Springer Representations on the Khovanov Springer Varieties

Authors

Heather M. Russell, University of RichmondFollow
Julianna Tymoczko

DOI

10.1017/S0305004111000132

Abstract

Springer varieties are studied because their cohomology carries a natural action of the symmetric group S_n and their top-dimensional cohomology is irreducible. In his work on tangle invariants, Khovanov constructed a family of Springer varieties X_n as subvarieties of the product of spheres (S²)ⁿ. We show that if X_n is embedded antipodally in (S²)ⁿ then the natural S_n-action on (S²)ⁿ induces an Sn-representation on the image of H_*(X_n). This representation is the Springer representation. Our construction admits an elementary (and geometrically natural) combinatorial description, which we use to prove that the Springer representation on H_*(X_n) is irreducible in each degree. We explicitly identify the Kazhdan-Lusztig basis for the irreducible representation of S_n corresponding to the partition (n/2, n/2).

Document Type

Post-print Article

Publication Date

2011

Publisher Statement

Copyright © 2011 Cambridge Philosophical Society

The definitive version is available at: http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8278141&fulltextType=RA&fileId=S0305004111000132

DOI: 10.1017/S0305004111000132

Full Citation:

Russell, Heather M., and Julianna Tymoczko. "Springer Representations on the Khovanov Springer Varieties." Mathematical Proceedings of the Cambridge Philosophical Society 151, no. 1 (2011): 59-81. doi:10.1017/S0305004111000132.

Recommended Citation

Russell, Heather M. and Tymoczko, Julianna, "Springer Representations on the Khovanov Springer Varieties" (2011). Department of Math & Statistics Faculty Publications. 111.
https://scholarship.richmond.edu/mathcs-faculty-publications/111