Q-data and representation theory of untwisted quantum affine algebras

TL;DR

This paper introduces Q-data, a new combinatorial framework that unifies the understanding of quantum affine algebras and their representations, generalizing previous results and providing new formulas and invariants.

Contribution

It develops the theory of Q-data for untwisted quantum affine algebras, generalizes the inverse quantum Cartan matrix, and applies these to representation theory and R-matrix denominators.

Findings

Derived a combinatorial formula for the inverse quantum Cartan matrix.

Provided an alternative description of block decompositions of modules.

Presented a unified formula for R-matrix denominators.

Abstract

For a complex finite-dimensional simple Lie algebra $\mathfrak{g}$, we introduce the notion of Q-datum, which generalizes the notion of a Dynkin quiver with a height function from the viewpoint of Weyl group combinatorics. Using this notion, we develop a unified theory describing the twisted Auslander-Reiten quivers and the twisted adapted classes introduced in [O.-Suh, J. Algebra, 2019] with an appropriate notion of the generalized Coxeter elements. As a consequence, we obtain a combinatorial formula expressing the inverse of the quantum Cartan matrix of $\mathfrak{g}$, which generalizes the result of [Hernandez-Leclerc, J. Reine Angew. Math., 2015] in the simply-laced case. We also find several applications of our combinatorial theory of Q-data to the finite-dimensional representation theory of the untwisted quantum affine algebra of $\mathfrak{g}$. In particular, in terms of Q-data and the inverse of the quantum Cartan matrix, (i) we give an alternative description of the block decomposition results due to [Chari-Moura, Int. Math. Res. Not., 2005] and [Kashiwara-Kim-O.-Park, arXiv:2003.03265], (ii) we present a unified (partially conjectural) formula of the denominators of the normalized R-matrices between all the Kirillov-Reshetikhin modules, and (iii) we compute the invariants $\Lambda(V,W)$ and $\Lambda^\infty(V, W)$ introduced in [Kashiwara-Kim-O.-Park, Compos. Math., 2020] for each pair of simple modules $V$ and $W$.