出现在SIE网格表示中的正交多项式
Orthogonal Polynomials appearing in SIE grid representations
摘要：
我们在这篇文章中显示如何正交多项式出现在某些表示的网格形箭图。在简短介绍了箭图的一般概念和它们的线性算子表示之后，我们定义了SIE箭图表示的概念：所有从箭图中的电路产生的内在自同态都充当标量乘子。然后，我们提出了几个引理，以确保这种SIE属性的一个ESTA表示。梯形换位子条件和某些图的可交换性“上至标量倍数”起着核心作用。然后将该理论应用于三个例子。大量的计算表明，相关的Laguerre，Legendre-Gegenbauer多项式和二项分布适合的网格形状的SIE箭图的框架。可以看出，这种代数观点是正交多项式和特殊函数的基础。
英文摘要：
We show in this article how orthogonal polynomials appear in certain representations of grid shaped quivers. After a short introduction into the general notion of quivers and their representations by linear operators we define the notion of an SIE quiver representation: All intrinsic endomorphisms arising from circuits in the quiver act as scalar multipliers. We then present several lemmas that ensure this SIE property of a quiver representation. Ladder commutator conditions and certain diagram commutativity "up to scalar multiples" play a central role. The theory will then be applied to three examples. Extensive calculations shows how Associated Laguerre, Legendre--Gegenbauer polynomials and binomial distributions fit into the framework of grid shaped SIE quivers. One can see, that this algebraic point of view is foundational for orthogonal polynomials and special functions.
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