arXiv subjects

Itamar Stein

Publications and source records attributed to Itamar Stein.

12 recordsLinked to original sources

arXiv2026

The representation theory of the wreath product of a finite group with the monoid of all partial functions on a finite set as an EI-category algebra

Let $G$ be a finite group. We provide a description of the ordinary quiver of the complex monoid algebra of the wreath product $G \wr \mathrm{PT}_n$, where $\mathrm{PT}_n$ denotes the monoid of all partial functions on an $n$-element set. This description depends on the multiplicities of simple $G$-modules appearing in the decomposition of tensor products of simple $G$-modules. We also prove that the global dimension of this algebra is $n-1$. Both results are obtained by analyzing the associated Ehresmann EI-category related to the monoid. Finally, we describe the quiver of the algebra of the wreath product of $G$ with the submonoid of all order-preserving partial functions.

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arXiv2025

Representation theory of monoids consisting of order-preserving functions and order-reversing functions on an n-set

Let $\operatorname{OD}_{n}$ be the monoid of all order-preserving functions and order-reversing functions on the set $\{1,\ldots,n\}$. We describe a quiver presentation for the monoid algebra $\Bbbk\operatorname{OD}_{n}$ where $\Bbbk$ is a field whose characteristic is not 2. We show that the quiver consists of two straightline paths, one with $n-1$ vertices and one with $n$ vertices, and that all compositions of consecutive arrows are equal to $0$. As part of the proof we obtain a complete description of all homomorphisms between induced left Sch\"utzenberger modules of $\Bbbk\operatorname{OD}_{n}$. We also define $\operatorname{COD}_{n}$ to be a covering of $\operatorname{OD}_{n}$ with an artificial distinction between order-preserving and order-reversing constant functions. We show that $\operatorname{COD}_{n}\simeq\operatorname{O}_{n}\rtimes\mathbb{Z}_{2}$ where $\operatorname{O}_{n}$ is the monoid of all order-preserving functions on the set $\{1,\ldots,n\}$. Moreover, if $\Bbbk$ is a field whose characteristic is not $2$ we prove that $\Bbbk\operatorname{COD}_{n}\simeq\Bbbk\operatorname{O}_{n}\times\Bbbk\operatorname{O}_{n}$. As a corollary, we deduce that the quiver of $\Bbbk\operatorname{COD}_{n}$ consists of two straightline paths with n vertices, and that all compositions of consecutive arrows are equal to $0$.

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arXiv2024

The algebra of the monoid of order-preserving functions on an $n$-set and other reduced $E$-Fountain semigroups

With every reduced $E$-Fountain semigroup $S$ which satisfies the generalized right ample condition we associate a category with zero morphisms $\mathcal{C}(S)$. Under some assumptions we prove an isomorphism of $\Bbbk$-algebras $\Bbbk S\simeq\Bbbk_{0}\mathcal{C}(S)$ between the semigroup algebra and the contracted category algebra where $\Bbbk$ is any commutative unital ring. This is a simultaneous generalization of a former result of the author on reduced E-Fountain semigroups which satisfy the congruence condition, a result of Junying Guo and Xiaojiang Guo on strict right ample semigroups and a result of Benjamin Steinberg on idempotent semigroups with central idempotents. The applicability of the new isomorphism is demonstrated with two well-known monoids which are not members of the above classes. The monoid of order-preserving functions on an $n$-set and the monoid of binary relations with demonic composition.

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arXiv2023

Algebras of reduced $E$-Fountain semigroups and the generalized ample identity II

We study the generalized right ample identity, introduced by the author in a previous paper. Let $S$ be a reduced $E$-Fountain semigroup which satisfies the congruence condition. We can associate with $S$ a small category $\mathcal{C}(S)$ whose set of objects is identified with the set $E$ of idempotents and its morphisms correspond to elements of $S$. We prove that $S$ satisfies the generalized right ample identity if and only if every element of $S$ induces a homomorphism of left $S$-actions between certain classes of generalized Green's relations. In this case, we interpret the associated category $\mathcal{C}(S)$ as a discrete form of a Peirce decomposition of the semigroup algebra. We also give some natural examples of semigroups satisfying this identity.

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arXiv2021

Algebras of Reduced $E$-Fountain Semigroups and the Generalized Ample Identity

Let $S$ be a reduced $E$-Fountain semigroup. If $S$ satisfies the congruence condition, there is a natural construction of a category $\mathcal{C}$ associated with $S$. We define a $\Bbbk$-module homomorphism $φ:\Bbbk S\to\Bbbk\mathcal{C}$ (where $\Bbbk$ is any unital commutative ring). With some assumptions, we prove that $φ$ is an isomorphism of $\Bbbk$-algebras if and only if some weak form of the right ample identity holds in $S$. This gives a unified generalization for a result of the author on right restriction $E$-Ehresmann semigroups and a result of Margolis and Steinberg on the Catalan monoid.

