Osaka Journal of Mathematics

A Block Refinement of the Green-Puig Parameterization of the Isomorphism Types of Indecomposable Modules

Morton E. Harris

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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 $\Lambda(G,P,X)$.

Article information

Osaka J. Math., Volume 56, Number 2 (2019), 229-236.

First available in Project Euclid: 3 April 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20C20: Modular representations and characters


Harris, Morton E. A Block Refinement of the Green-Puig Parameterization of the Isomorphism Types of Indecomposable Modules. Osaka J. Math. 56 (2019), no. 2, 229--236.

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