Related papers: Setting the scene for Betti characters
Let $S$ be the polynomial ring over a field $K$ in a finite set of variables, and let $ \mathfrak{m}$ be the graded maximal ideal of $S$. It is known that for a finitely generated graded $S$-module $M$ and all integers $k\gg 0$, the module…
Given a finite-dimensional module, $V$, for a finite-dimensional, complex, semi-simple Lie algebra $\lie g$ and a positive integer $m$, we construct a family of graded modules for the current algebra $\lie g[t]$ indexed by simple $\CC\lie…
This text, Chapter 23 in the "AutoMathA" handbook, is devoted to the study of rational subsets of groups, with particular emphasis on the automata-theoretic approach to finitely generated subgroups of free groups. Indeed, Stallings'…
Given a finite index subgroup $\Gamma$ of ${\rm{PSL}}_2(\Bbb{Z})$, we investigate Belyi functions on the corresponding modular curve $X(\Gamma)$ by introducing two methods for constructing such functions. Numerous examples have been worked…
In [4], Dunwoody defined resolutions for finitely presented group actions on simplicial trees, that is, an action of the group on a tree with smaller edge and vertex stabilizers. He, moreover, proved that the size of the resolution is…
We present a library autgradalg.lib for the free computer algebra system Singular to compute automorphisms of integral, finitely generated $\mathbb{C}$-algebras that are graded pointedly by a finitely generated abelian group. It implements…
Finite index subgroups of the modular group are of great arithmetic importance. Farey symbols, introduced by Ravi Kulkarni in 1991, are a tool for working with these groups. Given such a group $\Gamma$, a Farey symbol for $\Gamma$ is a…
One aspect of the Langlands program for linear groups is lifting of characters, which relates virtual representations on a group $G$ with those on an endoscopic group for $G$. The goal of this paper is to extend this theory to nonlinear…
We construct an action of a free resolution of the Frobenius properad on the differential forms of a closed oriented manifold. As a consequence, the forms of a manifold with values in a semi-simple Lie algebra have an additional structure…
A part of Grothendieck's program for studying the Galois group $G_{\mathbb Q}$ of the field of all algebraic numbers $\overline{\mathbb Q}$ emerged from his insight that one should lift its action upon $\overline{\mathbb Q}$ to the action…
We prove that finitely generated free metabelian groups $\Psi_n$ are profinitely rigid in the absolute sense: they are distinguished by their finite quotients among all finitely generated residually finite groups. The proof is based on a…
We classify (up to quasi-isomorphism) the free differential modules whose homology is equal to a given module $M$ by developing a theory for deforming an arbitrary free complex into a differential module. We use an iterative approach to…
Given an arbitrary field k and an arithmetic sequence of positive integers m_0<...<m_n, we consider the affine monomial curve parameterized by X_0=t^{m_0},...,X_n=t^{m_n}. In this paper, we conjecture that the Betti numbers of its…
The {\tt Macaulay2} package {\tt RandomMonomialIdeals} provides users with a set of tools that allow for the systematic generation and study of random monomial ideals. It also introduces new objects, Sample and Model, to allow for…
The purpose of this paper is to discuss representations of high order $C^0$ finite element spaces on simplicial meshes in any dimension. When computing with high order piecewise polynomials the conditioning of the basis is likely to be…
The Weyl closure is a basic operation in algebraic analysis: it converts a system of differential operators with rational coefficients into an equivalent system with polynomial coefficients. In addition to encoding finer information on the…
This paper gives a sharp upper bound for the Betti numbers of a finitely generated multigraded $R$-module, where $R=\Bbbk [x_{1},...,x_{m}]$ is the polynomial ring over a field $\Bbbk$ in $m$ variables. The bound is given in terms of the…
A recent result of Eisenbud-Schreyer and Boij-S\"oderberg proves that the Betti diagram of any graded module decomposes as a positive rational linear combination of pure diagrams. When does this numerical decomposition correspond to an…
The focus of this paper is the study of the moduli space of representations of fundamental groupoids of surfaces $\Sigma$ with boundaries with values in $G:=GL_n(\mathbb C)$. In absence of marked points on the boundary, this moduli space is…
The category of crossed complexes gives an algebraic model of the category of $CW$-complexes and cellular maps. We explain basic results on crossed complexes which allow the computation of free crossed resolutions of graph products of…