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Mathematische Zeitschrift

, Volume 292, Issue 3–4, pp 1387–1430 | Cite as

Two-block Springer fibers of types C and D: a diagrammatic approach to Springer theory

  • Catharina Stroppel
  • Arik WilbertEmail author
Article

Abstract

We explain an elementary topological construction of the Springer representation on the homology of (topological) Springer fibers of types C and D in the case of nilpotent endomorphisms with two Jordan blocks. The Weyl group and component group actions admit a diagrammatic description in terms of cup diagrams which appear in the definition of arc algebras of types B and D. We determine the decomposition of the representations into irreducibles and relate our construction to classical Springer theory. As an application we obtain presentations of the cohomology rings of all two-block Springer fibers of types C and D. Moreover, we deduce explicit isomorphisms between the Kazhdan-Lusztig cell modules attached to the induced trivial module and the irreducible Specht modules in types C and D.

Keywords

Springer fiber Action on cohomology Springer theory Diagram algebras Flag varieties Betti numbers 

Mathematics Subject Classification

Primary 14M15 17B08 17B10 Secondary 05E10 20C08 20F36 

Notes

Acknowledgements

The authors thank Michael Ehrig and Daniel Tubbenhauer for many helpful comments and interesting discussions and Daniel Juteau and Martina Lanini for raising the question how to make the action of the component group explicit in our construction. Parts of this article appeared in the second author’s PhD thesis under the supervision of the first author.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of BonnBonnGermany
  2. 2.School of Mathematics and StatisticsUniversity of MelbourneMelbourneAustralia

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