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2016 | Book

1 The Structure Theory of Finite Monoids Read chapter Read first chapter

Representation Theory of Finite Monoids

Author: Benjamin Steinberg

Publisher: Springer International Publishing

Book Series : Universitext

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This first text on the subject provides a comprehensive introduction to the representation theory of finite monoids. Carefully worked examples and exercises provide the bells and whistles for graduate accessibility, bringing a broad range of advanced readers to the forefront of research in the area. Highlights of the text include applications to probability theory, symbolic dynamics, and automata theory. Comfort with module theory, a familiarity with ordinary group representation theory, and the basics of Wedderburn theory, are prerequisites for advanced graduate level study. Researchers in algebra, algebraic combinatorics, automata theory, and probability theory, will find this text enriching with its thorough presentation of applications of the theory to these fields.

Prior knowledge of semigroup theory is not expected for the diverse readership that may benefit from this exposition. The approach taken in this book is highly module-theoretic and follows the modern flavor of the theory of finite dimensional algebras. The content is divided into 7 parts. Part I consists of 3 preliminary chapters with no prior knowledge beyond group theory assumed. Part II forms the core of the material giving a modern module-theoretic treatment of the Clifford –Munn–Ponizovskii theory of irreducible representations. Part III concerns character theory and the character table of a monoid. Part IV is devoted to the representation theory of inverse monoids and categories and Part V presents the theory of the Rhodes radical with applications to triangularizability. Part VI features 3 chapters devoted to applications to diverse areas of mathematics and forms a high point of the text. The last part, Part VII, is concerned with advanced topics. There are also 3 appendices reviewing finite dimensional algebras, group representation theory, and Möbius inversion.

Table of Contents

Frontmatter

Elements of Monoid Theory

Frontmatter
1 The Structure Theory of Finite Monoids
Abstract
This chapter contains those elements of the structure theory of finite monoids that we shall need for the remaining chapters. It also establishes some notation that will be used throughout. More detailed sources for finite semigroup theory include [KRT68, Eil76, Lal79, Alm94, RS09]. Introductory books on the algebraic theory of semigroups, in general, are [CP61, CP67, Hig92, How95]. A detailed study of some of the most important transformation monoids can be found in [GM09]. In this book, all semigroups and monoids will be finite except for endomorphism monoids of vector spaces and free monoids. On a first reading, it may be advisable to skip the proofs in this chapter.
Benjamin Steinberg
2 -trivial Monoids
Abstract
This chapter studies \(\mathcal{R}\)-trivial monoids, that is, monoids where Green’s relation \(\mathcal{R}\) is the equality relation. They form an important class of finite monoids, which have quite recently found applications in the analysis of Markov chains; see Chapter 14 The first section of this chapter discusses lattices and prime ideals of arbitrary finite monoids, as they shall play an important role in both the general theory and in the structure theory of \(\mathcal{R}\)-trivial monoids. The second section concerns \(\mathcal{R}\)-trivial monoids and, in particular, left regular bands, which are precisely the regular \(\mathcal{R}\)-trivial monoids.
Benjamin Steinberg
3 Inverse Monoids
Abstract
An important class of regular monoids is the class of inverse monoids. In fact, many semigroup theorists would assert that inverse monoids form the most important class of monoids outside of groups. They abstract the notion of partial symmetry in much the same way that groups abstract the notion of symmetry. For a detailed discussion of this viewpoint, see Lawson [Law98]. From the perspective of this book they provide a natural class of monoids whose representation theory we can understand as well as that of groups. Namely, we shall see in Chapter 9 that the algebra of an inverse monoid can be explicitly decomposed as a direct product of matrix algebras over the group algebras of its maximal subgroups (one per \(\mathcal{J}\)-class).
Benjamin Steinberg

