SMRI Algebra and Geometry Online: Vazirani -- From representations of the rational Cherednik algebra to parabolic Hilbert schemes via the Dunkl-Opdam

SMRI Algebra and Geometry Online 

’From representations of the rational Cherednik algebra to parabolic Hilbert schemes
via the Dunkl-Opdam subalgebra’

Monica Vazirani (University of California, Davis) 

Thursday, 14th April, 10:00am - 11:30am (AEST) Register:
https://uni-sydney.zoom.us/meeting/register/tZAkf-qupj0oE9RKU8A9NcwUnEnoxXMHm6TT After
registering, you will be sent a confirmation email ~24 hours prior to the seminar.  

Abstract: Young diagrams and standard tableaux on them parameterize irreducible
representations of the symmetric group and their bases, respectively.  There is a
similar story for the double affine Hecke algebra (DAHA) taking periodic tableaux, or
for the rational Cherednik algebra (a.k.a.  rational DAHA) with appropriate
modifications.  This construction of the basis makes use of an alternate presentation of
the rational DAHA and the basis diagonalizes the action of its Dunkl-Opdam subalgebra.
We make use of the combinatorics to construct explicit maps between standard modules
parameterized by hooks, thus recovering the BGG resolution of the simple module
parameterized by the trivial hook.  

We can also describe this simple module using the geometry of parabolic Hilbert schemes
of points on plane curve singularities.  The ``tableau" basis that diagonalizes the
Dunkl-Opdam subalgebra is the basis of equivariant homology that comes from torus fixed
points.  

This is joint work with Eugene Gorsky and José Simental.  

Biography: Monica Vazirani is a professor at UC Davis.  She received her PhD from UC
Berkeley, after which she had an NSF postdoc she spent at UC San Diego and UC Berkeley,
as well as postdoctoral positions at MSRI and Caltech.  Dr.  Vazirani’s research
interests center on the representation theory of algebras related to the symmetric group
and how to express algebraic phenomena via the combinatorics of partitions, tableaux,
crystal graphs and parking functions.  

Note: These seminars will be recorded, including participant questions (participants
only when asking questions), and uploaded to the SMRI YouTube Channel
https://www.youtube.com/c/SydneyMathematicalResearchInstituteSMRI 

Other upcoming SMRI events can be found here:
https://mathematical-research-institute.sydney.edu.au/news-events/