Related papers: Cyclotomic and simplicial matroids

Motivated by the notion of the inverse $Z$-polynomial introduced by Ferroni, Matherne, Stevens, and Vecchi, we study the equivariant inverse $Z$-polynomial of a matroid equipped with a finite group. We prove that the coefficients of the…

For a simplicial complex ${\mathcal C}$ denote by $\beta({\mathcal C})$ the minimal number of edges from ${\mathcal C}$ needed to cover the ground set. If ${\mathcal C}$ is a matroid then for every partition $A_1, \ldots, A_m$ of the ground…

It has recently been shown that infinite matroids can be axiomatized in a way that is very similar to finite matroids and permits duality. This was previously thought impossible, since finitary infinite matroids must have non-finitary…

We show that the basis graph of an even delta-matroid is Hamiltonian if it has more than two vertices. More strongly, we prove that for two distinct edges $e$ and $f$ sharing a common end, it has a Hamiltonian cycle using $e$ and avoiding…

We introduce and study a family of polytopes which can be seen as a generalization of the permutahedron of type $B_d$. We highlight connections with the largest possible diameter of the convex hull of a set of points in dimension $d$ whose…

An arithmetic matroid is weakly multiplicative if the multiplicity of at least one of its bases is equal to the product of the multiplicities of its elements. We show that if such an arithmetic matroid can be represented by an integer…

This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a)…

For a root of unity $\zeta$ of odd prime order, we restrict coefficients of non-semisimple quantum representations of mapping class groups associated with the small quantum group $\mathfrak{u}_\zeta \mathfrak{sl}_2$ from $\mathbb{Q}(\zeta)$…

We show that if the ground set of a matroid can be partitioned into $k\ge 2$ bases, then for any given subset $S$ of the ground set, there is a partition into $k$ bases such that the sizes of the intersections of the bases with $S$ may…

Let G be a finite connected graph. The Kirchhoff polynomial of G is a certain homogeneous polynomial whose degree is equal to the first betti number of G. These polynomials appear in the study of electrical circuits and in the evaluation of…

Let $n$ be an even positive integer with at most three distinct prime factors and let $\ze_n$ be a primitive $n$-th root of unity. In this study, we made an attempt to find the lowest-degree $0,1$-polynomial $f(x) \in \Q[x]$ having at least…

Using the cyclotomic identity we compute sums over d-tuples of monic polynomials in F_q[x] weighted by the multiplicity of their irreducible factors. As consequences we determine explicit expressions for the number of d-tuples of…

Building on the work [18], where some standard basis for the queer $q$-Schur superalgebra $\mathcal{Q}_q(n,r;R)$ is defined by a labelling set of matrices and their associated double coset representatives, we investigate the matrix…

In this paper, we study the principal specialization of monomial symmetric polynomials and investigate the special values of these polynomials at \[ \zeta_{(n,k)} := ( 1, \zeta_n, \zeta_n^2, \dots, \zeta_n^{kn-1} ), \] where $\zeta_n$ is a…

Four circulant codes form a special class of $2$-generator, index $4$, quasi-cyclic codes. Under some conditions on their generator matrices they can be shown to be self-dual. Artin primitive root conjecture shows the existence of an…

It is shown that matroid theory may provide a natural mathematical framework for a duality symmetries not only for quantum Yang-Mills physics, but also for M-theory. Our discussion is focused in an action consisting purely of the…

Let $n\geq 8$ be divisible by 4. The Clifford-cyclotomic gate set $\mathcal{G}_n$ is the universal gate set obtained by extending the Clifford gates with the $z$-rotation $T_n = \mathrm{diag}(1,\zeta_n)$, where $\zeta_n$ is a primitive…

It is known that every matrix of order n over the maximal order in an algebraic number eld is a sum of k-th powers in various cases if a discriminant condition is satis ed. It has been proved by Wadikar and Katre that for every matrix of…

Let $\zeta_k$ be a $k$-th primitive root of unity, $m\geq\phi(k)+1$ an integer and $\Phi_k(X)\in\mathbb Z [X]$ the $k$-th cyclotomic polynomial. In this paper we show that the pair $(-m+\zeta_k,\mathcal N)$ is a canonical number system,…

A matroid is $\text{GF}(q)$-regular if it is representable over all proper superfields of the field $\text{GF}(q)$. We show that, for highly connected matroids having a large projective geometry over $\text{GF}(q)$ as a minor, the property…