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JOURNAL OF
ALGEBRAIC
COMBINATORICS

  Editors-in-chief: C. A. Athanasiadis, T. Lam, A. Munemasa, H. Van Maldeghem
ISSN 0925-9899 (print) • ISSN 1572-9192 (electronic)
 

Enumeration of symmetry classes of alternating sign matrices and characters of classical groups

Soichi Okada
Nagoya University Graduate School of Mathematics Furo-cho, Chikusa-ku Nagoya 464-8602 Japan Furo-cho, Chikusa-ku Nagoya 464-8602 Japan

DOI: 10.1007/s10801-006-6028-3

Abstract

An alternating sign matrix is a square matrix with entries 1, 0 and  - 1 such that the sum of the entries in each row and each column is equal to 1 and the nonzero entries alternate in sign along each row and each column. To some of the symmetry classes of alternating sign matrices and their variations, G. Kuperberg associate square ice models with appropriate boundary conditions, and give determinant and Pfaffian formulae for the partition functions. In this paper, we utilize several determinant and Pfaffian identities to evaluate Kuperberg's determinants and Pfaffians, and express the round partition functions in terms of irreducible characters of classical groups. In particular, we settle a conjecture on the number of vertically and horizontally symmetric alternating sign matrices (VHSASMs).

Pages: 43–69

Keywords: keywords alternating sign matrix; classical group character; determinant; Pfaffian

Full Text: PDF

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