# What is a Specht module?

I'm studying the structure of the Specht module for $S_n$ and I would like to know if there is some generalizations of this structure for Weyls groups or Coxeter groups.

Also, I'm interest to know about category-theoretics way of study this module. I just know one article about this subject:

http://www.math.uni-bonn.de/people/stroppel/Specht.pdf

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I assume you know the definition. Is this a "what is a Specht module, really?" question, or "what are analogues of a Specht module for other groups?" question? –  David Roberts Feb 26 '11 at 3:56
"what is a Specht module, really???" and "what are analogues of a Specht module for other groups?" are indeed my questions here :) –  Yannic Feb 26 '11 at 5:29

## 1 Answer

The short answer to your question about Specht modules for other types than the symmetric group is "yes". The long answer is that you have to dig into the extensive literature built up around cyclotomic Hecke algebras and the like. One place to look is the arXiv, where the papers of Andrew Mathas (Sydney) and his collaborators from recent years can be found. A recent example is here, but there are many related papers by other people including Meinolf Geck and his collaborators.

Though I haven't followed all of these developments closely, my perception is that the emphasis tends to shift from the finite Coxeter groups such as $S_n$ to their Iwahori-Hecke algebras and related module categories. Combinatorics is a major subtheme throughout, as well as categorification.

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Kleshchev's survey article, front.math.ucdavis.edu/0909.4844, contains a good account of the (almost) current state of knowledge. It also gives a good explanation of what cyclotomic Hecke algebras have to do with anything. –  David Hill Feb 28 '11 at 19:46