representation theory | Delta Epsilons

Divisibility theorems for group representations II October 14, 2009

Posted by Akhil Mathew in algebra, representation theory.
Tags: Clifford's theorem, Frobenius reciprocity, induction, restriction
2 comments

So last time we proved that the dimensions of an irreducible representation divide the index of the center. Now to generalize this to an arbitrary abelian normal subgroup.

There are first a few basic background results that I need to talk about.

Induction

Given a group ${G}$ and a subgroup ${H}$ (in fact, this can be generalized to a non-monomorphic map ${H \rightarrow G}$ ), a representation of ${G}$ yields by restriction a representation of ${H}$ . One obtains a functor ${\mathrm{Res}^G_H: Rep(G) \rightarrow Rep(H)}$ . This functor has an adjoint, denoted by ${\mathrm{Ind}_H^G: Rep(H) \rightarrow Rep(G)}$ . (more…)

A quick lemma on group representations September 23, 2009

Posted by Akhil Mathew in algebra, representation theory.
Tags: faithful representation, regular representation, symmetric group, tensor powers
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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.
Tags: Specht modules, symmetric group, tableau, Young diagrams
12 comments

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…)

More Lie algebra constructions July 28, 2009

Posted by Akhil Mathew in algebra, representation theory.
Tags: bilinear forms, Lie algebras, Lie groups, representation theory
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The ultimate aim in the series on Lie algebras I am posting here is to cover the representation theory of semisimple Lie algebras. To get there, we first need to discuss some technical tools—for instance, invariant bilinear forms.

Generalities on representations

Fix a Lie algebra ${L}$ . Given representations ${V_1, V_2}$ , we clearly have a representation ${V_1 \oplus V_2}$ ; given a morphism of representations ${V_1 \rightarrow V_2}$ , i.e. one which respects the action of ${L}$ , the kernel and image are themselves representations.

Proposition 1 The category ${Rep(L)}$ of finite-dimensional representations of ${L}$ is an abelian category.

(more…)

Lie’s Theorem II July 27, 2009

Posted by Akhil Mathew in algebra, representation theory.
Tags: eigenvalues, Lie algebras, Lie's theorem, linear algebra, representation theory, solvability
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Yesterday I was talking about Lie’s theorem for solvable Lie algebras. I went through most of the proof, but didn’t finish the last step. We had a solvable Lie algebra ${L}$ and an ideal ${I \subset L}$ such that ${I}$ was of codimension one.

There was a finite-dimensional representation ${V}$ of ${L}$ . For ${\lambda \in I^*}$ , we set

$\displaystyle V_\lambda := \{ v \in V: Yv = \lambda(Y) v, \ \mathrm{all} \ Y \in I \}.$

We assumed ${V_\lambda \neq 0}$ for some ${\lambda}$ by the induction hypothesis. Then the following then completes the proof of Lie’s theorem, by the “fundamental calculation:”

Lemma 1 If ${V_\lambda \neq 0}$ , then ${\lambda([L,I])=0}$ .

(more…)

Lie’s Theorem I July 26, 2009

Posted by Akhil Mathew in algebra, representation theory.
Tags: derived series, Lie algebras, Lie's theorem, solvability
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I talked a bit earlier about nilpotent Lie algebras and Engel’s theorem. There is an analog for solvable Lie algebras, and the corresponding Lie’s theorem.

So, first the definitions. Solvability is similar to nilpotence in that one takes repeated commutators, except one uses the derived series instead of the lower central series.

In the future, fix a Lie algebra ${L}$ over an algebraically closed field ${k}$ of characteristic zero.

Definition 1 The derived series of ${L}$ is the descending filtration ${D_n}$ defined by ${D_0 := L, D_n := [D_{n-1}, D_{n-1}]}$ . The Lie algebra ${L}$ is solvable if ${D_M=0}$ for some ${M}$ .

For instance, a nilpotent Lie algebra is solvable, since if ${\{C_n\}}$ is the lower central series, then ${D_n \subset C_n}$ for each ${n}$ .

