Classification of prime modules of quantum affine algebras corresponding to 2-column tableaux
Nick Early, Jian-Rong Li
TL;DR
This work classifies prime finite-dimensional modules of $U_q(igl[\, angleigl)$ corresponding to 2-column semistandard Young tableaux in type $A$, by reducing primeness to the noncrossing/non-weakly-separated structure of a tableau’s 1-column components under a conjectural tensor-product criterion. It establishes that a 2-column tableau is prime exactly when its two 1-column constituents are noncrossing and not weakly separated, and provides an explicit count $a_{k,n,2}-b_{k,n}$ for such tableaux, with detailed enumeration for ${\\mathbb{C}}[ ext{Gr}(4,8)]$ and ${\mathbb{C}}[ ext{Gr}(5,10)]$. The paper also describes a promotion mechanism and derives concrete prime tableaux and modules in these cases, illustrating the interplay between quantum affine representation theory, Hernandez–Leclerc’s category, and Grassmannian cluster algebras. Finally, it offers a conjectural sufficient condition for primeness of tableaux with more than two columns and discusses the limitations of the condition through concrete examples, guiding future classification efforts across larger column counts.
Abstract
Finite dimensional simple modules of quantum affine algebras of type A correspond to semistandard Young tableaux of rectangular shapes. In this paper, we classify all prime modules corresponding to 2-column semistandard Young tableaux, up to a conjectural property. Moreover, we give a conjectural sufficient condition for a module corresponding to a tableau with more than two columns to be prime.