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SUMMARY:Tom Baird (Memorial University of Newfoundland)
DTSTART:20201001T120000Z
DTEND:20201001T140000Z
DTSTAMP:20260423T005758Z
UID:AGNTISTA/11
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/AGNTISTA/11/
 ">E-polynomials of character varieties for real curves</a>\nby Tom Baird (
 Memorial University of Newfoundland) as part of Algebraic Geometry and Num
 ber Theory seminar - ISTA\n\n\nAbstract\nGiven a Riemann surface $\\Sigma$
  denote by $$M_n(\\mathbb{F}) := Hom_{\\xi}( \\pi_1(\\Sigma)\, GL_n(\\math
 bb{F}))/GL_n(\\mathbb{F})$$ 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 tha
 t the fixed point variety $M_n^{\\tau}(\\mathbb{F})$ can be identified wit
 h the character variety of ``real representations" for the orbifold fundam
 ental group $\\pi_1(\\Sigma\, \\tau)$. When $\\mathbb{F} = \\mathbb{C}$\, 
 $M_n(\\mathbb{C})$ is a complex symplectic manifold and $M_n^{\\tau}(\\mat
 hbb{C})$ embeds as a complex Lagrangian submanifold (or ABA-brane).\nBy co
 unting 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 si
 milar methods can be used to calculate the E-polynomial of $M_n^\\tau(\\ma
 thbb{F}_q)$ using the representation theory of $GL_n(\\mathbb{F}_q)$.  We 
 express our formula as a generating function identity involving the plethy
 stic logarithm of a product of sums over Young diagrams. The Pieri's formu
 la for multiplying Schur polynomials arises in an interesting way.\n\nThis
  is joint work with Michael Lennox Wong.\n
LOCATION:https://researchseminars.org/talk/AGNTISTA/11/
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