An independent set axiomatization for symplectic matroids

有秩辛 matroid 的几何

The geometry of ranked symplectic matroids

本文研究有秩辛 matroid 的几何性质，证明其与普通 matroid 具有相似特性，并探讨其 matroid 多边形与 Bergman fan 的关系，同时部分验证了 Mason 猜想。

度量图论的一阶逻辑公理化

First-order logic axiomatization of metric graph theory

该研究类比Tarski的欧氏几何公理化，为度量图论中多类核心图建立带介于关系的一阶逻辑公理化，同时证明部分组合/拓扑定义的相关图类无法实现此类公理化。

δ-拟阵的秩函数与不变量

Rank functions and invariants of delta-matroids

本文通过秩函数公理化δ-拟阵，研究其秩生成函数，并利用洛伦兹多项式理论证明相关取值的对数凹性。

组合旗排列

Combinatorial flag arrangements

本文提出 matricube 作为拟阵的推广，用于组合公理化向量空间中的初始旗排列，并建立其秩函数、闭集、回路、独立集公理体系及对偶概念，同时阐明其与置换阵列和拟阵的联系。

Quantum Codes and Symplectic Matroids

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 …

Oriented Lagrangian Matroids

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.

Lagrangian Pairs and Lagrangian Orthogonal Matroids

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 $C_n$ …

关于辛Grassmannian的$T$-不变子簇及秩$2$辛拟阵在${\mathbb C}$上的可表示性

On $T$-invariant subvarieties of symplectic Grassmannians and representability of rank $2$ symplectic matroids over ${\mathbb C}$

本文研究了辛Grassmannian $\text{SpG}(2,2n)$的$T$-不变子簇，利用辛Coxeter拟阵给出其刻画，并借此完整刻画了在$\mathbb{C}$上可表示的秩$2$辛拟阵。

Lagrangian subspaces, delta-matroids and four-term relations

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 …