I was recently going through the literature on RR-field tadpole cancellation for fractional D-branes stuck at orbifold/orientifold singularities.

A clean review is in chapter 4 of Marchesano's thesis arXivhep-th/0307252.

There formula (4.9) gives the condition for tadpole cancellation of D4-branes at a G-orbifold singularity, and (4.15) is the analogue with an O8-plane intersecting, following Honecker's arXiv:0201037. Other cases like for D5-branes in (4.12) look slightly different, but always involve the same representation theoretic expressions -- namely the total characters of the linear representation corresponding to the brane configuration evaluated away from the neutral group element (as highlighted on top of p. 49 in Marchesano's text).

Now, the multiples of the regular representation (corresponding to the "mobile brane") always solve the homogeneous condition (4.9), and hence sums of the trivial rep cancelling the O-plane charge with any regular representation always solve the inhomogeneous condition (4.15). This is more or less highlighted on p. 49.

But it seems to me that these are in fact the *only* solutions. (!?)

This follows from a general representation-theoretic argument, which I have spelled out here. But it may also be checked explicitly in examples, which I have spelled out here.

So I am pretty sure this conclusion holds, but if you think I am overlooking anything, please drop me a note. If it holds, this would seem to be a fact worth highlighting. If anyone can point me to literature which makes this explicit, please let me know.