Upcoming Talks

Ist logo

E-polynomials of character varieties for real curves

Date: Thursday, October 1, 2020 14:00 - 15:30
Speaker: Tom Baird (Memorial University of Newfoundland)
Location: https://mathseminars.org/seminar/AGNTISTA
Series: Mathematics and CS Seminar
Host: Tamas Hausel

Given a Riemann surface $\Sigma$ denote by $$M_n(\mathbb{F}) := Hom_{\xi}( \pi_1(\Sigma), GL_n(\mathbb{F}))/GL_n(\mathbb{F})$$ the $\xi$-twisted character variety for $\xi \in \mathbb{F}$ a $n$-th root of unity.  An anti-holomorphic involution $\tau$ on $\Sigma$ induces an involution on $M_n(\mathbb{F})$ such that the fixed point variety $M_n^{\tau}(\mathbb{F})$ can be identified with the character variety of real representations" for the orbifold fundamental group $\pi_1(\Sigma, \tau)$. When $\mathbb{F} = \mathbb{C}$, $M_n(\mathbb{C})$ is a complex symplectic manifold and $M_n^{\tau}(\mathbb{C})$ embeds as a complex Lagrangian submanifold (or ABA-brane). By counting points of $M_n(\mathbb{F}_q)$ for finite fields $\mathbb{F}_q$, Hausel and Rodriguez-Villegas determined the E-polynomial of $M_n(\mathbb{C})$ (a specialization of the mixed Hodge polynomial). I will show how similar methods can be used to calculate the E-polynomial of $M_n^\tau(\mathbb{F}_q)$ using the representation theory of $GL_n(\mathbb{F}_q)$.  We express our formula as a generating function identity involving the plethystic logarithm of a product of sums over Young diagrams. The Pieri's formula for multiplying Schur polynomials arises in an interesting way. This is joint work with Michael Lennox Wong.


Qr image
Download ICS Download invitation
Back to eventlist