Why aren't representations of monoids studied so much?

Question

It seems to me like every book on representation theory leaps into groups right away, even though the underlying ideas, such as representations, convolution algebras, etc. don't really make explicit use of inverses. This leads to the fairly natural question of how much of representation theory still works for monoids (or even semigroups)?

I would imagine that irreducible representations probably exist in some form, since all you really need is to have some way to reduce monoid-modules over the complex numbers. Of course it could also be that monoid representation theory is in some form fatally flawed and reduces down to something boring for reasons that are not obvious to me at this point.

Answer 1

Certainly irreducible representations exist; one can still construct the monoid algebra of a monoid and consider modules over the algebra. But Maschke's theorem is false in general for finite monoids. Indeed, consider the monoid $M = \langle x | x^3 = x^2 \rangle$. Complex (for the sake of argument) representations of $M$ are the same as representations of the monoid algebra $\mathbb{C}[x]/(x^3 - x^2)$ and this algebra is not semisimple, so its finite-dimensional representations are not completely reducible. (Note that the usual proof of Maschke's theorem fails miserably; you can't average inner products over a monoid without inverses, and any unitary representation of a monoid has to factor through the Grothendieck group).

So part of the answer may just be that the representation theory of monoids is inherently more complicated. Although this isn't doesn't seem to be stressed much in textbooks, having inverses is a pretty important structural property of groups; it endows group algebras with an antipode and endows the category of representations of a group with duals.

Answer 2

Whenever you see people using Young tableaux to discuss representations of $GL_n$ they include an apologetic "we'll only consider polynomial representations, i.e. not $det^{-1}$". That is to say, they're really thinking about representations of the monoid $M_n$. But monoid representations (and monoids) aren't as fashionable as group representations, I guess. This case is special too in that $GL_n$ is dense in $M_n$, so the group rep determines the monoid rep.

Why do they want $M_n$ reps rather than $GL_n$? Because they really want reps of the whole category $\bf Vec$, i.e. functors ${\bf Vec} \to {\bf Vec}$, like "alt square". Then one can restrict such a functor to the single object ${\mathbb C}^n$ and its endomorphisms, obtaining a rep of $M_n$, and then further restrict to $GL_n$.

Answer 3

The representation theory of finite semigroups is an interesting blend of group representation theory and the representation theory of finite dimensional algebras. The subject is both old, going back to A.H. Clifford (from Clifford theory in group representation theory), and at the same time is in its infancy.

The reason why semigroup representation theory is not so well studied, in my opinion, lies in its origins. A description of the simple modules for a finite semigroup was given by work of Clifford, Munn and Ponizovsky in the forties and fifties. It was further clarified by Rhodes and Zalcstein and by Lallement and Petrich in the sixties. Roughly speaking the main theorem states that all irreducible representations of a finite semigroup can be constructed from irreducible representations of associated finite groups in a very explicit way. Sadly, this beautiful work was written up using very heavily the structure theory of finite semigroups, which is not widely known, and so the literature is virtually inaccessible to nonspecialists. The approach used here foreshadows the development of stratified and quasihereditary algebras by Cline, Parshall and Scott. In fact, in 1972, Nico computed a bound on the global dimension of the algebra of a finite von Neumann regula semigroup by finding a sequence of heredity ideals and discovering the bound it gives on global dimension years before the notion of heredity ideal was invented.

The character table of a finite semigroup was investigated in the sixties and seventies and shown to be invertible (although it is not orthogonal like in the group case). A method for writing a class function as a linear combination of irreducible characters was given that amalgamated the group situation with Möbius inversion in posets.

Progress on finite semigroup representation theory then more of less stalled for a number of years. I believe this was for two main reasons.

1. There was a lack of ready-made applications.
2. Semigroup algebras are almost never semisimple and the modern representation theory of quivers, etc. were only invented on the seventies. By then finite semigroup theorists were interested in other problems and they were mostly unaware of developments in the representation theory of finite dimensional algebras.

