On dimensions of block algebras

Abstract

Following a question by B. K¨ulshammer, we show that an inequality, due to Brauer, involving the dimension of a block algebra, has an analogue for source algebras, and use this to show that a certain case where this inequality is an equality can be characterised in terms of the structure of the source algebra, generalising a similar result on blocks of minimal dimensions. Let p be a prime and k an algebraically closed field of characteristic p. Let G be a finite group and B a block algebra of kG; that is, B is an indecomposable direct factor of kG as k-algebra. By a result of Brauer in [2], the dimension of B satisfies the inequality dimk(B) ≥ p2a−d · l(B) · u2 B where pa is the order of a Sylow-p-subgroup of G, pd is the order of a defect group of B, l(B) is the number of isomorphism classes of simple B-modules and uB is the unique positive integer such that pa−d · uB is the greatest common divisor of the dimensions of the simple B-modules. It is well-known that uB is prime to p. K¨ulshammer raised the question whether an equality could be expressed in terms of the structure of a source algebra of B, generalising the result in [3] on blocks of minimal dimension. We show that this is the case. The first observation is an analogue for source algebras of Brauer’s inequality. We keep the notation above and refer to [5] for block theoretic background material.