Minor summation formula and a proof of Stanley’s open problem |
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Authors: | Masao Ishikawa |
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Institution: | (1) Department of Mathematics, Faculty of Education, Tottori University Koyama, Tottori 680 8551, Japan |
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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.
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Keywords: | Schur functions Determinants Pfaffians Minor summation formula of Pfaffians |
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