Cells in modified $\imath$quantum groups of type AIII and related Schur algebras

TL;DR

This paper characterizes two-sided cells in modified $ extit{ extbf{AIII}}$-type $ extit{ extbf{imath}}$ quantum groups and related Schur algebras using combinatorial methods, extending known descriptions from type B to AIII.

Contribution

It provides a new combinatorial framework for understanding two-sided cells in $ extit{ extbf{AIII}}$-type $ extit{ extbf{imath}}$ quantum groups and related Schur algebras, building on type B descriptions.

Findings

Combinatorial description of two-sided cells in $ extit{ extbf{AIII}}$-type $ extit{ extbf{imath}}$ quantum groups.

Extension of cell descriptions to $ extit{ extbf{imath}}$-Schur and $ ilde{ extit{ extbf{imath}}}$-Schur algebras.

Connection established between $ extit{ extbf{AIII}}$ cells and $ extit{ extbf{j}}$-Schur algebra cells.

Abstract

We provide a combinatorial characterization of two-sided cells in modified $\imath$quantum groups of type AIII. Our approach is to lift a corresponding description of two-sided cells in $\jmath$-Schur algebras associated to Iwahori--Hecke algebras of type $B$. We further give a combinatorial description of two-sided cells in other two kinds of Schur-type algebras, namely $\imath$-Schur algebras and $\tilde{\imath}$-Schur algebras.