The number of flags in finite vector spaces: asymptotic normality and Mahonian statistics
We study the generalized Galois numbers which count flags of length r in N-dimensional vector spaces over finite fields. We prove that the coefficients of those polynomials are asymptotically Gaussian normally distributed as N becomes large. Furthermore, we interpret the generalized Galois numbers as weighted inversion statistics on the descent classes of the symmetric group on N elements and identify their asymptotic limit as the Mahonian inversion statistic when r approaches ∞. Finally, we apply our statements to derive further statistical aspects of generalized Rogers–Szegő polynomials, reinterpret the asymptotic behavior of linear q-ary codes and characters of the symmetric group acting on subspaces over finite fields, and discuss implications for affine Demazure modules and joint probability generating functions of descent-inversion statistics.
KeywordsGalois number Gaussian normal distribution MacMahon inversion statistic Rogers–Szegő polynomial Linear code Demazure module Symmetric group descent-inversion statistic
This work was supported by the Swiss National Science Foundation (grant PP00P2-128455), the National Centre of Competence in Research “Quantum Science and Technology,” the German Science Foundation (SFB/TR12, SPP1388, and grants CH 843/1-1, CH 843/2-1), and the Excellence Initiative of the German Federal and State Governments through the Junior Research Group Program within the Institutional Strategy ZUK 43.
The second author would like to thank Matthias Christandl for his kind hospitality at the ETH Zurich and Ryan Vinroot for helpful conversations.
We are indebted to the referees for pointing out an error in the proof of Theorem 3.5 in an earlier version of this article.
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