Timeline for Why aren't representations of monoids studied so much?
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12 events
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| Feb 19, 2014 at 17:12 | comment | added | Benjamin Steinberg | @BenWebster, I am aware of that. I was just trying to point out that in fact algebras of finite semigroups are much better behaved than typical finite dimensional algebras because they have a natural stratification coming from the ideal structure of the semigroup. | |
| Feb 19, 2014 at 11:41 | comment | added | Ben Webster♦ | @BenjaminSteinberg My intent wasn't to put down the representation theory of semi-groups, but rather just to point out (and I think you reinforced this) that not all non-semisimple algebras are non-semisimple in the same way (you can insert your own Tolstoy reference). So, having techniques to deal with one class (group rings over $\mathbb{F}_p$) may not tell you much about another, because you always want to use special properties of the class you consider. | |
| Jun 22, 2011 at 20:31 | comment | added | Benjamin Steinberg | @Ben, monoid algebras do have quite a bit more structure than a random finite dimensional algebra. In non-modular characteristic, they have a natural filtration $I_0>I_1>\cdots>I_k$ such that each quotient $I_j/I_{j+1}$ is either a heredity ideal (in the sense of Cline, Parshall and Scott) or has square zero. If the monoid is von Neumann regular each of these successive quotients is a heredity ideal and so the monoid algebra is quasi-hereditary and hence has finite global dimension, a property not enjoyed by modular group algebras. Many natural monoids (e.g. all maps on n letters) are regular. | |
| Aug 30, 2010 at 16:47 | comment | added | Mariano Suárez-Álvarez | @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 :) | |
| Aug 30, 2010 at 15:27 | comment | added | Richard Stanley | For much information on this subject, see A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, vol. 1. | |
| Aug 30, 2010 at 6:04 | comment | added | Ben Webster♦ | whereas it's very unclear whether monoid algebras are in any tangible way better behaved than a random finite dimensional algebra. | |
| Aug 30, 2010 at 6:03 | comment | added | Ben Webster♦ | @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. | |
| Aug 30, 2010 at 5:33 | comment | added | Yemon Choi | 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". | |
| Aug 30, 2010 at 5:01 | comment | added | Mariano Suárez-Álvarez | @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. | |
| Aug 30, 2010 at 4:56 | vote | accept | Mikola | ||
| Aug 30, 2010 at 4:56 | comment | added | David Corwin | If Maschke's theorem is false, does that mean we might be able to apply methods similar to those used in modular representation theory? | |
| Aug 30, 2010 at 4:50 | history | answered | Qiaochu Yuan | CC BY-SA 2.5 |