= Brauer tree path presentation
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Use a finite <tree> with cyclically ordered incident edges at each vertex, and multiplicities one except possibly at one vertex. Quiver vertices are tree edges; arrows follow each cyclic successor order, including loops at vertices of valency one. For an edge $i$ with endpoints $v,w$, write $C_{i,v}$ for the cycle around $v$. Impose $C_{i,v}^{m_v}=C_{i,w}^{m_w}$, $C_{i,v}^{m_v}\alpha_{i,v}=0$, and zero for any path that switches from one vertex-cycle to the other. The quotient of this <path algebra> is the basic <Brauer tree algebra>; redundant valency-one loops can be eliminated. Its indecomposable projective at $i$ has simple top and socle labelled by $i$, with two uniserial radical branches of lengths $m_v\operatorname{val}(v)$ and $m_w\operatorname{val}(w)$ meeting in that socle.
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