PhD

Abstract:

This dissertation considers the representation theory of Temperley-Lieb algebras, $\mathrm{TL}_n$ , along with some related cellular algebras, over positive and mixed characteristic fields.

The Temperley-Lieb algebras are planar diagrammatic algebras. They are generated by “$(n, n)$-diagrams”: planar pair matchings between n points. Multiplication is diagrammatic composition with closed loops resolving to linear factors of $\delta$. Their representation theory depends greatly on the characteristic of the underlying ring, as well as the choice of $\delta$.

This thesis shows that the representation theories of the Temperley-Lieb algebras in the semi-simple case, characteristic zero, or “unmixed” positive characteristics are special cases of the mixed-characteristic theory. We do this purely diagrammatically, without recourse to any Schur-Weyl dualities. We then find the generating idempotent of the principle projective module. In the semi-simple case this is the famed Jones-Wenzl idempotent. However, this celebrated object is not well defined in arbitrary characteristic and we extend the work of Burrull, Libedinsky and Sentinelli in the unmixed positive characteristic case to the remaining two characteristic classes.

Armed with these new elements, we consider truncations of $\mathrm{TL}_n$ by certain tensors of such generalised idempotents. These algebras are still cellular and can be considered as endomorphism rings of tensor products of indecomposable $U_q(\mathfrak{sl}_2)$-modules in the given characteristic. We determine the cellular data for a number of classes of such truncations and give indications of a general, if computationally intensive, approach for the remaining classes. Finally, we examine how some of these results may extend to “webs”, generalisations of Temperley-Lieb diagrams, and prove a conjecture by Elias.

@thesis{spencer2023phd,
      title={Modular Temperley-Lieb Theory},
      author={R. A. Spencer},
      year={2023},
      school={University of Cambridge},
}