Coxeter matroids. With illustrations by Anna Borovik
type: Item
instance of: scholarly article
title: Coxeter matroids. With illustrations by Anna Borovik
zbMATH Open document ID: 1050.52005
author: D. Massart
published in: Progress in Mathematics
publication date: 2003-10-20T00:00:00Z
review text: A \(n\)-dimensional convex polytope \(P\) is a Coxeter matroid polytope if all reflections in the mirrors of symmetries of all edges of \(P\) generate a finite group \(W\). (Note that \(W\) is a Coxeter group.) When \(W\) is the symmetric group \(S_n\) the associated Coxeter matroid polytope \(P\) is the matroid of the basis of a matroid \(M(P)\) [see \textit{I. M. Gelfand} and \textit{V. V. Serganova}, Usp. Mat. Nauk 42, No. 2, 107--134 (1987; Zbl 0629.14035)]. This book is devoted to Coxeter matroid polytopes and the related structures, the Coxeter matroids. Ordinary matroids are examples of Coxeter matroids. The content of the book is the following: Chapters 1 and 2 give elementary related concepts and results from matroid theory; for instance root systems, matroid polytopes, polytopes associated with flag matroids, maps of matroids, and (vectorial) representations of matroids. Chapters 3 and 4 study an important class of Coxeter matroids: the ones corresponding to the Coxeter group \(BC_n\) called symplectic matroids [resp. orthogonal matroids corresponding to the Coxeter group \(D_n\)]. A representation of this class of Coxeter matroids in the cube is a consequence of the Gelfand-Serganova theorem. The matroids of an important subclass of symplectic matroids are called Lagrangian matroids. This class is closed under the greedy algorithm and has an interesting geometric representation. (This class was introduced by \textit{A. Bouchet} and named by him \(\Delta\)-matroids [Discrete Math. 78, No.~1/2, 59--71 (1989; Zbl 0719.05019); and Discrete Appl. Math. 24, No.~1--3, 55--62 (1989; Zbl 0708.05014)].) Chapter 5 gives a self-contained introduction to Coxeter groups and associated notions needed to the characterizations of general Coxeter matroids presented in Chapter 6. Finally Chapter 7 is an introduction to buildings, the geometric objects introduced by Tits as generalizations of projective spaces. In the reviewer's opinion the theory of Coxeter matroids is attr…
reviewed by: Raul Cordovil
Mathematics Subject Classification ID: 52B40
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 05E15
Mathematics Subject Classification ID: 20E42
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 51F15
zbMATH DE Number: 1994511
zbMATH Keywords: matroids
zbMATH Keywords: Coxeter matroids
zbMATH Keywords: symplectic matroids
zbMATH Keywords: orthogonal matroids
zbMATH Keywords: Lagrangian matroids
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1994511