Modified affine Hecke algebras and quiver Hecke algebras of type $A$

TL;DR

This paper introduces modified affine Hecke algebras of type A, constructs their representations and bases, and explores their connections with quiver Hecke algebras, providing insights into their centers and isomorphisms.

Contribution

It defines new modified forms of affine Hecke algebras, constructs faithful representations, and establishes isomorphisms with quiver Hecke algebras, advancing understanding of their structure.

Findings

Faithful representations and standard bases constructed for the modified algebras.

Explicit descriptions of the centers of these algebras provided.

Proved equivalence of the center conjecture for cyclotomic quiver and Hecke algebras.

Abstract

We introduce some modified forms for the degenerate and non-degenerate affine Hecke algebras of type $A$. These are certain subalgebras living inside the inverse limit of cyclotomic Hecke algebras. We construct faithful representations and standard bases for these algebras and give some explicit description of their centers. We show that there are algebra isomorphisms between some generalized Ore localizations of these modified affine Hecke algebras and of the quiver Hecke algebras of type $A$. As an application, we show that the center conjecture for the cyclotomic quiver Hecke algebra of type $A$ holds if and only if the center conjecture for the cyclotomic Hecke algebra of type $A$ holds.