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 , who named them after H. S. M. Coxeter. 
 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 "M" of "W"/"P" that for every "w" in "W", "M" 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.