# Department Colloquium

**
Fri., Feb. 3, 2023 3:30 p.m.
**

**
Location: RIC 208
**

**Speaker:** Venkata Raghu Tej Pantangi, University of Regina

**Title: **Cameron–Liebler Sets in Permutation Groups (518 kB)

**Abstract: **

Let *G *≤ *S _{n} *be a transitive permutation group. Given

*i,j*∈ [

*n*] ={1, ...,

*n*}, denote by

*x*

_{i}_{→}

*the characteristic function of the set*

_{j}{*g *∈ *G *: *g *(*i *) = *j *}.

A Cameron–Liebler set (CL set) in *G *is a set which is represented by a Boolean function in the linear span of {*x _{i}*

_{→j}: (

*i ,j*) ∈ [

*n*]

^{2}}. These are analogous to Boolean degree 1 functions on the hypercube and to Cameron–Liebler line classes in

*PG*(3,

*q*). Sets of the form {

*g*:

*g*(

*i*) ∈

*X*} and {

*g*:

*i*∈

*g*(

*X*)} (for

*i*∈ [

*n*] and

*X*⊂ [

*n*]) are canonically occurring examples of CL sets. A result of Ellis et al. shows that all CL sets in

*S*are canonical.

_{n}In this talk, we will demonstrate many examples with “exotic” CL sets. Of special interest is an exotic CL set in

*PSL*(2

*; q*) with

*q*≡ 3 (mod 4), a 2-transitive group, just like

*S*. The talk is based on ongoing joint work with Jozefien D’haeseleer and Karen Meagher.

_{n}