Quantum alpha-determinants and q-deformed hypergeometric polynomials

TL;DR

This paper explores quantum alpha-determinants within quantum matrix algebras, revealing their structure through q-deformed hypergeometric polynomials and identifying conditions for irreducible submodules based on polynomial roots.

Contribution

It introduces a new framework linking quantum alpha-determinants to q-deformed hypergeometric polynomials, extending classical results to the quantum setting.

Findings

Polynomials describe irreducible decomposition of cyclic modules

Irreducible submodules correspond to roots of these polynomials

Polynomials are q-deformations of classical hypergeometric polynomials

Abstract

The quantum $\alpha$-determinant is defined as a parametric deformation of the quantum determinant. We investigate the cyclic $\mathcal{U}_q(\mathfrak{sl}_2)$-submodules of the quantum matrix algebra $\mathcal{A}_q(\mathrm{Mat}_2)$ generated by the powers of the quantum $\alpha$-determinant. For such a cyclic module, there exists a collection of polynomials which describe the irreducible decomposition of it in the following manner: (i) each polynomial corresponds to a certain irreducible $\mathcal{U}_q(\mathfrak{sl}_2)$-module, (ii) the cyclic module contains an irreducible submodule if the parameter is a root of the corresponding polynomial. These polynomials are given as a $q$-deformation of the hypergeometric polynomials. This is a quantum analogue of the result obtained in our previous work [K. Kimoto, S. Matsumoto and M. Wakayama, Alpha-determinant cyclic modules and Jacobi polynomials, to appear in Trans. Amer. Math. Soc.].