Coxeter Matroids

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Springer Science & Business Media, 11 de jul. 2003 - 266 pàgines

Matroids appear in diverse areas of mathematics, from combinatorics to algebraic topology and geometry. This largely self-contained work provides an intuitive and interdisciplinary treatment of Coxeter matroids, a new and beautiful generalization of matroids which is based on a finite Coxeter group.

Key topics and features:

* Systematic, clearly written exposition with ample references to current research

* Matroids are examined in terms of symmetric and finite reflection groups

* Finite reflection groups and Coxeter groups are developed from scratch

* The Gelfand-Serganova Theorem is presented, allowing for a geometric interpretation of matroids and Coxeter matroids as convex polytopes with certain symmetry properties

* Matroid representations and combinatorial flag varieties are studied in the final chapter

* Many exercises throughout

* Excellent bibliography and index

Accessible to graduate students and research mathematicians alike, Coxeter Matroids can be used as an introductory survey, a graduate course text, or a reference volume.

 

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Continguts

I
3
II
39
III
57
IV
83
V
103
VI
153
VII
201
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Pàgina vii - matroid is a combinatorial concept which arises from the elimination of continuous parameters from one of the most fundamental notions of mathematics: that of linear dependence of vectors. Indeed, let E be a finite set of vectors in a vector space
Pàgina vii - in 1936. He noticed that the set of linearly independent subsets of E has some very distinctive properties. In particular, if B is the set of maximal linearly independent subsets of E, then, by a
Pàgina viii - provided by its symmetric group acting on it. The symmetric group already lurks between the lines of the Exchange Property in the form of transpositions (a, b) responsible for the exchange of elements.
Pàgina viii - from even more general Coxeter matroids. Symplectic matroids are related to the geometry of vector spaces endowed with bilinear forms, although in a more intricate way than ordinary matroids to ordinary vector spaces.
Pàgina viii - permute the coordinate axes without changing their orientation; this action obviously preserves the n-cube [—1, 1]”. Thus ordinary matroids can be also understood as symplectic matroids, the latter becoming the most natural
Pàgina 1 - of matroids in terms of the Greedy Algorithm. It says, briefly, that for every linear ordering of the set of elements of the matroid, there is a unique maximal basis. But linear orderings of a finite set can be interpreted as its permutations. This brings the symmetric group into a pivotal role in matroid theory

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