International Mathematics Research Notices Advance Access originally published online on June 11, 2009
International Mathematics Research Notices (2009) 2009:4168-4182, doi:10.1093/imrn/rnp083 published on October 27, 2009
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Quantum Alpha-Determinants and q-Deformed Hypergeometric Polynomials
Department of Mathematical Sciences, University of the Ryukyus, 1 Senbaru, Nishihara-cho, Okinawa 903-0213 Japan and Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
Correspondence: Correspondence to be sent to: kimoto{at}math.u-ryukyu.ac.jp
The quantum 
-determinant is defined as a parametric deformation of the quantum determinant. We investigate the cyclic 
-submodules of the quantum matrix algebra 
generated by the powers of the quantum -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 
-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)].