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arxiv: 2606.28085 · v1 · pith:SKB7RZVNnew · submitted 2026-06-26 · 🧮 math.RT · math.CO· math.NT

Average divisibility in character tables of GL₂(mathbb{F}_q)

Anwesh Ray , Mishty Ray This is my paper

Pith reviewed 2026-06-29 01:53 UTC · model grok-4.3

classification 🧮 math.RT math.COmath.NT
keywords character tablesGL_2(F_q)divisibility in algebraic integersfinite groups of Lie typeequidistribution of argumentsrepresentation theory of finite groups
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The pith

In character tables of GL_2 over finite fields with q elements, the number of entries not divisible by any fixed prime ℓ is asymptotically q^4/2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that for odd prime powers q and any fixed prime ℓ, the count N_ℓ(q) of character-table entries of GL_2(F_q) that are not divisible by ℓ in the algebraic integers equals q^4/2 plus an error of size O_ε(q^{3+ε}). The same asymptotic holds for the number of zero entries. Consequently the overall proportion of entries not divisible by ℓ tends to 1/2 while the proportion among the nonzero entries tends to 1; this stands in contrast to the symmetric-group case where almost all entries become divisible by any fixed prime. The authors additionally prove that the arguments of the nonzero character values are equidistributed in [0, 2π] as q tends to infinity.

Core claim

Using the explicit parametrization of irreducible characters and conjugacy classes of GL_2(F_q), distinguishing split and non-split tori, the authors reduce the divisibility question to counting solutions of certain equations over finite fields. They obtain the main-term count q^4/2 for both the ℓ-free entries and the zero entries, with the stated error term, and deduce the two limiting proportions. They further establish equidistribution of arguments of the nonzero values on the unit circle.

What carries the argument

Explicit character table of GL_2(F_q) together with reduction of algebraic-integer divisibility to finite-field point counts on split and non-split tori.

If this is right

  • The proportion of all character-table entries not divisible by ℓ tends to 1/2 as q → ∞.
  • The proportion of nonzero entries not divisible by ℓ tends to 1 as q → ∞.
  • The arguments of the nonzero character values become equidistributed in the interval [0, 2π].
  • The same main-term count q^4/2 holds for the number of zero entries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The contrast with symmetric groups suggests that the average divisibility behavior may depend on whether the group is of Lie type or of symmetric type.
  • The equidistribution result supplies a quantitative version of the statement that nonzero character values are typically not real.
  • The reduction to finite-field equations may extend to give analogous counts for other groups of Lie type of fixed rank.

Load-bearing premise

The proof depends on the complete, explicit list of irreducible characters and conjugacy classes of GL_2(F_q) being known and on character values being algebraic integers whose divisibility can be read off from their expressions.

What would settle it

A sequence of odd prime powers q_n tending to infinity together with a fixed prime ℓ for which the actual count of ℓ-free entries differs from q_n^4/2 by more than C q_n^{3.1} for some constant C.

read the original abstract

Let $q$ range over odd prime powers and let $G_q=\mathrm{GL}_2(\mathbb{F}_q)$. Fix a prime number $\ell$. Motivated by work of Peluse and Soundararajan on Miller's conjecture for character tables of symmetric groups, we study the proportion of entries in the character table of $G_q$ which are not divisible by $\ell$, in the sense of divisibility in the ring of algebraic integers. We prove that $N_\ell(q)=\frac{q^4}{2}+O_\epsilon(q^{3+\epsilon})$ for every $\epsilon>0$, where $N_\ell(q)$ denotes the number of entries which are not divisible by $\ell$. We also show that the number of zero entries is $\frac{q^4}{2}+O_\epsilon(q^{3+\epsilon})$. Consequently, the proportion of all entries not divisible by $\ell$ tends to $1/2$, while the proportion of nonzero entries not divisible by $\ell$ tends to $1$. This differs significantly from the symmetric-group case, where almost every character-table entry is divisible by any fixed prime. We also prove an angular equidistribution result for the nonzero character values as $q\to\infty$. We show that the arguments become equidistributed in $[0,2\pi]$. This proves an analogue of Miller's question on the distribution of signs among the nonzero entries in character tables of symmetric groups.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that for G_q = GL_2(F_q) with q an odd prime power and fixed prime ℓ, the number N_ℓ(q) of character-table entries not divisible by ℓ (in the algebraic integers) equals q^4/2 + O_ε(q^{3+ε}) for every ε>0. The same asymptotic holds for the number of zero entries. Consequently the proportion of all entries not divisible by ℓ tends to 1/2 while the proportion among nonzero entries tends to 1. The arguments of the nonzero character values are shown to become equidistributed in [0,2π] as q→∞. The proofs use the explicit irreducible characters and conjugacy classes of GL_2(F_q), reduce divisibility to solution counts of auxiliary equations over F_q, and apply standard point-counting estimates.

Significance. If the results hold, the paper supplies a precise asymptotic for average ℓ-divisibility in the character table of GL_2(F_q) that differs markedly from the symmetric-group case, together with an equidistribution statement for arguments. The derivation proceeds directly from the group-theoretic data of GL_2 without parameter fitting, and the error term follows from standard estimates on finite-field equations; these features constitute a clear contribution to the statistical study of character tables of groups of Lie type.

minor comments (3)
  1. [§1] §1: The precise cardinality of the character table (number of irreducibles times number of classes) is used implicitly in the proportion statements but is not restated explicitly; adding the formula |Irr(G_q)| = |Cl(G_q)| = q(q-1) would clarify the normalization of the main term q^4/2.
  2. [§3] §3, after the statement of the main counting theorem: the dependence of the implied constant in O_ε(q^{3+ε}) on ℓ is not indicated; a brief remark on whether the constant is uniform in ℓ (for ℓ fixed) or grows with ℓ would be helpful for applications.
  3. [§4] §4, equidistribution argument: the discrepancy or rate of equidistribution for the arguments is not quantified in the statement; adding an explicit rate (even if weaker than the main term) would strengthen the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. No specific major comments appear in the report, so we have no points requiring point-by-point response or manuscript changes at this stage.

Circularity Check

0 steps flagged

Derivation self-contained from explicit GL_2 character table and finite-field point counting

full rationale

The paper invokes the known explicit parametrization of irreducible characters and conjugacy classes of GL_2(F_q) (split/non-split tori, algebraic-integer values) to reduce ℓ-divisibility to the number of solutions of auxiliary equations over F_q. The main term q^4/2 and error O_ε(q^{3+ε}) then follow from standard Weil-type estimates on those equations. No load-bearing step reduces the claimed asymptotic to a fitted parameter, a self-citation chain, or a definitionally equivalent quantity; the equidistribution argument for arguments is likewise independent. This is the normal case of a self-contained derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard classification of irreducible characters of GL_2(F_q) (Deligne-Lusztig or explicit formulas) and on the fact that character values lie in algebraic integers; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Irreducible characters of GL_2(F_q) take values in the ring of algebraic integers.
    Invoked when defining ℓ-divisibility of table entries.
  • domain assumption The conjugacy classes and character degrees of GL_2(F_q) admit an explicit parametrization by tori and characters of F_q^*.
    Required to reduce the divisibility count to finite-field equations.

pith-pipeline@v0.9.1-grok · 5799 in / 1608 out tokens · 21066 ms · 2026-06-29T01:53:05.963782+00:00 · methodology

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Reference graph

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