Minor summation formula and a proof of Stanley’s open problem
Minor summation formula and a proof of Stanley’s open problem
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DOI：
10.1007/s11139-007-9106-9
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发表时间：
2004-08
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通讯作者：
M. Ishikawa
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作者：
M. Ishikawa
在FPSAC'03的公开问题会议上，R. P. Stanley给出了一个关于Schur函数之和的公开问题。本文的目的是证明这个公开问题。证明包括三个步骤。在第一步中，我们用一个Pfuman来表示和，作为我们的小求和公式的应用（石川和和歌山，Linear Multilinear Algebra 39：285-305，1995）。在第二步中，我们证明了一个类似于柯西型恒等式的Pfiran恒等式，它推广了Sundquist的Pfiran恒等式（J. Algebr. 5：135-148，1996）。然后我们证明了Stanley在第4节中的公开问题。在本文的最后，我们提出了一定的推论，从这个身份涉及的大Schur函数和一些多项式所产生的Macdonald多项式，推广斯坦利的公开问题。
In the open problem session of the FPSAC’03, R.P. Stanley gave an open problem about a certain sum of the Schur functions. The purpose of this paper is to give a proof of this open problem. The proof consists of three steps. At the first step we express the sum by a Pfaffian as an application of our minor summation formula (Ishikawa and Wakayama in Linear Multilinear Algebra 39:285–305, 1995). In the second step we prove a Pfaffian analogue of a Cauchy type identity which generalizes Sundquist’s Pfaffian identities (J. Algebr. Comb. 5:135–148, 1996). Then we give a proof of Stanley’s open problem in Sect. 4. At the end of this paper we present certain corollaries obtained from this identity involving the Big Schur functions and some polynomials arising from the Macdonald polynomials, which generalize Stanley’s open problem.