Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: A conjectured formula for Fully Packed Loop configurations in a triangle. | A conjectured formula for Fully Packed Loop configurations in a triangle Paul Zinn-Justin LPTMS CNRS UMR 8626 Univ Paris-Sud 91405 Orsay Cedex France and LPTHE CNRS UMR 7589 Univ Pierre et Marie Curie-Paris 6 75252 Paris Cedex France pzinn @ Ipthe. jussieu. fr Submitted Jan 15 2010 Accepted Jul 30 2010 Published Aug 9 2010 Mathematics Subject Classification 05A15 Abstract We describe a new conjecture involving Fully Packed Loop counting which relates via the Razumov-Stroganov conjecture recent observations of Thapper to formulae in the Temperley-Lieb model of loops. PZJ was supported by EU Marie Curie Research Training Networks ENRAGE MRTN-CT-2004-005616 ENIGMA MRT-CT-2004-5652 ESF program MISGAM and ANR program GIMP ANR-05-BLAN-0029-01. iThe author wants to thank R. Langer for her participation in the early stages of this project J. Thap-per for sharing his numerical data as well as P. Di Francesco T. Fonseca and . Zuber for discussions. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R107 1 Contents 1 Introduction 3 2 Preliminaries 3 Bijections. 3 Order embedding . 4 Schur functions. 4 The involution . 5 Change of basis. 6 3 Statistical models of loops and Razumov-Stroganov conjecture 7 Counting of FPLs. 7 The Temperley-Lieb l loop model. 9 The Razumov-Stroganov conjecture. 10 4 The formula 10 Some properties . 11 Connection to FPLs. 12 Summation . 12 Special cases . 13 5 Analogues of Thapper s conjectures 14 The action of An. 14 Matrix identities. 15 The involution . 16 Recurrence . 17 Largest component. 18 The A s from the T s. 19 6 The T-generalization 19 A Existence of the matrix K 21 B The matrices P Pext 22 C Example of ground state entry of the Temperley-Lieb loop model 22 D Example of matrices A and C 23 THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R107 2 1 Introduction In the literature generated by the seminal papers 1 19 and revolving around the so-called Razumov-Stroganov .