(more…)

Engel’s Theorem and Nilpotent Lie Algebras July 23, 2009

Posted by Akhil Mathew in algebra, representation theory.
Tags: algebra, Engel's theorem, Lie algebras, linear algebra, nilpotent
1 comment so far

Now that I’ve discussed some of the basic definitions in the theory of Lie algebras, it’s time to look at specific subclasses: nilpotent, solvable, and eventually semisimple Lie algebras. Today, I want to focus on nilpotence and its applications.

Engel’s Theorem

To start with, choose a Lie algebra ${L \subset \mathfrak{gl} (V)}$ for some finite-dimensional ${k}$ -vector space ${V}$ ; recall that ${\mathfrak{gl} (V)}$ is the Lie algebra of linear transformations ${V \rightarrow V}$ with the bracket ${[A,B] := AB - BA}$ . The previous definition was in terms of matrices, but here it is more natural to think in terms of linear transformations without initially fixing a basis.

Engel’s theorem is somewhat similar in its statement to the fact that commuting diagonalizable operators can be simultaneously diagonalized.

(more…)

Why simple modules are often finite-dimensional II July 22, 2009

Posted by Akhil Mathew in algebra, representation theory.
Tags: algebra, finite-dimensional vector spaces, Nakayama's lemma, representation theory, simple modules
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I had a post a few days back on why simple representations of algebras over a field ${k}$ which are finitely generated over their centers are always finite-dimensional, where I covered some of the basic ideas, without actually finishing the proof; that is the purpose of this post.

So, let’s review the notation: ${k}$ is our ground field, which we no longer assume algebraically closed (thanks to a comment in the previous post), ${A}$ is a ${k}$ -algebra, ${Z}$ its center. We assume ${Z}$ is a finitely generated ring over ${k}$ , so in particular Noetherian: each ideal of ${Z}$ is finitely generated.

Theorem 1 (Dixmier, Quillen) If ${A}$ is a finite ${Z}$ -module, then any simple ${A}$ -module is a finite-dimensional ${k}$ -vector space.

(more…)

Representations of sl2, Part II July 18, 2009

Posted by Akhil Mathew in algebra, representation theory.
Tags: highest weights, Lie algebras, linear algebra, representation theory, sl2
1 comment so far

This post is the second in the series on ${\mathfrak{sl}_2}$ and the third in the series on Lie algebras. I’m going to start where we left off yesterday on ${\mathfrak{sl}_2}$ , and go straight from there to classification. Basically, it’s linear algebra.

Classification

We’ve covered all the preliminaries now and we can classify the ${\mathfrak{sl}_2}$ -representations, the really interesting material here. By Weyl’s theorem, we can restrict ourselves to irreducible representations. Fix an irreducible ${V}$ .

So, we know that ${H}$ acts diagonalizably on ${V}$ , which means we can write

$\displaystyle V = \bigoplus_\lambda V_\lambda$

where ${Hv_\lambda = \lambda v_{\lambda}}$ for each ${\lambda}$ , i.e. ${V_\lambda}$ is the ${H}$ -eigenspace.

(more…)

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Delta Epsilons

Divisibility theorems for group representations II October 14, 2009

A quick lemma on group representations September 23, 2009

Representations of the symmetric group September 20, 2009

More Lie algebra constructions July 28, 2009

Lie’s Theorem II July 27, 2009

Lie’s Theorem I July 26, 2009

Engel’s Theorem and Nilpotent Lie Algebras July 23, 2009

Why simple modules are often finite-dimensional II July 22, 2009

Representations of sl2, Part II July 18, 2009

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Delta Epsilons

Divisibility theorems for group representations II October 14, 2009

Divisibility theorems for group representations October 11, 2009

A quick lemma on group representations September 23, 2009

Representations of the symmetric group September 20, 2009

More Lie algebra constructions July 28, 2009

Lie’s Theorem II July 27, 2009

Lie’s Theorem I July 26, 2009

Engel’s Theorem and Nilpotent Lie Algebras July 23, 2009

Why simple modules are often finite-dimensional II July 22, 2009

Representations of sl2, Part II July 18, 2009

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