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Trace identities for quiver representations

Adrien Kassel, Thierry Lévy

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.

Trace identities for quiver representations

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.
Paper Structure (52 sections, 33 theorems, 158 equations, 5 figures)

This paper contains 52 sections, 33 theorems, 158 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Quivers, their linear representations, and the twisted Laplacian determinant
  3. Quiver representations
  4. Graphs versus quivers
  5. Twisted quiver Laplacian
  6. Kernel of the Laplacian
  7. Gauge-invariance
  8. Laplacian determinant
  9. Main results (I): abstract block matrices
  10. General block matrices
  11. Block matrices with scalar diagonal blocks
  12. Main results (II): the case of the Laplacian of a quiver representation
  13. Wilson loop expansion
  14. Discrete vector fields
  15. Prime cycles and Euler products
  16. ...and 37 more sections

Key Result

Proposition 1.1

The following equality holds: where the product is over cycles of $\sigma$, and ${\mathsf{b}}:{\llbracket n \rrbracket}\to{\llbracket p \rrbracket}$ is the map which to an index associates the index of the block to which it belongs.

Figures (5)

  • Figure 1: A quiver with integers on vertices representing the rank of the representation: it assigns to each arrow a matrix compatible with these dimensions (not shown).
  • Figure 2: A bidirected graph which has a sub-quiver consisting in the quiver drawn in Fig. \ref{['fig:quiver-rep']}.
  • Figure 3: An acyclic quiver.
  • Figure 4: A cycle-rooted-tree quiver.
  • Figure 5: Quiver with only two prime cycles: ${(\!(} 1,2,3,4 {)\!)}$ and ${(\!(} 5,6,7,8 {)\!)}$. Each vertex has outdegree $1$ except for vertices $2$ and $3$ which have outdegree $2$.

Theorems & Definitions (43)

  • Proposition 1.1: Proposition \ref{['prop:main']}
  • Corollary 1.2: Corollary \ref{['coro:main']}
  • Theorem 1.3: Theorem \ref{['thm:main']}
  • Proposition 1.4: Proposition \ref{['prop:Euler']}
  • Theorem 1.5: Corollary \ref{['coro:detDelta']}
  • Corollary 1.6: Theorem \ref{['thm:gauge-theory']}
  • Theorem 1.7: Theorem \ref{['thm:forman-zeilberger-vector-fields']}
  • Theorem 1.8: Theorem \ref{['thm:euler-product']}
  • Corollary 1.9: Theorem \ref{['thm:finite-euler']}
  • Proposition 2.1
  • ...and 33 more