Bounded core partitions and Borel-Weil-Bott

TL;DR

This paper develops new combinatorial formulas and bounds for computing Hodge numbers of twisted sheaves on Grassmannians, using core partitions, tableaux, and a q-analogue, extending classical theorems.

Contribution

It introduces a novel hook-product statistic on bounded partitions and improves bounds for positivity of Hodge numbers, connecting combinatorics with algebraic geometry.

Findings

Derived two effective formulas for Hodge numbers.

Established a combinatorial proof of Nakano vanishing theorem.

Extended Hodge number computations to a q-analogue.

Abstract

The Borel-Weil-Bott theorem can be used to decompose the cohomology of twisted sheaves of holomorphic forms on the complex Grassmannian into irreducible representations of the general linear group. By analyzing this decomposition, we provide two effective formulae for computing the associated Hodge numbers, and give examples in special cases. One of these involves a novel integer-valued hook-product statistic on bounded partitions, and the other is based on semistandard tableaux. We reformulate Snow's observation that the positivity of the Hodge numbers is equivalent to the existence of a partition satisfying certain properties, and improve known bounds on when this occurs. This involves a combinatorial proof of the Nakano vanishing theorem for the Grassmannian utilizing a map from core partitions to plane partitions. Finally, we extend our computation of the Hodge numbers of twisted holomorphic forms on the Grassmannian to a q-analogue.