Introduction

Whilst the

20th

century was still in its infancy, an article by F. G. Frobenius

was published in the Journal of the Berlin Science Academy, which contained a de-

scription of the irreducible complex characters of all symmetric groups [34]. Since

then, representation theory has evolved into a deep and sophisticated art, to the

point where most papers in the subject are incomprehensible to the multitude of

mathematicians. After all this development however, some basic questions remain

unanswered. Interrogate an expert on group representation theory over finite fields

and you will quite soon witness a shrug of the shoulders, and a protestation of igno-

rance. The irreducible characters of symmetric groups, which Frobenius so casually

exposed in characteristic zero, remain mysterious over fields of prime characteris-

tic. This monograph comprises a sequence of reflections surrounding the modular

representation theory of symmetric groups.

Our approach to the subject is homological, inspired by M. Brou´ e’s abelian

defect group conjecture [9], and encouraged by the proof of Brou´ e’s conjecture for

blocks of symmetric groups by J. Chuang, R. Kessar, and R. Rouquier.

The abelian defect group conjecture is the most homological of a menagerie

of general conjectures in modular representation theory, each of which predicts a

likeness between the representations of a finite group in characteristic l, and those

of its l-local subgroups. It stakes that the derived category of any block A of a

finite group is equivalent to the derived category of its Brauer correspondent B, so

long as the blocks have abelian defect groups. Such an equivalence should respect

the triangulated structure of the derived category, and therefore descend from a

two sided tilting complex of A-B-bimodules, by a theorem of J. Rickard [62]. Some

have postured to prove the conjecture by induction, and encountered the diﬃculty

of lifting an equivalence of stable categories to an equivalence of derived categories

[65]. Others have tried to prove the conjecture for particular examples, such as

symmetric groups.

In 1991, R. Rouquier observed a certain class of blocks of symmetric groups,

which he believed to possess a particularly simple structure. Indeed, Rouquier con-

jectured a beautiful structure theorem for such blocks of abelian defect, which was

subsequently proved by J. Chuang, and R. Kessar [11]. A corollary was a proof of

Brou´ e’s conjecture for this class of blocks, whose defect groups could be arbitrarily

large. In view of their history, these blocks should properly be called Rouquier, or

Chuang-Kessar blocks. We use the curt abbreviation “RoCK blocks”. Such blocks

can be defined in arbitrary defect, and in any species of type A representation

theory.

Chuang and Rouquier proved in a later work that all symmetric group blocks

of identical defect possess equivalent derived categories which, in conjunction with

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