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AN ALTERNATING SUM FORMULA FOR MULTIPLICITIES OF LOWER DEFECT GROUPS PETER SYMONDS 1. Introduction Let G be a finite group and let k be an algebraically closed field of prime characteristic p. Let 5be a block of kG, P a /^-subgroup of G and n a /(-element of G. Then the multiplicity of P as a lower defect group of B for the /^-section of n is defined [1, 2, 4] and is denoted by m (P). We shall prove the following induction theorem for m (P) in terms of the Brown complex A of chains of non-trivial /^-subgroups of G. THEOREM 1.1. Let B be a block of kG, P a non-trivial p-subgroup and n a p-element. Then aeG\A 6eBl(G ) P'~P n'~n ff b -B The first sum is over the simplices a of G\A and G is the stabiliser of a simplex of A above a (so it is well-defined up to conjugacy). The second sum is over the blocks b of kG such that the Brauer correspondent b is defined (in terms of central characters) and is equal to B. In fact, it is known that b is always defined for the subgroups G [3],
Bulletin of the London Mathematical Society – Wiley
Published: Mar 1, 1993
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