## Combinatorics Seminar: On statistic of irreducible components

Seminar | March 4 | 12:10-1 p.m. | 939 Evans Hall

Nicolai Reshetikhin, UC Berkeley

Department of Mathematics

For finite dimensional representations $V_1, \dots , V_m$ of a simple finite dimensional Lie algebra $\mathfrak g$ consider the tensor product $W=\otimes _{I=1}^m V_i^{\otimes N_i}$. The first result, which will be presented in the talk, is the asymptotic of the multiplicity of an irreducible representation $V_\lambda$ with the highest weight λ in this tensor product when $N_i=\tau _i/\epsilon , \lambda =\xi /\epsilon$ and $\epsilon \to 0$. Then we will discuss the asymptotical distribution of irreducible components with respect to the character probability measure $Prob(\lambda )=\frac {m_\lambda \chi _{V_\lambda }(e^t)}{\chi _W(e^t)}$. Here $\chi _V(e^t)$ is the character of representation $V$ evaluated on $e^t$ where $t$ is an element of the Cartan subalgebra of the split real form of the Lie algebra $\mathfrak g$. This is a joint work with O. Postnova.

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