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Blocks and virtually irreducible lattices

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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
Discipline: Mathematics
Degree Granting Institution: University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(s): Mathematics
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
Type: Text
Language: English
URI: http://hdl.handle.net/2142/19169
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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