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Title:Blocks and virtually irreducible lattices
Author(s):Ellers, Harald Erich Herbert
Doctoral Committee Chair(s):Dade, Everett C.
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
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.
Issue Date:1989
Rights Information:Copyright 1989 Ellers, Harald Erich Herbert
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9010852
OCLC Identifier:(UMI)AAI9010852

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