# Applications of the surjectivity of Brauer's decomposition map over arbitrary fields?

Recently I've been going over some of Serre's reformulation of Brauer theory with a student, following the influential treatment in Part III of Serre's lectures (revised 1971 French edition) later published as Springer GTM 42 (1977) Linear Representations of Finite Groups. This raises a question about the decomposition homomorphism relating Grothendieck groups in characteristic 0 and in characteristic $p>0$.

BACKGROUND: Originally Brauer relied on fairly concrete arguments involving "modular" (now called Brauer) characters, in order to compare the ordinary irreducible representations of a finite group $G$ over a "large enough" field $K$ of characteristic 0 with those over a suitable finite field $k$. Here "large enough" requires $K$ to contain all $m$th roots of unity, where $m$ is the exponent of $G$: this forces $K$ to be a splitting field for $G$ and all its subgroups, so irreducible representations are absolutely irreducible. Now take $K$ to be complete relative to a discrete valuation, with valuation ring $A$ and residue field $k$ of characteristic $p>0$ dividing $|G|$. Then $k$ is also "large enough" for $G$.

Serre follows the ideas of Swan and Giorgiutti, working with Grothendieck groups (including those defined by projective modules over $A$ and $k$) to formulate Brauer's main results in terms of the cde-triangle. Without assuming completeness, the process of reduction modulo the unique maximal ideal of $A$ yields a well-defined homomorphism $d:R_K(G) \rightarrow R_k(G)$ of Grothendieck groups (via non-canonical choices of $A[G]$-lattices in modules).

He views as the main theorem here the surjectivity of $d$ when the fields are large enough, or in other words the existence of a Brauer lifting of an irreducible representation over $k$ to a virtual representation over $K$. (This kind of lifting was exploited for example by Quillen in his proof of the Adams Conjecture.)

Following the first publication (in typescript format) of Serre's lectures, the improved 1971 edition includes a footnote in section 16.1 stating that Chevalley and Dress had independently proved the subjectivity of $d$ without the "large enough" hypothesis. Apparently Chevalley never published his proof, but Andreas Dress did publish a short note in J. Algebra 17 (1971), which I haven't yet looked at but have encouraged our German student to track down. This note doesn't seem to have a citation history and isn't mentioned by Curtis-Reiner in their later treatment following Serre in Methods of Representation Theory I. Which raises for me the question:

Are there any serious applications of the truth of Brauer's lifting theorem when the fields involved don't contain enough roots of unity?

One has to keep in mind here that irreducible representations of a finite group over a field can be much larger in dimension than those over a splitting field.

[UPDATE] I've managed to track down one paper which uses the result of Dress: C.T.C. Wall, "On rationality of modular representations", Bull. London Math. Soc. 5 (1973), 199-202. Basically Wall shows that the field of definition of an arbitrary modular representation of a finite group is determined by its Brauer character (or Quillen's version of Brauer lifting). But in his review, Lusztig observes that Wall's result follows more concretely from his own independent 1973 announcement here, with details in Lusztig's 1974 Annals of Mathematics Studies No. 81 where the Brauer lifting is more explicit than the character formulation of Quillen and doesn't rely on the argument of Dress.

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Nice question. I am eager to see the answer. – Joël Mar 1 '12 at 1:14