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Coxeter matroids. With illustrations by Anna Borovik. (English) Zbl 1050.52005
Progress in Mathematics (Boston, Mass.) 216. Boston, MA: Birkhäuser (ISBN 0-8176-3764-8/hbk). xii, 264 p. (2003).
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 I. M. Gelfand and 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 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 attractive and he expects interesting results in the next future.

52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
05B35 Combinatorial aspects of matroids and geometric lattices
05E15 Combinatorial aspects of groups and algebras (MSC2010)
20E42 Groups with a \(BN\)-pair; buildings
20F55 Reflection and Coxeter groups (group-theoretic aspects)
51F15 Reflection groups, reflection geometries