Irreducible Representations

Frontmatter
4 Recollement: The Theory of an Idempotent
Abstract
In this chapter we provide an account of the theory connecting the category of modules of a finite dimensional algebra A with the module categories of the algebras eAe and A∕AeA, for an idempotent e ∈ A, known as recollement  [BBD82, CPS88, CPS96]. We first learned of this subject from the monograph of Green [Gre80, Chapter 6]. A presentation much closer in spirit to ours is that of Kuhn [Kuh94b]. In the next chapter, we shall apply this theory to construct the irreducible representations of a finite monoid and in a later chapter we shall extend the results to finite categories.
Benjamin Steinberg
5 Irreducible Representations
Abstract
Clifford-Munn-Ponizovskiĭ theory, developed in Clifford [Cli42], Munn [Mun55, Mun57b, Mun60], and Ponizovskiĭ [Pon58] (and in further detail in [LP69, RZ91]), gives a bijection between equivalence classes of irreducible representations of a finite monoid and equivalence classes of irreducible representations of its maximal subgroups (taken one per regular \(\mathcal{J}\)-class). We follow here the approach of [GMS09], using the techniques of Chapter 4 Let us commence by introducing formally the notion of a representation of a monoid.
Benjamin Steinberg

Character Theory

Frontmatter
6 The Grothendieck Ring
Abstract
In this chapter, we introduce the Grothendieck ring of a finite monoid M over a field \(\mathbb{k}\). The main result is that the Grothendieck ring of M is isomorphic to the direct product of the Grothendieck rings of its maximal subgroups (one per regular \(\mathcal{J}\)-class). This result was first proved by McAlister for \(\mathbb{k} = \mathbb{C}\) in the language of virtual characters [McA72]. In Chapter 7, the results of this chapter will be used to study the ring of characters and the character table of a finite monoid. Throughout this chapter we hold fixed a finite monoid M and a field \(\mathbb{k}\).
Benjamin Steinberg
7 Characters and Class Functions
Abstract
In this chapter, we work exclusively over \(\mathbb{C}\), although most of the results hold in greater generality (cf. [MQS15], where the theory is worked out over an arbitrary field). We study the ring \(\mathop{\mathrm{Cl}}\nolimits (M)\) of class functions on a finite monoid M. It turns out that \(\mathop{\mathrm{Cl}}\nolimits (M)\cong \mathbb{C} \otimes _{\mathbb{Z}}\mathop{ G_{0}}\nolimits (\mathbb{C}M)\). The character table of a monoid is defined and shown to be invertible. In fact, it is block upper triangular with group character tables on the diagonal blocks. Inverting the character table allows us to determine, in principle, the composition factors of a representation directly from its character. The fundamental results of this chapter are due to McAlister [McA72] and, independently, to Rhodes and Zalcstein [RZ91].
Benjamin Steinberg

The Representation Theory of Inverse Monoids

Frontmatter
8 Categories and Groupoids
Abstract
In this chapter, we consider a generalization of monoid algebras that will be used in the next chapter to study inverse monoid algebras, namely the algebra of a small category. Further examples include incidence algebras of posets (cf. Appendix C) and path algebras of quivers. We show that the Clifford-Munn-Ponizovskiĭ theory applies equally well to categories. The parametrization of the simple modules for the algebra of a finite category given here could also be obtained from a result of Webb [Web07], reducing to the monoid case, and the Clifford-Munn-Ponizovskiĭ theory, but we give a direct proof. Since category algebras are contracted semigroup algebras, these results also follow from the original results of Munn and Ponizovskiĭ (cf. [CP61, Chapter 5]). A basic reference on category theory is Mac Lane [Mac98]. Category algebras were considered at least as far back as Mitchell [Mit72].
Benjamin Steinberg
9 The Representation Theory of Inverse Monoids
Abstract
In this chapter we develop the representation theory of inverse monoids following the approach of the author [Ste06, Ste08] using groupoid algebras and Möbius inversion. In fact, this work has a precursor in the work of Rukolaĭne [Ruk78, Ruk80], who used alternating sums of idempotents to achieve the same effect as Möbius inversion and used Brandt inverse semigroups instead of groupoids. The author only became aware of the work of Rukolaĭne after [Ste06, Ste08] were published. However, our more explicit approach lets one take advantage of the detailed knowledge of the Möbius function for a number of naturally occurring lattices. We will, for instance, exploit this for our simple character theoretic proof of Solomon’s computation [Sol02] of the tensor powers of the natural module for the symmetric inverse monoid.
Benjamin Steinberg

