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Daily Overview |
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Pl 6: Emmy Noether Lecture
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Global vs local in the representation theory of finite groups: conjectures and theorems University of Wuppertal
Local analysis in finite group theory consists, for a given group G and a prime number p, in
dealing with information about p-subgroups of G and their normalizers. This was one of the
main threads in the classification of finite simple groups (1955–1985), first with the prime 2 after
the Odd Order Theorem, then with other primes.
In the representation theory of finite groups, Brauer’s theory of defect subgroups for blocks in
characteristic p makes heavy use of the p-local structure of the groups involved. The conjectures by McKay (1971), Alperin (1986), Broué (1990), and Dade (1992) are all formulated in that language.
McKay’s conjecture was proved recently (Cabanes–Späth, 2013–2026) using the reduction to a
problem about simple groups (Isaacs–Malle–Navarro, 2007). It is now a theorem asserting that a finite group has as many irreducible complex characters of degree prime to p as the normalizer of a Sylow p-subgroup does.
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