Cyclotomic generating functions

TL;DR

This paper surveys cyclotomic generating functions (CGFs), exploring their algebraic, analytic, and asymptotic properties, and discusses their applications in combinatorics, representation theory, and probability, highlighting their normal distribution tendencies.

Contribution

It provides a comprehensive survey of CGFs, detailing their properties, examples, and conditions for existence, and extends known asymptotic normality results.

Findings

CGFs often have roots that are roots of unity or zero

CGFs are generally asymptotically normal

Conditions for the existence of CGFs are characterized

Abstract

It is a remarkable fact that for many statistics on finite sets of combinatorial objects, the roots of the corresponding generating function are each either a complex root of unity or zero. These and related polynomials have been studied for many years by a variety of authors from the fields of combinatorics, representation theory, probability, number theory, and commutative algebra. We call such polynomials \textbf{cyclotomic generating functions} (CGFs). With Konvalinka, we have studied the support and asymptotic distribution of the coefficients of several families of CGFs arising from tableau and forest combinatorics. In this paper, we survey general CGFs from algebraic, analytic, and asymptotic perspectives. We review some of the many known examples of CGFs in combinatorial representation theory; describe their coefficients, moments, cumulants, and characteristic functions; and give a variety of necessary and sufficient conditions for their existence arising from probability, commutative algebra, and invariant theory. As a sample result, we show that CGFs are ``generically'' asymptotically normal, generalizing a result of Diaconis on $q$-binomial coefficients using work of Hwang--Zacharovas. We include several open problems concerning CGFs.