On conjugation invariants in the dual Steenrod algebra

We investigate the canonical conjugation, $\chi$, of the mod $2$ dual Steenrod algebra, $\mathcal{A}_{*}$, with a view to determining the subspace, $\mathcal{A}_{*}^{\chi }$, of elements invariant under $\chi$. We give bounds on the dimension of this subspace for each degree and show that, after inverting $\xi _{1}$, it becomes polynomial on a natural set of generators. Finally we note that, without inverting $\xi _{1}$, $\mathcal{A}_{*}^{\chi }$ is far from being polynomial.