As a particular one parameter deformation of the quantum determinant, we introduce a quantum α-determinant detq(α) and study the Uq (gln)-cyclic module generated by it: We show that the multiplicity of each irreducible representation in this cyclic module is determined by a certain polynomial called the q-content discriminant. A part of the present result is a quantum counterpart for the result of Matsumoto and Wakayama [S. Matsumoto, M. Wakayama, Alpha-determinant cyclic modules of gln (C), J. Lie Theory 16 (2006) 393-405], however, a new distinguished feature arises in our situation. Specifically, we determine the degeneration of the multiplicities for 'classical' singular points and give a general conjecture for singular points involving semi-classical and quantum singularities. Moreover, we introduce a quantum α-permanent perq(α) and establish another conjecture which describes a 'reciprocity' between the multiplicities of the irreducible summands of the cyclic modules generated respectively by detq(α) and perq(α).
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory