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https://hdl.handle.net/2142/19169
Description
Title
Blocks and virtually irreducible lattices
Author(s)
Ellers, Harald Erich Herbert
Issue Date
1989
Doctoral Committee Chair(s)
Dade, Everett C.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
We use R. Knorr's theory of virtually irreducible lattices to study the blocks of a finite group.
Let G be a finite group and let p be a rational prime. Let R be a complete discrete valuation ring of characteristic zero with maximal ideal generated by $\pi$ and with p $\varepsilon$ $\pi$R. Let K be the field of fractions of R, and let R = R/$\pi$R. Assume that R is algebraically closed and that K is a splitting field for every subgroup of G.
Knorr showed that any indecomposable RG-lattice of height zero is virtually irreducible. We use this fact to generalize Brauer's Third Main Theorem on Blocks as follows.
Theorem (3.1). Let B be a block of RG, and let M be an indecomposable RG-lattice in B of height zero. Suppose that H is a subgroup of G and that b is an admissible block of RH. Then b$\sp{\rm G}$ = B if and only if b contains an indecomposable component of M$\sb{\rm H}$ of height zero.
We also prove the following connection between Brauer correspondence of blocks and induction of virtually irreducible lattices.
Theorem (5.2). Let H be a subgroup of G and let b be a block of RH. Suppose that there is a virtually irreducible RH-lattice U in b such that U$\sp{\rm G}$ = V $\oplus$ W with V virtually irreducible and U $\not\vert$ W$\sb{\rm H}$. Then b$\sp{\rm G}$ is defined and V is in b$\sp{\rm G}$.
Most admissible blocks contain a virtually irreducible lattice U as in Theorem (5.2); there is a finite extension S of R such that the following is true.
Theorem (5.11). Let H be a subgroup of G and let b be an admissible block of SH with defect group D. If every automorphism of D which preserves conjugacy classes is an inner automorphism, then there is a virtually irreducible SH-lattice in b with vertex D such that U$\sp{\rm G}$ = V $\oplus$ W with V virtually irreducible and U $\not\vert$ W$\sb{\rm H}$.
We also investigate the question: if B is a block of RG and if there is a block pair (D,b) in G with b$\sp{\rm G}$ = B, is there a virtually irreducible RG-lattice in B with vertex D? Theorem (5.11) gives a sufficient condition on D for this question to have an affirmative answer, provided we replace R by a certain finite extension. We give several more conditions of this kind. This is a partial converse to a theorem of Knorr.
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