Download e-book for iPad: Coxeter Matroids by Alexandre V. Borovik, Israel M. Gelfand, Neil White, A.

By Alexandre V. Borovik, Israel M. Gelfand, Neil White, A. Borovik

ISBN-10: 1461220661

ISBN-13: 9781461220664

ISBN-10: 1461274001

ISBN-13: 9781461274001

Matroids seem in diversified parts of arithmetic, from combinatorics to algebraic topology and geometry. This principally self-contained textual content offers an intuitive and interdisciplinary therapy of Coxeter matroids, a brand new and gorgeous generalization of matroids that's in response to a finite Coxeter group.

Key issues and features:

* Systematic, essentially written exposition with plentiful references to present research
* Matroids are tested by way of symmetric and finite mirrored image groups
* Finite mirrored image teams and Coxeter teams are constructed from scratch
* The Gelfand-Serganova theorem is gifted, making an allowance for a geometrical interpretation of matroids and Coxeter matroids as convex polytopes with sure symmetry properties
* Matroid representations in constructions and combinatorial flag kinds are studied within the ultimate chapter
* Many workouts throughout
* first-class bibliography and index

Accessible to graduate scholars and study mathematicians alike, "Coxeter Matroids" can be utilized as an introductory survey, a graduate path textual content, or a reference volume.

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Example text

6 it will suffice to prove that one of the edges Pi, Yj is w-positive. Assume the contrary; let all Pi -

Proof. l···km be a flag matroid with constituents M;, 1 ~ i ~ m. We have already seen that each M; is a matroid. Now suppose that C is a circuit of Mj which is not a union of circuits of M;, for some i < j. Therefore, there exists an element x E C such that no circuit D of M; satisfies xED and D ~ C. Let us now choose W E Sym n such that y >w x >w Z for all y E C" {x}, Z E [n] " C. It follows that the maximal basis of M j contains C " {x } but does not contain x, whereas the maximal basis of M; contains x.

1. Let :F be a set offlags of the same rank (k}, ... ,km ) on [n]. Then the following conditions are equivalent: (1) :F is aflag matroid. y: is a matroid polytope. (3) :F satisfies the Increasing Exchange Property. 26 1 Matroids and Flag Matroids Proof (1) implies (2). Assume first that:F is a flag matroid. :F, since each vertex 8F has kI coordinates equal to m, ki+I - ki coordinates equal to m - i, for 1 ~ i ~. m - 1, and n - k m coordinates equal to o. Let v be an edge with vertices 8F and8G, F, G E :F, that is not parallel to any root.

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Coxeter Matroids by Alexandre V. Borovik, Israel M. Gelfand, Neil White, A. Borovik

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