Representation type of finite quiver Hecke algebras of type $A^{(2)}_{2\ell}$

TL;DR

This paper investigates the structure and representation types of cyclotomic quiver Hecke algebras of type A^{(2)}_{2 extell}, providing formulas for their dimensions and criteria for their representation type, extending known results beyond classical cases.

Contribution

It introduces the first dimension formulas and representation type criteria for cyclotomic quiver Hecke algebras of this type, generalizing previous results for Iwahori-Hecke algebras.

Findings

Derived a formula for algebra dimensions.

Established a criterion for representation type.

Extended known results to new algebra classes.

Abstract

We study cyclotomic quiver Hecke algebras $R^{\Lambda_0}(\beta)$ in type $A^{(2)}_{2\ell}$, where $\Lambda_0$ is the fundamental weight. The algebras are natural $A^{(2)}_{2\ell}$-type analogue of Iwahori-Hecke algebras associated with the symmetric group, from the viewpoint of the Fock space theory developed by the first author and his collaborators. We give a formula for the dimension of the algebra, and a simple criterion to tell the representation type. The criterion is a natural generalization of Erdmann and Nakano's for the Iwahori-Hecke algebras. Except for the examples coming from cyclotomic Hecke algebras, no results of these kind existed for cyclotomic quiver Hecke algebras, and our results are the first instances beyond the case of cyclotomic Hecke algebras.