The Rhodes Radical

Frontmatter
10 Bi-ideals and R. Steinberg’s Theorem
Abstract
In this chapter, we prove R. Steinberg’s Theorem [Ste62] that the direct sum of the tensor powers of a faithful representation of a monoid yields a faithful representation of the monoid algebra. We also commence the study of a special family of ideals, called bi-ideals, which will be at the heart of the next chapter.The results of this chapter should more properly be viewed as about bialgebras, but we have chosen not to work at that level of generality in order to keep things more concrete. The approach we follow here is influenced by Passman [Pas14]. The bialgebraic approach was pioneered by Rieffel [Rie67].
Benjamin Steinberg
11 The Rhodes Radical and Triangularizability
Abstract
In this chapter we provide a correspondence between nilpotent bi-ideals and a certain class of congruences on a finite monoid. We characterize the largest nilpotent bi-ideal, which is called the Rhodes radical because it was first described by Rhodes [Rho69b] in the case of an algebraically closed field of characteristic zero. For simplicity, we only give complete details in characteristic zero. As an application, we characterize those monoids with a faithful representation by upper triangular matrices over an algebraically closed field \(\mathbb{k}\) (the general case will be left to the reader in the exercises). These are precisely the monoids with a basic algebra over \(\mathbb{k}\) and so it also characterizes these monoids. Our treatment of these topics is based, for the most part, on that of Almeida, Margolis, the author, and Volkov [AMSV09].
Benjamin Steinberg

Applications

Frontmatter
12 Zeta Functions of Languages and Dynamical Systems
Abstract
In this chapter, we apply the character theory of finite monoids to provide a proof of a theorem of Berstel and Reutenauer on the rationality of zeta functions of cyclic regular languages [BR90]. This generalizes the rationality of zeta functions of sofic shifts [Man71, LM95], an important result in symbolic dynamics. Background on free monoids, formal languages, and automata can be found in [Eil74, Eil76, Lot97, Lot02, BPR10, BR11].
Benjamin Steinberg
13 Transformation Monoids
Abstract
In this chapter we shall use the representation theory of finite monoids to study finite monoids acting on finite sets. Such actions play an important role in automata theory and we provide here some applications in this direction. In particular, we study connections with the popular Černý conjecture [Č64]; see [Vol08] for a survey. This chapter is primarily based upon the paper [Ste10b].
Benjamin Steinberg
14 Markov Chains
Abstract
A Markov chain is a stochastic process on a finite state space such that the system evolves from one state to another according to a prescribed probabilistic law. For example, card shuffling can be modeled via a Markov chain. The state space is all 52! orderings of a deck of cards. Each step of the Markov chain corresponds to performing a riffle shuffle to the deck.
Benjamin Steinberg

