| Description |
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 PConfn(C): PConfn(C) := {(x1, x2, . . . ,xn) 2 Cn | xi 6= xj} PConfn(C) has a natural Sn action by permuting its coordinates which gives the cohomology groups Hi(PConfn(C);Q) the structure of an Sn representation. The cohomology of PConfn(C) stabilizes as n tends toward infinity when viewed as a family of Sn representations. From previous work, there is an explicit description for Hi(PConfn(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 ! 1. We implement an algorithm which, given a Young Tableau, computes the cohomological degrees where the corresponding family of irreducible representations appears stably as n!1. 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. |
