Aarhus University Seal / Aarhus Universitets segl

Seminar by Daniel Tubbenhauer

Title: $\mathfrak{sl}_3$-web bases, categorification and invariants of links

2013.11.12 | Christine Dilling

Date Wed 27 Nov
Time 15:15 16:15
Location Aud. D3

The representation categories $\mathbf{Rep}({\mathbf U}_q(\mathfrak{sl}_n))$ of the quantum groups ${\mathbf U}_q(\mathfrak{sl}_n)$ are known to have a graphical and combinatorial presentation, the so-called $\mathfrak{sl}_n$-web categories or $\mathfrak{sl}_n$-spiders. These categories have connections to the $\mathfrak{sl}_n$-link polynomials and varies aspects of combinatorics.

For $n=2$ all suitable bases of these spaces are the same, i.e. the dual canonical basis is the web basis and it is the Satake basis. This is no longer true for $n>2$.

I show that the Kuperberg's $\mathfrak{sl}_3$-web basis is still special in some sense, i.e. I show that it is an intermediate crystal basis. In order to do this, I identify it via a growth algorithm (we call it Leclerc-Toffin algorithm) that is obtained through $q$-skew Howe duality.

Based on work of Khovanov, Brundan-Stroppel, Mackaay-Pan-T and Mackaay-Yonezawa (and others) the $\mathfrak{sl}_n$-web categories have "categorified'' analogons called $\mathfrak{sl}_n$-web algebras. These "foam''-algebras are related to the $\mathfrak{sl}_n$-link homologies, aspects of combinatorics and algebraic geometry.

Then I will explain how Leclerc-Toffin's algorithm can be categorified using an instance of categorified $q$-skew Howe duality giving a graded cellular basis of the $\mathfrak{sl}_3$-web algebra $K_S$. The approach I use should, up to some combinatorics, work for all $n>1$.

Moreover, I sketch the connection of the $\mathfrak{sl}_n$-web/foam categories to the quantum link polynomials/homologies and I discuss some possible applications of our results - although the details still need to be worked out.