On Poincare polynomials of shuffle algebra representations

TL;DR

This paper investigates the combinatorial structure of a two-parameter shuffle algebra, deriving its Hilbert series under various conditions, thus advancing understanding of its algebraic and representation-theoretic properties.

Contribution

It provides explicit Hilbert series formulas for the shuffle algebra with two parameters, including special cases where parameters satisfy specific relations.

Findings

Hilbert series for the algebra in generic cases

Hilbert series when parameters satisfy $q_1^a q_2^b=1$

Enhanced understanding of algebraic structure and combinatorial properties

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 $a, b \in \mathbb{N}$.