Extremal weight projectors II

TL;DR

This paper extends the construction of extremal weight projectors from sl(2) to gl(N), enabling categorification of torus skein algebras and symmetric polynomials, with applications to link homology.

Contribution

It generalizes extremal weight projectors to gl(N), providing new diagrammatic tools for categorification and representation theory.

Findings

Constructed diagrammatic idempotents for gl(N)

Categorified power-sum symmetric polynomials

Provided presentations of gl(N) representation categories

Abstract

In previous work, we have constructed diagrammatic idempotents in an affine extension of the Temperley-Lieb category, which describe extremal weight projectors for sl(2), and which categorify Chebyshev polynomials of the first kind. In this paper, we generalize the construction of extremal weight projectors to the case of gl(N) for N greater than or equal to 2, with a view towards categorifying the corresponding torus skein algebras via Khovanov-Rozansky link homology. As by-products, we obtain compatible diagrammatic presentations of the representation categories of gl(N) and its Cartan subalgebra, and a categorification of power-sum symmetric polynomials.