Searcharxiv⌕ Search

arXiv subjects

Benjamin Steinberg

Publications and source records attributed to Benjamin Steinberg.

At least 37 records · Page 2Linked to original sources
arXiv2023

A note on projections in étale groupoid algebras and diagonal preserving homomorphisms

Carlsen (Adv.~Math, 2018) showed that any $\ast$-homomorphism between Leavitt path algebras over $\mathbb Z$ is automatically diagonal preserving and hence induces an isomorphism of boundary path groupoids. His result works over conjugation-closed subrings of $\mathbb C$ enjoying certain properties. In this paper, we characterize the rings considered by Carlsen as precisely those rings for which every $\ast$-homomorphism of algebras of Hausdorff ample groupoids is automatically diagonal preserving. Moreover, the more general groupoid result has a simpler proof.

math.RA↗
arXiv2023

The modular representation theory of monoids

This paper develops the fundamentals of modular representation theory for finite monoids, introducing the decomposition matrix and exploring its connection to Brauer characters. We define modular characteristic and explain how the representation theory in nonmodular positive characteristic behaves like the characteristic zero theory by showing that one can lift from nonmodular characteristic $p$ all simple and projective indecomposable modules, as well as a quiver presentation of the basic algebra. As an application of the theory developed, we give a new proof of Glover's theorem that the monoid of $2\times 2$-matrices over $\mathbb F_p$ has infinite representation type over fields of characteristic $p$. We also investigate the relationship between nonsingularity of the Cartan matrix of a monoid algebra in characterstic zero and in positive characteristic. We show that for von Neumann regular monoids the Cartan matrix is always nonsingular and we show that if a monoid has aperiodic left (or right) stabilizers, then nonsingularity in characteristic zero implies nonsingularity in positive characteristic. Florian Eisele has recently shown that the Cartan matrix of a monoid algebra can be nonsingular in characteristic zero and singular in positive characteristic, disproving a conjecture of the author in an earlier version of this paper. A new conjecture is proposed, unifying the cases of regular monoids and monoids with aperiodic stabilizers.

math.RT↗
arXiv2023

On the minimal faithful degree of Rhodes semisimple semigroups

In this paper we compute the minimum degree of a faithful representation by partial transformations of a finite semigroup admitting a faithful completely reducible matrix representation over the field of complex numbers. This includes all inverse semigroups, and hence our results generalize earlier results of Easdown and Schein on the minimal faithful degree of an inverse semigroup. It also includes well-studied monoids like full matrix monoids over finite fields and the monoid of binary relations (i.e., matrices over the Boolean semiring). Our answer reduces the computation to considerations of permutation representations of maximal subgroups that are faithful when restricted to distinguished normal subgroups. This is analogous to (and inspired by) recent results of the second author on the minimal number of irreducible constituents in a faithful completely reducible complex matrix representation of a finite semigroup that admits one. To illustrate what happens when a finite semigroup does not admit a faithful completely reducible representation, we compute the minimum faithful degree of the opposite monoid of the full transformation monoid.

math.GR↗
arXiv2022

Finitely presented inverse semigroups with finitely many idempotents in each $\mathcal D$-class and non-Hausdorff universal groupoids

The complex algebra of an inverse semigroup with finitely many idempotents in each $\mathcal D$-class is stably finite by a result of Munn. This can be proved fairly easily using $C^*$-algebras for inverse semigroups satisfying this condition that have a Hausdorff universal groupoid, or more generally for direct limits of inverse semigroups satisfying this condition and having Hausdorff universal groupoids. It is not difficult to see that a finitely presented inverse semigroup with a non-Hausdorff universal groupoid cannot be a direct limit of inverse semigroups with Hausdorff universal groupoids. We construct here countably many non-isomorphic finitely presented inverse semigroups with finitely many idempotents in each $\mathcal D$-class and non-Hausdorff universal groupoids. At this time there is not a clear $C^*$-algebraic technique to prove these inverse semigroups have stably finite complex algebras.

math.GR↗
arXiv2022

Extensions of theorems of Gaschütz, Žmud$'$ and Rhodes on faithful representations

Gaschütz (1954) proved that a finite group $G$ has a faithful irreducible complex representation if and only if its socle is generated by a single element as a normal subgroup; this result extends to arbitrary fields of characteristic $p$ so long as $G$ has no nontrivial normal $p$-subgroup. Žmud$'$ (1956) showed that the minimum number of irreducible constituents in a faithful complex representation of $G$ coincides with the minimum number of generators of its socle as a normal subgroup; this result can also be extended to arbitrary fields of any characteristic $p$ such that $G$ has no nontrivial normal $p$-subgroup (i.e., over which $G$ admits a faithful completely reducible representation). Rhodes (1969) characterized the finite semigroups admitting a faithful irreducible representation over an arbitrary field as generalized group mapping semigroups over a group admitting a faithful irreducible representation over the field in question. Here, we provide a common generalization of the theorems of Žmud$'$ and Rhodes by determining the minimum number of irreducible constituents in a faithful completely reducible representation of a finite semigroup over an arbitrary field (provided that it has one). Our key tool for the semigroup result is a relativized version of Žmud$'$'s theorem that determines, given a finite group $G$ and a normal subgroup $N\lhd G$, what is the minimum number of irreducible constituents in a completely reducible representation of $G$ whose restriction to $N$ is faithful.

