Document Type
Article
Publication Date
3-2022
Publication Title
International Mathematics Research Notices
Abstract
Webs are planar graphs with boundary that describe morphisms in a diagrammatic representation category for 𝔰𝔩k . 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 (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, which 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 𝔰𝔩3 -webs ([ 33] and [ 29].) 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 𝔰𝔩2 -webs, our new partial order coincides with the tableau partial order on webs studied by the authors and others [ 12, 17, 29, 33]. We also prove that though the new partial order for 𝔰𝔩3 -webs is a refinement of the previously studied tableau order, the two partial orders do not agree for 𝔰𝔩3 .
Volume
2022
Issue
5
First Page
3371
Last Page
3416
DOI
10.1093/imrn/rnaa290
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Rights
Licensed to Smith College and distributed CC-BY under the Smith College Faculty Open Access Policy.
Recommended Citation
Russell, Heather M. and Tymoczko, Julianna, "The Transition Matrix Between the Specht and 𝔰𝔩3 Web Bases is Unitriangular With Respect to Shadow Containment" (2022). Mathematics Sciences: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/155
Comments
Peer reviewed accepted manuscript.