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Journal of Algebra


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.


Published in Journal of Algebra, Volume 575, 2021, pp. 159-191, by Elsevier. Available via doi: 10.1016/j.jalgebra.2021.01.036.

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