Coxeter Matroids

Portada
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.

 

Continguts

I
3
II
39
III
57
IV
83
V
103
VI
153
VII
201
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Frases i termes més freqüents

adjacent admissible ordering arbitrary Assume Axiom B₁ basis BN-pair Bruhat order building of type called Chapter circuit coedge collection of bases columns common panel concordant constituents contains convex hull Corollary corresponding cosets Coxeter complex Coxeter group Coxeter matroids defined Deletion denote dimension edge elements equivalent example Exercise finite Coxeter group flag matroid flag variety Gaussian scheme geodesic gallery geometric lattice graph group W H₁(S hence hyperplane Increasing Exchange Property intersection isotropic subspaces Lagrangian matroid Lagrangian orthogonal matroid Lemma linear matrix matroid map matroid of rank maximal chains Maximality Property mirrors Notice obviously ordinary matroids parabolic subgroup permutation Prove reflection group representation residues root system satisfies semimodular lattice simplicial complex spanned Strong Exchange Property subsets symmetric group Symn symplectic flag matroid symplectic matroid Theorem transposition unique vector space vertex vertices W-matroid w-maximal wall H

Passatges populars

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
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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
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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.
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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.
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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
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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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Informació bibliogràfica

TítolCoxeter Matroids
Volum 216 de Progress in Mathematics
AutorsAlexandre V. Borovik, Izrail M. Gelfand, Neil White
Il·lustracions deA. Borovik
Edicióil·lustrada
EditorSpringer Science & Business Media, 2003
ISBN0817637648, 9780817637644
Nre. de pàgines266 pàgines
  
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