Alternating Sign Matrices, Schubert and

Grothendieck Polynomials

Alain Lascoux

CNRS-Universit Marne la Vall e, France

Center for Combinatorics, Nankai University, China

Alain.Lascoux@univ-mlv.fr

Abstract. Full Text PDF

Alternating sign matrices (ASM) are matrices with entries which generalize permutations matrices. They have led to interesting enumeration properties, formulated in conjectures which attracted many combinatorialists since the work of Mills, Robins and Rumsey.

They are in bijection with ice configurations, and also with a variety of Young tableaux. This last interpretation shows that they are the vertices of a distributive lattice into which embeds the symmetric group. In my opinion, this is their most fundamental property, because it establishes a connection with geometry (flag manifolds and Schubert varieties), with the work of S.S.Chern (characteristic classes), with the work of C.N. Yang (Yang-Baxter equation) and with the description of the Ehresmann-Bruhat order on the symmetric group.

Motivated by geometry, but in a pure combinatorial manner, one can decompose the set of ASM into cells, each of which contains a unique permutation matrix. Taking now very elementary weights on the ASM (just the positions of entries, and the positions of some zeroes), one gets from each cell a polynomial which happens to be the Grothendieck polynomial associated to a Schubert variety. With a little variation on the weights, one obtains instead a Schubert polynomial.

These last polynomials are a non-symmetric generalization of Schur functions, but their most elementary definition is that they are the universal coefficients in the discrete version of the Taylor formula, when using Newton’s divided differences instead of derivatives.