TAILIEUCHUNG - Báo cáo toán học: "A Combinatorial Formula for Orthogonal Idempotents in the 0-Hecke Algebra of the Symmetric Group"

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 Combinatorial Formula for Orthogonal Idempotents in the 0-Hecke Algebra of the Symmetric Group. | A Combinatorial Formula for Orthogonal Idempotents in the 0-Hecke Algebra of the Symmetric Group Tom Denton Submitted Jul 28 2010 Accepted Jan 25 2011 Published Feb 4 2011 Mathematics Subject Classification 20C08 Abstract Building on the work of . Norton we give combinatorial formulae for two maximal decompositions of the identity into orthogonal idempotents in the 0-Hecke algebra of the symmetric group CH0 SN . This construction is compatible with the branching from SN-1 to SN. 1 Introduction The 0-Hecke algebra CH0 SN for the symmetric group SN can be obtained as the Iwahori-Hecke algebra of the symmetric group Hq SN at q 0. It can also be constructed as the algebra of the monoid generated by anti-sorting operators on permutations of N. P. N. Norton described the full representation theory of CH0 SN in 11 In brief there is a collection of 2N-1 simple representations indexed by subsets of the usual generating set for the symmetric group in correspondence with collection of 2N-1 projective indecomposable modules. Norton gave a construction for some elements generating these projective modules but these elements were neither orthogonal nor idempotent. While it was known that an orthogonal collection of idempotents to generate the indecomposable modules exists there was no known formula for these elements. Herein we describe an explicit construction for two different families of orthogonal idempotents in CH0 SN one for each of the two orientations of the Dynkin diagram for SN. The construction proceeds by creating a collection of 2N-1 demipotent elements which we call diagram demipotents each indexed by a copy of the Dynkin diagram with signs attached to each node. These elements are demipotent in the sense that for each element X there exists some number k N 1 such that X is idempotent for all j k. The collection of idempotents thus obtained provides a maximal orthogonal decomposition of the identity. An important feature of the 0-Hecke algebra is that it is the

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