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arXiv2021

Ehresmann Semigroups Whose Categories are EI and Their Representation Theory : Extended Version

We study simple and projective modules of a certain class of Ehresmann semigroups, a well-studied generalization of inverse semigroups. Let $S$ be a finite right (left) restriction Ehresmann semigroup whose corresponding Ehresmann category is an EI-category, that is, every endomorphism is an isomorphism. We show that the collection of finite right restriction Ehresmann semigroups whose categories are EI is a pseudovariety. We prove that the simple modules of the semigroup algebra $\Bbbk S$ (over any field $\Bbbk$) are formed by inducing the simple modules of the maximal subgroups of $S$ via the corresponding Schützenberger module. Moreover, we show that over fields with good characteristic the indecomposable projective modules can be described in a similar way but using generalized Green's relations instead of the standard ones. As a natural example, we consider the monoid $\mathcal{PT}_{n}$ of all partial functions on an $n$-element set. Over the field of complex numbers, we give a natural description of its indecomposable projective modules and obtain a formula for their dimension. Moreover, we find certain zero entries in its Cartan matrix.

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arXiv2020

Diamond Subgraphs in the Reduction Graph of a One-Rule String Rewriting System

In this paper, we study a certain case of a subgraph isomorphism problem. We consider the Hasse diagram of the lattice $M_{k}$ (the unique lattice with $k+2$ elements and one anti-chain of length $k$) and want to find the maximal $k$ for which it is isomorphic to a subgraph of the reduction graph of a given one-rule string rewriting system. We obtain a complete characterization for this problem and show that there is a dichotomy. There are one-rule string rewriting systems for which the maximal such $k$ is $2$ and there are cases where there is no maximum. No other intermediate option is possible.

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arXiv2019

Representation theory of order-related monoids of partial functions as locally trivial category algebras

In this paper we study the representation theory of three monoids of partial functions on an $n$-set. The monoid of all order-preserving functions (i.e., functions satisfying $f(x)\leq f(y)$ if $x\leq y$) the monoid of all order-decreasing functions (i.e. functions satisfying $f(x)\leq x$) and their intersection (also known as the partial Catalan monoid). We use an isomorphism between the algebras of these monoids and the algebras of some corresponding locally trivial categories. We obtain an explicit description of a quiver presentation for each algebra. Moreover, we describe other invariants such as the Cartan matrix and the Loewy length.

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arXiv2018

The global dimension of the algebra of the monoid of all partial functions on an $n$-set as the algebra of the EI-category of epimorphisms between subsets

We prove that the global dimension of the complex algebra of the monoid of all partial functions on an n-set is $n-1$ for all $n\geq 1$. This is also the global dimension of the complex algebra of the category of all epimorphisms between subsets of an $n$-set. In our proof we use standard homological methods as well as combinatorial techniques associated to the representation theory of the symmetric group. As part of the proof, we obtain a partial description of the Cartan matrix of these algebras.

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arXiv2017

Algebras of Ehresmann semigroups and categories

$E$-Ehresmann semigroups are a commonly studied generalization of inverse semigroups. They are closely related to Ehresmann categories in the same way that inverse semigroups are related to inductive groupoids. We prove that under some finiteness condition, the semigroup algebra of an $E$-Ehresmann semigroup is isomorphic to the category algebra of the corresponding Ehresmann category. This generalizes a result of Steinberg who proved this isomorphism for inverse semigroups and inductive groupoids and a result of Guo and Chen who proved it for ample semigroups. We also characterize $E$-Ehresmann semigroups whose corresponding Ehresmann category is an EI-category and give some natural examples. Erratum: Shoufeng Wang discovered an error in the main theorem of the paper. Wang observed that the function we suggest as an isomorphism is not a homomorphism unless the semigroup being discussed is left restriction. In order to fix our mistake we will add this assumption. Note that our revised result is still a generalization of earlier work of Guo and Chen, the author, and Steinberg .

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arXiv2016

The Littlewood-Richardson rule for wreath products with symmetric groups and the quiver of the category $F \wr {\bf FI}_n$

We give a new proof for the Littlewood-Richardson rule for the wreath product $F \wr S_{n}$ where $F$ is a finite group. Our proof does not use symmetric functions but more elementary representation theoretic tools. We also derive a branching rule for inducing the natural embedding of $F\wr S_{n}$ to $F\wr S_{n+1}$. We then apply the generalized Littlewood-Richardson rule for computing the ordinary quiver of the category $F \wr {\bf FI}_{n}$ where ${\bf FI}_{n}$ is the category of all injective functions between subsets of an $n$-element set.

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arXiv2015

The representation theory of the monoid of all partial functions on a set and related monoids as EI-category algebras

The (ordinary) quiver of an algebra $A$ is a graph that contains information about the algebra's representations. We give a description of the quiver of $\mathbb{C}PT_{n}$, the algebra of the monoid of all partial functions on $n$ elements. Our description uses an isomorphism between $\mathbb{C}PT_{n}$ and the algebra of the epimorphism category, $E_{n}$, whose objects are the subsets of $\{1,\ldots, n\}$ and morphism are all total epimorphisms. This is an extension of a well known isomorphism of the algebra of $IS_{n}$ (the monoid of all partial injective maps on $n$ elements) and the algebra of the groupoid of all bijections between subsets of an $n$-element set. The quiver of the category algebra is described using results of Margolis, Steinberg and Li on the quiver of EI-categories. We use the same technique to compute the quiver of other natural transformation monoids. We also show that the algebra $\mathbb{C}PT_{n}$ has three blocks for $n>1$ and we give a natural description of the descending Loewy series of $\mathbb{C}PT_{n}$ in the category form.

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