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Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians

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Abstract

For each integer \(k\ge 4\), we describe diagrammatically a positively graded Koszul algebra \(\mathbb {D}_k\) such that the category of finite dimensional \(\mathbb {D}_k\)-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type \(\mathrm{D}_k\) or \(\mathrm{B}_{k-1}\), constructible with respect to the Schubert stratification. The algebra is obtained by a (non-trivial) “folding” procedure from a generalized Khovanov arc algebra. Properties such as graded cellularity and explicit closed formulas for graded decomposition numbers are established by elementary tools.

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Notes

  1. see, e.g., [19, 9.4] with the notation \(\tilde{\Omega }_\lambda \) there.

  2. This condition arises also in the context of blob algebras as studied for instance in [32], and of generalized Temperley–Lieb algebras [21]. It is derived naturally from the theory in [7] by a “folding” procedure, see [30, 5.2].

  3. In case C has no dots, this agrees with the obvious notion of (anti)clockwise.

  4. The artificially looking choice of only allowing admissible surgeries between admissible stacked circle diagrams becomes transparent and consistent with [7] if we use the approach from [30]. Rewriting the dotted cup diagrams in terms of symmetric cup diagrams, as in [30], turns neighbored cup–cap pairs into nested ones, and our admissibility assures that we only use those that are applicable for surgery in the sense of [7].

  5. This definition of outer circle makes sense when we work with symmetric diagrams as in [30] where it would turn just into a circle which is not nested inside any other circle.

  6. We tried to clarify the misleading formulation of the analog of (R-5) (iii) in [4].

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Acknowledgments

We like to thank Jon Brundan, Ngau Lam, Antonio Sartori, and Vera Serganova for many helpful discussions and Daniel Tubbenhauer for comments on the manuscript. We are in particular grateful to the referee for his/her extremely careful reading of the manuscript and for several helpful suggestions.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Bonn, Endenicher Allee 60, Room 1.003, 53115, Bonn, Germany

    Michael Ehrig

  2. Mathematisches Institut, Universität Bonn, Endenicher Allee 60, Room 4.007, 53115, Bonn, Germany

    Catharina Stroppel

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  1. Michael Ehrig
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  2. Catharina Stroppel
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Correspondence to Catharina Stroppel.

Additional information

M.E. was financed by the DFG Priority program 1388. This material is based on work supported by the National Science Foundation under Grant No. 0932078 000, while the authors were in residence at the MSRI in Berkeley, California.

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Ehrig, M., Stroppel, C. Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians. Sel. Math. New Ser. 22, 1455–1536 (2016). https://doi.org/10.1007/s00029-015-0215-9

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