In the eighties and early nineties, there was some renewed investigation of the representation theory of finite semigroups, mostly due to Putcha, Okninski, Renner and their collaborators. In particular, connections with quivers and quasihereditary algebras and other aspects of modern representation theory were made.

The past decade has seen a renaissance in the subject of semigroup representation theory, spurred on by probabilists and algebraic combinatorialists. Bidigare, Hanlon, Rockmore, Diaconis and Brown, to name a few, have shown that a number of random walks are much more easily analyzed using semigroup representation than using group representation theory. For instance, it is nearly trivial to compute eigenvalues for the riffle shuffle and the top-to-random shuffle using semigroup theory. It is more difficult to use the representation theory of the symmetric group. Moreover, the diagonalizabilty of these walks is not explained by group theory, but it is explained by semigroup theory.

Also Bidigare's observation that Solomon's descent algebra associated to a finite Coxeter group is a subalgebra of a hyperplane face semigroup algebra has been important to people in algebraic combinatorics. There are also applications of semigroup theory to automata theory in particular in connection to the notorious Cerny conjecture on synchronizing automata.

In the last year, a half-dozen papers on semigroup representation theory have appeared on the ArXiv, many by nonsemigroup theorists. I expect that the trend will continue. We now know how to compute the quiver for a large class of finite semigroups, describe in semigroup theoretic terms projective indecomposable modules and for some classes we have techniques for computing global dimension. Semigroups with basic algebras over a given field have been described.

What is needed is a book covering all this for the general public!

Edit. Our new paper gives a close connection between monoid representation theory, poset topology and Leray numbers of simplicial complexes with classifying spaces of small categories thrown in. If browsing this paper doesn't convince you that monoid representation theory has something to it, then I don't know what will.

Edit. (2/18/14) Since this question just got bumped, let me add the new paper http://arxiv.org/abs/1401.4250 which gives a general introduction to Markov chains and semigroup representation theory and new examples.

Edit(4/1/15). Since this question just got bumped again, let me add that I am in the process of writing a book on the representation theory of monoids. In a sense I started writing this book because of this question (which was the first MO question I ever answered). Hopefully the book will be an answer to this question. I will make a link available shortly from my blog page.

Edit. (Question was bumped again) The book is published, and called "Representation theory of finite monoids." Link to the book at publisher's page

Answer 4

One standard example of a monoid is the monoid $\mathbb N$ of natural numbers. The monoid ring $\mathbb C[\mathbb N]$ is equal to the polynomial ring $\mathbb C[T]$; the study of this ring and its modules (which are just representations of $\mathbb N$) is a pretty standard topic in algebra. More generally, if the ideal $I \subset \mathbb C[T_1,\ldots,T_n]$ is generated by differences of pairs of monomials $m_i(\underline{T}) - m_i'(\underline{T})$ ($i = 1,\ldots,s$), then the quotient $\mathbb C[T_1,\ldots,T_n]/I$ is equal to the monoid ring of the associated commutative monoid $$\langle T_1,\ldots, T_n \, | \, m_i(\underline{T}) = m_i'(\underline{T}) (i = 1,\ldots,s)\rangle.$$ (In his answer, Qiaochu gives the example of $\mathbb C[T]/(T^2-T^3)$.) These kind of rings come up quite a bit in the study of toric geometry and log geometry; they are particularly nice examples of affine rings. Representations of the corresponding monoids are just modules over these rings.

However, when people study these rings and their modules, they tend to use the language of commutative algebra and algebraic geometry (and also quite a lot of combinatorial language, in the context of toric geometry). For the reasons noted in other answers, there doesn't seem to be that much advantage to using a more representation-theoretic viewpoint, because these rings are less special (among all rings) than group rings are.

Answer 5

Representations of special sorts of monoids are studied quite a lot: one examples is quivers, whose representation theory is more or less the same as that of their associated monoid.

