Represented Coxeter matroids of types $C_n$ and $D_n$, that is, symplectic and orthogonal matroids arising from totally isotropic subspaces of symplectic or (even-dimensional) orthogonal spaces, may also be represented in buildings of type $C_n$ and $D_n$, respectively. Indeed, the particular buildings involved are those arising from the flags or oriflammes, respectively, of totally isotropic subspaces. There are also buildings of type $B_n$ arising from flags of totally isotropic subspaces in odd-dimensional orthogonal space. Coxeter matroids of type $B_n$ are the same as those of type $C_n$ (since they depend only upon the reflection group, not the root system). However, buildings of type $B_n$ are distinct from those of the other types. The matroids representable in odd dimensional orthogonal space (and therefore in the building of type $B_n$) turn out to be a special case of symplectic (flag) matroids, those whose top component, or Lagrangian matroid, is a union of two Lagrangian orthogonal matroids. These two matroids are called a Lagrangian pair, and they are the combinatorial manifestation of the ``fork'' at the top of an oriflamme (or of the fork at the end of the Coxeter diagram of $D_n$).
Here we give a number of equivalent characterizations of Lagrangian pairs, and prove some rather strong properties of them.
In this paper we extend the theory of oriented matroids to Lagrangian orthogonal matroids and their representations, and give a completely natural transformation from a representation of a classical oriented matroid to a representation of the same oriented matroid considered as a Lagrangian orthogonal matroid. Classical concepts of orientation and this extension may both be thought of as stratifications of thin Schubert cells into unions of connected components.
We introduce the concept of orientation for Lagrangian matroids represented in the flag variety of maximal isotropic subspaces of dimension N in the real vector space of dimension 2N+1. The paper continues the study started in math.CO/0209100.
In this paper we present a definition of oriented Lagrangian symplectic matroids and their representations. Classical concepts of orientation and this extension may both be thought of as stratifications of thin Schubert cells into unions of connected components. The definitions are made first in terms of a combinatorial axiomatisation, and then again in terms of elementary geometric properties of the Coxeter matroid polytope. We also generalise the concept of rank and signature of a quadratic form to symplectic Lagrangian matroids in a surprisingly natural way.
The Lagrangian geometry of matroids was introduced in [ADH20] through the construction of the conormal fan of a matroid M. We used the conormal fan to give a Lagrangian-geometric interpretation of the h-vector of the broken circuit complex of M: its entries are the degrees of the mixed intersections of certain convex piecewise linear functions $γ$ and $δ$ on the conormal fan of M. By showing that the conormal fan satisfies the Hodge-Riemann relations, we proved Brylawski's conjecture that this h-vector is a log-concave sequence. This sequel explores the Lagrangian combinatorics of matroids, fu…
The aim of the paper is to clarify the nature of combinatorial structures associated with maps on closed compact surfaces. We prove that maps give rise to Lagrangian matroids representable in a setting provided by cohomology of the surface with punctured points. Our proof is very elementary. We further observe that the greedy algorithm has a natural interpretation in this setting, as a `peeling' procedure which cuts the (connected) surface into a closed ring-shaped peel, and that this procedure is local.
For a map $\mathcal M$ cellularly embedded on a connected and closed orientable surface, the bases of its Lagrangian (also known as delta-) matroid $Δ(\mathcal M)$ correspond to the bases of a Lagrangian subspace $L$ of the standard orthogonal space $\mathbb{Q}^E\oplus\mathbb{Q}^{E^*}$, where $E$ and $E^*$ are the edge-sets of $\mathcal M$ and its dual map. The Lagrangian subspace $L$ is said to be a representation of both $\mathcal M$ and $Δ(\mathcal M)$. Furthermore, the bases of $Δ(\mathcal M)$, when understood as vertices of the hypercube $[-1,1]^n$, induce a polytope $\mathbf P(Δ(\mathcal…
Finite order invariants (Vassiliev invariants) of knots are expressed in terms of weight systems, that is, functions on chord diagrams satisfying the four-term relations. Weight systems have graph analogues, so-called $4$-invariants of graphs, i.e. functions on graphs that satisfy the four-term relations for graphs. Each $4$-invariant determines a weight system. The notion of weight system is naturally generalized for the case of embedded graphs with an arbitrary number of vertices. Such embedded graphs correspond to links; to each component of a link there corresponds a vertex of an embedded …
We introduce the conormal fan of a matroid M, which is a Lagrangian analog of the Bergman fan of M. We use the conormal fan to give a Lagrangian interpretation of the Chern-Schwartz-MacPherson cycle of M. This allows us to express the h-vector of the broken circuit complex of M in terms of the intersection theory of the conormal fan of M. We also develop general tools for tropical Hodge theory to prove that the conormal fan satisfies Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. The Lagrangian interpretation of the Chern-Schwartz-MacPherson cycle of M, when com…
We give a semi-small orthogonal decomposition of the Chow ring of a matroid M. The decomposition is used to give simple proofs of Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations for the Chow ring, recovering the main result of [AHK18]. We also show that a similar semi-small orthogonal decomposition holds for the augmented Chow ring of M.
In [RW19] Rietsch and Williams relate cluster structures and mirror symmetry for type A Grassmannians Gr(k, n), and use this interaction to construct Newton-Okounkov bodies and associated toric degenerations. In this article we define a cluster seed for the Lagrangian Grassmannian, and prove that the associated Newton-Okounkov body agrees up to unimodular equivalence with a polytope obtained from the superpotential defined by Pech and Rietsch on the mirror Orthogonal Grassmannian in [PR13].
The correspondence between linear codes and representable matroids is well known. But a similar correspondence between quantum codes and matroids is not known. We show that representable symplectic matroids over a finite field $\mathbb{F}_q$ correspond to $\mathbb{F}_q$-linear quantum codes. Although this connection is straightforward, it does not appear to have been made earlier in literature. The correspondence is made through isotropic subspaces. We also show that the popular Calderbank-Shor-Steane (CSS) codes are essentially the homogenous symplectic matroids while the graph states, which …
In this article, we make a close analysis on quantum multiplication operators on the quantum cohomology rings of Lagrangian and orthogonal Grassmannians, and give an explicit description on all simultaneous eigenvectors and the corresponding eigenvalues for these operators. As a result, we show that Conjecture $\mathcal{O}$ of Galkin, Golyshev and Iritani holds for these manifolds.
We obtain a combinatorial expression for the coefficients of the boundary map of real isotropic and odd orthogonal Grassmannians providing a natural generalization of the formulas already obtained for Lagrangian and maximal isotropic Grassmannians. The results are given in terms of the classification into four types of covering pairs among the Schubert cells when identified with signed $k$-Grassmannian permutations. It turns out that these coefficients only depend on the positions changed over each pair of permutations. As an application, we give an orientability criterion, exhibit a symmetry …
We introduce the Symplectic Grassmann codes as projective codes defined by symplectic Grassmannians, in analogy with the orthogonal Grassmann codes introduced in [4]. Note that the Lagrangian-Grassmannian codes are a special class of Symplectic Grassmann codes. We describe the weight enumerator of the Lagrangian--Grassmannian codes of rank $2$ and $3$ and we determine the minimum distance of the line Symplectic Grassmann codes.
