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Representations of the Symmetric Group from Geometry

Year: 2023

Presenter Name: Emil Geisler

Throughout this abstract, \C represents the complex numbers, \Q represents the rational numbers, and S_n represents the symmetric group with n elements. Representation stability was introduced to study mathematical structures which stabilize when viewed from a representation theoretic framework. The instance of representation stability studied in this project is that of ordered complex configuration space, denoted PConf_n(\C): PConf_n(\C) := { (x_1, x_2, ..., x_n) | x_i != x_j } PConf_n(\C) has a natural S_n action by permuting its coordinates which gives the cohomology groups H^i(PConf_n(\C);\Q) the structure of an S_n representation. The cohomology of PConf_n(\C) stabilizes as n tends toward infinity when viewed as a family of S_n representations. From previous work, there is an explicit description for H^i(PConf_n(\C);\Q) as a direct sum of induced representations for any i, n, but this description does not explain the behavior of families of irreducible representations as n tends toward infinity.
We implement an algorithm which, given a Young Tableau, computes the cohomological degrees where the corresponding family of irreducible representations appears stably as n tends to infinity. Previously, these values were known for only a few Young Tableaus and cohomological degrees. Using this algorithm, results have been found for all Young Tableau with up to 8 boxes and certain Tableau with more, which has led us to conjectures based on the data collected.
University / Institution: University of Utah
Type: Poster
Format: In Person
Presentation #B82
SESSION B (10:45AM-12:15PM)
Area of Research: Science & Technology
Faculty Mentor: Sean Howe