Some determinants in the semi-stable Langlands program

We evaluate some determinants involving harmonic numbers that are needed in order to provide solutions to certain matrix equations occurring in our earlier paper \cite{CG24}. That paper determined the mod $p$ reductions of all two-dimensional semi-stable representations $V_{k,\mathcal{L}}$ of the Galois group of $\mathbb{Q}_p$ of weights $3 \leq k \leq p+1$ and $\mathcal{L}$-invariants $\mathcal{L}$ for primes $p \geq 5$. The present paper computes these determinants with the aid of two computer packages.