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On syzygies of highest weight orbits

A. L. Gorodentsev·A. S. Khoroshkin·A. N. Rudakov
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Abstract

We consider the graded space $R$ of syzygies for the coordinate algebra $A$ of projective variety $X=G/P$ embedded into projective space as an orbit of the highest weight vector of an irreducible representation of semisimple complex Lie group $G$. We show that $R$ is isomorphic to the Lie algebra cohomology $H=H^\bdot(\Lt,\CC)$, where $\Lt$ is graded Lie subalgebra of the graded Lie s-algebra $L=L_1\oplus\Lt$ Koszul dual to $A$. We prove that the isomorphism identifies the natural associative algebra structures on $R$ and $H$ coming from their Koszul and Chevalley DGA resolutions respectively. For subcanonically embedded $X$ a Frobenius algebra structure on the syzygies is constructed. We illustrate the results by several examples including the computation of syzygies for the Plücker embeddings of grassmannians $\Gr(2,N)$.

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A. L. Gorodentsev, A. S. Khoroshkin, A. N. Rudakov. 2007-02-03. On syzygies of highest weight orbits. https://arxiv.org/abs/math/0602316

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