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
1
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