Skew Howe duality and q-Krawtchouk polynomial ensemble

2022/08/22 by Nazarov, Anton, Nikitin, Pavel, Sarafannikov, Daniil
#05E05 #17B10 #22E46 #33C45 #60B10 #60G55 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2208.10331

Abstract

We consider the decomposition into irreducible components of the exterior algebra \bigwedge(ℂⁿ⊗ (ℂ^k)^*) regarded as a GL_n× GL_k module. Irreducible GL_n× GL_k representations are parameterized by pairs of Young diagrams (λ,λ'), where λ' is the complement conjugate diagram to λ inside the n× k rectangle. We set the probability of a diagram as a normalized specialization of the character for the corresponding irreducible component. For the principal specialization we get the probability that is equal to the ratio of the q-dimension for the irreducible component over the q-dimension of the exterior algebra. We demonstrate that this probability distribution can be described by the q-Krawtchouk polynomial ensemble. We derive the limit shape and prove the central limit theorem for the fluctuations in the limit when n,k tend to infinity and q tends to one at comparable rates.

Related

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