Jucys-Murphy Elements and Unitary Matrix Integrals
Jucys-Murphy Elements and Unitary Matrix Integrals
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DOI：
10.1093/imrn/rnr267
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发表时间：
2009-05
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通讯作者：
Sho Matsumoto;Jonathan Novak
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文献类型：
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作者：
Sho Matsumoto;Jonathan Novak
本文研究酉群上的多项式积分与对称函数在Jucys-Murphy元上的共轭类展开式之间的关系。我们的主要结果是一个显式的单项对称函数的类展开的顶部系数的Jucys-Murphy元素，从我们恢复的多项式积分的一阶渐近$\U（N）$为$N \rightarrow \infty$。我们的结果类扩展包括一个类似的Macdonald的结果的顶部连接系数的类代数，推广的Stanley和Olshanski的结果多项式的内容统计的Plancherel随机分区，和一个确切的公式的多重性类的全周期在一个完整的对称函数的Jucys-Murphy元素的扩展。后者导致一个新的组合解释的Carlitz Riordan中心阶乘数。
In this paper, we study the relationship between polynomial integrals on the unitary group and the conjugacy class expansion of symmetric functions in Jucys-Murphy elements. Our main result is an explicit formula for the top coefficients in the class expansion of monomial symmetric functions in Jucys-Murphy elements, from which we recover the first order asymptotics of polynomial integrals over $\U(N)$ as $N \rightarrow \infty$. Our results on class expansion include an analogue of Macdonald's result for the top connection coefficients of the class algebra, a generalization of Stanley and Olshanski's result on the polynomiality of content statistics on Plancherel-random partitions, and an exact formula for the multiplicity of the class of full cycles in the expansion of a complete symmetric function in Jucys-Murphy elements. The latter leads to a new combinatorial interpretation of the Carlitz-Riordan central factorial numbers.