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SUMMARY:Apoorva Khare (Indian Institute of Science)
DTSTART:20220602T060000Z
DTEND:20220602T070000Z
DTSTAMP:20260423T021507Z
UID:ARCSIN/21
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ARCSIN/21/">
 Higher order Verma modules\, and a positive formula for all highest weight
  modules - talk 2</a>\nby Apoorva Khare (Indian Institute of Science) as p
 art of ARCSIN - Algebra\, Representations\, Combinatorics and Symmetric fu
 nctions in INdia\n\n\nAbstract\nWe study weights of highest weight modules
  $V$ over a Kac-Moody algebra $\\mathfrak{g}$ (one may assume this to be $
 \\mathfrak{sl}_n$ throughout the talk\, without sacrificing novelty). We b
 egin by recalling the notation\, "higher order holes"\, and "higher order 
 Verma modules" (along with their universal property\, via examples). We th
 en recall our result from last time: the weights of any highest weight mod
 ule equal the weights of its "higher order" Verma cover.\n\nNext\, we defi
 ne the higher order category $\\mathcal{O}^\\mathcal{H}$\, and recall some
  properties in the zeroth and first order cases (work of Bernstein–Gelfa
 nd–Gelfand and Rocha-Caridi)\, and end by explaining that in the higher 
 order cases\, (a) the category $\\mathcal{O}^\\mathcal{H}$ still has enoug
 h projectives and injectives\; (b) BGG reciprocity does not always hold "o
 n the nose"\, yet (c) the "Cartan matrix" (of simple multiplicities in pro
 jective covers) is still symmetric over $\\mathfrak{g} = \\mathfrak{sl}_2^
 {\\oplus n}$.\n
LOCATION:https://researchseminars.org/talk/ARCSIN/21/
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