  The spectrum of conformal weights for the CFT describing the two-dimensional
critical $Q$-state Potts model (or its close cousin, the dense loop model) has
been known for more than 30 years. However, the exact nature of the
corresponding $\hbox{Vir}\otimes\overline{\hbox{Vir}}$ representations has
remained unknown up to now. Here, we solve the problem for generic values of
$Q$. This is achieved by a mixture of different techniques: a careful study of
"Koo--Saleur generators" [arXiv:hep-th/9312156], combined with measurements of
four-point amplitudes, on the numerical side, and OPEs and the four-point
amplitudes recently determined using the "interchiral conformal bootstrap" in
[arXiv:2005.07258] on the analytical side. We find that null-descendants of
diagonal fields having weights $(h_{r,1},h_{r,1})$ (with $r\in \mathbb{N}^*$)
are truly zero, so these fields come with simple
$\hbox{Vir}\otimes\overline{\hbox{Vir}}$ ("Kac") modules. Meanwhile, fields
with weights $(h_{r,s},h_{r,-s})$ and $(h_{r,-s},h_{r,s})$ (with
$r,s\in\mathbb{N}^*$) come in indecomposable but not fully reducible
representations mixing four simple $\hbox{Vir}\otimes\overline{\hbox{Vir}}$
modules with a familiar "diamond" shape. The "top" and "bottom" fields in these
diamonds have weights $(h_{r,-s},h_{r,-s})$, and form a two-dimensional Jordan
cell for $L_0$ and $\bar{L}_0$. This establishes, among other things, that the
Potts-model CFT is logarithmic for $Q$ generic. Unlike the case of non-generic
(root of unity) values of $Q$, these indecomposable structures are not present
in finite size, but we can nevertheless show from the numerical study of the
lattice model how the rank-two Jordan cells build up in the infinite-size
limit.
