Alpha-determinant cyclic modules and Jacobi polynomials
Alpha-determinant cyclic modules and Jacobi polynomials
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DOI：
10.1090/s0002-9947-09-04860-0
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发表时间：
2007-10
影响因子：
1.3
通讯作者：
Kazufumi Kimoto;Sho Matsumoto;M. Wakayama
中科院分区：
文献类型：
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作者：
Kazufumi Kimoto;Sho Matsumoto;M. Wakayama
对于正整数n和l，研究了由α-行列式det（α）（X）的l次幂生成的循环u（gln）-模.这个循环模同构于n阶张量空间Sl（n）n的对称l阶张量空间的n阶张量空间Sl（n）n，其中除了n个例外值α。如果α是例外的，则循环模等价于Sl（n）n的一个真子模，即循环模中几个不可约子表示的重数小于Sl（n）n中的重数.循环模的每个同构分支的退化由一个矩阵来描述，该矩阵的大小由Kostka数给出，其元素是α中具有有理系数的多项式。特别地，当n = 2时，我们完全确定矩阵。在这种情况下，矩阵变成标量，本质上由经典雅可比多项式给出。此外，我们证明了这些多项式是酉的。在附录中，我们考虑了（Onl，Onl）的球面Fourier变换的一种变形作为分析同样问题的主要工具，并利用Gelfand对（O2 l，O2 l）的带状球面函数描述了n = 2的情况.
For positive integers n and l, we study the cyclic u(gl n )-module generated by the l-th power of the α-determinant det (α) (X). This cyclic module is isomorphic to the n-th tensor space S l (ℂ n ) ⊗n of the symmetric l-th tensor space of ℂ n for all but finitely many exceptional values of α. If α is exceptional, then the cyclic module is equivalent to a proper submodule of S l (ℂ n ) ⊗n , i.e. the multiplicities of several irreducible subrepresentations in the cyclic module are smaller than those in S l (ℂ n ) ⊗n . The degeneration of each isotypic component of the cyclic module is described by a matrix whose size is given by a Kostka number and whose entries are polynomials in α with rational coefficients. In particular, we determine the matrix completely when n = 2. In this case, the matrix becomes a scalar and is essentially given by a classical Jacobi polynomial. Moreover, we prove that these polynomials are unitary. In the Appendix, we consider a variation of the spherical Fourier transformation for (O nl , O n l ) as a main tool for analyzing the same problems, and describe the case where n = 2 by using the zonal spherical functions of the Gelfand pair (O 2l , O 2 l ).