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SUMMARY:Nicolai Reshetikhin (University of California\, Berkeley)
DTSTART;VALUE=DATE-TIME:20211214T140000Z
DTEND;VALUE=DATE-TIME:20211214T150000Z
DTSTAMP;VALUE=DATE-TIME:20211209T081055Z
UID:IAMP_seminars/77
DESCRIPTION:Title: On the asymptotic behavior of character measures in large tensor po
wers of finite dimensional representations of simple Lie algebras\nby
Nicolai Reshetikhin (University of California\, Berkeley) as part of One w
orld IAMP mathematical physics seminar\n\n\nAbstract\nLet $V$ be a finite
dimensional representation of a simple Lie algebra and $H$ be an positive
element of its Cartan subalgebra ("magnetic field"). On the space $V^{\\ot
imes N}$ we have a natural density matrix $N\\exp(-H)$ where $H$ acts diag
onally: $H(x\\otimes y\\otimes z\\dots)=Hx\\otimes y\\otimes z\\dots+x\\ot
imes Hy\\otimes z\\dots+x\\otimes y\\otimes Hz\\dots+\\dots$. The space $V
^{\\otimes N}$ decomposes into a direct\nsum of irreducible subrepresentat
ions:\n\\[\nV^{\\otimes N}\\simeq \\oplus_{\\lambda} V_\\lambda^{\\oplus m
(\\lambda\, N)}\n\\]\nwhere $\\lambda$ is the highest weight of the repres
entation $V_\\lambda$ and $m(\\lambda\, N)$\nis its multiplicity in the te
nsor product. The character distribution assigns the probability \n\\[\np_
{\\lambda}(N\,H)=\\frac{m(\\lambda\, N)Tr_{V_\\lambda}(e^{-H)})}{(Tr_V(e^{
-H}))^N}\n\\]\nto each $\\lambda$ in the decomposition of the tensor produ
ct.\n\nOne of the natural problems for this distribution is to find its as
ymptotic \nin the limit $N\\to \\infty$ and $\\lambda\\to \\infty$ in the
appropriate way.\nWhen $H=0$ the character distribution becomes a uniform
distribution.\nIn this case\, in such generality the asymptotic was studi
ed by Ph. Biane\, 1993 by T. Tate and S. Zelditch\, 2004. For tensor power
s of vector representations the asymptotic was derived by S. Kerov in 1986
. When $H$ is generic\, i.e. when $H$ is strictly inside of the principal
Weyl chamber\nit was computed by O.Postnova and N.R. in 2018. This talk is
based on a joint work with O. Postnova and V. Serganova (to appear on the
arxiv).\n
LOCATION:https://researchseminars.org/talk/IAMP_seminars/77/
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