tailieunhanh - Báo cáo toán học: " A Theory of Transformation Monoids: Combinatorics and Representation Theory"

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 Theory of Transformation Monoids: Combinatorics and Representation Theory. | A Theory of Transformation Monoids Combinatorics and Representation Theory Benjamin Steinberg School of Mathematics and Statistics Carleton University Ottawa Ontario Canada bsteinbg@ Submitted May 1 2010 Accepted Nov 18 2010 Published Dec 3 2010 Mathematics Subject Classification 20M20 20M30 20M35 Abstract The aim of this paper is to develop a theory of finite transformation monoids and in particular to study primitive transformation monoids. We introduce the notion of orbitals and orbital digraphs for transformation monoids and prove a monoid version of D. Higman s celebrated theorem characterizing primitivity in terms of connectedness of orbital digraphs. A thorough study of the module or representation associated to a transformation monoid is initiated. In particular we compute the projective cover of the transformation module over a field of characteristic zero in the case of a transitive transformation or partial transformation monoid. Applications of probability theory and Markov chains to transformation monoids are also considered and an ergodic theorem is proved in this context. In particular we obtain a generalization of a lemma of P. Neumann from the theory of synchronizing groups concerning the partition associated to a transformation of minimal rank. The author was supported in part by NSERC THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R164 1 Contents 1 Introduction 3 2 Actions of monoids on sets 4 M -sets. 4 Green-Morita theory. 8 3 Transformation monoids 13 The minimal ideal. 13 Wreath products . 18 4 Finite 0-transitive transformation monoids 20 5 Primitive transformation monoids 23 6 Orbitals 27 Digraphs and cellular morphisms . 28 Orbital digraphs. 30 7 Transformation modules 32 The subspace of M-invariants. 33 The augmentation submodule. 35 Partial transformation modules. 37 8 A brief review of monoid representation theory 39 9 The projective cover of a transformation module 41 The .