On multiplicities of irreducibles in large tensor product of representations of simple Lie algebras

Abstract

In this paper we study the asymptotic of multiplicities of irreducible representations in large tensor products of finite dimensional representations of simple Lie algebras and their statistics with respect to Plancherel and character probability measures. We derive the asymptotic distribution of irreducible components for the Plancherel measure, generalizing results of Biane and Tate and Zelditch. We also derive the asymptotic of the character measure for generic parameters and an intermediate scaling in the vicinity of the Plancherel measure. It is interesting that the asymptotic measure is universal and after suitable renormalization does not depend on which representations were multiplied but depends significantly on the degeneracy of the parameter in the character distribution.

Probability measure near the boundary of tensor power decomposition for so(2n+1)

“…This result can be interpreted as a random walk with a critical drift [5]. Connection to random walks is easy to understand in the sl 2 -case for the probability measure defined on the weight diagram of N -th tensor power of fundamental representation L ω : p(λ) = dimV λ (dimL ω ) N , where dimV λ is the dimension of the weight subspace in the representation (L ω ) ⊗N .…”

“…The case g = so 2n+1 , N → ∞, n = const was studied in [3] and [4]. The central limit regime and the regime of large deviations for character measure (5), with n fixed, N → ∞, were considered in [5] by O.Postnova and N.Reshetikhin. They also suggested that in the regime near the boundary the character measure will converge to some Poisson type process.…”

Limit shape for infinite rank limit of non simply-laced Lie algebras of series so(2n+1)

“…In the paper [30] an asymptotic formula for tensor product decomposition coefficients was obtained. The character measure was studied in the case of Lie albegras of fixed rank n in the papers [26]. In a separate publication we will consider the limit n, N → ∞ for the character measure.…”

“…The asymptotic behavior of this measure was studied in three regimes: N → ∞ with n fixed, N → ∞, n → ∞ with N/n fixed and N, n → ∞ with N/n 2 fixed. The first case was studied [14] and later generalized to all simple Lie algebras in [25,26,30]. For the second case Kerov discovered that Vershik-Kerov-Logan-Shepp limit shape of Young diagrams with respect to the Plancherel measure on S N as N → ∞ also appears as the limit shape with respect to this measure.…”

Large deviation principle for moment map estimation

“…In [23], Tate and Zelditch analysed the multiplicities in high tensor powers and proved a central limit theorem as well as large deviation results by relating them to the (much simpler) weight multiplicity asymptotics. Moving away from the tracial state to positive operators arising from the representation of the complexified group, Postnova and Reshetikhin [20] generalised these asymptotic formulas to character distributions.…”