[geometry-ml:04561] セミナーの周知 Nicolai Reshetikhin (University of California, Berkeley)

[sumio yamada]

みなさま、

学習院大物理学科の田崎晴明先生から、以下の案内をいただきましたので、本mlで共有させていただきます。

山田澄生
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Suject: IAMP seminar / Nicolai Reshetikhin / Dec. 14, 23:00 (JST)

One World Mathematical Physics Seminar は、コロナ禍を受けて IAMP (International
Association of Mathematical Physics) が企画しているオンラインのセミナーシリーズです。
これまでのセミナーは全て録画され以下で公開されています。
https://www.youtube.com/channel/UCxeiraz5cxmNXe2NN8qpbZQ/videos

12/14 日（火曜）の日本時間深夜 11 時からのセミナーでは、量子群の提唱など様々な業績で知られる Nicolai Reshetikhin
氏が登壇します。主催者から送られたアブストラクトとリンク情報を以下に転載します。ご興味のある方の参加を歓迎いたします。

田崎晴明

============

Nicolai Reshetikhin (University of California, Berkeley)

On the asymptotic behavior of character measures in large tensor
powers of finite dimensional representations of simple Lie algebras

Let $V$ 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 $V^{\otimes N}$ we have a natural density matrix
$N\exp(-H)$where $H$ acts diagonally: $H(x\otimes y\otimes z\dots)=H
x\otimes y\otimes z\dots +x\otimes H y\otimes z\dots+x\otimes y\otimes
H z\dots$. The space $V^{\otimes N}$ decomposes into a direct
sum of irreducible subrepresentations:
\[
V^{\otimes N}\simeq \oplus_{\lambda} V_\lambda^{\oplus m(\lambda, N)}
\]
where $\lambda$ is the highest weight of the representation
$V_\lambda$ and $m(\lambda, N)$
is its multiplicity in the tensor product. The character distribution
assigns the probability
\[
p_{\lambda}(N,H)=\frac{m(\lambda,
N)Tr_{V_\lambda}(e^{-H)})}{(Tr_V(e^{-H}))^N}
\]
to each $\lambda$ in the decomposition of the tensor product.

One of the natural problems for this distribution is to find its asymptotic
in the limit $N\to \infty$ and $\lambda\to \infty$ in the appropriate way.
When $H=0$ the character distribution becomes a uniform distribution.
In this case, in such generality  the asymptotic was studied by Ph.
Biane, 1993 by T. Tate and S. Zelditch, 2004. For tensor powers 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 it 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).


Livestream link will appear here 15 min before the talk:
https://researchseminars.org/talk/IAMP_seminars/77/

The schedule of upcoming and past seminars can be found here:
http://www.iamp.org/page.php?page=page_seminar

Best regards,
Jan Dereziński, Marcello Porta, Kasia Rejzner and Daniel Ueltschi

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