21
$\begingroup$

I'm teaching a class on the representation theory of finite groups at the advanced undergrad level. One of the things I'd like to talk about, or possibly have a student do any independent project on is applications of finite groups in statistics and data analysis. Unfortunately, the only book I know on the subject is Persi Diaconis's Group representations in probability and statistics, which is lovely but nowhere close to the level that I would expect an undergrad to read (or for that matter, anyone from a field outside math). Are there any books or articles people know of which provide a gentler introduction?

Improve this question
$\endgroup$
1

5 Answers 5

Reset to default
14
$\begingroup$

I have a chapter on this in my book Representation theory of finite groups. Sorry for the self promotion. It is intended for advanced undergrads. I basically focus on the abelian case, giving the upper bound lemma on convergence rates and the description of the eigenvalues for this case only. I do one explicit computation (I don't have the book in front of me right now, but I think I do it for the lazy random walk on the hypercube). Also, if memory serves there is an exercise on what to do if the probability measure is constant on conjugacy classes for non abelian groups, but maybe that was in the section on eigenvalues of Cayley graphs.

An alternative is Harmonic Analysis on Finite Groups. Representation Theory, Gelfand Pairs and Markov Chains by Tullio Ceccherini-Silberstein, Fabio Scarabotti, and Filippo Tolli, but I find it no easier than Diaconis except that it may expect more of an algebra background than a probability background.

Improve this answer
$\endgroup$
6
  • $\begingroup$ I just post an answer with a reference to your book $\endgroup$
    – Yannic
    Commented Aug 30, 2013 at 15:49
  • $\begingroup$ By the way, to set your mind at ease about the self-promotion, our library already had a copy of your book, so there was no financial gain. I don't know if getting to the probability stuff makes sense, but the Fourier transform section may make a good project. $\endgroup$
    – Ben Webster ♦
    Commented Aug 30, 2013 at 17:31
  • $\begingroup$ You can do the probability chapter right after the Fourier analysis one. I only put it at the end because it is optional. $\endgroup$
    – Benjamin Steinberg
    Commented Aug 30, 2013 at 18:55
  • 2
    $\begingroup$ Just to follow up, your book ended up being very useful for one of my students doing a project on random walks. $\endgroup$
    – Ben Webster ♦
    Commented Dec 5, 2013 at 17:30
  • 1
    $\begingroup$ @BenWebster, I'm glad to hear it. $\endgroup$
    – Benjamin Steinberg
    Commented Dec 5, 2013 at 19:56
8
$\begingroup$

Recently, Benjamin Steinberg write "Representation Theory of Finite Groups: An Introductory Approach". Is a modern and very nice written course. It include Fourier analysis on finite groups and probability and random walks on groups.

I would also recommend "Introduction to Representation Theory" by Pavel Etingof et. al. It come from lectures notes that Etingof writed for MIT undergraduate students. It contain a bunch of subjects that students can take for a project (mostly abstract subjects, like Schur-Weyl duality, quiver representations, category theory, homological algebra).

"Representations and character of groups", by Gordon James and Martin Liebeck is another nice reference for an undergraduate course. The last chapter contain applications to molecular vibration.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Steinberg's book is also referenced (by Steinberg!) in the accepted answer. $\endgroup$
    – LSpice
    Commented Mar 27, 2023 at 18:07
7
$\begingroup$

This 74-page paper in Journal of Machine Learning Research (by Huang, Guestrin, and Guibas) — Fourier Theoretic Probabilistic Inference over Permutations — is an amazingly useful and undergrad-friendly intro to the representation theory of the symmetric group (in a very surprising venue). Also, the fact that its applications are to machine learning and statistics looks very appropriate for OP's question.

Improve this answer
$\endgroup$
4
$\begingroup$

There is a virtually forgotten (alas) very clear and accessible memoir Group representations and applied probability by Hannan which significantly (1965) predates Diaconis.

Improve this answer
$\endgroup$
3
$\begingroup$

The book Probability on Discrete Structures contains a chapter called "Random walks on finite groups" by Laurent Saloff-Coste which predates Benjamin Steinberg's book.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .