TAILIEUCHUNG - Báo cáo toán học: "ombinatorial Constructions of Weight Bases: The Gelfand-Tsetlin Basis"

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: Combinatorial Constructions of Weight Bases: The Gelfand-Tsetlin Basis. | Combinatorial Constructions of Weight Bases The Gelfand-Tsetlin Basis Patricia Hersh1 Department of Mathematics North Carolina State University Raleigh NC 27695 United States plhersh@ Cristian Lenart2 Department of Mathematics and Statistics State University of New York at Albany Albany NY 12222 United States lenart@ Submitted Oct 20 2009 Accepted Feb 15 2010 Published Feb 22 2010 Mathematics Subject Classification 17B10 05E10 Abstract This work is part of a project on weight bases for the irreducible representations of semisimple Lie algebras with respect to which the representation matrices of the Chevalley generators are given by explicit formulas. In the case of slra the celebrated Gelfand-Tsetlin basis is the only such basis known. Using the setup of supporting graphs developed by Donnelly we present a new interpretation and a simple combinatorial proof of the Gelfand-Tsetlin formulas based on a rational function identity all the known proofs use more sophisticated algebraic tools . A constructive approach to the Gelfand-Tsetlin formulas is then given based on a simple algorithm for solving certain equations on the lattice of semistandard Young tableaux. This algorithm also implies certain extremal properties of the Gelfand-Tsetlin basis. 1 Introduction This work is related to combinatorial constructions of weight bases for the irreducible representations of semisimple Lie algebras on which the action of the Chevalley generators 1Partially supported by National Science Foundation grants DMS-0500638 DMS-0757935 and DMS-1002636. 2Partially supported by National Science Foundation grants DMS-0403029 and DMS-0701044. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R33 1 is made explicit. We will use the setup introduced by Donnelly 3 5 6 . The main idea is to encode a weight basis into an edge-colored ranked poset called a supporting graph whose Hasse diagram has its edges labeled with two complex coefficients. This structure is known as a .

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