*Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer* by Charles W. Curtis is not a book I'd recommend to many people, but there are certain segments of the mathematical community for whom this book is a must-read.

Curtis' book is the fifteenth in a series of history of mathematics books published jointly by the American Mathematical Society and the London Mathematical Society. Its goal is to acquaint the reader with the history of the theory of representations of finite groups through a detailed analysis of the major papers of the founders of the discipline (Frobenius, Burnside, Schur, and Brauer) and other mathematicians. The book follows the development of this twentieth century field of mathematics in more-or-less chronological order, beginning with a chapter on the pre-requisite mathematics of the nineteenth century. A chapter each is then alloted to the turn-of-the-century work of Frobenius and Burnside and an additional two chapters apiece are devoted to the later work of Schur and Brauer.

There are parts of the book that will appeal to any reader with an interest in the history of mathematics. For instance, there are biographies of each of the major contributors at the beginnings of the chapters, and numerous smaller biographical sketches are scattered throughout the text. There are also revealing glimpses into the mathematical culture of the early twentieth century found in the numerous citations from the correspondence between these men. In one example, describing an exchange between Ferdinand Frobenius and Adolf Hurwitz, Curtis notes:

Adolf Hurwitz's letter was an eye-opener for Frobenius, as he had associated Hurwitz with the mathematical school of Felix Klein at Gottingen. Hurwitz had been a student of Klein, and his early work was strongly influenced by Klein's program to continue the geometric approach to complex function theory inaugurated by Riemann. Frobenius was skeptical of the big-picture geometric approach of Klein, and his reply to Hurwitz, in a letter dated 3 February 1896, contains the passage:

If you were emerging from a school, in which one amuses oneself more with rosy images than hard ideas and if, to my joy, you are also gradually becoming emancipated from that, then . . . old loves don't rust [alte Liebe rostet nicht]. Please take this joke facetiously.

Curtis' accounts of the major upheavals in the lives of German mathematicians caused by the anti-semitism of the Nazi regime are also quite engaging.

But overall this is probably not the sort of book that will appeal to mathematical generalists with only an interest in reading about the history of mathematics. Additionally, serious historians of mathematics are bound to view this book as a perfect example of what Ivor Grattan-Guinness once referred to as the tendency of mathematicians to "view history as the record of a 'royal road to me' — that is, an account of how a particular modern theory arose out of older theories instead of an account of those older theories in their own right." (For a more historical account, Curtis refers his readers to a series of articles by Thomas Hawkins in the *Archive for History of Exact Science*.) Indeed, at its heart this book is a mathematics text aimed at developing a deep understanding of the early twentieth century mathematical results which gave rise to modern representation theory through a thorough (though modern) account of some of the seminal works. But be warned: Although Curtis states in his preface that "[t]he mathematical parts of the book are intended to be accessible to students as well as professional mathematicians and others with a basic knowledge of algebra", those with only a passing interest in algebra or representation theory — especially those with only a limited undergraduate background in algebra — are bound to find many of the mathematical passages to be a challenge.

Consider, for instance, Chapter V, *Polynomial Representations of* GL_{n}(**C**), where Curtis discusses the representation theory of GL_{n}(**C**), as developed independently by Jacques Deruyts and Issai Schur. He begins with a brief discussion of Deruyts' work. As Curtis acknowledges, understanding Deruyts' work is not straightforward because Deruyts wrote in the long-forgotten language of 19th century invariant theory. Nevertheless, he barrels through the necessary material for understanding Deruyts' version of the representation theory of GL_{n}(**C**) — including an introduction to algebraic forms, invariants, and covariants as well as (in modern terms) left and right G-actions and tensor products of polynomial algebras — in about four pages. (More details can be found in a 1980 work of J.A. Green, which Curtis follows closely.)

His treatment of Schur is more leisurely and detailed, but it is still not an easy road to travel. Following his custom in the rest of the book, Curtis focuses on specific papers and their results. The first paper by Schur is his dissertation, *Uber eine Klasse von Matrizen, die sich einer gegebenen*. The essential idea in Schur's proof was to establish a correspondence between the representations of GL_{n}(**C**) and Sym(n), but along the way he passed through the Schur algebra and its representations. It's not a particularly complicated construction, but the trip from the elementary definition of a polynomial representation to the complete characterization of representations of GL_{n}(**C**) in terms of representations of Sym(n) takes about six pages. A novice will be breathless at the end. Moreover, twentieth century algebraic notation is in full force throughout. For instance, Curtis restates one of Schur's results as "an equivalence of categories from the category of homogeneous polynomial representations of [GL_{n}(**C**)] of order m to the category of finite dimensional left modules for the Schur algebra..." As in the remainder of the book, some proofs are those given by the original authors — dressed up in modern notation — but just as often modern proofs are given instead. At other times, Curtis is content to present a brief sketch of a proof — leaving details to the reader — or to cite a reference where a proof may be found. Again, to understand most of it, an interested reader should have a good graduate course in abstract algebra under his or her belt.

But while someone unfamiliar with the language of algebra and representation theory may find this a difficult book, graduate students just starting out on a research program in algebra or representation theory would benefit immensely from it. In a typical research field in mathematics today there are numerous introductory texts which sift through the countless research papers in the field and present students with a state-of-the-art understanding of the logical development of the discipline. However, in mathematics, like every other academic discipline, this understanding has been refined in a slow exchange of ideas taking place over a long period of time. The ability to see through the muddle of ideas and pick out what is most important is the hallmark of a successful mathematical researcher. But how are research students supposed to develop this instinct for distilling out relevant ideas, especially if they have never seen a serious example in their own research field and, indeed, have only a hazy understanding of where the most important ideas in their discipline came from? Most textbooks contain the archeological remains of this process of development (with names like "Schur's Lemma" and "Maschke's Theorem" adorning their major results) and, sandwiched in between the latest and greatest proofs, some authors will note the existence of this development in brief casual asides. But, for the most part, this vital enculturation at the graduate level is left to casual conversations at the departmental coffee hour.

Curtis has produced a rare but necessary sort of book that fills this need for a more formal introduction to the historical mathematical background of the twentieth-century mathematics usually seen first in graduate school. Hopefully, as the twenty-first century begins, other researchers and graduate educators will follow Curtis' lead and begin to piece together the many threads mathematics followed in the twentieth century into a coherent story that can easily be passed on to the next generation of mathematicians.

Andrew Leahy is an Assistant Professor of Mathematics at Knox College in Galesburg, IL.