Orthogonal Matroids
Journal of Algebraic Combinatorics
Orthogonal Matroids
ANDREW VINCE NEIL WHITE 0
0 Department of Mathematics, University of Florida , Gainesville, FL 32611 , USA
The notion of matroid has been generalized to Coxeter matroid by Gelfand and Serganova. To each pair (W, P) consisting of a finite irreducible Coxeter group W and parabolic subgroup P is associated a collection of objects called Coxeter matroids. The (ordinary) matroids are the special case where W is the symmetric group (the An case) and P is a maximal parabolic subgroup. This generalization of matroid introduces interesting combinatorial structures corresponding to each of the finite Coxeter groups. Borovik, Gelfand and White began an investigation of the Bn case, called symplectic matroids. This paper initiates the study of the Dn case, called orthogonal matroids. The main result (Theorem 2) gives three characterizations of orthogonal matroid: algebraic, geometric, and combinatorial. This result relies on a combinatorial description of the Bruhat order on Dn (Theorem 1). The representation of orthogonal matroids by way of totally isotropic subspaces of a classical orthogonal space (Theorem 5) justifies the terminology orthogonal matroid. Matroids, introduced by Hassler Whitney in 1935, are now a fundamental tool in combinatorics with a wide range of applications ranging from the geometry of Grassmannians to combinatorial optimization. In 1987 Gelfand and Serganova [9, 10] generalized the matroid concept to the notion of Coxeter matroid. To each finite Coxeter group W and parabolic subgroup P is associated a family of objects called Coxeter matroids. Ordinary matroids correspond to the case where W is the symmetric group (the An case) and P is a maximal parabolic subgroup. This generalization of matroid introduces interesting combinatorial structures corresponding to each of the finite Coxeter groups. Borovik, Gelfand and White [2] began an investigation of the Bn case, called symplectic matroids. The term “symplectic” comes from examples constructed from symplectic geometries. This paper initiates the study of the Dn case, called orthogonal matroid because of examples constructed from orthogonal geometries. The first goal of this paper is to give three characterizations of orthogonal matroids: algebraic, geometric and combinatorial. This is done in Sections 3, 4 and 6 (Theorem 2) after preliminary results in Section 2 concerning the family Dn of Coxeter groups. The ∗The work of Neil White on this paper was partially supported at UMIST under funding of EPSRC Visiting Fellowship GR/M24707.
orthogonal matroid; Coxeter matroid; Coxeter group; Bruhat order
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Introduction
algebraic description is in terms of left cosets of a parabolic subgroup P in Dn. The Bruhat
order on Dn plays a central role in the definition. The geometric description is in terms of
a polytope obtained as the convex hull of a subset of the orbit of a point in Rn under the
action of Dn as a Euclidean reflection group. The roots of Dn play a central role in the
definition. The combinatorial description is in terms of k-element subsets of a certain set
and flags of such subsets. The Gale order plays a central role in the definition. Section 5
gives a precise description of the Bruhat order on both Bn and Dn in terms of the Gale order
on the corresponding flags (Theorem 1). A fourth characterization, in terms of oriflammes,
holds for an important special case (Theorem 3 of Section 6).
Section 7 concerns the relationship between symplectic and orthogonal matroids. Every
orthogonal matroid is a symplectic matroid. Necessary and sufficient conditions are provided
when a Lagrangian symplectic matroid is orthogonal (Theorem 4). More generally, the
question remains open.
Section 8 concerns the representation of orthogonal matroids and, in particular, justifies
the term orthogonal. Just as ordinary matroids arise from subspaces of projective spaces,
symplectic and orthogonal matroids arise from totally isotropic subspaces of symplectic
and orthogonal spaces, respectively (Theorem 5).
2. The Coxeter group Dn
We give three descriptions of the family Dn of Coxeter groups: (
1
) in terms of generators
and relations; (
2
) as a permutation group; and (
3
) as a reflection group in Euclidean space.
Presentation in terms of generators and relations. A Coxeter group W is defined in terms
of a finite set S of generators with the presentation
s ∈ S | (ss )mss = 1 ,
where mss is the order of ss , and mss = 1 (hence each generator is an involution). The
cardinality of S is called the rank of W . The diagram of W is the graph where each
generator is represented by a node, and nodes s and s are joined by an edge labeled mss
whenever mss ≥ 3. By convention, the label is omitted if mss = 3. A Coxeter system
is irreducible if its diagram is a connected graph. A reducible Coxeter group is the direct
product of the Coxeter groups corresponding to the connected components of its diagram.
The (...truncated)