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SUMMARY:Infinite sequences via Lie algebra actions for oligomorphic groups
  [Oberseminar Darstellungstheorie]
DTSTART:20260424T120000Z
DTEND:20260424T140000Z
DTSTAMP:20260420T222600Z
UID:indico-event-1223@math-events.uni-bonn.de
DESCRIPTION:Speakers: Zbigniew Wojciechowski (Technische Universität Dres
 den)\n\nMany integer sequences arise by counting G-orbits on the set of n-
 element subsets of a set X\, for a group G acting on X. For finite X\, Sta
 nley proved that these finite sequences increase towards the middle using 
 an action of the Lie algebra sl2. For infinite sets X\, and hence infinite
  sequences\, Cameron provided an argument for monotonicity. He first ident
 ifies orbits with a vector space basis of a certain commutative k-algebra 
 H*\, called the orbit algebra. He then considers the operator which forms 
 the product with the constant 1-function on X\, and proves its injectivity
 . In this paper\, we generalize Stanley's approach to oligomorphic groups\
 , and in particular extend Cameron's operator to a full sl2-action on H*.\
 nMost crucially\, we define for every oligomorphic permutation group G the
  X-th tensor power of the standard representation of the general linear Li
 e algebra gl_d\, generalizing work of Entova-Aizenbud. This space carries 
 natural commuting actions of G and gl_d\, the latter depending on a Harman
 –Snowden measure on G. In the case d = 2\, Cameron's result implies that
  H* decomposes into a direct sum of Verma modules\, which provides a repre
 sentation-theoretic explanation for monotonicity. A highlight of the talk 
 is an sl2-representation whose weight space dimensions are given by the Fi
 bonacci numbers.\n\nhttps://math-events.uni-bonn.de/event/1223/
LOCATION:MPIM\, Vivatsgasse\,  7 - Lecture Hall (Max Planck Institute for 
 Mathematics)
URL:https://math-events.uni-bonn.de/event/1223/
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