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

作者：Lagarias Jeffrey C^*

来源：Journal of Number Theory, 2016, 161: 311-342.

DOI：10.1016/j.jnt.2015.09.003

• Finite field
• Polynomial factorization
• Symmetric group

For each n >= 1 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 = q = p(f) this measure describes splitting probabilities of monic degree n polynomials over F-q [X], conditioned on being square-free. It studies these measures in the case z = 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 F-1". It additionally studies the case z = -1, which also has a representation theoretic interpretation.

• 出版日期2016-4