# Easier reference for material like Diaconis's “Group representations in probability and statistics”

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 Perse 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?

## 4 Answers

I have a chapter on this in my book Steinberg - 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 Chain by Tullio Ceccherini-Silberstein, Fabio Scarabotti, andFilippo Tolli, but I find it no easier than Diaconis except that it may expect more of an algebra background than a probability background.

• I just post an answer with a reference to your book – Yannic Aug 30 '13 at 15:49
• 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. – Ben Webster Aug 30 '13 at 17:31
• You can do the probability chapter right after the Fourier analysis one. I only put it at the end because it is optional. – Benjamin Steinberg Aug 30 '13 at 18:55
• Just to follow up, your book ended up being very useful for one of my students doing a project on random walks. – Ben Webster Dec 5 '13 at 17:30
• @BenWebster, I'm glad to hear it. – Benjamin Steinberg Dec 5 '13 at 19:56

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.

This 74-page paper in Journal of Machine Learning Research (by Huang, Guestrin, and Guibas) --

http://www.jmlr.org/papers/volume10/huang09a/huang09a.pdf

-- 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.

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.