Quantum $α$-determinant cyclic modules of $\mathcal{U}_q(\mathfrak{gl}_n)$

Details Quantum $α$-determinant cyclic modules of $\mathcal{U}_q(\mathfrak{gl}_n)$ Quantum $α$-determinant cyclic modules of $\mathcal{U}_q(\mathfrak{gl}_n)$

2006-06-06

arxiv.org/abs/math/0606123v2

J. Algebra 313 (2007), 922--956.

Mathematics

Representation Theory

27 pages, 1 figure

Scientific paper

10.1016/j.jalgebra.2006.12.015

As a particular one parameter deformation of the quantum determinant, we introduce a quantum $\alpha$-determinant and study the $\mathcal{U}_q(\mathfrak{gl}_n)$-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 (2005), however, a new distinguished feature arises in our situation.

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