Poincaré Series of Quantum Spaces Associated to Hecke Operators

Abstract

We study the Poincaré 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 Poincaré series of the corresponding matrix bialgebra is also considered. Using an old result on Polyá frequency sequence, we show that the Poincaré 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.