Minor summation formula and a proof of Stanley's open problem

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Abstract

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.

Original languageEnglish
Pages (from-to)211-234
Number of pages24
JournalRamanujan Journal
Volume16
Issue number2
DOIs
Publication statusPublished - Jul 1 2008

Keywords

  • Determinants
  • Minor summation formula of Pfaffians
  • Pfaffians
  • Schur functions

ASJC Scopus subject areas

  • Algebra and Number Theory

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