A family of measures on symmetric groups and 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

30 pages

References in corpus (3)

• Mapping F_1-land:An overview of geometries over the field with one element
• Projective geometry over F_1 and the Gaussian binomial coefficients
• Splitting Behavior of $S_n$-Polynomials

Cited by in corpus (3)

• Polynomial splitting measures and cohomology of the pure braid group
• Splitting Behavior of $S_n$-Polynomials
• Notes on Cardinal's Matrices