Alpha-determinant cyclic modules and Jacobi polynomials

TL;DR

This paper investigates the structure of cyclic modules generated by the alpha-determinant power, revealing how their decomposition varies with alpha and connecting special cases to Jacobi polynomials, including their unitarity.

Contribution

It characterizes the module structure for generic and exceptional alpha values, linking degenerations to Kostka polynomials and Jacobi polynomials, and provides explicit matrices for the case n=2.

Findings

For generic alpha, the cyclic module is isomorphic to a tensor space.

Exceptional alpha values cause drastic changes in module decomposition.

The special case n=2 relates the module to classical Jacobi polynomials, which are shown to be unitary.

Abstract

We study the cyclic $U(\mathfrak{gl}_n)$-module generated by the $l$-th power of the $\alpha$-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 the cyclic module. The degeneration of each isotypic component of the cyclic module is described by a matrix whose size is given by a Kostka number and entries are polynomials in $alpha$ with rational coefficients. As a special case, we determine the matrix in a full of the detail for the case where $n=2$; the matrix becomes a scalar and is essentially given by the classical Jacobi polynomial. Moreover, we prove that these polynomials are unitary.