Document Type
Article
Publication Date
7-1-2011
Publication Title
Mathematical Proceedings of the Cambridge Philosophical Society
Abstract
Springer varieties are studied because their cohomology carries a natural action of the symmetric group Sn and their top-dimensional cohomology is irreducible. In his work on tangle invariants, Khovanov constructed a family of Springer varieties Xn as subvarieties of the product of spheres (S2)n. We show that if Xn is embedded antipodally in (S2)n then the natural Sn-action on (S2)n induces an Sn-representation on the image of H*(Xn). 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*(Xn) is irreducible in each degree. We explicitly identify the Kazhdan-Lusztig basis for the irreducible representation of Sn corresponding to the partition (n/2, n/2).
Volume
151
Issue
1
First Page
59
Last Page
81
DOI
10.1017/S0305004111000132
ISSN
03050041
Recommended Citation
Russell, Heather M. and Tymoczko, Julianna S., "Springer Representations on the Khovanov Springer Varieties" (2011). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/105
Comments
Peer reviewed accepted manuscript.