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

Abstract: 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: $H(x\otimes y\otimes z\dots)=Hx\otimes y\otimes z\dots+x\otimes Hy\otimes z\dots+x\otimes y\otimes Hz\dots+\dots$. 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).

mathematical physics

Audience: researchers in the discipline

( video )

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