= Uniserial branch of a Brauer tree algebra
{title2=$1\leq q\leq m_v\operatorname{val}(v)$}
Let $i$ be an edge at $v$. Kill the other radical branch of its indecomposable <projective module>. The resulting <uniserial module> has length $m_v\operatorname{val}(v)$ and its factors follow the cyclic edge order beginning at $i$. Quotienting its radical powers gives all shorter such modules. The edge succession uniquely fixes the path, and its nonzero arrow maps can be rescaled to one, giving uniqueness up to isomorphism. At an exceptional leaf this reduces to a nilpotent single Jordan chain; the other leaf loop, when present, is a socle path and vanishes in these shorter quotients.
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