Modular representation theory: Difference between revisions

Revision as of 12:17, 18 January 2016

Revision as of 12:17, 18 January 2016

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In modular representation theory, while Maschke's theorem does not hold

when the characteristic divides the group order, the group algebra may be decomposed as the direct sum of a maximal collection of two-sided ideals known as '''blocks''' (when the field ''K'' has characteristic 0, or characteristic coprime to the group order), there is also such a decomposition of the group algebra ''K''[''G''] as a sum of blocks (one for each isomorphism type of simple module), but the situation is relatively transparent (at least when ''K'' is sufficiently large): each block is a full matrix algebra over ''K'', the endomorphism ring of the vector space underlying the associated simple module).

To obtain the blocks, the identity element of the group ''G'' is decomposed as a sum of primitive [[idempotent]]s