A topological construction for all two-row Springer varieties

“…This vector space carries a natural S 2n -action coming from the category of U q (sl 2 ) representations (described further below). The resulting web representation is irreducible and in fact is the same as the irreducible S 2n -representation on the classic Specht module of shape (n, n), whose basis of Specht vectors is naturally indexed by the standard Young tableaux of shape (n, n) [24,28,29].…”

Section: Introductionmentioning

confidence: 99%

The Transition Matrix between the Specht and Web Bases Is Unipotent with Additional Vanishing Entries

Russell

Tymoczko

2017

International Mathematics Research Notices

View full text Add to dashboard Cite

We compare two important bases of an irreducible representation of the symmetric group: the web basis and the Specht basis. The web basis has its roots in the Temperley-Lieb algebra and knot-theoretic considerations. The Specht basis is a classic algebraic and combinatorial construction of symmetric group representations which arises in this context through the geometry of varieties called Springer fibers. We describe a graph that encapsulates combinatorial relations between each of these bases, prove that there is a unique way (up to scaling) to map the Specht basis into the web representation, and use this to recover a result of Garsia-McLarnan that the transition matrix between the Specht and web bases is upper-triangular with ones along the diagonal. We then strengthen their result to prove vanishing of certain additional entries unless a nesting condition on webs is satisfied. In fact we conjecture that the entries of the transition matrix are nonnegative and are nonzero precisely when certain directed paths exist in the web graph.

show abstract

“…This conjecture was proven independently by Wehrli [Weh09] and Russell-Tymoczko [RT11,Appendix] using results contained in [CK08]. The constructions and results were generalized to all two-row Springer fibers of type A in [Rus11]. In this article we define topological models for all two-row Springer fibers associated with the even orthogonal (type D) and the symplectic group (type C) and prove that they are homeomorphic to their corresponding Springer fiber.…”

Section: Introductionmentioning

confidence: 92%

Topology of two-row Springer fibers for the even orthogonal and symplectic group

Wilbert

2017

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

We construct an explicit topological model (similar to the topological Springer fibers appearing in work of Khovanov and Russell) for every two-row Springer fiber associated with the even orthogonal group and prove that the respective topological model is homeomorphic to its corresponding Springer fiber. This confirms a conjecture by Ehrig and Stroppel concerning the topology of the equal-row Springer fiber for the even orthogonal group. Moreover, we show that every two-row Springer fiber for the symplectic group is homeomorphic (even isomorphic as an algebraic variety) to a connected component of a certain two-row Springer fiber for the even orthogonal group.

show abstract

A topological construction for all two-row Springer varieties

Cited by 19 publications

References 18 publications

Euler Characteristic of Springer Fibers

Euler Characteristic of Springer Fibers

The Transition Matrix between the Specht and Web Bases Is Unipotent with Additional Vanishing Entries

Topology of two-row Springer fibers for the even orthogonal and symplectic group

Contact Info

Product

Resources

About