Gaussian-scheme representability of Coxeter matroids: the conjectured equality of geometric and combinatorial flag varieties for every finite Coxeter group, and the symplectic case

For every finite Coxeter group $W$, Borovik, Gelfand and White introduced a combinatorial flag variety $\tilde{\Omega}_W$ — the simplicial complex of chains of Coxeter matroids — together with a notion of representability by Gaussian schemes, a combinatorial abstraction of Gaussian elimination, and proved in type A that every matroid on $n$ points is representable in this sense. They conjectured that the same holds for every finite Coxeter group, which would identify the Gaussian-representable part $\Omega_W$ with the whole combinatorial flag variety, and listed the symplectic type $BC_n$ — where the Coxeter matroids are symplectic matroids — as the leading instance. The conjecture asks whether every Coxeter matroid is representable by a Gaussian scheme over its Coxeter group; in particular, whether every symplectic flag matroid is representable in $\tilde{\Omega}_{BC_n}$. The originating school's follow-ups treat only Lagrangian (maximal-rank) representations in types B, C and D; the quotient and circuit theory for symplectic matroids required by the foreseen lattice-theoretic approach remains underdeveloped; and the modern framework of matroids over tracts and the field with one element (Jarra–Lorscheid, Lorscheid–Thas) re-establishes the type-A case and lists the other Dynkin types as open problems. Neither a proof nor a counterexample is known.

Background

A matroid is the standard combinatorial abstraction of linear dependence: a finite ground set $E$ together with a family of subsets (bases, or independent sets) satisfying exchange axioms that the maximal linearly independent subsets of any vector configuration satisfy; a flat of a matroid is a subset closed under the matroid's closure operator. A flag matroid is a sequence $M_1,\dots,M_s$ of matroids on the same ground set, of strictly increasing rank, such that each $M_i$ is a quotient of $M_{i+1}$ (equivalently, every flat of $M_i$ is a flat of $M_{i+1}$); flag matroids relate to flags of subspaces as matroids relate to single subspaces. A Coxeter group $W$ is a group generated by involutions $r_1,\dots,r_n$ (the standard generators) with relations $(r_ir_j)^{m_{ij}}=1$; the symmetric group $\mathrm{Sym}_n$ (type $A_{n-1}$) and the group of signed permutations (type $BC_n$) are the classical examples. A standard parabolic subgroup of $W$ is a subgroup generated by a subset of the standard generators. The (strong) Bruhat order is the standard partial order on $W$ in which $u\le v$ when $v$ can be reached from $u$ by multiplying by reflections so as to increase length.

A Coxeter matroid for $W$ and a standard parabolic subgroup $P$ is a map $\mu\colon W\to W/P$ into the coset space satisfying the matroid inequality $\mu(u)\le_v\mu(v)$ for all $u,v\in W$, read in the Gale order on cosets (the partial order on $W/P$ induced by the Bruhat order on minimal coset representatives, shifted by $v$). For the trivial subgroup $P=1$ this is a map $\mu\colon W\to W$ satisfying $w^{-1}\mu(u)\le w^{-1}\mu(w)$ for all $u,w\in W$ in Bruhat order, called a $W$-matroid or complete flag matroid; the two levels are cryptomorphic, since every Coxeter matroid $\mu\colon W\to W/P$ determines its underlying complete flag matroid $\mu^*\colon W\to W$ defined by $\mu^*(w)=\max_w\mu(w)$, and every complete flag matroid decomposes into constituents attached to the maximal standard parabolic subgroups. Introduced by Gelfand and Serganova as a uniform generalization of matroids, the notion recovers ordinary matroids on $[n]$ as the Coxeter matroids for $W=\mathrm{Sym}_n$ and a maximal standard parabolic subgroup, and complete flag matroids on $[n]$ as the $\mathrm{Sym}_n$-matroids. For $W$ of type $BC_n$, the resulting matroids are the symplectic matroids, introduced under that name by Borovik, Gelfand and White in 1998 and first studied by Bouchet in 1987 as symmetric matroids: matroids on the doubled ground set $E=[n]\cup[n]^*$ whose bases are admissible (they contain at most one of each pair $i,i^*$) and satisfy a symmetric exchange axiom; the maximal-rank ones are called Lagrangian matroids.

Borovik, Gelfand and White associated to each finite Coxeter group $W$ of rank $n$ the combinatorial flag variety $\tilde{\Omega}_W$: the simplicial complex whose simplices are the chains of pairwise concordant Coxeter matroids attached to distinct maximal standard parabolic subgroups of $W$, where two such matroids $\mu,\nu$ are concordant when the cosets $\mu(w)$ and $\nu(w)$ intersect for every $w\in W$. For $W=\mathrm{Sym}_n$ this is the order complex of the poset of matroids on $[n]$ under the quotient (matroid quotient) order; its maximal simplices (chambers) are indexed by complete flag matroids, and it contains a copy of the Coxeter complex of $W$ (the simplicial complex encoding the combinatorics of the parabolic subgroups of $W$; the constant maps $W\to\{w\}$ form such a subcomplex). Borovik, Gelfand and Stone proved that $\tilde{\Omega}_W$ is shellable for every finite Coxeter group $W$ — a conjecture of White; shellable means its maximal simplices admit an ordering in which each one intersects the union of the earlier ones in entire faces, so the complex is homotopy equivalent to a wedge (one-point union) of $(n-1)$-spheres. They also proved $\operatorname{rank}H_{n-1}(\tilde{\Omega}_W,\mathbb{Z})\ge|W|$, and stated, with the detailed proof deferred to a paper that never appeared, the sharper type-A bound $\operatorname{rank}H_{n-2}(\tilde{\Omega}_n,\mathbb{Z})\ge (n!)^2/(2^{n-1}n)+n!$.

