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

For each n≥1n≥1 this paper considers a one-parameter family of complex-valued measures on the symmetric group SnSn, depending on a complex parameter z  . For parameter values z=q=pfz=q=pf this measure describes splitting probabilities of monic degree n   polynomials over Fq[X]Fq[X], conditioned on being square-free. It studies these measures in the case z=1z=1, and shows that they have an interesting internal structure having a representation theoretic interpretation. These measures may encode data relevant to the hypothetical “field with one element F1F1”. It additionally studies the case z=−1z=−1, which also has a representation theoretic interpretation.