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

By Soichi Okada. 2004

111 citations recorded by the source corpus.

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).

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/198595bd902e3a571db0f24c963bda34d0a24b7e

Citations in the corpus

• Advanced Determinant Calculus: A Complement (2005, unknown): “ −j+i pj−i + p−(j−i) = (−1)(n2) 2n Y 1≤i<j≤n (p2(j−i) − p−2(j−i))2 Y 1≤i,j≤2n+1 2∤i, |j (qpj−i − q−1p−(j−i)) Y 1≤i,j≤n (p j−i+p ( ))(pj+i +p−(j+i)) . (5.12) Subsequent to Kuperberg’s work, Okada [126] related Kuperberg’s determinants and Pfaffians to characters of classical groups, by coming up wi”