Counterexample to Conjectured Su ( N ) Character Asymptotics Tatsuya Tate And

The purpose of this note is to give a counterexample to the conjectured large N asymptotics of character values χ R (U) of irreducible characters of SU (N), which appears in papers of Gross-Matytsin [M, GM] and Kazakov-Staudacher-Wynter [KW, KSW, KSW2, KSW3]). Asymptotics of characters are important in the large N limit of Y M 2 (2D Yang-Mills theory). Our counterexample consists of one special sequence of elements a N ∈ SU (N) for which the conjectured asymptotics on χ R N (a N) fail for any relevant sequence χ R N of irreducible characters. It is not clear at present how widespread in SU (N) the failure is. To state the conjecture and the counterexample, we will need some notation. We recall that irreducibles of SU (N) are parametrized by their highest weights λ or equivalently by Young diagrams with ≤ N − 1 rows. To facilitate comparison with [GM], we will use a further parametrization of representations R of SU (N) by their shifted highest weights = λ + ρ N , ρ N = half the sum of the positive roots. (1) The components of the shifted highest weight are then strictly decreasing ∞ > 1 > 2 > · · · > N > −∞ (cf. (13) for the explicit formula). To a shifted highest weight we associate the probability measure on R defined by