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# Mathematics > Combinatorics

# Title: Differential graded algebras for trivalent plane graphs and their representations

(Submitted on 14 Oct 2021)

Abstract: To any trivalent plane graph embedded in the sphere, Casals and Murphy associate a differential graded algebra (dg-algebra), in which the underlying graded algebra is free associative over a commutative ring. Our first result is a generalization of the Casals--Murphy dg-algebra to non-commutative coefficients. In generalizing, we prove various functoriality properties which did not appear in the commutative setting, notably including changing the chosen face at infinity of the graph. Our second result is to prove that rank $r$ representations of this dg-algebra, over a field $\mathbb{F}$, correspond to colorings of the faces of the graph by elements of the Grassmannian $\operatorname{Gr}(r,2r;\mathbb{F})$ so that bordering faces are transverse, up to the natural action of $\operatorname{PGL}_{2r}(\mathbb{F})$. Underlying the combinatorics, the dg-algebra is a computation of the fully non-commutative Legendrian contact dg-algebra for Legendrian satellites of Legendrian 2-weaves, though we do not prove as such in this paper. The graph coloring problem verifies that for Legendrian 2-weaves, rank $r$ representations of the Legendrian contact dg-algebra correspond to constructible sheaves of microlocal rank $r$. This is the first such verification of this conjecture for an infinite family of Legendrian surfaces.

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