The ring of $\omega$-invariant symmetric functions in characteristic 2

We provide a simple presentation by generators and relations of the ring of $\omega$-invariant symmetric functions over the field ${\mathbb{F}}_2$. Here, $\omega$ denotes the standard involution on the ring of symmetric functions, interchanging the elementary symmetric functions with the complete homogeneous symmetric functions. Along the way, we prove several important properties of this involution in the specific setting of characteristic 2.