Abstract
In this paper, we study equivariant vector bundles on partial flag varieties arising from Schur functors. We show that a partial flag variety with three or more steps does not admit an Ulrich bundle of this form with respect to the minimal ample class. We classify Ulrich bundles of this form on two-step flag varieties F(1,n−1;n), F(2,n−1;n), F(2,n−2;n), F(k,k+1;n) and F(k,k+2;n). We give a conjectural description of the two-step flag varieties which admit such Ulrich bundles.
Highlights
In this paper, we study equivariant vector bundles on partial flag varieties arising from Schur functors
The question we study in this paper is to determine which Schur bundles on partial flag varieties are Ulrich bundles with respect to the minimal ample class
The sum of the Schubert divisors corresponds to a line bundle O(1) that defines a projectively normal embedding of F (k1, . . . , kr; n) [R]
Summary
The partial flag variety can be realized as an iterated Grassmannian bundle. The degree of the partial flag variety F The partition λ is a concatenation λ = (λ1| · · · |λr+1), where λi has length ki − ki−1, and where Sλi denotes the Schur functor of type λi Such bundles can be characterized as the pushforwards of line bundles on complete flag varieties. Adding a constant integer to all the entries of λ corresponds to tensoring the bundle Eλ with a power of the trivial line bundle and does not change its isomorphism class. The famous Borel-Weil-Bott Theorem computes the cohomology of line bundles on the complete flag variety, which allows us to calculate the cohomology of the Schur bundle Eλ [W]. The rank of the vector bundle Eλ is given by r+1 ks − ks−1 + j − i
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