Symplectic matroids. (English) Zbl 0919.05012
A Coxeter matroid, also called WP-matroid, is a subset \(M\) of the set of cosets \(W/P\) of a Coxeter group \(W\) modulo a parabolic subgroup \(P\) such that \(M\) satisfies a certain maximality condition. More precisely, for each \(w\in W\) there is an element \(x\in M\) such that for all \(y\in M\) the element \(w^{-1}y\) precedes or equals \(w^{-1}x\) in the strong Bruhat order, compare A. V. Borovik, I. Gelfand, and N. White [Adv. Math. 120, No. 2, 258-264, Art. No. 0038 (1996; Zbl 0858.05025)]. In this context an ordinary matroid is a Coxeter matroid corresponding to the symmetric group, see N. L. White [Contemp. Math. 197, 401-409 (1996; Zbl 0862.05021)]. A symplectic matroid is a Coxeter matroid in which the Coxeter group is the group of symmetries of the \(n\)-cube. In this paper the authors examine the basic algebraic and geometric properties of symplectic matroids.
Reviewer: Herman J.Servatius (Worcester)
MSC:
| 05B35 | Combinatorial aspects of matroids and geometric lattices |
Keywords:
Coxeter matroid; symplectic matroid; totally isotropic subspace; 2-matroid; symmetric matroid; root system; flag matroidReferences:
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