Related papers: Monomial Bases for Broken Circuit Complexes

For finite nilpotent groups $G$ and $G^{\prime}$, and a $G$-adapted ring $S$ (the rational integers, for example), it is shown that any isomorphism between the centers of the group rings $SG$ and $SG^{\prime}$ is monomial, i.e., maps class…

Given a set of forms f={f_1,...,f_m} in R=k[x_1,...,x_n], where k is a field of characteristic zero, we focus on the first syzygy module Z of the transposed Jacobian module D(f), whose elements are called differential syzygies of f. There…

We consider the Burnside ring $A(\mathcal{F})$ of $\mathcal{F}$-stable $S$-sets for a saturated fusion system $\mathcal{F}$ defined on a $p$-group $S$. It is shown by S. P. Reeh that the monoid of $\mathcal{F}$-stable sets is a free…

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Assume that $G$ is a graph with edge ideal $I(G)$. We prove that the modules $S/\overline{I(G)^k}$ and…

Associated to a simple undirected graph G is a simplicial complex whose faces correspond to the independent sets of G. We call a graph G shellable if this simplicial complex is a shellable simplicial complex in the non-pure sense of…

Motivated by the Gray code interpretation of Hamiltonian cycles in Cayley graphs, we investigate the existence of Hamiltonian cycles in tope graphs of hyperplane arrangements, with a focus on simplicial, reflection, and supersolvable…

Let $G$ be a finite group and $H$ a normal subgroup of prime index $p$. Let $V$ be an irreducible ${\mathbb F}H$-module and $U$ a quotient of the induced ${\mathbb F}G$-module $V\kern-3pt\uparrow$. We describe the structure of $U$, which is…

Linear structural equation models are multivariate statistical models encoded by mixed graphs. In particular, the set of covariance matrices for distributions belonging to a linear structural equation model for a fixed mixed graph $G=(V,…

It is proved that any one-to-one edge map f from a 3-connected graph G onto a graph H, G and H possibly infinite, satisfying f(C) is a circuit in H whenever C is a circuit in G is induced by a vertex isomorphism. This generalizes a result…

We compute the rank of the group of central units in the integral group ring $\Z G$ of a finite strongly monomial group $G$. The formula obtained is in terms of the strong Shoda pairs of $G$. Next we construct a virtual basis of the group…

The machinery of zonotopal algebra is linked with two particular polytopes: the Stanley-Pitman polytope and the regular simplex $\mathfrak{Sim}_n(t_1,...,t_n)$ with parameters $t_1,...,t_n\in \mathbb{R}_+^n$, defined by the inequalities…

We introduce the notion of set-decomposition of a normal G-flat chain. We show that any normal rectifiable $G$-flat chain admits a decomposition in set-indecomposable sub-chains. This generalizes the decomposition of sets of finite…

Let $R$ be a unital ring satisfying the invariant basis number property, that every stably free $R$-module is free, and that the complex of partial bases of every finite rank free module is Cohen--Macaulay. This class of rings includes…

If $R$ is a commutative ring, $M$ a compact $R$-oriented manifold and $G$ a finite graph without loops or multiple edges, we consider the graph configuration space $M^G$ and a Bendersky-Gitler type spectral sequence converging to the…

We say that finite groups are isospectral if they have the same sets of orders of elements. It is known that every nonsolvable finite group $G$ isospectral to a finite simple group has a unique nonabelian composition factor, that is, the…

Let J be a strongly stable monomial ideal in P=k[X0,...,Xn] and let BSt(J) be the family of all the homogeneous ideals in P such that the set N(J) of all the monomials that do not belong to J is a k-vector basis of the quotient P/I. We show…

Let $A$ be a Dedekind domain of characteristic zero such that its localization at every maximal ideal has mixed characteristic with finite residue field. Let $R=A[X_1,\ldots, X_n]$ be a polynomial ring and $I=(a_1U_1, \ldots, a_c…

In this short note we give an elementary description of the linear part of the minimal free resolution of a Stanley-Reisner ring of a simplicial complex $\Delta$. Indeed, the differentials in the linear part are simply a compilation of…

Let $\sigma$ be a simple involution of an algebraic semisimple group $G$ and let $H$ be the subgroup of $G$ of points fixed by $\sigma$. If the restricted root system is of type $A$, $C$ or $BC$ and $G$ is simply connected or if the…

An algorithm and its first implementation in C# are presented for assembling arbitrary quantum circuits on the base of Hadamard and Toffoli gates and for constructing multivariate polynomial systems over the finite field Z_2 arising when…