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A large dihedral symmetry of the set of alternating sign matrices. (English) Zbl 0956.05015
Electron. J. Comb. 7, No. 1, Research paper R37, 13 p. (2000); printed version J. Comb. 7, No. 2 (2000).
There is a well-known bijection between alternating sign matrices and 6-vertex models which are directed graphs on a square lattice with in-degree and out-degree two at each vertex and prescribed directions for the edges that do not connect to a vertex within the square. These, in turn, are in bijection with undirected graphs on a square lattice with 2-colored edges: We alternately color vertices black or white and then color blue each directed edge out of a black vertex, green each directed edge out of a white vertex. Since each interior vertex has two incident blue edges and two incident green edges, the blue edges define paths connecting black vertices along the boundary. Let \(\pi_B\) be the set of pairs of black boundary vertices joined by blue paths, \(\pi_G\) the set of pairs of white boundary vertices joined by green paths, and let \(\ell\) be the total number of blue or green cycles. Let \(A_n(\pi_B,\pi_G,\ell)\) be the number of alternating sign matrices with these pairings and total number of cycles. Let \(\pi'_B\) be the pairing obtained by replacing each black boundary vertex by the next black boundary vertex as we travel clockwise around the boundary, \(\pi'_G\) the pairing obtained by replacing each white vertex by the next white boundary vertex in the counter-clockwise direction. The author proves by direct bijection that \( A_n(\pi_B,\pi_G,\ell) = A_n(\pi'_B,\pi'_G,\ell)\).

MSC:
05A19 Combinatorial identities, bijective combinatorics
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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