奇异$α$ -行列式的不变量理论
Invariant theory for singular $α$-determinants
摘要：
从不可约分解的观点来看，当$GL_n$.本文证明了$-\frac1k$ -行列式具有与普通行列式相似的性质。根据这个事实，我们可以定义一个新的（相对）不变量，称为花环行列式。利用Howe意义下的$(GL_m, GL_n)$ -对偶性，得到了一个花环行列式的表达式，它是由相应的常次行列式的某种线性组合表示的，并用适当的矩形图作了标号.我们还研究了类似于范德蒙德行列式的一个花环行列式，并在花环行列式的框架下研究了对称函数如Schur函数.此外，我们检查的系数，我们称之为$(n,k)$ -符号出现在线性表达式的花环行列式的关系的带状球函数的一个年轻的子群的对称群$S_{nk}$。
英文摘要：
From the irreducible decompositions' point of view, the structure of the cyclic $GL_n$-module generated by the $α$-determinant degenerates when $α=\pm \frac1k (1\leq k\leq n-1)$. In this paper, we show that $-\frac1k$-determinant shares similar properties which the ordinary determinant possesses. From this fact, one can define a new (relative) invariant called a wreath determinant. Using $(GL_m, GL_n)$-duality in the sense of Howe, we obtain an expression of a wreath determinant by a certain linear combination of the corresponding ordinary minor determinants labeled by suitable rectangular shape tableaux. Also we study a wreath determinant analogue of the Vandermonde determinant, and then, investigate symmetric functions such as Schur functions in the framework of wreath determinants. Moreover, we examine coefficients which we call $(n,k)$-sign appeared at the linear expression of the wreath determinant in relation with a zonal spherical function of a Young subgroup of the symmetric group $S_{nk}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
