Quantum affine $\frak{gl}_n$ via Hecke algebras

TL;DR

This paper constructs a new algebra using affine Hecke algebras and Schur algebras, proving it is isomorphic to the quantum loop algebra of rak{gl}_n, offering a purely algebraic and combinatorial approach.

Contribution

It provides a new algebraic and combinatorial construction of the quantum affine rak{gl}_n, independent of geometric methods, and presents a novel description of the Ringel--Hall algebra of a cyclic quiver.

Findings

Established an explicit isomorphism to the quantum loop algebra of rak{gl}_n

Developed a basis, generators, and multiplication formulas for the new algebra

Presented a new presentation of the Ringel--Hall algebra of a cyclic quiver

Abstract

We use the Hecke algebras of affine symmetric groups and their associated Schur algebras to construct a new algebra through a basis, and a set of generators and explicit multiplication formulas of basis elements by generators. We prove that this algebra is isomorphic to the quantum enveloping algebra of the loop algebra of $\mathfrak {gl}_n$. Though this construction is motivated by the work \cite{BLM} by Beilinson--Lusztig--MacPherson for quantum $\frak{gl}_n$, our approach is purely algebraic and combinatorial, independent of the geometric method which seems to work only for quantum $\mathfrak{gl}_n$ and quantum affine $\mathfrak{sl}_n$. As an application, we discover a presentation of the Ringel--Hall algebra of a cyclic quiver by semisimple generators and their multiplications by the defining basis elements.