Poincar\'e Series of Quantum Spaces Associated to Hecke Operators

TL;DR

This paper investigates the Poincaré series of quantum spaces linked to Hecke operators, revealing their rational nature with specific root and pole characteristics, and establishing bounds on the rank of even Hecke operators.

Contribution

It provides a detailed analysis of the Poincaré series for quantum spaces associated with Hecke operators, including their rational form and root-pole structure, which was not previously known.

Findings

Poincaré series are always rational functions with negative roots and positive poles.

The rank of an even Hecke operator is greater than the dimension of the vector space.

The Poincaré series of the matrix bialgebra associated to these quantum spaces also exhibit rationality.

Abstract

We study the Poincar\'e series of the quantum spaces associated to a Hecke operator, i.e., a Yang-Baxter operator satisfying the equation $(x+1)(x-q)=0$. The Poincar\'e series of the corresponding matrix bialgebra is also considered. Using an old result on Poly\'a frequency sequence, we show that the Poincar\'e series of quantum spaces are always rational functions having negative roots and positive poles. In particular, we show that the rank of an even Hecke operator should be rational functions having negative roots and positive poles. In particular, we show that the rank of an even Hecke operator should be greater than the dimension of the vector space it is acting on.