An introduction to $(G,c)$-bands

TL;DR

This paper introduces $(G,c)$-bands, connecting cluster structures with quantum affine algebra representations, and verifies a conjecture relating $q$-characters to a discrete Miura transformation.

Contribution

It constructs a discrete analogue of the difference Miura transformation using $(G,c)$-bands and proves its relation to $q$-characters of quantum affine algebra representations.

Findings

Constructed a discrete Miura transformation analogue.

Verified the conjecture relating $q$-characters to the transformation.

Connected cluster structures with quantum affine algebra representations.

Abstract

We give an introduction to our results on cluster structures for schemes of $(G,c)$-bands emphasizing their connections with seminal works of Frenkel and Reshetikhin in the 90's. In particular we construct using $(G,c)$-bands a discrete analogue of the difference Miura transformation of the loop group $LG$, and we show that it calculates the $q$-characters of the finite-dimensional representations of the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ of the same $A$, $D$, $E$ type as $G$, thus verifying a conjecture of Frenkel and Reshetikhin.