Cyclotomic quiver Hecke algebras and Hecke algebra of G(r,p,n)

TL;DR

This paper presents a new presentation for fixed subalgebras of cyclotomic quiver Hecke algebras under quiver automorphisms and extends known isomorphisms to the Hecke algebra of type G(r,p,n).

Contribution

It generalizes Brundan and Kleshchev's isomorphism to include fixed subalgebras and provides explicit presentations and isomorphisms for cyclotomic Hecke algebras of type G(r,p,n).

Findings

Presented a new presentation for fixed subalgebras of cyclotomic quiver Hecke algebras.

Extended isomorphisms to the Hecke algebra of type G(r,p,n).

Explicit isomorphism realizing Morita equivalence between Ariki-Koike algebras.

Abstract

Given a quiver automorphism with nice properties, we give a presentation of the fixed subalgebra of the associated cyclotomic quiver Hecke algebra. Generalising an isomorphism of Brundan and Kleshchev between the cyclotomic Hecke algebra of type G(r,1,n) and the cyclotomic quiver Hecke algebra of type A, we apply the previous result to find a presentation of the cyclotomic Hecke algebra of type G(r,p,n) which looks very similar to the one of a cyclotomic quiver Hecke algebra. In the meanwhile, we give an explicit isomorphism which realises a well-known Morita equivalence between Ariki-Koike algebras.