An Erd\H{o}s-Ko-Rado result for some principal series representations

TL;DR

This paper extends the Erd ext{"o}s-Ko-Rado theorem to certain principal series representations of ngf3f2f2(q), identifying maximum product sizes of cross-1-intersecting subsets using eigenvalue and representation theory methods.

Contribution

It determines the maximum product size of cross-1-intersecting subsets within specific principal series representations of ngf3f2f2(q).

Findings

Calculated maximum ngf3f2f2(q) subset products for cross-1-intersecting sets.

Applied eigenvalue techniques and representation theory in the proof.

Extended Erd ext{"o}s-Ko-Rado results to new algebraic structures.

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