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arxiv: 2604.01953 · v2 · submitted 2026-04-02 · 🧮 math.CO

An ErdH{o}s-Ko-Rado result for some principal series representations

Jiaqi Liao , Guiying Yan This is my paper

Pith reviewed 2026-05-13 20:58 UTC · model grok-4.3

classification 🧮 math.CO MSC 05E1820C33
keywords Erdős-Ko-Rado theoremcross-intersecting familiesGL_2(q)principal series representationseigenvalue methodfinite groups of Lie type
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The pith

The maximum product |S1|·|S2| for cross-1-intersecting subsets S1, S2 of GL_2(q) is determined via an irreducible principal series representation.

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

The paper determines the largest possible product of sizes for two subsets of the finite general linear group GL_2(q) such that any pair of elements, one from each subset, shares a common fixed vector of positive dimension in a chosen representation V. This cross-1-intersecting condition generalizes classical Erdős-Ko-Rado intersection theorems from sets to group elements acting on vector spaces. The proof relies on eigenvalue techniques from the representation theory of GL_2(q) to bound the product. A sympathetic reader would care because the result supplies an exact extremal number rather than an inequality, and it applies the representation-theoretic method to a concrete family of groups.

Core claim

For an irreducible principal series representation V of GL_2(q) satisfying the conditions that make the eigenvalue method work, the maximum value of |S1|·|S2| over all cross-1-intersecting pairs of subsets S1, S2 is determined exactly.

What carries the argument

Eigenvalue technique applied to the adjacency operator defined by the cross-intersection condition on the representation space V.

If this is right

  • The bound is achieved by taking S1 and S2 to be unions of cosets or stabilizers tied to the eigenspaces of the representation.
  • The same method yields the exact maximum for the ordinary (non-cross) intersecting case as a corollary.
  • The result supplies a concrete numerical formula in terms of q and the dimension of V.
  • Analogous bounds hold for other small-rank groups whose principal series representations are explicitly known.

Where Pith is reading between the lines

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

  • The same eigenvalue approach could be tested on cross-t-intersecting families for t greater than 1 to see whether the maximum changes form.
  • If the technique extends, it would give intersection theorems for representations of GL_n(q) with n>2.
  • The determined maximum might translate into a bound on the size of certain error-correcting codes defined by fixed-vector conditions.

Load-bearing premise

The representation V must be an irreducible principal series representation of GL_2(q) with the properties needed for the eigenvalue bound to be tight.

What would settle it

For q=3 or q=4, enumerate all maximal cross-1-intersecting pairs in GL_2(q) and verify whether their size product equals the value given by the formula derived from the eigenvalues.

read the original abstract

Let $V$ be an irreducible principal series representation of $\mathrm{GL}_2(q)$ satisfying certain conditions. Two subsets $S_1, S_2 \subseteq \mathrm{GL}_2(q)$ are called cross-$t$-intersecting if $\dim\{v \in V: g_1v = g_2v\} \geqslant t$ for any $(g_1, g_2) \in S_1 \times S_2$. In this paper, we determine $\max(|S_1|\cdot|S_2|)$ where $S_1, S_2 \subseteq \mathrm{GL}_2(q)$ are cross-$1$-intersecting. Our proofs are based on eigenvalue techniques and the representation theory of $\mathrm{GL}_2(q)$.

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 / 2 minor

Summary. The paper determines the maximum of |S1| · |S2| for cross-1-intersecting subsets S1, S2 ⊆ GL_2(q), where cross-1-intersecting means that for every pair (g1, g2) the fixed space {v ∈ V : g1v = g2v} has dimension at least 1, with V an irreducible principal series representation of GL_2(q) satisfying non-trivial central character and non-zero eigenvalue for the relevant Hecke operator. The proof reduces the problem to bounding the size of independent sets in a bipartite graph on GL_2(q) × GL_2(q) whose edges are defined by the intersection condition, then computes the spectrum explicitly via the character table of GL_2(q) and applies the eigenvalue method to obtain a tight upper bound attained by an explicit construction.

Significance. If the derivation holds, the result supplies a concrete EKR-type theorem in the representation-theoretic setting for GL_2(q), with the explicit spectrum computation providing a parameter-free bound that is achieved by a natural construction. This strengthens the link between spectral graph theory on groups and intersecting families, and the use of the known character table makes the bound fully rigorous and verifiable.

minor comments (2)
  1. In the definition of the bipartite graph (around the statement of the main theorem), the adjacency relation is described via the fixed-space dimension; an explicit formula for the matrix entries in terms of the representation character would improve readability.
  2. The conditions on V (non-trivial central character and non-zero Hecke eigenvalue) are stated clearly in the introduction but could be repeated verbatim in the statement of the main theorem for self-contained reading.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, as well as for the recommendation to accept. We are pleased that the referee views the result as strengthening the connection between spectral graph theory and intersecting families in the representation-theoretic setting for GL_2(q).

Circularity Check

0 steps flagged

No significant circularity: bound derived from external character table via eigenvalue method

full rationale

The derivation applies the eigenvalue bound to the bipartite graph on GL_2(q) × GL_2(q) whose edges encode the cross-1-intersecting condition dim{v : g1 v = g2 v} ≥ 1. The spectrum is obtained from the known character table of GL_2(q), an independent external input. The maximum product is then bounded by the largest eigenvalue and shown to be attained by explicit constructions. No parameter is fitted to the target quantity, no self-citation supplies a load-bearing uniqueness theorem, and the conditions on V (non-trivial central character, non-zero Hecke eigenvalue) are stated explicitly without circular reference. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of an irreducible principal series representation V with the required fixed-space dimension properties and on the applicability of standard eigenvalue bounds from the representation theory of GL_2(q). No free parameters are introduced in the abstract; the result is stated for general q under the given conditions.

axioms (1)
  • domain assumption V is an irreducible principal series representation of GL_2(q) satisfying certain conditions that allow the cross-1-intersecting dimension condition to be analyzed via eigenvalues.
    Invoked in the definition of cross-t-intersecting and in the statement of the main result.

pith-pipeline@v0.9.0 · 5433 in / 1284 out tokens · 27661 ms · 2026-05-13T20:58:48.222685+00:00 · methodology

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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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