α-行列式循环模与雅可比多项式
Alpha-determinant cyclic modules and Jacobi polynomials
摘要：
研究了由$α$ -行列式的$l$次幂生成的循环$U(\mathfrak{gl}_n)$ -模.当$l$是一个非负整数时，对于$alpha$的有限例外值，证明了这个循环模同构于$\mathbb{C}^n$的对称$l$次张量空间的$n$次张量空间$(S^l(\mathbb{C}^n))^{\otimes n}$.如果$alpha$是例外的，那么模的结构会发生剧烈的变化，也就是说，一些作为$(S^l(\mathbb{C}^n))^{\otimes n}$分解的不可约分量的不可约表示在循环模的分解中消失了。循环模的每个同构分量的退化由一个矩阵来描述，该矩阵的大小由Kostka数给出，并且条目是$alpha$中具有有理系数的多项式。作为一种特殊情况，我们确定的矩阵在一个完整的细节的情况下，$n=2$，矩阵成为一个标量，基本上是由经典的雅可比多项式。此外，我们证明了这些多项式是酉的。
英文摘要：
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 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.
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