Cells in modified $\mathrm{ı}$quantum groups of type AIII and related Schur algebras

We provide a combinatorial characterization of two-sided cells in modified $\mathrm{ı}$quantum groups of type AIII. Our approach is to lift a corresponding description of two-sided cells in $\mathrm{ȷ}$-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 $\mathrm{ı}$-Schur algebras and $\tilde{\mathrm{ı}}$-Schur algebras.