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April 2019 A Block Refinement of the Green-Puig Parameterization of the Isomorphism Types of Indecomposable Modules
Morton E. Harris
Osaka J. Math. 56(2): 229-236 (April 2019).

Abstract

Let $p$ be a prime integer, let $\mathcal{O}$ be a commutative complete local Noetherian ring with an algebraically closed residue field $k$ of charateristic $p$ and let$G$ be a finite group. Let $P$ be a $p$-subgroup of $G$ and let $X$ be an indecomposable $\mathcal{O} P$-module with vertex $P$. Let $\Lambda (G,P,X)$ denote a set of representatives for the isomorphism classes of indecomposable $\mathcal{O} G$-modules with vertex-source pair $(P,X)$ (so that $\Lambda(G,P,X)$ is a finite set by the Green correspondence). As mentioned in [5, Notes on Section~26], L. Puig asserted that a defect multiplicity module determined by $(P,X)$ can be used to obtain an extended parameterization of $\Lambda(G,P,X)$. In [5, Proposition 26.3], J. Thévenaz completed this program under the hypotheses that $X$ is $\mathcal{O}$-free. Here we use the methods of proof of [5, Theorem 26.3] to show that the $\mathcal{O}$-free hypothesis on $X$ is superfluous. (M. Linckelmann has also proved this, cf. [3]). Let $B$ be a block of $\mathcal{O} G$. Then we obtain a corresponding paramaterization of the $(\mathcal{O} G)B$-modules in $\Lambda(G,P,X)$.

Citation

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Morton E. Harris. "A Block Refinement of the Green-Puig Parameterization of the Isomorphism Types of Indecomposable Modules." Osaka J. Math. 56 (2) 229 - 236, April 2019.

Information

Published: April 2019
First available in Project Euclid: 3 April 2019

zbMATH: 07080081
MathSciNet: MR3934972

Subjects:
Primary: 20C20

Rights: Copyright © 2019 Osaka University and Osaka City University, Departments of Mathematics

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Vol.56 • No. 2 • April 2019
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