Intersection cohomology of singular quiver moduli

[pbelmans]

Ports the intersection cohomology computation from HodgeDiamonds.jl into QuiverTools.

When some proper subdimension vector has the slope of d, there are properly semistable representations, the moduli space is singular, and poincare_polynomial / hodge_diamond no longer apply. Meinhardt--Reineke (MR4000572) identify the Donaldson--Thomas invariants of the quiver with the intersection cohomology of M^ss(Q, d), which makes it computable from the motives of the semistable stacks that motive already provides. What is left to add is the plethystic logarithm.

New exported methods, all in src/Hodge.jl:

• intersection_poincare_polynomial(M)
• intersection_betti_numbers(M)
• intersection_hodge_diamond(M)

Everything is a rational function in the Lefschetz class, so the half powers appearing in the theorem only need a square root w of it, with L^{1/2} = -w.

The theorem needs the stability parameter to be generic for the slope of d (the antisymmetrized Euler form vanishes on the dimension vectors of that slope) and stable representations to exist; both are checked and raise an ArgumentError otherwise. That the result is a polynomial in w^2 is checked too, which validates the whole computation.

For a dimension vector that is primitive in the monoid of its slope the plethystic logarithm is its own leading term, so these agree with the ordinary invariants on smooth moduli spaces. The tests route several smooth cases through the new code path on purpose, and check the reflection functor identities M(a, b) = M(b, m*b - a) for the m-Kronecker quiver on singular moduli spaces, Poincaré duality, and the error paths.