. Corollaries to our main theorem include a combinatorial interpretation
of various objects such as the coeffficients of Ramanujan's fifth order mock
theta functions , Kostka polynomials for hook
partitions and quotients of Chebyshev polynomials. We also get a combinatorial
interpretation of the graded multiplicities in a level one flag of a local Weyl
module associated to the simple Lie algebras of type
Macdonald Polynomials and level two Demazure modules for affine
We define a family of symmetric polynomials indexed by a pair of dominant integral weights. The polynomial is the specialized Macdonald polynomial and we prove that is the graded character of a level two Demazure module associated to the affine Lie algebra . Under suitable conditions on (which includes the case when or ) we prove that is Schur positive and give explicit formulae for them in terms of Macdonald polynomials
Macdonald Polynomials and Level Two Demazure Modules for Affine sln+1
We define a family of symmetric polynomials Gν,λ(z1, ...,zn+1, q) indexed by a pair of dominant integral weights for a root system of type An. The polynomial Gν,0(z, q) is the specialized Macdonald polynomial Pν(z, q, 0) and is known to be the graded character of a level one Demazure module associated to the affine Lie algebra sln+1. We prove that G0,λ(z, q) is the graded character of a level two Demazure module for sln+1. Under suitable conditions on (ν, λ) (which apply to the pairs (ν, 0) and (0, λ)) we prove that Gν,λ(z, q) is Schur positive, i.e., it can be written as a linear combination of Schur polynomials with coefficients in Z+[q]. We further prove that Pν(z, q, 0) is a linear combination of elements G0,λ(z, q) with the coefficients being essentially products of q-binomials. Together with a result of K. Naoi, a consequence of our result is an explicit formula for the specialized Macdonald polynomial associated to a non-simply laced Lie algebra as a linear combination of the level one Demazure characters of the non-simply laced algebra