We propose a generalisation for the notion of the centre of an algebra in the setup of algebras graded by an arbitrary abelian group $G$. Our generalisation, which we call the $G$-centre, is designed to control the endomorphism category of the grading shift functors. We show that the $G$-centre is preserved by gradable derived equivalences given by tilting modules. We also discuss links with existing notions in superalgebra theory.

As a natural generalisation of $q$-Schur algebras associated with the Hecke algebra ${\mathcal{H}}_{r,R}$ (of the symmetric group), we introduce the queer $q$-Schur superalgebra associated with the Hecke–Clifford superalgebra ${\mathcal{H}}_{r,R}^{\mathsf{c}}$, which, by definition, is the endomorphism algebra of the induced ${\mathcal{H}}_{r,R}^{\mathsf{c}}$-module from certain $q$-permutation modules over ${\mathcal{H}}_{r,R}$. We will describe certain integral bases for these superalgebras in terms of matrices and will establish the base-change property for them. We will also identify the queer $q$-Schur superalgebras with the quantum queer Schur superalgebras investigated in the context of quantum queer supergroups and provide a constructible classification of their simple polynomial representations over a certain extension of the field $\mathbb{C}(\mathbf{v})$ of complex rational functions.

This paper examines a systematic method of constructing a pair of (inter-related) root systems for arbitrary Coxeter groups from a class of nonstandard geometric representations. This method can be employed to construct generalizations of root systems for a large family of linear groups generated by involutions. We then give a characterization of Coxeter groups, among these groups, in terms of such paired root systems. Furthermore, we use this method to construct and study the paired root systems for reflection subgroups of Coxeter groups.

Kleinewillinghöfer classified Laguerre planes with respect to linearly transitive groups of central automorphisms. Polster and Steinke investigated two-dimensional Laguerre planes and their so-called Kleinewillinghöfer types. For some of the feasible types the existence question remained open. We provide examples of such planes of type II.A.2, which are based on certain two-dimensional Laguerre planes of translation type. With these models only one type, I.A.2, is left for which no two-dimensional Laguerre planes are known yet.

We study the numbers of involutions and their relation to Frobenius–Schur indicators in the groups $\text{SO}^{\pm }(n,q)$ and $\unicode[STIX]{x1D6FA}^{\pm }(n,q)$. Our point of view for this study comes from two motivations. The first is the conjecture that a finite simple group $G$ is strongly real (all elements are conjugate to their inverses by an involution) if and only if it is totally orthogonal (all Frobenius–Schur indicators are 1), and we observe this holds for all finite simple groups $G$ other than the groups $\unicode[STIX]{x1D6FA}^{\pm }(4m,q)$ with $q$ even. We prove computationally that for small $m$ this statement indeed holds for these groups by equating their character degree sums with the number of involutions. We also prove a result on a certain twisted indicator for the groups $\text{SO}^{\pm }(4m+2,q)$ with $q$ odd. Our second motivation is to continue the work of Fulman, Guralnick, and Stanton on generating functions and asymptotics for involutions in classical groups. We extend their work by finding generating functions for the numbers of involutions in $\text{SO}^{\pm }(n,q)$ and $\unicode[STIX]{x1D6FA}^{\pm }(n,q)$ for all $q$, and we use these to compute the asymptotic behavior for the number of involutions in these groups when $q$ is fixed and $n$ grows.

We prove the existence of common fixed points for monotone contractive and monotone nonexpansive semigroups of nonlinear mappings acting in Banach spaces equipped with partial order. We also discuss some applications to differential equations and dynamical systems.