The Alternating Sign Matrix Polytope

The Birkhoff (permutation) polytope, Bn, consists of the n × n nonnegative doubly stochastic matrices, has dimension (n− 1)2, and has n2 facets. A new analogue, the alternating sign matrix polytope, ASMn, is introduced and characterized. Its vertices are the Qn−1 j=0 (3j+1)! (n+j)! n × n alternating sign matrices. It has dimension (n− 1)2, has 4[(n− 2)2 +1] facets, and has a simple inequality description. Its face lattice and projection to the permutohedron are also described. Résumé. Le polytope Bn de permutation (aussi dit de Birkhoff) consiste en les matrices double stochastiques n × n non négatives. Il est de dimension (n − 1)2, et a n2 facettes. Un nouvel analogue, le polytope des matrices à signe alternant, ASMn, est présenté et caractérisé. Ses sommets sont les Qn−1 j=0 (3j+1)! (n+j)! matrices n×n à signe alternant. Il est de dimension (n− 1)2, a 4[(n− 2)2 +1] facettes, et est décrit par une inégalité simple. Le treillis de ses faces et sa projection sur le permutoèdre sont également décrits. 1. Background and Summary The Birkhoff (permutation) polytope Bn is defined as the convex hull of n-by-n permutation matrices. Its dimension is (n− 1), it has n! vertices, and has n facets (each facet is made up of all doubly stochastic matrices with a 0 in a specified entry) [9]. Many analogous polytopes have been studied which are subsets of Bn. In contrast, the alternating sign matrix polytope ASMn is formed by taking the convex hull of n-by-n alternating sign matrices, which is a set of matrices containing the permutations. Thus Bn is contained in ASMn. Definition 1.1. Alternating sign matrices (ASMs) are square matrices with the following properties [7]: • entries ∈ {0, 1,−1} • the entries in each row and column sum to 1 • nonzero entries in each row and column alternate in sign Permutation matrices, then, are the alternating sign matrices whose entries are nonnegative. The connection between these two sets of matrices, though, is much deeper. There exists a partial ordering on alternating sign matrices that is a distributive lattice. This lattice contains as a subposet the Bruhat order on the symmetric group, and in fact, it is the smallest lattice that does so (i.e. it is the MacNeille completion of the Bruhat order) [5]. Given this close relationship between permutations and ASMs it is natural to hope for something relating their polytopes. The dimension of ASMn is (n − 1) because the last entry in each row and column must be precisely what is needed to make that row or column sum equal 1. ASMn has 4[(n− 2) + 1] facets and its vertices are the alternating sign matrices (proofs in section 3), whose count is given by [4]: