The transition matrix between the Specht and $\mathfrak{sl}_3$ web bases is unitriangular with respect to shadow containment

TL;DR

This paper introduces new combinatorial tools called band diagrams and shadows to analyze webs in representation theory, proving the unitriangularity of the change-of-basis matrix between Specht and web bases for -webs.

Contribution

It defines band diagrams and shadows, and uses them to prove the unitriangularity of the Specht-web basis change-of-basis matrix for -webs, resolving an open conjecture.

Findings

The change-of-basis matrix between Specht and web bases for -webs is unitriangular.

Introduces band diagrams and shadows as new combinatorial structures for webs.

Establishes a new partial order on webs that refines existing orders.

Abstract

Webs are planar graphs with boundary that describe morphisms in a diagrammatic representation category for $\mathfrak{sl}_k$. They are studied extensively by knot theorists because braiding maps provide a categorical way to express link diagrams in terms of webs, producing quantum invariants like the well-known Jones polynomial. One important question in representation theory is to identify the relationships between different bases; coefficients in the change-of-basis matrix often describe combinatorial, algebraic, or geometric quantities (like, e.g., Kazhdan-Lusztig polynomials). By "flattening" the braiding maps, webs can also be viewed as the basis elements of a symmetric-group representation. In this paper, we define two new combinatorial structures for webs: band diagrams and their one-dimensional projections, shadows, that measure depths of regions inside the web. As an application, we resolve an open conjecture that the change-of-basis between the so-called Specht basis and web basis of this symmetric-group representation is unitriangular for $\mathfrak{sl}_3$-webs. We do this using band diagrams and shadows to construct a new partial order on webs that is a refinement of the usual partial order. In fact, we prove that for $\mathfrak{sl}_2$-webs, our new partial order coincides with the tableau partial order on webs studied by the authors and others. We also prove that though the new partial order for $\mathfrak{sl}_3$-webs is a refinement of the previously-studied tableau order, the two partial orders do not agree for $\mathfrak{sl}_3$.