Quantum α-determinant cyclic modules of Uq (gln)

JOURNAL OF ALGEBRA

卷:313

Article

Kimoto, Kazufumi ; Wakayama, Masato

关键词: alpha-determinant;    quantum group;    Iwahori-Hecke algebra;    q-Young symmetrizer;    cyclic module;    irreducible decomposition;    elementary divisors;    content polynomial;    Kostka number;    partition function;   

DOI  :  10.1016/j.jalgebra.2006.12.015

来源: Elsevier

【 摘 要 】

As a particular one parameter deformation of the quantum determinant, we introduce a quantum alpha-determinant det(q)((alpha)) and study the Uq(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 [S. Matsumoto, M. Wakayama, Alpha-determinant cyclic modules of gl(n) (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 alpha-permanent per(q)((alpha)) and establish another conjecture which describes a 'reciprocity' between the multiplicities of the irreducible summands of the cyclic modules generated respectively by det(q)((alpha)) and per(q)((alpha)). (c) 2007 Elsevier Inc. All rights reserved.

【 授权许可】

Free