The representability relevant here is combinatorial, not linear. Throughout, 'Gaussian scheme', the category $\operatorname{GAUSS}_W$, and representability by a Gaussian scheme carry the definitions of the sources [1] and [5] (the 2000 paper and the Coxeter Matroids monograph), which fix the class of Gaussian schemes over $W$ and the representation relation; the statement below is to be read in that sense. Borovik, Gelfand and White introduced Gaussian schemes — the objects of a category $\operatorname{GAUSS}_W$ abstracting the process of Gaussian elimination, with the Coxeter group $W$ in place of the general linear group — and, in the 2000 paper and its companion on representations of matroids in semimodular lattices (a semimodular lattice is a lattice satisfying the covering condition that the lattice of flats of a matroid satisfies), proved the striking type-A theorem that every matroid on $n$ points is representable in $\tilde{\Omega}_n$ in this sense: a combinatorial reformulation of representability by vector configurations that, unlike classical linear representability, captures all matroids, including those representable over no field. Their monograph records the aspiration (Section 7.14) that the resulting complexes are, in many aspects, "more suitable candidates for the role of a 'universal' combinatorial geometry of type $W$ over the field of 1 element", tying the program to the idea of a geometry over the field with one element $\mathbb{F}_1$. This tie has since been made precise in the theory of matroids over hyperfields and tracts (algebraic structures generalizing fields in which addition is multivalued): Jarra and Lorscheid identify the type-A combinatorial flag variety with the set of rational points over the Krasner hyperfield $\mathbb{K}$ (the two-element hyperfield, in which $1+1=\{0,1\}$) of their moduli space of flag matroids, expressing the type-A theorem as the surjectivity of a natural map of moduli spaces, and Lorscheid and Thas embed the story in $\mathbb{F}_1$-geometry via band schemes and crowds (the scheme-like and group-like objects of that theory: a band is a multiplicative monoid with a distinguished null set, and a crowd is a set with an identity and a ternary law generalizing group multiplication and inversion).

In their closing section, Borovik, Gelfand and White listed six open problems. The most structural is the representability conjecture: whether every $W$-matroid over any finite Coxeter group $W$ is representable by some Gaussian scheme, which would imply $\Omega_W=\tilde{\Omega}_W$ (the Gaussian-representable part fills the entire combinatorial flag variety); its leading instance, listed first among their questions, is the symplectic case — whether every symplectic flag matroid is representable in $\tilde{\Omega}_{BC_n}$ — with two foreseen approaches: a developed theory of quotients for symplectic matroids, in which circuits are more intricate than in the classical case, or a better understanding of the category $\operatorname{GAUSS}_W$. The remaining questions — a MacPhersonian-style study of the flag complex of oriented matroids (oriented matroids are matroids enriched with base orientations, abstracting real vector configurations; the MacPhersonian is the poset of rank-$k$ oriented matroids under weak order), an orientation theory for symplectic matroids beyond the Lagrangian rank, the $\mathrm{Sym}_n$-module structure of $H_{n-2}(\Omega_n)$, and the connectivity of $\tilde{\Omega}_W$ for infinite Coxeter groups — are separate research directions rather than parts of one objective.

Problem Statement

Conjecture (Borovik-Gelfand-White, 2000). Let $W$ be a finite Coxeter group. Every $W$-matroid (Coxeter matroid) is representable by some Gaussian scheme over $W$; equivalently, the subcomplex of the combinatorial flag variety $\tilde{\Omega}_W$ spanned by Gaussian-scheme-representable chains coincides with the entire complex — in the source's notation, $\Omega_W=\tilde{\Omega}_W$ for every finite Coxeter group $W$.

The task is to prove or refute this conjecture. The leading open instance, listed first among the source's questions, is the symplectic case $W$ of type $BC_n$ (the hyperoctahedral group of signed permutations): is every symplectic flag matroid representable in $\tilde{\Omega}_{BC_n}$?

The conjecture is quantified over all finite Coxeter groups $W$ and, for each, every $W$-matroid (complete flag matroid) over $W$, whose constituents run through every maximal standard parabolic subgroup and every rank; in the leading symplectic instance this means every symplectic flag matroid of every rank, not only the maximal (Lagrangian) rank treated in the follow-up literature. A counterexample is a $W$-matroid that no Gaussian scheme over $W$ represents; such a $W$-matroid may be specified by its values as a map $W\to W$, or by an equivalent cryptomorphic description such as the admissible bases of a symplectic matroid together with its Coxeter group. Both outcomes are consistent with current knowledge: classical linear representability genuinely fails in type $BC_n$ (not even all rank-2 symplectic matroids are $\mathbb{C}$-representable), while the combinatorial Gaussian notion in type A captures every matroid, including those representable over no field.

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