Alpha-determinant cyclic modules of $gl_n(C)$

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

As a particular one parameter deformation of the quantum determinant, we introduce a quantum $α$-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.

Quantum alpha-determinants and q-deformed hypergeometric polynomials

The quantum $α$-determinant is defined as a parametric deformation of the quantum determinant. We investigate the cyclic $\mathcal{U}_q(\mathfrak{sl}_2)$-submodules of the quantum matrix algebra $\mathcal{A}_q(\mathrm{Mat}_2)$ generated by the powers of the quantum $α$-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 $\mathcal{U}_q(\mathfrak{sl}_2)$-module, (ii) the cyclic module contains an irreducible submodule if the parameter is…

Alpha-determinant cyclic modules and Jacobi polynomials

We study the cyclic $U(\mathfrak{gl}_n)$-module generated by the $l$-th power of the $α$-determinant. When $l$ is a non-negative integer, for all but finite exceptional values of $alpha$, one shows that this cyclic module is isomorphic to the $n$-th tensor space $(S^l(\mathbb{C}^n))^{\otimes n}$ of the symmetric $l$-th tensor space of $\mathbb{C}^n$. If $alpha$ is exceptional, then the structure of the module changes drastically, i.e. some irreducible representations which are the irreducible components of the decomposition of $(S^l(\mathbb{C}^n))^{\otimes n}$ disappear in the decomposition of…

单项对称多项式的主特殊化与循环群的群行列式

Principal Specialization of Monomial Symmetric Polynomials and Group Determinants of Cyclic Groups

研究单项对称多项式在单位根处的主特殊化，给出显式公式，并证明这些特殊值自然出现在循环群行列式的k次幂展开中，推广了Ore公式，同时确定了循环群群积和k次幂的项数。

Averages of alpha-determinants over permutations

We show that certain weighted average of the alpha-determinant of a $kn$ by $kn$ matrix of the form $A\otimes1_{1,k}$, the Kronecker product of a $kn$ by $n$ matrix $A$ and $1$ by $k$ all one matrix $1_{1,k}$, over permutations of $kn$ letters is reduced to the $k$-wreath determinant of $A$ up to constant. The constant is exactly given by the modified content polynomial for the Young diagram $(k^n)$. As a corollary, we give a `determinantal' formula for certain functions on the symmetric groups which are invariant under the left and right translation by a Young subgroup, especially the values …

Integer group determinants for ${\rm C}_{2}^{4}$

We determine all possible values of the integer group determinant of ${\rm C}_{2}^{4}$, where ${\rm C}_{2}$ is the cyclic group of order $2$.