斜线性与多重线性拟阵的Von Staudt构造

$q$-matroids 的表示性 via 排序度码

Representability of $q$-matroids via rank-metric codes

本文研究了 $q$-matroids 的多线性表示性，通过排序度码的自然可除性条件定义了 $m$-多线性表示性，并证明了非平凡均匀 $q$-matroids 无法纯多线性表示，且非帕普斯 $q$-matroid 的块大小至少为 9。

拟阵的熵表示与几乎多重线性表示

On entropic and almost multilinear representability of matroids

本文通过将群论中的不可判定问题归约到拟阵表示问题，证明了判定拟阵是否具有熵表示或几乎多重线性表示是不可判定的，并由此得出条件独立蕴涵问题及理想秘密共享方案存在性问题的不可判定性。

Representability of Matroids by c-Arrangements is Undecidable

For a natural number $c$, a $c$-arrangement is an arrangement of dimension $c$ subspaces satisfying the following condition: the sum of any subset of the subspaces has dimension a multiple of $c$. Matroids arising as normalized rank functions of $c$-arrangements are also known as multilinear matroids. We prove that it is algorithmically undecidable whether there exists a $c$ such that a given matroid has a $c$-arrangement representation, or equivalently whether the matroid is multilinear. It follows that certain network coding problems are also undecidable. In the proof, we introduce a general…

Tableaux in the Whitney Module of a Matroid

The Whitney module of a matroid is a natural analogue of the tensor algebra of the exterior algebra of a vector space that takes into account the dependencies of a matroid. In this paper we indicate the role that tableaux can play in describing the Whitney module. We will use our results to describe a basis of the Whitney module of a certain class of matroids known as freedom matroids (also known as Schubert, or shifted matroids). The doubly multilinear submodule of the Whitney module is a representation of the symmetric group. We will describe a formula for the multiplicity of hook shapes in …

Representing some non-representable matroids

We extend the notion of representation of a matroid to algebraic structures that we call skew partial fields. Our definition of such representations extends Tutte's definition, using chain groups. We show how such representations behave under duality and minors, we extend Tutte's representability criterion to this new class, and we study the generator matrices of the chain groups. An example shows that the class of matroids representable over a skew partial field properly contains the class of matroids representable over a skew field. Next, we show that every multilinear representation of a ma…