math.RT↗
arXiv2021

Reconstruction of twisted Steinberg algebras

We show how to recover a discrete twist over an ample Hausdorff groupoid from a pair consisting of an algebra and what we call a quasi-Cartan subalgebra. We identify precisely which twists arise in this way (namely, those that satisfy the local bisection hypothesis), and we prove that the assignment of twisted Steinberg algebras to such twists and our construction of a twist from a quasi-Cartan pair are mutually inverse. We identify the algebraic pairs that correspond to effective groupoids and to principal groupoids. We also indicate the scope of our results by identifying large classes of twists for which the local bisection hypothesis holds automatically.

math.RA↗
arXiv2021

Factoring the Dedekind-Frobenius determinant of a semigroup

The representation theory of finite groups began with Frobenius's factorization of Dedekind's group determinant. In this paper, we consider the case of the semigroup determinant. The semigroup determinant is nonzero if and only if the complex semigroup algebra is Frobenius, and so our results include applications to the study of Frobenius semigroup algebras. We explicitly factor the semigroup determinant for commutative semigroups and inverse semigroups. We recover the Wilf-Lindström factorization of the semigroup determinant of a meet semilattice and Wood's factorization for a finite commutative chain ring. The former was motivated by combinatorics and the latter by coding theory over finite rings. We prove that the algebra of the multiplicative semigroup of a finite Frobenius ring is Frobenius over any field whose characteristic doesn't divide that of the ring. As a consequence we obtain an easier proof of Kovács's theorem that the algebra of the monoid of matrices over a finite field is a direct product of matrix algebras over group algebras of general linear groups (outside of the characteristic of the finite field).

math.RT↗
arXiv2021

On the simplicity of Nekrashevych algebras of contracting self-similar groups

Nekrashevych algebras of self-similar group actions are natural generalizations of the classical Leavitt algebras. They are discrete analogues of the corresponding Nekrashevych $C^\ast$-algebras. In particular, Nekrashevych, Clark, Exel, Pardo, Sims and Starling have studied the question of simplicity of Nekrashevych algebras, in part, because non-simplicity of the complex algebra implies non-simplicity of the $C^\ast$-algebra. In this paper we give necessary and sufficient conditions for the Nekrashevych algebra of a contracting group to be simple. Nekrashevych algebras of contracting groups are finitely presented. We give an algorithm which on input the nucleus of the contracting group, outputs all characteristics of fields over which the corresponding Nekrashevych algebra is simple. Using our methods, we determine the fields over which the Nekrashevych algebras of a number of well-known contracting groups are simple including the Basilica group, Gupta-Sidki groups, GGS-groups, multi-edge spinal groups, Šunić groups associated to polynomials (this latter family includes the Grigorchuk group, Grigorchuk-Erschler group and Fabrykowski-Gupta group) and self-replicating spinal automaton groups.

math.RA↗
arXiv2021

The quiver of an affine monoid

If $R$ is a finite commutative ring, then the affine monoid of $R$ is the monoid of all affine mappings $x\mapsto ax+b$ on $R$. Alternatively, it is the semidirect product of the multiplicative monoid of $R$ with the additive group of $R$. In this paper we compute the Gabriel quiver of the complex algebra of the affine monoid of any finite commutative ring.

math.RT↗
arXiv2021

Twists, crossed products and inverse semigroup cohomology

Twisted étale groupoid algebras have been studied recently in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper, we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups and we show that they are classified by Lausch's second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists. We also define a novel notion of inverse semigroup crossed product, generalizing skew inverse semigroup rings, and prove that twisted Steinberg algebras of Hausdorff ample groupoids are instances of inverse semigroup crossed products. The cocycle defining the crossed product is the same cocycle that classifies the twist in Lausch cohomology.

