Vol. 40, No. 1, 1972

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Some results on blocks over local fields

Wayland M. Hubbart

Vol. 40 (1972), No. 1, 101–109

Let F be an unequal characteristic local field. The aim of this paper is to outline a block form of the Cartan-Brauer modular decomposition theory which incorporates the notion of defect groups. The irreducible F-representations of a finite group G are associated with blocks in the group algebra over the residue field F. The defect groups of a block to which an irreducible F-representation T belongs are shown to coincide with the defect groups of the block to which any absolutely irreducible constituent of T belongs. A result on the Schur index of an absolutely irreducible representation belonging to a block of defect zero is proven which yields an analogue to the Brauer-Nesbitt Theorem on blocks of defect zero. The number of F-blocks of highest defect is shown to be equal to the number of p-regular F-conjugacy classes of highest defect.

Mathematical Subject Classification 2000
Primary: 20C20
Received: 14 May 1970
Published: 1 January 1972
Wayland M. Hubbart