The weak parking-space conjecture

Armstrong, Reiner and Rhoades conjectured in 2012 that the noncrossing and algebraic parking spaces of an irreducible finite real reflection group have the same character under the reflection group and Coxeter rotation. We prove their weak conjecture in types $E_7$ and $E_8$, its last two open real types, completing the conjecture. The proof expresses fixed parking functions as intersections of conjugacy classes with cosets of pointwise parabolic subgroups. Complete finite certificates verify these intersections and the noncrossing interval, using exact arithmetic. Together with the previously established types, the result gives the conjectured complex representation isomorphism for every irreducible finite real reflection group. The assertion concerns the original parking spaces, with Fuss parameter $1$.

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