TAILIEUCHUNG - Báo cáo toán học: "Around the Razumov–Stroganov conjecture: proof of a multi-parameter sum rule"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Around the Razumov–Stroganov conjecture: proof of a multi-parameter sum rule. | Around the Razumov-Stroganov conjecture proof of a multi-parameter sum rule P. Di Francesco Service de Physique Theorique de Saclay CEA DSM SPhT URA 2306 du CNRS F-91191 Gif sur Yvette Cedex France P. Zinn-Justin LIFR-MIIP Independent University 119002 Bolshoy Vlasyevskiy Pereulok 11 Moscow Russia and Laboratoire de Physique Theorique et Modèles Statistiques UMR 8626 du CNRS Universite Paris-Sud Bâtiment 100 F-91405 Orsay Cedex France Submitted Nov 9 2004 Accepted Dec 21 2004 Published Jan 11 2005 Mathematics Subject Classification Primary 05A19 Secondary 52C20 82B20 Abstract We prove that the sum of entries of the suitably normalized groundstate vector of the 0 1 loop model with periodic boundary conditions on a periodic strip of size 2n is equal to the total number of n X n alternating sign matrices. This is done by identifying the state sum of a multi-parameter inhomogeneous version of the 0 1 model with the partition function of the inhomogeneous six-vertex model on a n X n square grid with domain wall boundary conditions. 1. Introduction Alternating Sign Matrices ASM . matrices with entries 0 1 1 such that 1 and 1 s alternate along each row and column possibly separated by arbitrarily many 0 s and such that row and column sums are all 1 have attracted much attention over the years and seem to be a Leitmotiv of modern combinatorics hidden in many apparently unrelated problems involving among others various types of plane partitions or the rhombus tilings of domains of the plane see the beautiful book by Bressoud 1 and references therein . The intrusion first of physics and then of physicists in the subject was due to the fundamental remark that the ASM of size n X n may be identified with configurations of the six-vertex model that consist of putting arrows on the edges of a n X n square grid subject to the ice rule there are exactly two incoming and two outgoing arrows at each vertex of the grid with so-called domain wall boundary conditions.

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