# What is the definition of Loewy diagram?

I had seen an exercise:

Write the Loewy-diagram of the direct indecomposable modules of the $$A = K\Gamma / I$$ algebra, where \begin{align*} \Gamma: 1 \underset{\beta}{\overset{\alpha}{\rightleftarrows}} 2 \underset{\delta}{\overset{\gamma}{\rightleftarrows}} 3, \quad I = (\alpha \beta, \beta \alpha - \gamma \delta, \delta \gamma). \end{align*}

Unfortunately I couldn't find the definition, or a way to compute, of Loewy-diagram.

Could someone point me to a resource explaining it?

• This is actually a good question. You see the term Loewy-diagram often but I am also not aware of a formal definition in the literature. It probably just means the decomposition of the Loewy factors $J^iM/J^{i+1}M$ into simple moduels of a module $M$. – Mare Oct 29 '19 at 12:09

## 1 Answer

A somewhat formal definition of the Loewy structure (which is the same as how Mare described, and I'm sure this is the Loewy-diagram you're looking for) can be found in Benson's treatise in Lecture Notes in Mathematics. See Appendix, page 174.