Related papers: Combinatorial representations
We give a combinatorial characterization of generic frameworks that are minimally rigid under the additional constraint of maintaining symmetry with respect to a finite order rotation or a reflection. To establish these results we develop a…
Generalizing a theorem of the first two authors and Geelen for planes, we show that, for a real-representable matroid $M$, either the average hyperplane-size in $M$ is at most a constant depending only on its rank, or each hyperplane of $M$…
A common theme of enumerative combinatorics is formed by counting functions that are polynomials evaluated at positive integers. In this expository paper, we focus on four families of such counting functions connected to hyperplane…
A commutative algebra over a field gives rise to a representation of the category of finite sets and surjective maps. We consider the restriction of this representation to the subcategory of sets of cardinality at most $r$. For each $r$, we…
We are interested in representations and characterizations of lattice polynomial functions f:L^n -> L, where L is a given bounded distributive lattice. In companion papers [arXiv 0901.4888, arXiv 0808.2619], we investigated certain…
There are many similarities between the theories of matroids and $q$-matroids. However, when dealing with the direct sum of $q$-matroids many differences arise. Most notably, it has recently been shown that the direct sum of representable…
We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation category: for example, we describe the Grothendieck group (including the effective cone), classify…
This article studies two notions of generalized matroid representations motivated by algorithmic information theory and cryptographic secret sharing. The first (entropic representability) involves discrete random variables, while the second…
Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets.…
In this paper we determine the representation type of some algebras of infinite matrices continuously controlled at infinity by a compact metrizable space. We explicitly classify their finitely presented modules in the finite and tame…
In this short note we show that every connected reductive simply-connected algebraic group of rank $>1$ over the complex numbers has infinitely many pairs of irreducible representations which are not related by an automorphism of the…
In this article, we study polymatroids that are representable by means of linear restricted rank-metric codes, namely, by subspaces of the space of alternating, symmetric, or Hermitian square matrices endowed with the rank metric. More…
Molecular graphs generally contain subgraphs (known as groups) that are identifiable and significant in composition, functionality, geometry, etc. Flat latent representations (node embeddings or graph embeddings) fail to represent, and…
We use representation theory to construct spaces of matrices of constant rank. These spaces are parametrized by the natural representation of the general linear group or the symplectic group. We present variants of this idea, with more…
2-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some 2-level…
Inspired by notorious combinatorial optimization problems on graphs, in this paper we consider a series of related problems defined using a metric space and topology determined by a graph. Particularly, we present the Independent Set,…
We discuss permutation representations which are obtained by the natural action of $S_n \times S_n$ on some special sets of invertible matrices, defined by simple combinatorial attributes. We decompose these representations into…
We consider two problems that appear at first sight to be unrelated. The first problem is to count certain diagrams consisting of noncrossing arcs in the plane. The second problem concerns the weak order on the symmetric group. Each…
We demonstrate that graph-based models are fully capable of representing higher-order interactions, and have a long history of being used for precisely this purpose. This stands in contrast to a common claim in the recent literature on…
For a rational matrix function R of one variable in general position, the matrix functions R(x)/R(y) and R(y)\R(x) of two variables are considered. For these matrix functions of two variables, representations which are analogous to the…