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The problem is open, and not because nobody tried. For instance, it is known that the number of similarity classes in $M_n(\mathbf Z/p^2 \mathbf Z)$ is equal to the number of simultaneous conjugacy cl …

Define a standard bitableau of size $n$ to be a pair $(P_1, P_2)$ of standard tableaux of total size $n$ such that each of the integers $1,\dotsc, n$ occurs exactly once in either tableau. Then $I_2( …

$K\pi^\lambda K$ has a transitive right action of $K$. The stabilizer of $K\pi^\lambda$ for this action is $K\cap \pi^{-\lambda}K\pi^\lambda$. Thus, $K\pi^\lambda K = \coprod_x K\pi^\lambda x$ as $x$ …

If you have $P$, I think you can recover $Q$ as $(D^{-1}PD)^{-1}$. Therefore, you are looking for invertible integer matrices $P$ such that $D^{-1}PD$ is also invertible (i.e., $P\in GL_n(\mathbf Z)\c …

Let $\mathbf A_T$ denote the restricted direct product $\mathbf R\times \prod'_{p\in T}\mathbf Q_p$ (relative to the subgroups $\mathbf Z_p$, $p\in T$). The OP asked whether the Pontryagin dual of $S …