论文标题
非对称麦克唐纳超级分析
Nonsymmetric Macdonald Superpolynomials
论文作者
论文摘要
A型Hecke代数在反交换变量中的多项式空间上有表示。 Luque和作者[Sém。洛萨。组合。 66（2012），艺术。 B66b，68页，Arxiv：1106.0875]构建的非对称MacDonald多项式在Hecke代数的任意模块中采用值。在本文中，将两个思想结合在一起，以定义和研究非对称的麦克唐纳多项式，以上述反向交易多项式中的值，换句话说，是超多个国家的多项式。模块，它们的正交碱基及其属性首先得出。就标准的Young Tableau表示形式而言，这些模块对应于Hook Tableaux。介绍了Dunkl-Luque理论和特定应用的细节。多项式上有一个内部产物，麦克唐纳多项式是相互正交的。确定该产品的平方规范。通过使用贝克和福雷斯特的技术[Ann。梳子。 3（1999），159-170，arxiv：q-alg/9707001]对称麦克唐纳多项式的对称性是由非对称理论建立的。在这里，“对称”是指在Hecke代数意义上，而不是古典群体的意义。对于最小对称多项式的平方规范，有一个简洁的公式，以及抗对称多项式的一些公式。对于对称和抗对称多项式，当在特殊点评估多项式时，都会分解。
There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [Sém. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875] constructed nonsymmetric Macdonald polynomials taking values in arbitrary modules of the Hecke algebra. In this paper the two ideas are combined to define and study nonsymmetric Macdonald polynomials taking values in the aforementioned anti-commuting polynomials, in other words, superpolynomials. The modules, their orthogonal bases and their properties are first derived. In terms of the standard Young tableau approach to representations these modules correspond to hook tableaux. The details of the Dunkl-Luque theory and the particular application are presented. There is an inner product on the polynomials for which the Macdonald polynomials are mutually orthogonal. The squared norms for this product are determined. By using techniques of Baker and Forrester [Ann. Comb. 3 (1999), 159-170, arXiv:q-alg/9707001] symmetric Macdonald polynomials are built up from the nonsymmetric theory. Here "symmetric" means in the Hecke algebra sense, not in the classical group sense. There is a concise formula for the squared norm of the minimal symmetric polynomial, and some formulas for anti-symmetric polynomials. For both symmetric and anti-symmetric polynomials there is a factorization when the polynomials are evaluated at special points.
