Let E be a Frobenius category. Let E denote its stable category. The shift functor on E induces, by pointwise application, an inner shift functor on the category of acyclic complexes with entries in E. Shifting a complex by 3 positions yields an outer shift functor on this category. Passing to the quotient modulo split acyclic complexes, Heller remarked that inner and outer shift become isomorphic, via an isomorphism satisfying still a further compatibility. Moreover, Heller remarked that a choice of such an isomorphism determines a Verdier triangulation on E, except for the octahedral axiom. We generalise the notion of acyclic complexes such that the accordingly enlarged version of Heller's construction includes octahedra.

The elementary divisors of the Gram matrices of Specht modules S λ over the symmetric group are determined for two-row partitions and for two-column partitions λ. More precisely, the subquotients of the Jantzen filtration are calculated using Schaper's formula. Moreover, considering a general partition λ of n at a prime p > n − λ 1 , the only possible non trivial composition factor of S λ Fp is induced by the morphism of Carter and Payne, as shown by means of Kleshchev's modular branching rule. This enables the Jantzen filtration to be calculated in this case as well.

We generalize certain parts of the theory of group rings to the twisted case. Let G be a finite group acting (possibly trivially) on a field L of characteristic coprime to the order of the kernel of this operation. Let K J L be the fixed field of this operation and let S be a discrete valuation ring with field of fractions K, with maximal ideal generated by p and with integral closure T in L. We compute the colength of T o G in a maximal order in L o G. In the case when S=pS is finite, we compute the S=pS-dimension of the center of T o G=JacðT o GÞ. If this quotient is split semisimple, this yields a formula for the number of simple T o G-modules, generalizing Brauer's formula.Brought to you by | University of Birmingham Authenticated Download Date | 5/28/15 11:01 PM

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