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 Uq(sl2)-submodules of the quantum matrix algebra Aq(Mat2) 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 Uq(sl2)-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.].