正交辛李超代数、Koszul对偶和C-N复形的完全交模拟
Orthosymplectic Lie superalgebras, Koszul duality, and a complete intersection analogue of the Eagon-Northcott complex
摘要：
本文研究矩阵空间中一类二次完全交Littlewood簇的极大子式的理想。我们给出了一个大类的模的尺规作图法，包括这个理想的所有权力，并表明，他们有一个线性的自由分解在完全的交叉和他们的Koszul对偶是无限维不可约表示的正交辛李超代数。我们计算了作用在这个自由分解上的上同调算子的代数。证明了长为2的复形簇中的极大子式理想的幂的类似结果，并证明了它们的Koszul对偶是一般线性李超代数的无限维不可约表示.这概括了工作的阿金，Józefiak，Pragacz，韦曼，和作者的决议行列式理想的多项式环的设置完全交叉，并提供了一个新的连接之间的代表性的经典李超代数和交换代数。作为一个有趣的应用，我们证明了一类可约齐性丛的上同调辛和正交格拉斯曼和2步旗品种可以计算的Borel-Weil-Bott定理的模拟。
英文摘要：
We study the ideal of maximal minors in Littlewood varieties, a class of quadratic complete intersections in spaces of matrices. We give a geometric construction for a large class of modules, including all powers of this ideal, and show that they have a linear free resolution over the complete intersection and that their Koszul dual is an infinite-dimensional irreducible representation of the orthosymplectic Lie superalgebra. We calculate the algebra of cohomology operators acting on this free resolution. We prove analogous results for powers of the ideals of maximal minors in the variety of length 2 complexes when it is a complete intersection, and show that their Koszul dual is an infinite-dimensional irreducible representation of the general linear Lie superalgebra. This generalizes work of Akin, Józefiak, Pragacz, Weyman, and the author on resolutions of determinantal ideals in polynomial rings to the setting of complete intersections and provides a new connection between representations of classical Lie superalgebras and commutative algebra. As a curious application, we prove that the cohomology of a class of reducible homogeneous bundles on symplectic and orthogonal Grassmannians and 2-step flag varieties can be calculated by an analogue of the Borel-Weil-Bott theorem.
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