Sums of binomial coefficients evaluated at $\alpha \in \overline{\mathbb Q},$ and applications

TL;DR

This paper investigates the structure of additive monoids generated by binomial coefficients evaluated at algebraic numbers, revealing conditions under which these form rings and providing explicit descriptions, with applications to Deligne categories.

Contribution

It characterizes when the monoid $R_+( ext{alpha})$ is a ring for algebraic $ ext{alpha}$ and offers explicit descriptions, advancing understanding of binomial coefficient structures in algebraic number contexts.

Findings

$R_+( ext{alpha})$ is a ring iff $ ext{alpha}$ is algebraic and not a non-negative integer

All algebraic integers in $ ext{Q}( ext{alpha})$ are contained in $R_+( ext{alpha})$

Explicit descriptions of $R_+( ext{alpha})$ for quadratic algebraic numbers and roots of unity

Abstract

The additive monoid $R_+(x)$ is defined as the set of all nonnegative integer linear combinations of binomial coefficients $\binom{x}{n}$ for $n \in \mathbb Z_+$. This paper is concerned with the inquiry into the structure of $R_+(\alpha)$ for complex numbers $\alpha.$ Particularly interesting is the case of algebraic $\alpha$ which are not non-negative integers. This question is motivated by the study of functors between Deligne categories $\textrm{Rep}(S_t)$ (and also $\textrm{Rep}(\textrm{GL}_t)$) for $t \in \mathbb C\backslash \mathbb Z_+$. We prove that this object is a ring if and only if $\alpha$ is an algebraic number that is not a nonnegative integer. Furthermore, we show that all algebraic integers generated by $\alpha,$ i.e. all elements of $\mathcal O_{\mathbb Q(\alpha)},$ are also contained in this ring. We also give two explicit representations of $R_+(\alpha)$ for both algebraic integers and general algebraic numbers $\alpha.$ One is in terms of inequalities for the valuations with respect to certain prime ideals and the other is in terms of explicitly constructed generators. We show how these results work in the context of the study of symmetric monoidal functors between Deligne categories in positive characteristic. Moreover, this leads to a particularly simple description of $R_+(\alpha)$ for both quadratic algebraic numbers and roots of unity.