Web and Polytabloid Bases

Date of Award

5-2023

Document Type

Thesis

Degree Name

M.S.

Committee Chair

Arik Wilbert, Ph.D.

Advisor(s)

Andrei Pavelescu, Cornelius Pillen, Martin Frank

Abstract

We analyze the little-understood inverse map of a function introduced by Russell and Tymoczko in their 2019 paper "The Transition Matrix Between the Specht and Web Bases is Unipotent with Additional Vanishing Entries." The function introduced in this paper maps form the vector space generated by the polytabloid basis to the vector space generated by web basis. Additionally, these vector spaces come with an action of the symmetric group, and thus the function in the Russell-Tymoczko theorem is equivariant. Moreover, the inverse function is also equivariant. Before defining key characteristics of the web and polytabloid bases and developing the relationship between the two bases, we familiarize ourselves with basic representation theoretic concepts. Most importantly, we analyze the examples of mappings of the inverse function. Drawing from these examples, we formalize our main conjecture. The inverse images of the web basis elements are expressed in terms of the polytabloid basis; and-notably-the coefficients associated with the inverse images are either -1, 0, or 1.

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