Advanced Topics

Frontmatter
15 Self-injective, Frobenius, and Symmetric Algebras
Abstract
In this chapter we characterize regular monoids with a self-injective algebra. It turns out that the algebra of such a monoid is a product of matrix algebras over group algebras and hence is a symmetric algebra. The results of this chapter are new to the best of our knowledge.
Benjamin Steinberg
16 Global Dimension
Abstract
An important homological invariant of a finite dimensional algebra A is its global dimension. It measures the maximum length of a minimal projective resolution of an A-module. Nico obtained a bound on the global dimension of \(\mathbb{k}M\) for a regular monoid M in “good” characteristic [Nic71, Nic72]. Essentially, Nico discovered that \(\mathbb{k}M\) is a quasi-hereditary algebra in the sense of Cline, Parshall, and Scott [CPS88], more than 15 years before the notion was defined, and proved the corresponding bound on global dimension. Putcha was the first to observe that \(\mathbb{k}M\) is quasi-hereditary [Put98]. Here we provide a direct approach to Nico’s theorem, without introducing the machinery of quasi-hereditary algebras, although it is hidden under the surface (compare with [DR89]). Our approach follows the ideas in [MS11] and uses the results of Auslander, Platzeck, and Todorov [APT92] on idempotent ideals in algebras. In this chapter, we do not hesitate to assume familiarity with notions from homological algebra, in particular, with properties of the \(\mathop{\mathrm{Ext}}\nolimits\)-functor. Standard references for homological algebra are [Wei94, HS97, CE99], although everything we shall need can be found in [Ben98] or the appendix of [ASS06].
Benjamin Steinberg
17 Quivers of Monoid Algebras
Abstract
In this chapter we provide a computation of the quiver of a left regular band algebra, a result of Saliola [Sal07], and of a \(\mathcal{J}\)-trivial monoid algebra, a result of Denton, Hivert, Schilling, and Thiéry [DHST11]. These are special cases of the results of Margolis and the author [MS12a], computing the quiver of an arbitrary rectangular monoid algebra. However, the latter result is much more technical and beyond the scope of this text. We also describe the projective indecomposable modules for \(\mathcal{R}\)-trivial monoid algebras as partial transformation modules. This result is from the paper of Margolis and the author [MS12a] and again generalizes earlier results of Saliola [Sal07] for left regular bands and of Denton, Hivert, Schilling, and Thiéry [DHST11] for \(\mathcal{J}\)-trivial monoids.
Benjamin Steinberg
18 Further Developments
Abstract
This chapter highlights some further developments in the representation theory of finite monoids whose detailed treatment is beyond the scope of this text. No proofs are presented.
Benjamin Steinberg
Appendix A Finite Dimensional Algebras
Abstract
This appendix reviews the necessary background material from the theory of finite dimensional algebras. Standard texts covering most of this subject matter are [CR88, Lam91, Ben98, ASS06]. Readers familiar with this material are urged to skim this chapter or skip it entirely. Very few proofs will be given here, as the results can be found in the references. Let us remark that, unlike the case of group algebras, monoid algebras are seldom semisimple, even over the complex numbers. This forces us to use more of the theory of finite dimensional algebras than would be encountered in a first course on the ordinary representation theory of finite groups. Some of the more elaborate tools, like projective covers, are not used except for in the final chapters of the text. Most of the text uses nothing beyond Wedderburn theory.
Benjamin Steinberg
Appendix B Group Representation Theory
Abstract
This appendix surveys those aspects of the representation theory of finite groups that are used throughout the text. The final section provides a brief overview of the representation theory of the symmetric group in characteristic zero. Monoid representation theory very much builds on group representation theory and so one should have a solid foundation in the latter subject before attempting to master the former. Good references for the representation theory of finite groups are [Isa76, Ser77, CR88]. See also [Ste12a]. We mostly omit proofs here, although we occasionally do provide a sketch or complete proof for convenience of the reader.
Benjamin Steinberg
Appendix C Incidence Algebras and Möbius Inversion
Abstract
The purpose of this appendix is to familiarize the reader with an important combinatorial technique that is used throughout the text. The theory of incidence algebras and Möbius inversion for posets was developed by Rota [Rot64] and can be considered as part of the origins of algebraic combinatorics. It provides a highly conceptual generalization of the principle of inclusion-exclusion. A thorough introduction, including techniques for computing the Möbius function of a poset and connections with algebraic topology, can be found in Stanley’s classic text [Sta97, Chapter 3]. Throughout this appendix, P will denote a finite poset.
Benjamin Steinberg
Backmatter
Metadata
Title
Representation Theory of Finite Monoids
Author
Benjamin Steinberg
Copyright Year
2016
Electronic ISBN
978-3-319-43932-7
Print ISBN
978-3-319-43930-3
DOI
https://doi.org/10.1007/978-3-319-43932-7