Events for 04/19/2019 from all calendars

Algebra and Combinatorics Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Anton Dochtermann, Texas State University

Title: Exposed circuits, linear quotients, and chordal clutters

Abstract: Chordal graphs are widely studied combinatorial objects, with various characterizations and applications. They also appear in commutative algebra in the context of Froberg’s theorem, which says that a graph G is chordal if and only if the edge ideal of its complement has a linear resolution. Recently Culbertson, Guralnik, and Stiller give a new characterization of chordal graphs in terms of what they call ‘edge-erasures’. We show that these moves are in fact equivalent to a linear quotient ordering on the ideal, leading to an algebraic proof of their result. We consider higher-dimensional analogues and show that linear quotients for more general circuit ideals of d-clutters can be characterized in terms of removing `exposed circuits' in the complement clutter. Here a circuit is exposed if it is uniquely contained in a maximal clique, reminiscent of the free faces of simple homotopy theory. This leads to a notion of a higher-dimensional ‘chordal clutter’ which borrows from commutative algebra, related to other constructions from the literature. We discuss some applications including a proof of a special case of (a generalization of) Simon’s conjecture, which posits that the k-skeleta of a simplex are extendably shellable.