非对称Jack多项式的超多项式形式
A Superpolynomial Version of Nonsymmetric Jack Polynomials
摘要：
超多项式由交换变量和反交换变量组成。通过把反交换变量看作是对称群的模，向量值非对称Jack多项式理论可以被特殊化为超多项式。该理论与Desrosiers，Mathieu和Lapointe（Nucl.Phys.B606，2001）在几篇论文中介绍和研究的超对称杰克多项式有很大不同。向量值杰克多项式出现在有理切雷德尼克代数的标准模中，由Griffeth（T.A.M.S. 362，2010）对于复反射群的族G（n，p，N）。在目前的情况下，有一个反交换多项式的正交基，它对应于在对称群的杨氏表示中产生的钩子表。然后，通过专门化Luque和作者（SIGMA 7，2011）的论文中建立的机器，使用该基础来构造非对称Jack多项式。有一个内积，这些多项式形成一个正交基，并明确找到平方范数。超对称多项式是作为包含在一个子模中的非对称杰克多项式的线性组合获得的;这是基于Baker和Forrester的想法（Ann.Comb.3，1999）。得到了阶次超对称多项式的Poincaré级数，并利用某些极小多项式给出了Poincaré级数的解释.有一个简短的讨论反对称多项式和应用的波函数的Calogero-Moser量子模型的循环。
英文摘要：
Superpolynomials consist of commuting and anti-commuting variables. By considering the anti-commuting variables as a module of the symmetric group the theory of vector-valued nonsymmetric Jack polynomials can be specialized to superpolynomials. The theory significantly differs from the supersymmetric Jack polynomials introduced and studied in several papers by Desrosiers, Mathieu and Lapointe (Nucl. Phys. B606, 2001). The vector-valued Jack polynomials arise in standard modules of the rational Cherednik algebra and were originated by Griffeth (T.A.M.S. 362, 2010) for the family G(n,p,N) of complex reflection groups. In the present situation there is an orthogonal basis of anti-commuting polynomials which corresponds to hook tableaux arising in Young's representations of the symmetric group. The basis is then used to construct nonsymmetric Jack polynomials by specializing the machinery set up in a paper by Luque and the author (SIGMA 7,2011). There is an inner product for which these polynomials form an orthogonal basis, and the squared norms are explicitly found. Supersymmetric polynomials are obtained as linear combinations of the nonsymmetric Jack polynomials contained in a submodule; this is based on an idea of Baker and Forrester (Ann. Comb. 3, 1999). The Poincaré series for supersymmetric polynomials graded by degree is obtained and is interpreted in terms of certain minimal polynomials. There is a brief discussion of antisymmetric polynomials and an application to wavefunctions of the Calogero-Moser quantum model on the circle.
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