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Cox rings of moduli of quasi-parabolic principal bundles and the \(K\)-Pieri rule. (English) Zbl 1328.05197

Summary: We study a toric degeneration of the Cox ring of the moduli of quasi-principal \(\mathrm{SL}_m(\mathbb{C})\) bundles on a marked projective line in the case where the parabolic data is chosen in the stabilizer of the highest weight vector in \(\mathbb{C}^m\) or its dual representation \(\land^{m-1}(\mathbb{C}^m)\). 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 A. M. Castravet and J. Tevelev [Compos. Math. 142, No. 6, 1479–1498 (2006; Zbl 1117.14048)] and S. Sturmfels and Z. Xu [J. Eur. Math. Soc. (JEMS) 12, No. 2, 429–459 (2010; Zbl 1202.14053)]. Along the way we describe a relationship between the Cox ring and a classical invariant ring studied by H. Weyl [The classical groups, their invariants and representations. Princeton. New Jersey: Univ. Press a. London: Humphrey Milford Oxford University Press (1939; Zbl 0020.20601)].

MSC:

05E10 Combinatorial aspects of representation theory
13C14 Cohen-Macaulay modules
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
14H10 Families, moduli of curves (algebraic)
14Q99 Computational aspects in algebraic geometry
13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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