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

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

Matroids look in diversified parts of arithmetic, from combinatorics to algebraic topology and geometry. This mostly self-contained textual content presents an intuitive and interdisciplinary remedy of Coxeter matroids, a brand new and lovely generalization of matroids that is according to a finite Coxeter group.

Key subject matters and features:

* Systematic, in actual fact written exposition with considerable references to present research
* Matroids are tested when it comes to 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 yes symmetry properties
* Matroid representations in structures and combinatorial flag kinds are studied within the ultimate chapter
* Many workouts throughout
* very good bibliography and index

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

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Coxeter Matroids

Matroids seem in diversified parts of arithmetic, from combinatorics to algebraic topology and geometry. This mostly self-contained textual content presents an intuitive and interdisciplinary therapy of Coxeter matroids, a brand new and lovely generalization of matroids that is in response to a finite Coxeter crew. Key subject matters and features:* Systematic, essentially written exposition with plentiful references to present learn* Matroids are tested when it comes to symmetric and finite mirrored image teams* Finite mirrored image teams and Coxeter teams are constructed from scratch* The Gelfand-Serganova theorem is gifted, taking into consideration a geometrical interpretation of matroids and Coxeter matroids as convex polytopes with yes symmetry homes* Matroid representations in structures and combinatorial flag types are studied within the ultimate bankruptcy* Many workouts all through* first-class bibliography and indexAccessible to graduate scholars and examine mathematicians alike, "Coxeter Matroids" can be utilized as an introductory survey, a graduate path textual content, or a reference quantity.

Additional info for Coxeter Matroids

Sample text

Vertices of r are elements in [n] and two vertices a and b are connected by a (non-oriented) edge if and only if the transposition (a, b) belongs to T. Now let [n] = S1 U •.. U Sd be the partition of [n] into the vertex sets of connected 28 1 Matroids and Flag Matroids components of I"' and T = Tl U ... U Td the corresponding partition of the set of edges. Note that the set 1'; can be empty if the corresponding subgraph component S; consists of one vertex. It will be convenient for us to assume that the empty subset in Sym n generates the identity subgroup 1.

4. Prove that graphic matroids are representable. Hint: Orient the edges of the graph in an arbitrary way. Take the vector space over a field K spanned by the vertices VI, ••• , Vn of the graph G and associate the vector Vi - V j to an edge with the starting vertex Vi and the end vertex V j . 5~* (Borovik, Gelfand and Stone) of Exercise 4 has a natural topological interpretation. Let us view graphs as finite I-dimensional C W -complexes, edges being I-cells and vertices O-cells. Assume now that a graph G is embedded as a closed subset into an (n - 2)connected oriented manifold S of dimension ~ 2.

The mirror of the reflection t is the hyperplane EA,B perpendicular to the vector VA,B = OB -OA. Therefore the intersection X of the mirrors of all reflections t E T(I3) = is the orthogonal complement to the subspace Y spanned by the direction vectors vA, B of the edges of fl.. Obviously the dimension of the subspace Y is exactly the dimension of the polytope fl.. In particular, dim X = dim V - dim Y = dim V - dim fl.. On the other hand, the space X is obviously the subspace of the fixed points for the action of the group U in the space V.

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