Answer 6

One implicit aspect of the question is the intrinsic interest of studying monoid representations. This is addressed by Qiaochu's answer and comments on it. But I'd emphasize more the role of applicability. Semigroups (including monoids) have been developed especially in functional analysis, in the setting of semigroups of operators. It's not clear to me whether standard ideas of representation theory are helpful here, or whether representations of monoids will have natural applications elsewhere.

On the other hand, groups and group actions have been a major motivator for the development of group representations: symmetry questions are ubiquitous. There have been widespread applications of classical theories involving finite groups or Lie groups in geometry, combinatorics, number theory, mathematical physics, chemistry, etc. Increasingly some less classical theories (including Kac-Moody algebras, "quantum groups", $p$-adic and modular representations) have similarly become influential.

It's also true that mathematicians find considerable intrinsic beauty in Frobenius-Schur character theory for finite groups, Cartan-Weyl theory for compact Lie groups, and more recent work of numerous people on representation theory of reductive Lie groups or algebraic groups over various fields. This is independent of applicability. Little of comparable breadth or depth has so far emerged for monoids, except perhaps for the systematic work of Mohan Putcha and Lex Renner on structure and representations of linear algebraic monoids.

Answer 7

A short note (relevant to Jim's) is that representations of very specific families of monoids do get studied quite a bit. In studying random walks on groups (which is where I first really encountered representations), for example, representations of the whole group are used to make very precise calculations, and people are starting to use representations of monoids to make these sorts of calculations for less nice walks.

As pointed out, this is generally quite a bit harder to do. Quick googling gave http://mathstat.carleton.ca/~bsteinbg/pubs/mobius2.pdf, but it seems to give less than a perfect analogue (though he does seem to do lots of more standard work in representations of monoids). There are other examples that make this look more like the classical group case, but unfortunately I can't find any at the moment...

Answer 8

There is a very useful surveys of Donald B. McAlister:

Representations of semigroups by linear transformations. Semigroup Forum, 1971, v.2, N 1, 189-263,

Representations of semigroups by linear transformations: Part II. Semigroup Forum, 1971, v.2, N 1, 283-320

Answer 9

In the previous answers, almost everyone stressed on the reasons why have the representations of monoids not been studied widely, which probably answers your question.

But I would like to mention a couple of interesting cases where the structure of monoid algebras are fully understood (over $\mathbb{C}$ at least).

1.Brauer Algebras, which are introduced by Richard Brauer.

2.Temperley-lieb Algebras, which are quotients of Hecke algebras.

3.Partition Algebra, which are introduced and studied by Paul Martin.

All of the above examples are part of a wider branch which is called Diagram Algebras.

Answer 10

One of the reasons is that it is not natural for the theory of monoids to consider representations by full linear transformations of a vector space. In the case of quivers (mentioned above), the representation is by partial linear transformations (between proper subspaces), for example. Then it is very rare that the set of irreducible representations is finite. That set is very interesting (I think) but not studied at all. In general, the set of all possible representations of a monoid in a given vector space by partial linear maps is very interesting and basically unknown even for the simplest monoids. Especially important is the case of inverse semigroups and representations by partial isomorphisms.

Answer 11

In my experience, monoids, inverse semigroups and so on lack quotients, orbits and averaging over orbits.

This make them less powerful than groups, where one easily can consider quotients like $G/H$ between groups and $X/G$ between space and group.

Answer 12

Maybe representations of monoids in general just aren't a very natural class of things to consider. Several answers discuss how good theories result by specializing to restricted classes of monoids. Conversely, I can think of at least two natural generalizations to consider:

• $k$-Representations of a monoid $M$ are representations of the monoid algebra $k[M]$. Representations of rings are the same thing as modules, which are of course well-studied.

• A monoid is just a category with one object, and representations of categories are also widely studied (they're usually called representations of "quivers with relations").

Studying general monoid representations basically amounts the "intersection" of these two subjects -- module theory and quiver-with-relation representations. My hunch is that there's not much more to say -- I would guess that just about any general statement about monoid representations generalizes to at least one of the more general classes -- a statement about modules, or a statement about quiver-with-relation representations.