# Why are they called Specht Modules?

I know that the simple modules of $\mathbb{C}S_n$ are called $\it{Specht}$ $\it{Modules}$, and they are named after the German Mathematician Wilhelm Specht because he studied them, but I think these modules were studied before him, for example by Frobenius .

Could any one please tell me what is the Specht's contribution to this area that made people name these modules after him?

**I was reluctant about asking this question here or in MathsStackExchange, but finally I posted it here, if you think it is too naive to be here, please let me know I will happily move it to there.

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I personally think the question is fine. It's specific, and topic of interest to research mathematicians. –  Ben Webster Feb 17 at 23:36
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## 1 Answer

The question is interesting though perhaps not strictly "research-level". Terminology in mathematics develops a bit haphazardly, and sometimes things get misleading names. In this case the work of Specht around 1935 did place the representations of symmetric groups in the then-modern setting of module theory. But the notion of "Specht module" seems to have emerged around 1970 in the rapidly developing work on representations of the groups in prime characteristic. Work of M.H. Peel and especially work of Gordon James popularized the notion. In particular, James exploited the fact that the characteristic 0 Specht modules have a fairly natural reduction mod $p$ for any prime $p$. This is somewhat analogous to the algebfaic group situation, where "Weyl modules" come by such reductions and then have a unique distinguished composition factor.

By now the literature on symmetric group representations in prime characteristic is quite extensive, with the term "Specht module" being ubiquitous. In the classical characteristic 0 theory, no such language is usually needed. Anyway, one might make a case for the terminology "James module" here, but it's too late for that. Specht himself had no special influence on the modular theory.

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Thank you very much for your answer. –  math137 Feb 17 at 23:06
Gene Murphy tried to popularise the use of "James module" for the irreducible quotient of the Specht modules. Unfortunately it never caught on. –  Andrew Feb 18 at 2:02
I usually think of the Specht module as the particular integral form from, for example, James' book, and its reduction mod p. I've often wondered about finding other interesting forms and what the reductions mod p look like. –  daveh Feb 18 at 13:55
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