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Báo cáo toán học: "On the Doubly Refined Enumeration of Alternating Sign Matrices and Totally Symmetric Self-Complementary Plane Partitions Tiago Fonseca"
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Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: On the Doubly Refined Enumeration of Alternating Sign Matrices and Totally Symmetric Self-Complementary Plane Partitions Tiago Fonseca. | On the Doubly Refined Enumeration of Alternating Sign Matrices and Totally Symmetric Self-Complementary Plane Partitions Tiago Fonseca LPTHE CNRS UMR 7589 Univ Pierre et Marie Curie-Paris6 75252 Paris Cedex France fonseca@ lpthe.jussieu.fr Paul Zinn-Justiny LPTMS CNRS UMR 8626 Univ Paris-Sud 91405 Orsay Cedex France and LPTHE CNRS UMR 7589 Univ Pierre et Marie Curie-Paris6 75252 Paris Cedex France pzinn @ lpthe.jussieu.fr Submitted Mar 26 2008 Accepted Jun 5 2008 Published Jun 13 2008 Abstract We prove the equality of doubly refined enumerations of Alternating Sign Matrices and of Totally Symmetric Self-Complementary Plane Partitions using integral formulae originating from certain solutions of the quantum Knizhnik-Zamolodchikov equation. The authors thank N. Kitanine for discussions and J.-B. Zuber for a careful reading of the manuscript. yPZJ 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. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R81 1 Contents 1 Introduction 2 2 The models 3 2.1 Alternating Sign Matrices. 3 2.2 6-Vertex model. 4 2.3 Totally Symmetric Self-Complementary Plane Partitions. 4 2.4 Non-Intersecting Lattice Paths. 6 3 The conjecture 8 3.1 ASM generating function . 8 3.2 NILP generating function. 9 3.3 The conjecture. 10 4 The proof 10 4.1 ASM counting as the partition function of the 6-Vertex model. 10 4.2 Integral formula for refined ASM counting. 13 4.3 Integral formula for refined NILP counting. 16 4.4 Equality of integral formulae . 18 A Formulating the conjecture directly in terms of TSSCPPs 20 A.1 Extending the theorem. 20 A. 2 The conjecture in terms of TSSCPPs . 22 B Properties of the 6-Vertex model partitionfunction 23 B. 1 Korepin recursion relation. 24 B.2 Cubic root of unity case. 26 C The space of polynomials satisfying the wheel condition 27 D An antisymmetrization formula 29 D.1 The general case. 29 D.2 .