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Let A be a self-injective connected Nakayama algebra. What is the Loewy length of any indecomposable projective A-module?

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I don't understand what you are asking, really. As Julian observes, this depends on the algebra... (and not on the particular indecomposable projective, by the way!) but: what kind of answer were you expecting? –  Mariano Suárez-Alvarez Feb 9 '12 at 19:43

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The Loewy length can be arbitrary. See e.g. [Assem, Simson, Skowronski: Elements of the Representation Theory of Associative Algebras 1] Proposition V.3.8:

$A$ basic, self-injective, connected Nakayama, then $A\cong kQ/I$, where $Q$ is an oriented cycle and $I=(kQ^+)^h$, where $(kQ^+)$ is the ideal spanned by all the arrows.

The Loewy length of each projective indecomposable should then be $h$.

This certainly also holds true for non-basic algebras if you replace isomorphism by Morita equivalence.

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If you want a group-theoretical example, take the group algebra of the cyclic group of order $p$ over a field of characteristic $p$. Then this has Loewy length $p$. –  Julian Kuelshammer Feb 9 '12 at 17:06

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