On odd unitary Steinberg group

TL;DR

This paper investigates the algebraic structure of odd unitary Steinberg groups over rings with pseudo-involution, focusing on computing their Schur multipliers, which are key invariants in group theory.

Contribution

It provides the first computation of the Schur multipliers for odd unitary Steinberg groups and their elementary subgroups, advancing understanding of their algebraic properties.

Findings

Computed Schur multipliers of odd unitary Steinberg groups

Determined Schur multipliers of elementary subgroups

Enhanced understanding of algebraic invariants in unitary groups

Abstract

Let $R$ be a ring with pseudo-involution, $\mathfrak L$ be an odd form parameter, $\mathrm U(2n,\,R,\,\mathfrak L)$ be an odd hyperbolic unitary group, $\mathrm{EU}(2n,\,R,\,\mathfrak L)$ be it elementary subgroup and $\mathrm{StU}(2n,\,R,\,\mathfrak L)$ be an odd unitary Steinberg group. We compute the Schur multipliers of $\mathrm{StU}(2n,\,R,\,\mathfrak L)$ and $\mathrm{EU}(R,\,\mathfrak L)$.