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Symmetry classes of alternating-sign matrices under one roof. (English) Zbl 1010.05014

The author continues his work from [Int. Math. Res. Not. 1996, 139-150 (1996; Zbl 0859.05027)] where he derived Zeilberger’s alternating sign matrix (ASM) theorem from the Izergin-Korepin determinant for a partition function for square ice with domain wall boundary. Here he uses similar techniques to enumerate three symmetry classes of ASMs, namely those which possess a vertical reflection, half turn or quarter turn symmetry. In doing so he confirms conjectures of Mills and Robbins. Following this, he introduces numerous further symmetry classes of ASMs and of matrices obtained by relaxing the definition of ASMs slightly. He obtains enumerations for some of these classes, but leaves a number of questions and conjectures open.

MSC:

05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
15B36 Matrices of integers
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
05A15 Exact enumeration problems, generating functions

Citations:

Zbl 0859.05027
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