Abstract
Given a representation of a compact Lie group and a state we define a probability measure on the coadjoint orbits of the dominant weights by considering the decomposition into irreducible components. For large tensor powers and independent copies of the state we show that the induced probability distributions converge to the value of the moment map. For faithful states we prove that the measures satisfy the large deviation principle with an explicitly given rate function.
Highlights
This paper is concerned with probability distributions related to decompositions of tensor power representations into irreducibles
In the context of random walks, it was shown in [13] that counting the multiplicities of irreducible representations in certain tensor power representations is equivalent to enumerating the number of walks for “reflectable” walk types, conditioned on staying within a Weyl chamber
This class of random walks was introduced by Gessel and Zeilberger in [12] as a generalisation of the classical ballot problem [2, 25] to finite reflection groups
Summary
This paper is concerned with probability distributions related to decompositions of tensor power representations into irreducibles. Given a representation π of a compact Lie group K on a finite dimensional Hilbert space H as well as a positive operator ρ on H with unit trace (a state), for every n the values Tr Pλρ⊗n determine a probability distribution on the set of dominant weights λ of K. Related work has been done independently by Cole Franks and Michael Walter [11]
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