Average divisibility in character tables of $\mathrm{GL}_2(\mathbb{F}_q)$

Technical Summary: Average Divisibility in Character Tables of $GL_2(\mathbb{F}_q)$

Problem Statement
This paper investigates the arithmetic properties of character tables for the family of finite general linear groups $G_q = GL_2(\mathbb{F}_q)$, where $q$ ranges over odd prime powers. Specifically, the authors study the proportion of entries in the character table that are not divisible by a fixed prime $\ell$, considering divisibility within the ring of algebraic integers. This inquiry is motivated by recent work of Peluse and Soundararajan on Miller's conjecture for symmetric groups $S_N$, which established that as $N \to \infty$, almost every entry in the character table of $S_N$ is divisible by any fixed prime $\ell$. The central question is whether a similar phenomenon occurs for $GL_2(\mathbb{F}_q)$ as $q \to \infty$, or if the behavior differs significantly. Additionally, the paper addresses the distribution of arguments (angles) of the nonzero character values in the complex plane.

Methodology
The authors employ a structural approach based on the classification of irreducible complex representations and conjugacy classes of $GL_2(\mathbb{F}_q)$.

1. Classification: The irreducible representations are categorized into four families: one-dimensional, principal series, Steinberg twists, and cuspidal. Similarly, conjugacy classes are divided into four families: scalar, split regular semisimple, elliptic, and non-semisimple (Jordan).
2. Block Decomposition: The character table is analyzed as a $4 \times 4$ block matrix. The authors observe that the rows corresponding to one-dimensional and Steinberg representations, and the columns corresponding to scalar and Jordan classes, are "small" (size $O(q)$), contributing a density of zero to the total table size ($O(q^4)$).
3. Dominant Blocks: The asymptotic behavior is determined by the two large blocks where the character values are generally nonzero:
   • Principal series representations evaluated on split regular semisimple classes.
   • Cuspidal representations evaluated on elliptic classes.
     The other two large blocks (Principal series on elliptic classes, and Cuspidal on split regular semisimple classes) consist entirely of zeros.
4. Divisibility Analysis: For the dominant nonzero blocks, the character values take the form of sums of roots of unity (e.g., $\chi_1(a)\chi_2(b) + \chi_1(b)\chi_2(a)$). The authors utilize elementary properties of cyclotomic polynomials and roots of unity (specifically Lemma 2.2 regarding the divisibility of $1+\zeta$) to count how often these sums are divisible by $\ell$. They show that such divisibility is a rare event, occurring only for a sparse set of parameters.
5. Angular Distribution: To address the distribution of arguments, the authors apply a variant of Weyl's criterion. They analyze the exponential sums of the character arguments, utilizing the action of the character group of $\mathbb{F}_q^\times$ on the set of representations to show that the sums vanish asymptotically, implying equidistribution.

Key Results
The paper establishes the following quantitative and asymptotic results:

1. Divisibility Asymptotics:
   Let $N_\ell(q)$ be the number of entries in the character table of $G_q$ not divisible by $\ell$, and let $M_0(q)$ be the number of zero entries. The total number of entries is $(q^2-1)^2$. The authors prove:
   $$N_\ell(q) = \frac{q^4}{2} + O_\epsilon(q^{3+\epsilon})$$
   $$M_0(q) = \frac{q^4}{2} + O_\epsilon(q^{3+\epsilon})$$
   Consequently, the proportion of all entries not divisible by $\ell$ tends to $1/2$:
   $$\lim_{q \to \infty} \frac{N_\ell(q)}{(q^2-1)^2} = \frac{1}{2}$$
   Furthermore, the proportion of nonzero entries not divisible by $\ell$ tends to 1:
   $$\lim_{q \to \infty} \frac{N_\ell(q)}{(q^2-1)^2 - M_0(q)} = 1$$
   This result holds for any fixed prime power $\ell^r$ as well.

2. Angular Equidistribution:
   Let $P_q(I)$ denote the proportion of nonzero character values whose arguments lie in an interval $I \subseteq [0, 2\pi]$. The authors prove that as $q \to \infty$, these arguments become equidistributed:
   $$\lim_{q \to \infty} P_q(I) = \frac{b-a}{2\pi}$$
   for any interval $I = [a, b]$.

Significance and Claims
The paper claims to demonstrate a fundamental divergence between the arithmetic statistics of character tables for symmetric groups ($S_N$) and general linear groups ($GL_2(\mathbb{F}_q)$).

• Contrast with Symmetric Groups: While almost every entry in the character table of $S_N$ is divisible by a fixed prime $\ell$ as $N \to \infty$, the limit for $GL_2(\mathbb{F}_q)$ is exactly $1/2$. The authors attribute this difference to the fact that for $GL_2(\mathbb{F}_q)$, the "divisible" entries are predominantly the zero entries, whereas the nonzero entries are almost never divisible by $\ell$.
• Analogue of Miller's Question: The angular equidistribution result is presented as an analogue to Miller's question regarding the distribution of signs (positive vs. negative) among nonzero entries in symmetric group character tables. Since character values for $GL_2(\mathbb{F}_q)$ are generally complex, the "sign" question is replaced by an angular distribution question, which the authors resolve affirmatively.
• Methodological Distinction: The authors note that their proof relies on the explicit classification of representations and conjugacy classes for $GL_2(\mathbb{F}_q)$, differing from the combinatorial partition methods used in the study of symmetric groups.

The paper concludes by posing open questions regarding the extension of these results to $GL_n(\mathbb{F}_q)$ for fixed $n$ with $q \to \infty$, and for fixed $q$ with $n \to \infty$, as well as for other families of finite groups of Lie type.