论文标题
非对称插孔多项式的超级物质版本
A Superpolynomial Version of Nonsymmetric Jack Polynomials
论文作者
论文摘要
超级分类式由通勤和反交易变量组成。通过将反交换变量视为对称组的模块，可以将矢量值的非对称插孔多项式理论专门针对超级单位。该理论与Desrosiers，Mathieu和Lapointe的几篇论文中引入和研究的超对称插孔多项式有显着不同（Nucl。Phys。B606，2001）。矢量值插孔多项式出现在理性Cherednik代数的标准模块中，并由Griffeth（T.A.M.S. 362，2010）起源于复杂反射组的家族G（N，P，N）。在当前情况下，反交换多项式的正交基础与对称组的Young表示相对应。然后，该基础用于构建非对称插孔多项式，通过专门修复Luque和作者的纸张中设置的机械（Sigma 7,2011）。这些多项式形成正交的基础，有一个内部产物，并明确找到平方规范。获得超对称多项式作为子模块中包含的非对称插孔多项式的线性组合。这是基于贝克和福雷斯特的想法（Ann。Comb。3，1999）。获得了按程度分级的超对称多项式的Poincaré序列，并根据某些最小多项式来解释。简短地讨论了反对称多项式，并应用了圆上Calogero-Moser量子模型的波形。
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.
