Upgrades of $p$-divisors

A $p$-divisor on a normal variety $Y$ is a divisor satisfying some positivity properties, where the usual integral or rational coefficients have been replaced by polyhedral coefficients.
K. Altmann and J. Hausen have shown that there is a correspondence between $p$-divisors and affine $T$-varieties, i.e. normal varieties together with some effective torus action.
Given some $T$-variety $X$, one can try to “upgrade” the torus action by considering some larger torus acting on $X$. I will discuss how this upgrading procedure changes the corresponding $p$-divisor. Time permitting, I will present an application dealing with the $p$-divisors of Cox rings of certain $T$-varieties.
This is joint work with R. Vollmert.