Symmetry Classes of Alternating Sign Matrices
By D. P. Robbins. 2000
80 citations recorded by the source corpus.
An alternating sign matrix is a square matrix satisfying (i) all entries are equal to 1, -1 or 0; (ii) every row and column has sum 1; (iii) in every row and column the non-zero entries alternate in sign. The 8-element group of symmetries of the square acts in an obvious way on square matrices. For any subgroup of the group of symmetries of the square we may consider the subset of matrices invariant under elements of this subgroup. There are 8 conjugacy classes of these subgroups giving rise to 8 symmetry classes of matrices. R. P. Stanley suggested the study of those alternating sign matrices in each of these symmetry classes. We have found evidence suggesting that for six of the symmetry classes there exist simple product formulas for the number of alternating sign matrices in the class. Moreover the factorizations of certain of their generating functions point to rather startling connections between several of the symmetry classes and cyclically symmetric plane partitions.
Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering
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Citations in the corpus
- Enumeration of symmetry classes of alternating sign matrices and characters of classical groups (2004, background): “our point of view, the remaining conjectures (see [ 12 ]) on HTSASMs and DSASMs are reformulated as follows:”
- Symmetry classes of alternating-sign matrices under one roof (2000, methodology): “) UU (4n+4;x) A(2) HT (4n;x,−1)=(−x)nA(1) QT (4n;x)2 A(2) OO (8n;x,−1)=(−x)nA(1) UO (8n;x,1)Ae(1) UO (8n;x,1). Many of the factorizations in Theorem 4 were conjectured experimentally by David Robbins [30]; the formula for AU(2n;x,y) was conjectured by Cohn and Propp [6]. Theorem 5. The generating fu”