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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.

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This book, especially its introduction and references, seems pretty relevant: amazon.com/… –  David Corwin Aug 30 '10 at 4:35
I recommend looking at the paper that Jianrong Li linked to (arxiv.org/abs/0712.2076). I would have said that there's nothing special about representations of finite semigroups versus arbitrary finite-dimensional algebras, but the paper gives a description of the irreducible representations purely in terms of the structure of the semigroup. –  arsmath Dec 29 '10 at 14:15
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11 Answers 11

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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.

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@Davidac897, the reasons for non-semisimplicity of reps of monoids (even in characteristic zero) are different to those of the non-semisimplicity of reps of finite groups in the modular situation. So the methods used to deal with the two are not the same, in most cases. –  Mariano Suárez-Alvarez Aug 30 '10 at 5:01
One slightly banal but important point is that it is very difficult to say anything meaningful about monoids in full generality, without restricting to some sub-class of particular interest. A look at a standard textbook on semigroup theory, such as Howie's, will hopefully convince the reader that without inverses things get rapidly more complicated and admit many more "pathologies". –  Yemon Choi Aug 30 '10 at 5:33
@Davidac897 - To expand on what Mariano said: a lot of the reason that modular representation theory works as well is it does is that group algebras over positive characteristic fields are still special beasts: they are symmetric, Hopf, and the special point of a very canonical deformation which is generically semi-simple (you should think of the p-adics as a deformation of the characteristic p situation), and preserves all of these structures. –  Ben Webster Aug 30 '10 at 6:03
whereas it's very unclear whether monoid algebras are in any tangible way better behaved than a random finite dimensional algebra. –  Ben Webster Aug 30 '10 at 6:04
@Ben, monoid algebras have a very special basis! Lots and lots of work have gone into trying to determine which finite dimensional algebras have such multiplicative bases, and they are more or less quite non-random :) –  Mariano Suárez-Alvarez Aug 30 '10 at 16:47
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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 $GL_n$.

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This just cleared up a lot of things for me about the representation theory of GL_n. –  Qiaochu Yuan Sep 2 '10 at 4:57
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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 the 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) then group rings are.

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I think the reason the classical representation theory point of view isn't helpful for the study of toric algebras is that modules of interest are very far from being simple and, in fact, have positive Krull dimension. Many tools in commutative algebra (and also in the theory of operator algebras in the functional analysis setting) have been developed specifically to apply to such "large" modules. –  Victor Protsak Aug 31 '10 at 2:35
Dear Victor, I agree. On the other hand, one can sometimes think of positive Krull dimension modules as a kind of "direct integral" of their fibres over the points of their support, which is just to say, the divisions between module theory as in commutative algebra, and spectral theory and its applications in representation theory, maybe aren't as great as they are conventionally made out to be. –  Emerton Aug 31 '10 at 2:57
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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.

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The application to random walks sounds interesting; any good references for this? –  Qiaochu Yuan Jun 22 '11 at 3:29
Look at the paper K. Brown, Semigroups, rings, and Markov chains. J. Theoret. Probab. 13 (2000), 871–938 and the references therein. –  Benjamin Steinberg Jun 22 '11 at 3:41
The intro to my paper arxiv.org/pdf/1101.0416.pdf with Margolis can be viewed as an extended version of my answer for anyone interested. –  Benjamin Steinberg Jun 30 '11 at 2:40
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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.

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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.

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It may be worth pointing out that in most cases I know of, semigroups of operators are just images of the positive real line. So in that corner of analysis, it's not clear that "representation theory of the real half-line" is that useful a viewpoint. –  Yemon Choi Sep 6 '10 at 21:33
On the other hand, there has been interest in recent years in certain algebras of operators generated by representations of special classes of monoid (polycyclic monoids; inverse semigroups; monoids arising from dynamics) so perhaps that is what Jim's answer is referring to –  Yemon Choi Sep 6 '10 at 21:35
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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...

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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

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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.

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When considering actions of a monoid/semigroup on a set, I understand the desire for using partial maps, but when considering linear actions on a vector space, any partial linear map can be extended by 0 to a non-partial linear map. E.g. in the case of quivers, this extension by 0 preserves all of the compositions, etc. For linear actions, is there a potentially useful or interesting way in which the viewpoint of partial maps make you think about it differently? –  Joshua Grochow Jul 8 at 15:37
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In the previous answers almost every one 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 interesting cases were the structure of monoid-algebras are fully understood (over the field of complexes).

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

2.Partition Algebras, which are invented and studied by Paul Martin.

Both of the above examples are part of a wider brach which is called Diagram Algebras.

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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.

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That is correct partially, but when one makes them into a monoid algebras then the quotients can be talked about. Indeed quotients play vital role in the study some types of monoid algebras, for example Partition Algebras. –  math137 Feb 18 at 23:22
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