Quantum $\frak {gl}_\infty$, infinite $q$-Schur algebras and their representations

TL;DR

This paper explores the structure and representations of the quantum infinite general linear group, providing realizations, examining homomorphisms to infinite q-Schur algebras, and analyzing polynomial representations.

Contribution

It introduces a BLM-type realization for quantum infinity and its images, and constructs a completion algebra to extend homomorphisms to infinite q-Schur algebras.

Findings

The homomorphism infinityinfinity to infinity,rinfinity is not surjective for any r.

A BLM-type realization for the image infinity,r is established.

A completion algebra infinity is constructed to extend homomorphisms.

Abstract

In this paper, we investigate the structure and representations of the quantum group ${\mathbf{U}(\infty)}=\mathbf U_\upsilon(\frak{gl}_\infty)$. We will present a realization for $\mathbf{U}(\infty)$, following Beilinson--Lusztig--MacPherson (BLM) \cite{BLM}, and show that the natural algebra homomorphism $\zeta_r$ from $\mathbf{U}(\infty)$ to the infinite $q$-Schur algebra ${\boldsymbol{\mathcal S}}(\infty,r)$ is not surjective for any $r\geq 1$. We will give a BLM type realization for the image $\mathbf{U}(\infty,r):=\zeta_r(\mathbf{U}(\infty))$ and discuss its presentation in terms of generators and relations. We further construct a certain completion algebra $\hat{\boldsymbol{\mathcal K}}^\dagger(\infty)$ so that $\zeta_r$ can be extended to an algebra epimorphism $\tilde\zeta_r:\hat{\boldsymbol{\mathcal K}}^\dagger(\infty)\to{\boldsymbol{\mathcal S}}(\infty,r)$. Finally we will investigate the representation theory of ${\bf U}(\infty)$, especially the polynomial representations of ${\bf U}(\infty)$.