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SUMMARY:Miriam Norris (King's College London)
DTSTART:20220302T160000Z
DTEND:20220302T170000Z
DTSTAMP:20260418T064544Z
UID:LNTS/63
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/LNTS/63/">La
 ttice graphs for representations of $GL_3(\\F_p)$</a>\nby Miriam Norris (K
 ing's College London) as part of London number theory seminar\n\n\nAbstrac
 t\nIn a recent paper Le\, Le Hung\, Levin and Morra proved a generalisatio
 n of Breuil's Lattice conjecture in dimension three. This involved showing
  that lattices inside representations of $GL_3(\\F_p)$ coming from both a 
 global and a local construction coincide. Motivated by this we consider th
 e following graph. For an irreducible representation $\\tau$ of a group $G
 $ over a finite extension $K$ of $\\Q_p$ we define a graph on the $\\mathc
 al{O}_K$-lattices inside $\\tau$ whose edges encapsulate the relationship 
 between lattices in terms of irreducible modular representations of $G$ (o
 r Serre weights in the context of the paper by Le et al.). \n\nIn this tal
 k\, I will demonstrate how one can apply the theory of graduated orders an
 d their lattices\, established by Zassenhaus and Plesken\, to understand t
 he lattice graphs of residually multiplicity free representation over suit
 ably large fields in terms of a matrix called an exponent matrix. Furtherm
 ore I will explain how I have been able to show that one can determine the
  exponent matrices for suitably generic representation go $GL_3(\\F_p)$ al
 lowing us to construct their lattice graphs.\n
LOCATION:https://researchseminars.org/talk/LNTS/63/
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