Enumeration of Symmetry Classes of Alternating Sign Matrices and Characters of Classical Groups
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 determinanat 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).
 Publication:

arXiv Mathematics eprints
 Pub Date:
 August 2004
 arXiv:
 arXiv:math/0408234
 Bibcode:
 2004math......8234O
 Keywords:

 Mathematics  Combinatorics;
 05A15 (primary);
 05E05;
 15A15 (secondary)
 EPrint:
 21 pages