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

TL;DR

This paper introduces a family of measures on symmetric groups linked to finite field polynomials and explores their structure and implications for the concept of the field with one element, revealing new representation-theoretic insights.

Contribution

It defines a new family of measures on symmetric groups parameterized by complex numbers and connects them to finite field polynomial splitting probabilities and the hypothetical field with one element.

Findings

Measures have a representation-theoretic interpretation at z=1 and z=-1.

Special case z=1 relates to splitting probabilities over finite fields.

The measures may shed light on the structure of the field with one element.

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