Eigenvalue Fluctuations of Symmetric Group Permutation Representations on k-tuples and k-subsets

2018/10/28 by Benjamin K. Tsou, Tsou, Benjamin
Mathematics · #11K06 #15B52 #60B15 #60B20 #60C05 #60F05 #FOS: Mathematics #Limits and Structures in Graph Theory #Probability (math.PR) #Random Matrices and Applications #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1810.11904

openalex publication_date 2018/10/28 · openalex created_date 2018/11/02 · openalex updated_date 2026/07/28

Abstract

Let the term k-representation refer to the permutation representations of the symmetric group \mathfrakS_n on k-tuples and k-subsets as well as the S^(n-k,1k) irreducible representation of \mathfrakS_n. Endow \mathfrakS_n with the Ewens distribution and let α and β be linearly independent irrational numbers over ℚ. Then for fixed k > 1 we show that as n → ∞, the normalized count of the number of eigenangles in a fixed interval (α, β) of a k-representation evaluated at a random element σ∈ \mathfrakS_n converges weakly to a compactly supported distribution. In particular, we compute the limiting moments and moreover provide an explicit formula for the limiting density when k = 2 and the Ewens parameter θ= 1 (uniform probability measure). This is in contrast to the k = 1 case where it has been shown previously that the distribution is asymptotically Gaussian.

Citations

• On the number of eigenvalues of modified permutation matrices in\n mesoscopic intervals

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