Cox rings of moduli of quasi-parabolic principal bundles and the K-Pieri rule

We study a toric degeneration of the Cox ring of the moduli of quasi-principal SLm(C)SLm(C) bundles on a marked projective line in the case where the parabolic data is chosen in the stabilizer of the highest weight vector in CmCm or its dual representation ⋀m−1(Cm)⋀m−1(Cm). The result of this degeneration is an affine semigroup algebra which is naturally related to the combinatorics of the K-Pieri rule from Kac–Moody representation theory. We find that this algebra is normal and Gorenstein, with a quadratic square-free Gröbner basis. This implies that the Cox ring is Gorenstein and Koszul for generic choices of markings, and generalizes results of Castravet, Tevelev and Sturmfels, Xu. Along the way we describe a relationship between the Cox ring and a classical invariant ring studied by Weyl.