On the Largest intersecting set in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi> GL </mml:mi> <mml:mn>2</mml:mn> </mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> and some of its subgroups

Ahanjideh, Milad

doi:10.5802/crmath.320

Comptes Rendus. Mathématique

2022

DOI: 10.5802/crmath.320

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On the Largest intersecting set in GL 2 (q) and some of its subgroups

Milad Ahanjideh¹

Abstract: Let q be a power of a prime number and V be a 2-dimensional column vector space over a finite field F q . Assume that SL 2 (V ) < G ≤ GL 2 (V ). In this paper we prove an Erdős-Ko-Rado theorem for intersecting sets of G and we show that every maximum intersecting set of G is either a coset of the stabilizer of a point or a coset of G 〈w〉 , whereIt is also shown that every intersecting set of G is contained in a maximum intersecting set.

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Intersection theorems for finite general linear groups

ERNST¹,

SCHMIDT²

2023

Math. Proc. Camb. Phil. Soc.

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A subset Y of the general linear group $\text{GL}(n,q)$ is called t-intersecting if $\text{rk}(x-y)\le n-t$ for all $x,y\in Y$ , or equivalently x and y agree pointwise on a t-dimensional subspace of $\mathbb{F}_q^n$ for all $x,y\in Y$ . We show that, if n is sufficiently large compared to t, the size of every such t-intersecting set is at most that of the stabiliser of a basis of a t-dimensional subspace of $\mathbb{F}_q^n$ . In case of equality, the characteristic vector of Y is a linear combination of the characteristic vectors of the cosets of these stabilisers. We also give similar results for subsets of $\text{GL}(n,q)$ that intersect not necessarily pointwise in t-dimensional subspaces of $\mathbb{F}_q^n$ and for cross-intersecting subsets of $\text{GL}(n,q)$ . These results may be viewed as variants of the classical Erdős–Ko–Rado Theorem in extremal set theory and are q-analogs of corresponding results known for the symmetric group. Our methods are based on eigenvalue techniques to estimate the size of the largest independent sets in graphs and crucially involve the representation theory of $\text{GL}(n,q)$ .

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Intersection theorems for finite general linear groups

ERNST¹,

SCHMIDT²

2023

Math. Proc. Camb. Phil. Soc.

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show abstract

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On the Largest intersecting set in GL 2 (q) and some of its subgroups

Cited by 1 publication

References 14 publications

Intersection theorems for finite general linear groups

Intersection theorems for finite general linear groups

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