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The conjecture of Alperin we consider has to do with modular representations of a finite group G over an algebraically closed field k of characteristic p. CONJECTURE 1 (Alperin [2]). The number of weights for G equals the number of simple kG-modules. Alperin makes his definition of a weight for G in [2]. There now exist various equivalent forms of this conjecture, and we will work with the one which appeared first after Alperin's original version. We let np(G) denote the number of nonprojective simple kG-modules, and \Delta the simplicial complex of chains of nonidentity p-subgroups of G (see [10]). CONJECTURE 2 (Kn&quot;orr-Robinson [5]). For all finite groups G, np(G) = X oe2\Delta /G (-1)dim oe np(Goe)

Year: 1990

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