Macdonald Polynomials and level two Demazure modules for affine ${\mathfrak{sl}}_{n+1}$

Abstract

We define a family of symmetric polynomials $G_{\nu ,\lambda }(z_1,\cdots ,z_{n+1},q)$ indexed by a pair of dominant integral weights. The polynomial $G_{\nu ,0}(z,q)$ is the specialized Macdonald polynomial and we prove that $G_{0,\lambda }(z,q)$ is the graded character of a level two Demazure module associated to the affine Lie algebra ${\widehat{\mathfrak{sl}}}_{n+1}$. Under suitable conditions on $(\nu ,\lambda )$ (which includes the case when $\nu =0$ or $\lambda =0$) we prove that $G_{\nu ,\lambda }(z,q)$ is Schur positive and give explicit formulae for them in terms of Macdonald polynomials.

Version 2; many typos corrected and proofs streamlined

Cited by in corpus (1)

• A note on the fusion product decomposition of Demazure modules