Algebraic Groups and their Representations by Leonard Scott (auth.), R. W. Carter, J. Saxl (eds.)
By Leonard Scott (auth.), R. W. Carter, J. Saxl (eds.)
This quantity includes 19 articles written by way of audio system on the complicated examine Institute on 'Modular representations and subgroup constitution of al gebraic teams and similar finite teams' held on the Isaac Newton Institute, Cambridge from twenty third June to 4th July 1997. We recognize with gratitude the monetary aid given by means of the NATO technological know-how Committee to let this ASI to happen. beneficiant monetary aid used to be additionally supplied by means of the ecu Union. we're additionally happy to recognize money given by way of EPSRC to the Newton Institute which have been used to aid the assembly. it's a excitement to thank the Director of the Isaac Newton Institute, Professor Keith Moffatt, and the employees of the Institute for his or her devoted paintings which did lots to additional the luck of the assembly. The editors desire to thank Dr. Ross Lawther and Dr. Nick Inglis such a lot warmly for his or her assist in the construction of this quantity. Dr. Lawther particularly made a useful contribution in getting ready the amount for submission to the publishers. eventually we want to thank the celebrated audio system on the ASI who agreed to put in writing articles for this quantity in response to their lectures on the meet ing. we are hoping that the amount will stimulate extra major advances within the concept of algebraic groups.
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Relations Between the Defining and Nondefining Characteristic, and to Symmetric Groups. Quantum and Finite-Dimensional Algebras Beyond the conjectures of James and Lusztig, one would want the characters of irreducible modules for the smaller primes, in both defining and nondefining characteristic. Some previous discussion of analogies between the defining and nondefining characteristic problems has been given in Dipper [26, 27], but our focus here is more on actual relationships. Surprisingly, for any prime p whatsoever, the complete solution of the nondefining characteristic problem in type A breaks naturally into two subproblems, one of which is the defining characteristic problem!
E. amI E #,MI . The case j = 1 in (2) gives a short exact sequence Note that the first term here may be identified with B and the second term with BS (by the assumption ii». The last term is isomorphic to Bt E9 T for some tEN and some torsion B-module T. Clearly, the images of m2, ... ,ms in MIMI ® A B generate this module. Therefore, t+r~s-l where r denotes the minimal set of generators for T. However, by tensoring (3) with Q(v]/(¢p) we see that t = s -1. It follows that T = 0 and that (3) is split.
The last decades have seen a lot of activity focused on finding ways to solve this problem. In 1979 G. Lusztig [17] came up with a conjecture which, in the form of an algorithm, tells what the answer should be for p ~ 2h - 3, h being the Coxeter number for G. A little reformulation allows us to get the bound on p down to p ~ h (see [16]) but for p < h nobody has so far been able to formulate a conjecture which would give all irreducible characters. e. larger than some unknown integer depending only on the root data associated with G).