Let $G$ be a finite group and $\Sigma_g$ a closed Riemann surface of genus $g$. Then Mednykh's formula states

$\frac{\left|\mathrm{Hom}(\pi_1(\Sigma_g),G)\right|}{\left|G\right|} = \frac{1}{\left|G\right|^{\chi(\Sigma_g)}}\sum_{V}\left(\dim V\right)^{\chi(\Sigma_g)}$

where the sum on the right is over irreducible complex representations of $G$.

Is there an analog in modular representation theory? For example we can fix the problem in the case $g=0,1$: the number of irreducible representations equals to number of $p$-regular conjugacy classes, the vector $x=(\text{dim} V_i)^t$ satisfies $x^tCx=|G|$ where $C=D^tD$ is the Cartan matrix.


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