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A quick lemma on group representations September 23, 2009

Posted by Akhil Mathew in algebra, representation theory.
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So, since I’ll be talking about the symmetric group a bit, and since I still don’t have enough time for a deep post on it, I’ll take the opportunity to cover a quick and relevant lemma in group representation theory (referring as usual to the past blog post as background).

A faithful representation of a finite group {G} is one where different elements of {G} induce different linear transformations, i.e. {G \rightarrow Aut(V)} is injective. The result is

Lemma 1 If {V} is a faithful representation of {G}, then every simple representation of {G} occurs as a direct summand in some tensor power {V^{\otimes p}}.   (more…)

Representations of the symmetric group September 20, 2009

Posted by Akhil Mathew in algebra, combinatorics, representation theory.
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I’ve now decided on future plans for my posts. I’m going to alternate between number theory posts and posts on other subjects, since I lack the focus have too many interests to want to spend all my blogging time on one area.

For today, I’m going to take a break from number theory and go back to representation theory a bit, specifically the symmetric group. I’m posting about it because I don’t understand it as well as I would like. Of course, there are numerous other sources out there—see for instance these lecture notes, Fulton and Harris’s textbook, Sagan’s textbook, etc.  Qiaochu Yuan has been posting on symmetric functions and may be heading for this area too, though if he does I’ll try to avoid overlapping with him; I think we have different aims anyway, so this should not be hard.  (more…)