Nonsymmetric Askey-Wilson polynomials and Q-polynomial distance-regular graphs
Nonsymmetric Askey-Wilson polynomials and Q-polynomial distance-regular graphs
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DOI：
10.1016/j.jcta.2016.11.006
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发表时间：
2015-09
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影响因子：
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通讯作者：
Jae-Ho Lee
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文献类型：
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作者：
Jae-Ho Lee
在他著名的定理(1982)中，Douglas Leonard从q-多项式距离正则图的对偶性质刻画了Askey方案中的q-racah多项式及其相关项。在这篇文章中，我们考虑了上述情形下q-racah多项式的一个非对称(或Laurent)形式。设Γ表示一个含有Delsarte团C的q-多项式距离正则图。假设Γ有q-raca型。确定一个顶点x∈C.我们根据到x和C的路径长度距离来划分Γ的顶点集.对于(C1Hecke，C1)型泛双仿射Hecke代数HˆQ，在这个划分中对应于单元的特征向量的线性跨度具有不可约的模结构.从这个模中，我们自然地得到了一个有限的正交Laurent多项式序列。我们利用H-ˆQ-模和Leonard系理论证明了这些多项式的正交性。将HˆQ改为HˆQ−1，我们展示了我们的Laurent多项式如何与非对称的Askey-Wilson多项式相联系，从而我们的Laurent多项式可以被看作是非对称的Q-Racah多项式。
Abstract In his famous theorem (1982), Douglas Leonard characterized the q-Racah polynomials and their relatives in the Askey scheme from the duality property of Q-polynomial distance-regular graphs. In this paper we consider a nonsymmetric (or Laurent) version of the q-Racah polynomials in the above situation. Let Γ denote a Q-polynomial distance-regular graph that contains a Delsarte clique C. Assume that Γ has q-Racah type. Fix a vertex x∈ C. We partition the vertex set of Γ according to the path-length distance to both x and C. The linear span of the characteristic vectors corresponding to the cells in this partition has an irreducible module structure for the universal double affine Hecke algebra H ˆ q of type (C 1∨, C 1). From this module, we naturally obtain a finite sequence of orthogonal Laurent polynomials. We prove the orthogonality relations for these polynomials, using the H ˆ q-module and the theory of Leonard systems. Changing H ˆ q by H ˆ q− 1 we show how our Laurent polynomials are related to the nonsymmetric Askey–Wilson polynomials, and therefore how our Laurent polynomials can be viewed as nonsymmetric q-Racah polynomials.