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An Erdős-Ko-Rado result for some principal series representations

Jiaqi Liao, Guiying Yan

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 $\mathrm{GL}_2(q)$.

An Erdős-Ko-Rado result for some principal series representations

Abstract

Let be an irreducible principal series representation of satisfying certain conditions. Two subsets are called cross--intersecting if for any . In this paper, we determine where are cross--intersecting. Our proofs are based on eigenvalue techniques and the representation theory of .

Paper Structure

This paper contains 14 sections, 12 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. History
  3. Our main result
  4. Structure of the paper
  5. Notation
  6. Background
  7. Representation theory
  8. Cayley Graphs
  9. Hoffman’s bound
  10. Proof of Theorem \ref{['main_thm']}
  11. Strategy
  12. Find the derangements
  13. Compute the eigenvalues
  14. Run the linear programming

Key Result

Theorem 1.1

Fix an odd prime power $q$ and factor $q - 1 = \ell \cdot m$ where $\ell$ is an odd prime with $\ell \nmid m$. Let $\beta: \mathrm{GL}_2(q) \rightarrow \mathrm{GL}(V_{m, 0})$ be the irreducible principal series representation. If two subsets $S_1, S_2 \subseteq \mathrm{GL}_2(q)$ are cross-$1$-inters

Theorems & Definitions (21)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 3.1: Babai MR546860; Diaconis and Shahshahani MR626813; Roichiman MR1400314
  • Theorem 3.2: Hoffman MR284373
  • Lemma 4.1: Table 12.4 in MR1878556
  • Remark 4.2
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • ...and 11 more