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

Jeffrey C. Lagarias

Published 2015-05-21Version 1

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^k a prime power, this measure describes splitting probabilities for monic degree n polynomials over the finite field with p^k 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."