## Osaka Journal of Mathematics

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

Morton E. Harris

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

#### Article information

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

Dates
First available in Project Euclid: 3 April 2019

https://projecteuclid.org/euclid.ojm/1554278421

Mathematical Reviews number (MathSciNet)
MR3934972

Zentralblatt MATH identifier
07080081

Subjects
Primary: 20C20: Modular representations and characters

#### Citation

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. https://projecteuclid.org/euclid.ojm/1554278421

#### References

• W. Feit: The representation theory of finite groups, North-Holland, Amsterdam, 1982.
• M.E. Harris: A block refinement of the Green–Puig correspondences in finite group modular representation theory, Int. J. Algebra 5 (2011), 315–323.
• M. Linckelmann: private communication of notes for a forthcoming book.
• H. Nagao and Y. Tsushima: Representations of finite groups, Academic Press, San Diego, 1989.
• J. Thévenaz: $G$-algebras and modular representation theory, Oxford University Press, New York, 1995.