Generating basis webs for $\SL_n$

TL;DR

This paper constructs a basis of webs for the invariant space of tensor products of minuscule representations of SL_n, generalizing previous results and establishing a basis via the geometric Satake correspondence.

Contribution

It introduces a new set of webs for SL_n that form a basis for the invariant space, extending Westbury's work to all n and connecting to the geometric Satake correspondence.

Findings

The web set forms a basis for the invariant space.

There is an upper unitriangular change of basis to the Satake basis.

The construction generalizes previous cases for n=2,3 and n≥4.

Abstract

Given a simple algebraic group $G$, a web is a directed trivalent graph with edges labelled by dominant minuscule weights. There is a natural surjection of webs onto the invariant space of tensor products of minuscule representations. Following the work of Westbury, we produce a set of webs for $\SL_n$ which form a basis for the invariant space via the geometric Satake correspondence. In fact, there is an upper unitriangular change of basis to the Satake basis. This set of webs agrees with previous work in the cases $n=2,3$ and generalizes the work of Westbury in the case $n\geq 4$.