Oct 1, 2020

E-polynomials of character varieties for real curves

Date: October 1, 2020 | 2:00 pm – 3:30 pm
Speaker: Tom Baird, Memorial University of Newfoundland
Location: https://mathseminars.org/seminar/AGNTISTA

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.

More Information:

Date:
October 1, 2020
2:00 pm – 3:30 pm

Speaker:
Tom Baird, Memorial University of Newfoundland

Location:
https://mathseminars.org/seminar/AGNTISTA

Contact:

Birgit Oosthuizen-Noczil

Email:
birgit.oosthuizen-noczil@ist.ac.at

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