On Poincare polynomials of shuffle algebra representations

Abstract

In this paper we study some combinatorial properties of the shuffle algebra $S_{q_1,q_2}$ with two complex parameters and, in particular, obtain the Hilbert series for $S_{q_1,q_2}/I$, where $I$ is the ideal generated by $z-\lambda$ for both generic $q_1,q_2$ and satisfying the relation $q_1^a{q}_2^b=1$ with $a,b\in \mathbb{N}$.