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arXiv · 2603.22156

Trace identities for quiver representations

Adrien Kassel·Thierry Lévy
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Abstract

We give an expression for the determinant of the twisted Laplacian associated with any linear representation of a finite quiver in terms of traces of the holonomy of its cycles. To establish this expression, we prove a general identity for the determinant of a block matrix in terms of traces of products of its blocks. We give two proofs, one purely enumerative and one using generating series. In the special case of a finite graph equipped with a vector bundle and a connection, the twisted Laplacian determinant admits a combinatorial interpretation as a weighted count of tuples of oriented cycle-rooted spanning forests, where the weights involve traces of holonomies along cycles formed by combining the edges of the forests.

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Adrien Kassel, Thierry Lévy. 2026-03-23. Trace identities for quiver representations. https://arxiv.org/abs/2603.22156

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