Blocks as orders over a \(p\)-adic ring

Florian Eisele (City)

Frank Adams 1,

Let \(\mathcal{O}\) be an extension of the \(p\)-adic integers. Let \(k\) denote its residue field and \(K\) its field of fractions. We consider a fixed \(p\)-block of a finite group \(G\). An interesting problem in the modular representation theory of finite groups is
the determination of the basic algebra of such a block, which is defined to be the unique smallest (i. e. lowest-dimensional) \(k\)-algebra whose module category is equivalent to that of the block. I will talk about how to approach this problem
from an integral point of view, i. e. by looking at the corresponding block of \(\mathcal{O}G\) as an \(O\)-order in a semisimple \(K\)-algebra. I will present some results, in particular on "tame blocks", and an approach to this problem using derived equivalences.

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