A braid-like presentation of the integral Steinberg group of type $C_2$

TL;DR

This paper presents a novel braid-like presentation of the integral Steinberg group of type C2, connecting it to braid groups and providing a new perspective on symplectic modular groups.

Contribution

It introduces a braid group quotient description of the Steinberg group of type C2 and derives a new braid-like presentation of the symplectic modular group Sp_4(Z).

Findings

Steinberg group of type C2 is a quotient of B_6 by one relation.

New braid-like presentation of Sp_4(Z) derived.

Provides algebraic insights linking braid groups and symplectic groups.

Abstract

We show that the Steinberg group $\text{St}(C_2,{\mathbb Z})$ associated with the Lie type $C_2$ and with integer coefficients can be realized as a quotient of the braid group $B_6$ by one relation. As an application we give a new braid-like presentation of the symplectic modular group $\text{Sp}_4({\mathbb Z})$.