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 ����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 (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 ����3 -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 ����2 -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 ����3 -webs is a refinement of the previously studied tableau order, the two partial orders do not agree for ����3 .
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.
Version
Author's Accepted Manuscript
Recommended Citation
Russell, Heather M. and Tymoczko, Julianna, "The Transition Matrix Between the Specht and ����3 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.