math.RA↗
arXiv2020

Simplicity of inverse semigroup and étale groupoid algebras

In this paper, we prove that the algebra of an étale groupoid with totally disconnected unit space has a simple algebra over a field if and only if the groupoid is minimal and effective and the only function of the algebra that vanishes on every open subset is the null function. Previous work on the subject required the groupoid to be also topologically principal in the non-Hausdorff case, but we do not. Furthermore, we provide the first examples of minimal and effective but not topologically principal étale groupoids with totally disconnected unit spaces. Our examples come from self-similar group actions of uncountable groups. More generally, we show that the essential algebra of an étale groupoid (the quotient by the ideal of functions vanishing on every open set), is simple if and only if the groupoid is minimal and topologically free, generalizing to the algebraic setting a recent result for essential $C^*$-algebras. The main application of our work is to provide a description of the simple contracted inverse semigroup algebras, thereby answering a question of Munn from the seventies. Using Galois descent, we show that simplicity of étale groupoid and inverse semigroup algebras depends only on the characteristic of the field and can be lifted from positive characteristic to characteristic $0$. We also provide examples of inverse semigroups and étale groupoids with simple algebras outside of a prescribed set of prime characteristics.

math.RA↗
arXiv2020

Decidability of flow equivalence and isomorphism problems for graph C*-algebras and quiver representations

We note that the deep results of Grunewald and Segal on algorithmic problems for arithmetic groups imply the decidability of several matrix equivalence problems involving poset-blocked matrices over Z. Consequently, results of Eilers, Restorff, Ruiz and Sørensen imply that isomorphism and stable isomorphism of unital graph C*-algebras (including the Cuntz-Krieger algebras) are decidable. One can also decide flow equivalence for shifts of finite type, and isomorphism of Z-quiver representations (i.e., finite diagrams of homomorphisms of finitely generated abelian groups).

math.OA↗
arXiv2020

The Karoubi envelope of the mirage of a subshift

We study a correspondence associating to each subshift $\mathcal X$ of $A^{\mathbb Z}$ a subcategory of the Karoubi envelope of the free profinite semigroup generated by $A$. The objects of this category are the idempotents in the mirage of $\mathcal X$, that is, in the set of pseudowords whose finite factors are blocks of $\mathcal X$. The natural equivalence class of the category is shown to be invariant under flow equivalence. As a corollary of our proof, we deduce the flow invariance of the profinite group that Almeida associated to each irreducible subshift. We also show, in a functorial manner, that the isomorphism class of the category is invariant under conjugacy. Finally, we see that the zeta function of $\mathcal X$ is naturally encoded in the category. These results hold, with obvious translations, for relatively free profinite semigroups over many pseudovarieties, including all of the form $\overline{\mathsf H}$, with $\mathsf H$ a pseudovariety of groups.

math.GR↗
arXiv2020

Free inverse monoids are not $FP_2$

We give a topological proof that a free inverse monoid on one or more generators is neither of type left-$FP_2$ nor right-$FP_2$. This strengthens a classical result of Schein that such monoids are not finitely presented as monoids.

math.GR↗
arXiv2019

Lamplighter groups, bireversible automata and rational series over finite rings

We realize lamplighter groups $A\wr \mathbb Z$, with $A$ a finite abelian group, as automaton groups via affine transformations of power series rings with coefficients in a finite commutative ring. Our methods can realize $A\wr \mathbb Z$ as a bireversible automaton group if and only if the $2$-Sylow subgroup of $A$ has no multiplicity one summands in its expression as a direct sum of cyclic groups of order a power of $2$.

math.GR↗
arXiv2019

A Lyndon's identity theorem for one-relator monoids

For every one-relator monoid $M = \langle A \mid u=v \rangle$ with $u, v \in A^*$ we construct a contractible $M$-CW complex and use it to build a projective resolution of the trivial module which is finitely generated in all dimensions. This proves that all one-relator monoids are of type ${\rm FP}_\infty$, answering positively a problem posed by Kobayashi in 2000. We also apply our results to classify the one-relator monoids of cohomological dimension at most $2$, and to describe the relation module, in the sense of Ivanov, of a torsion-free one-relator monoid presentation as an explicitly given principal left ideal of the monoid ring. In addition, we prove the topological analogues of these results by showing that all one-relator monoids satisfy the topological finiteness property ${\rm F}_\infty$, and classifying the one-relator moniods with geometric dimension at most $2$. These results give a natural monoid analogue of Lyndon's Identity Theorem for one-relator groups.

math.GR↗
arXiv2019

Random walks on rings and modules

We consider two natural models of random walks on a module $V$ over a finite commutative ring $R$ driven simultaneously by addition of random elements in $V$, and multiplication by random elements in $R$. In the coin-toss walk, either one of the two operations is performed depending on the flip of a coin. In the affine walk, random elements $a \in R,b \in V$ are sampled independently, and the current state $x$ is taken to $ax+b$. For both models, we obtain the complete spectrum of the transition matrix from the representation theory of the monoid of all affine maps on $V$ under a suitable hypothesis on the measure on $V$ (the measure on $R$ can be arbitrary).

math.CO↗