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DTSTAMP:20260404T020028Z
UID:1601553600@ist.ac.at
DTSTART:20201001T140000
DTEND:20201001T153000
DESCRIPTION:Speaker: Tom Baird\nhosted by Tamas Hausel\nAbstract: Given a R
 iemann surface $\\Sigma$ denote by $$M_n(\\mathbb{F}) := Hom_{\\xi}( \\pi_
 1(\\Sigma)\, GL_n(\\mathbb{F}))/GL_n(\\mathbb{F})$$ the $\\xi$-twisted cha
 racter variety for $\\xi \\in \\mathbb{F}$ a $n$-th root of unity.  An an
 ti-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 $\\math
 bb{F} = \\mathbb{C}$\, $M_n(\\mathbb{C})$ 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 fie
 lds $\\mathbb{F}_q$\, Hausel and Rodriguez-Villegas determined the E-polyn
 omial of $M_n(\\mathbb{C})$ (a specialization of the mixed Hodge polynomia
 l). I will show how similar methods can be used to calculate the E-polynom
 ial 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 identi
 ty involving the plethystic logarithm of a product of sums over Young diag
 rams. The Pieri's formula for multiplying Schur polynomials arises in an i
 nteresting way. This is joint work with Michael Lennox Wong.
LOCATION:https://mathseminars.org/seminar/AGNTISTA\, ISTA
ORGANIZER:birgit.oosthuizen-noczil@ist.ac.at
SUMMARY:Tom Baird: E-polynomials of character varieties for real curves
URL:https://talks-calendar.ista.ac.at/events/2868
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