Coincident root loci and Jack and Macdonald polynomials for special values of the parameters
Coincident root loci and Jack and Macdonald polynomials for special values of the parameters
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参数特殊值的重合根轨迹以及 Jack 和 Macdonald 多项式
DOI：
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发表时间：
2004
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通讯作者：
A. Veselov
中科院分区：
文献类型：
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作者：
M. Kasatani;T. Miwa;A. Sergeev;A. Veselov
本文研究了由至少有两个二重根的多项式构成的重合根轨迹，并给出了对称多项式代数中相应理想的一个线性基，该线性基是由具有特殊参数α =-2.作为推论，我们给出了这个理想的Hilbert-Poincar`e级数的一个显式公式，以及作为特殊Jack多项式的最小次数的生成元。
对对称多项式在双移位对角线上为零和Macdonald多项式在t^2 q = 1时为零的情形也作了推广。我们也给出了类似的结果插值杰克多项式。
We consider the coincident root loci consisting of the polynomials with at least two double roots andpresent a linear basis of the corresponding ideal in the algebra of symmetric polynomials in terms of the Jack polynomials with special value of parameter $alpha = -2.$ As a corollary we present an explicit formula for the Hilbert-Poincar`e series of this ideal and the generator of the minimal degree as a special Jack polynomial.
A generalization to the case of the symmetric polynomials vanishing on the double shifted diagonals and the Macdonald polynomials specialized at $t^2 q = 1$ is also presented. We also give similar results for the interpolation Jack polynomials.