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arXiv · 1505.05745

A family of measures on symmetric groups and the field with one element

Jeffrey C. Lagarias
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Abstract

For each positive integer n this paper considers a one-parameter family of complex-valued measures on the symmetric group S_n, depending on a complex parameter z. For parameter values z=p^f a prime power, this measure describes splitting probabilities for monic degree n polynomials over the finite field with p^f elements, conditioned on being square-free. It studies these measure in the special case z=1, and shows they have an interesting internal structure having a representation theoretic interpretation. These measures may provide data relevant to the hypothetical "field with one element ${\mathbb F_1}$". It additionally studies the case z=-1, which also has a representation-theoretic interpretation

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Jeffrey C. Lagarias. 2018-01-17. A family of measures on symmetric groups and the field with one element. https://doi.org/10.1016/j.jnt.2015.09.003

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