p -Blocks Relative to a Character of a Normal Subgroup II
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p-Blocks Relative to a Character of a Normal Subgroup II Noelia Rizo Abstract. Let p be a prime number, let G be a finite group, let N be a normal subgroup of G, and let θ be a G-invariant irreducible character of N . In Rizo (J Algebra 514:254–272, 2018), we introduced a canonical partition of the set Irr(G|θ) of irreducible constituents of the induced character θG , relative to the prime p. We call the elements of this partition the θ-blocks. In this paper, we construct a canonical basis of the complex space of class functions defined on {x ∈ G | xp ∈ N }, which supersedes previous non-canonical constructions. This allows us to define θ-decomposition numbers in a natural way. We also prove that the elements of the partition of Irr(G|θ) established by these θ-decomposition numbers are the θ-blocks. Mathematics Subject Classification. Primary 20D, Secondary 20C15. Keywords. Brauer p-blocks, Brauer’s first main theorem, Block orthogonality, Central extensions.
1. Introduction Let p be a prime number, and let M be a maximal ideal of the ring R of algebraic integers in C containing p. For every finite group G, and only de pending on M , R. Brauer constructed a basis of the space cf(Gp ) of complex class functions defined on the set Gp of the p-regular elements of G, namely the set IBr(G) of the irreducible Brauer characters of G. Suppose now that N is a fixed normal subgroup of G, and consider G0 = {g ∈ G | gp ∈ N }, where gp is the p-part of g ∈ G, and let cf(G0 ) denote the space of complex class functions defined on G0 . If χ is any complex class function defined on G, then χ0 denotes the restriction of χ to the set G0 . If N is a p-group, Navarro constructed in Ref. [6] a canonical basis IBr(G, N ) of cf(G0 ) (depending only on M ) satisfying that whenever χ ∈ The author acknowledges support by Ministerio de Ciencia e Innovaci´ on PID2019103854GB-I00 and FEDER funds.
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N. Rizo
Irr(G), then: χ0 =
MJOM
dχϕ ϕ
ϕ∈IBr(G,N )
for some uniquely defined non-negative integers dχϕ . The N -projective characters dχϕ χ Φϕ = χ∈Irr(G)
were previously studied by K¨ ulshammer and Robinson in Ref. [4]. In the not-so-widely available [7], Navarro constructed a similar basis of cf(G0 ) for any normal subgroup N of G, where N was not necessarily a p-group. However, his methods did not allow him to prove that this basis was canonical (that is, depending only on M ). In the first main result of this paper, we construct such a canonical basis. When studying the complex space cf(G0 ), it is well known that standard Clifford reductions (that we shall review below) allow us to fix a G-invariant character θ ∈ Irr(N ) and only consider the subspace cf(G0 |θ) which is the span of {χ0 | χ ∈ Irr(G|θ)}, where, as usual, Irr(G|θ) is the set of irreducible constituents of the induced character θG . Theorem A. Suppose that p is a prime, G is a finite group, N is a normal subgroup of G, and θ ∈ Irr(N ) is G-invariant. Then, there is a canonical basis IBr(G|θ) of the space cf(G0 |θ) (depending only on the
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