Representations of Finite Groups

I have very fond memories of the late Chitikila Musili from my days at UCLA back in the 1970s. I had the great good fortune to take linear algebra from him, as well as a first course in abstract algebra, and he was largely responsible for installing a true appreciation for mathematical structure in me at a very early stage of my education — and this way of looking at mathematics has only grown more pervasive in my professional life since. To boot, Professor Musili was a strong supporter of my goals of entering the ranks of professionals, noting that I had a real algebraic bent: a kudo I have always cherished. When I learnt of his death (at a shockingly young age) some time ago I was greatly saddened. May he rest in peace.

What struck me most about Professor Musili’s lectures was the clarity with which he developed the themes to be covered in his courses. I remember vividly his careful and crystal clear development of what he called (Emmy) Noether’s theorem, i.e. the first isomorphism theorem, which, in the context of linear algebra, he used as a springboard for deriving the rank-nullity theorem. It was gorgeous even to my neophyte’s eyes, and he made it all so clear that even at that early stage of my mathematical life I was made to appreciate this result couched in rather more abstract terms than is usually the case in a first course in linear algebra. Matrices were not placed in the foreground, but, instead, the homomorphisms were, and this made quite an impact on me from the start. I also recall Professor Musili hitting the topic of change of basis with unstinting use of commutative diagrams: again, a wonderful thing to me, whom Professor Musili had already persuaded to revel in more and more abstraction.

Certainly, then, Professor’s Musili was a master of clear exposition who ordered his presentation very effectively so as to have the main ideas and the various proofs emerge with proper backlighting and in proper detail. And he coupled this clarity with concision and elegance, making for a wonderful learning experience from beginning to end.

The same can be said about his writings. The book under review, dealing with a sine qua non in the education on any mathematician, is a pure pleasure for me to leaf through, given the memories it evokes of the author’s wonderful lecturing style, and I claim that any one who reads it will come to the same conclusion: this is a fine way to learn this material.

The arrangement of the book is the logical one, given the fourfold goal Professor Musili presents in the Preface, viz.,

to give an elementary introduction to the theory of ordinary representations of finite groups … accessible to students with just one semester exposure to the rudiments of Linear Algebra, Groups, Rings, and Modules … to do the theory rather explicitly for the important case of the symmetric group … with as little of the technical preparation as possible … to do the same for … the Alternating group … [and] the Hyperoctahedral groups … to prepare the reader … to look around and ahead so as to explore a lot more interesting frontiers that are beyond the scope of this book.

Thus, in bijective correspondence with these goals, the book is split into four parts, respectively on the structure of semisimple rings, representations of finite groups per se, representations of the symmetric and alternating groups, and then representations of the indicated hyperoctahedral groups. One result is that Burnside theory occurs in the middle of Part II, which also contains coverage of Frobenius reciprocity, Mackey’s irreducibility criterion, and even the Wigner-Mackey “method of little groups.” Part III sports Young tableaux and Part IV continues with this theme, and many others besides.

Representations of Finite Groups also comes equipped with a host of exercise sets which are all very carefully crafted. These are generally greater in length than what one finds in other texts on these subjects, making for a terrific opportunity for the student to master this material at an uncommonly deep level early in the game. This again reminds me of Professor Musili’s class back in the 1970s: his exercises were tailor made to educate us in thinking along the right lines and to give us a good deal of facility attainable only by getting our hands really dirty. And the benefits were palpable, of course, resonating well into the future.

It is worth noting, too, that the exercises in the book are supplemented by sections titled “True/False Statements,” which conspire (successfully, I would project) to train the reader in the kind of careful discernment needed in such a subject as the representation theory of finite groups. Here is a sampling from p. 65ff., the rules of the game being to decide not just T vs. F, but also to determine whether an additional hypothesis can turn an F into a T (this being the PT option: “partially true”); and then T’s are to be proven, F’s are to be dealt a counterexample, and PT’s are to be given the right additional hypothesis.

1. A vector space is semi-simple
10. An Artinian simple ring is a matrix ring over a division ring.
20. Tensor product (over Z) of division rings is a simple ring.
30. A central simple algebra over K is M_n(K) for some n.  

Sporty, no?

All in all, Chitikila Musili’s Representations of Finite Groups, is a wonderful text for learning the subject of its title. And it is a great experience for me, personally, to encounter once again Professor Musili’s terrific pedagogical style, even if it is only in print and not face to face.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.