Leading term strandings for webs

Abstract

A web is a plane graph encoding an invariant vector in a tensor product of fundamental representations of a quantum group. A stranding of an ${\mathfrak{sl}}_n$ web is a system of colored oriented curves recording one monomial of the vector it encodes. This article focuses on identifying and constructing leading term strandings, those recording the leading term of a web's vector with respect to a lexicographic order on monomials. We show that every open strand of a leading term stranding is clockwise, which constrains the boundary data of such strandings enough to yield a sufficient criterion for a set of webs to form a web basis. From a row-strict tableau, we construct a web with a prescribed leading term, and the resulting webs form a web basis, giving a non-recursive construction of Fontaine's ${\mathfrak{sl}}_n$ web bases. For ${\mathfrak{sl}}_3$ webs with no flat vertices, we identify a leading term stranding using the depths of the faces of the web. Finally, we show a leading term stranding for any ${\mathfrak{sl}}_3$ web can be reached from an arbitrary stranding via a sequence of operations called strand reversals.

26 pages, 14 figures