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 Do. 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 boolean AND Z(D)]. As applications, we prove that all Cartan invariants for B are divisible by [Z(D) : D-0 boolean AND Z(D)], that if Out(D) is a p-group (and D 0 1), then the number of irreducible characters of height zero in B is divisible by p and that if Z(D)not less than or equal to Do, then the block B is of Lefschetz type (see [R. Knorr, G.R. Robinson, Some remarks on a conjecture of Alperin, J. London Math. Soc. (2) 39 (1) (1989) 48-60]). (C) 2008 Elsevier Inc. All rights reserved.