Cluster structures on $SL_n/SO_n$

We construct cluster structures on the strata $\mathring{S}_w$ of a stratification of the symmetric space $SL_n/SO_n$ over $\mathbb{C}$. To accomplish this, we study foldings of upper cluster algebras and show that the cluster structures on $SL_n/SO_n$ can be obtained from those on $SL_n$ via folding. As a corollary, we show that these cluster structures are compatible with the De Concini Poisson structures, and we construct cluster structures on the variety of symmetric matrices $\text{Sym}_n$.