Consider a semisimple, connected, simply-connected algebraic group G over an algebraically closed field k of characteristic p ) 0. One can ลฝ . construct for each dominant weight a Weyl module โฌ with that highest weight whose character is given by Weyl's character formula.
ลฝ .
ลฝ .
Although not in general simple, โฌ has a simple head L , and all simple modules arise in this manner.
w ลฝ . ลฝ .x Knowledge of the decomposition numbers d s โฌ : L for ลฝ . and ''small'' i.e., p-restricted is equivalent to calculating the characters ลฝ of the corresponding simple modules and hence by Steinberg's tensor . product theorem to determining the characters of all the simples . Consequently, much work has been undertaken to try to determine these numbers, concentrating mainly on the case when p is large enough to be able to consider the Lusztig conjecture. Indeed, for sufficiently large w x primes the d are now known by the work of Andersen et al. 1 .
Although in principle all decomposition numbers can be determined ลฝ from those for p-restricted weights via character calculations using the . tensor product theorem and Weyl's character formula this is not straightforward in practice. Further, it is often more convenient to know decomposition numbers rather than characters; for example, when relating representations of the general linear and symmetric groups via Ringel duality only the former can be translated between the two categories.
We shall consider the situation where is ''large'' and give an elementary algorithm for calculating decomposition numbers given those for all p 2 -restricted weights. If we regard Steinberg's tensor product theorem as an algorithm for determining large characters from smaller ones, then this 1 Supported by EPSRC Grant M22536 and EC Grant FMRX-CT97-0100.