Action of degenerate Bethe operators on representations of the symmetric group

Abstract

Degenerate Bethe operators are elements defined by explicit sums in the center of the group algebra of the symmetric group. They are useful on account of their relation to the Gelfand-Zetlin algebra and the Young-Jucys-Murphy elements, both of which are important objects in the Okounkov-Vershik approach to the representation theory of the symmetric group. We examine all of these results over the course of the thesis.

Degenerate Bethe operators are a new, albeit promising, topic. Therefore, we include proofs for previously-unproven basic aspects of their theory. The primary contribution of this thesis, however, is the computation of eigenvalues and eigenvectors of all the degenerate Bethe operators in sizes 4 and 5, as well as many in size 6. For each partition $\bld{\lambda} \vdash k$ we compute the operators $B_{\ell j}$, where $\ell + j \leq k$, and give the eigenvalues and their corresponding eigenvectors in terms of standard Young tableaux of shape $\bld{\lambda}$. The number of terms in the degenerate Bethe operators grows very rapidly so we used a program written in the computer algebra system \texttt{SAGE} to compute the eigenvalue-eigenvector pair data. From this data, we observed a number of patterns that we have formalized and proven, although others remain conjectural. All of the data computed is collected in an appendix to this thesis.