On the focal defect group of a block, characters of height zero, and lower defect group multiplicities

We discuss the focal subgroup of the defect group D of a p-block B, which we refer to as the focal defect group, and denote by D 0 . We note that (the character group) of D / D 0 acts (in a defect (or height) preserving fashion) on irreducible characters in B, and prove that the action on irreducibl...

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Bibliographic Details
Published in:Journal of algebra 2008-09, Vol.320 (6), p.2624-2628
Main Author: Robinson, Geoffrey R.
Format: Article
Language:English
Subjects:
Group characters
Modular representations
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Summary:We discuss the focal subgroup of the defect group D of a p-block B, which we refer to as the focal defect group, and denote by D 0 . We note that (the character group) of D / D 0 acts (in a defect (or height) preserving fashion) on irreducible characters in B, and prove that the action on irreducible characters of height zero is semi-regular. We also prove that all orbits under this action have length divisible by [ Z ( D ) : D 0 ∩ Z ( D ) ] . As applications, we prove that all Cartan invariants for B are divisible by [ Z ( D ) : D 0 ∩ Z ( D ) ] , that if Out ( D ) is a p-group (and D ≠ 1 ), then the number of irreducible characters of height zero in B is divisible by p and that if Z ( D ) ⩽̸ D 0 , then the block B is of Lefschetz type (see [R. Knörr, G.R. Robinson, Some remarks on a conjecture of Alperin, J. London Math. Soc. (2) 39 (1) (1989) 48–60]).
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2008.04.032