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SUMMARY:Apoorva Khare (Indian Institute of Science)
DTSTART;VALUE=DATE-TIME:20220602T060000Z
DTEND;VALUE=DATE-TIME:20220602T070000Z
DTSTAMP;VALUE=DATE-TIME:20220816T035559Z
UID:ARCSIN/21
DESCRIPTION:Title:
Higher order Verma modules\, and a positive formula for all highest weight
modules - talk 2\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 Bernsteinâ€“Gelfa
ndâ€“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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