2025 Theses Doctoral

SL2-plethysms and Cartan categorification

Martinez Ruiz, Alvaro Luis

The existing categorifications of quantum groups by Khovanov-Lauda and Rouquier, and in particular Lauda's categorification of quantum sl₂, decategorify to their idempotented forms. The key missing ingredient for categorifying the full non-idempotented quantum 𝔰𝔩₂ is a suitable categorification of its Cartan subalgebra. In this dissertation, we introduce a presentation for Lusztig's ℤ[𝑞,𝑞⁻¹]-form of 𝔰𝔩₂ based on Lusztig's elements [^K;c_t].

First, we establish a multiplication formula for these elements that is integral and positive. Next, we propose a heuristic using this presentation that has been highly effective in predicting characteristic-independent isomorphisms between SL₂-plethysms. We develop methods to prove these results and find several new infinite families of SL₂-plethystic isomorphisms.

In the last part, we reinterpret these methods diagrammatically and propose a diagrammatic categorification of the Cartan subalgebra based on a thickening 𝑇𝑇𝐿 of the Temperley-Lieb category. We show how to lift some identities in the Cartan as direct sum decompositions in 𝑇𝑇𝐿, and we outline a potential approach to categorifying the full non-idempotented quantum 𝔰𝔩₂.