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An Alternating Sum Formula for Multiplicities of Lower Defect Groups

An Alternating Sum Formula for Multiplicities of Lower Defect Groups 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], http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

An Alternating Sum Formula for Multiplicities of Lower Defect Groups

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References (6)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/25.2.121
Publisher site
See Article on Publisher Site

Abstract

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],

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Mar 1, 1993

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