Doubly-refined enumeration of Alternating Sign Matrices and determinants of 2-staircase Schur functions

Philippe Biane (1) , Andrea Sportiello (2) , Luigi Cantini (3)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 LIPN - Laboratoire d'Informatique de Paris-Nord
3 LPTM - UMR 8089 - Laboratoire de Physique Théorique et Modélisation

We prove a determinantal identity concerning Schur functions for 2-staircase diagrams lambda=(ln+l',ln,l(n-1)+l',l(n-1),...,l+l',l,l',0). When l=1 and l'=0 these functions are related to the partition function of the 6-vertex model at the combinatorial point and hence to enumerations of Alternating Sign Matrices. A consequence of our result is an identity concerning the doubly-refined enumerations of Alternating Sign Matrices.

Mots clés: Alternating sign matrices; Schur functions; compound determinants

Domaines: Théorie des représentations [math.RT]

Philippe Biane, Andrea Sportiello, Luigi Cantini. Doubly-refined enumeration of Alternating Sign Matrices and determinants of 2-staircase Schur functions. Seminaire Lotharingien de Combinatoire, 2012, 65 (B65f), pp.1-25. ⟨hal-00822921⟩