Coxeter matroid

In mathematics, Coxeter matroids are generalization of matroids depending on a choice of a Coxeter group W and a parabolic subgroup P. Ordinary matroids correspond to the case when P is a maximal parabolic subgroup of a symmetric group W. They were introduced by Gelfand and Serganova (1987, 1987b), who named them after H. S. M. Coxeter.

Borovik, Gelfand & White (2003) give a detailed account of Coxeter matroids.

Definition

Suppose that W is a Coxeter group, generated by a set S of involutions, and P is a parabolic subgroup (the subgroup generated by some subset of S). A Coxeter matroid is a subset of W/P that for every w in W contains a unique minimal element with respect to the w-Bruhat order.

Relation to matroids

Suppose that the Coxeter group W is the symmetric group S_n and P is the parabolic subgroup S_k×S_n–k. Then W/P can be identified with the k-element subsets of the n-element set {1,2,...,n} and the elements w of W correspond to the linear orderings of this set. A Coxeter matroid consists of k elements sets such that for each w there is a unique minimal element in the corresponding Bruhat ordering of k-element subsets. This is exactly the definition of a matroid of rank k on an n-element set in terms of bases: a matroid can be defined as some k-element subsets called bases of an n-element set such that for each linear ordering of the set there is a unique minimal base in the Gale ordering of k-element subsets.

References

Borovik, Alexandre V.; Gelfand, I. M.; White, Neil (2003), Coxeter matroids, Progress in Mathematics, 216, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-3764-4, MR 1989953, doi:10.1007/978-1-4612-2066-4
Gelfand, I. M.; Serganova, V. V. (1987), "On the general definition of a matroid and a greedoid", Doklady Akademii Nauk SSSR (in Russian), 292 (1): 15–20, ISSN 0002-3264, MR 871945
Gelfand, I. M.; Serganova, V. V. (1987b), "Combinatorial geometries and the strata of a torus on homogeneous compact manifolds", Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, 42 (2): 107–134, ISSN 0042-1316, MR 0898623, doi:10.1070/RM1987v042n02ABEH001308 – English translation in Russian Mathematical Surveys 42 (1987), no. 2, 133–168

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