Quantum Alpha-Determinants and q-Deformed Hypergeometric Polynomials

The quantum α-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 α-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 not a root of the corresponding polynomial. These polynomials are given as a q-deformation of the hypergeometric polynomials. This is a quantum analog of the result obtained in our previous work [Kimoto, K., S. Matsumoto, and M. Wakayama. “Alpha-determinant cyclic modules and Jacobi polynomials.” Transactions of the American Mathematical Society (forthcoming)].