## Abstract

As a particular one parameter deformation of the quantum determinant, we introduce a quantum α-determinant det_{q}^{(α)} and study the U_{q} (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 α-permanent per_{q}^{(α)} and establish another conjecture which describes a 'reciprocity' between the multiplicities of the irreducible summands of the cyclic modules generated respectively by det_{q}^{(α)} and per_{q}^{(α)}.

Original language | English |
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Pages (from-to) | 922-956 |

Number of pages | 35 |

Journal | Journal of Algebra |

Volume | 313 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jul 15 2007 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

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