$PI$-eigenfunctions of the Star graphs

Abstract

We consider the symmetric group ${\mathrm{Sym}}_n,n⩾2$, generated by the set $S$ of transpositions $(1i),2⩽i⩽n$, and the Cayley graph $S_n=Cay({\mathrm{Sym}}_n,S)$ called the Star graph. For any positive integers $n⩾3$ and $m$ with $n>2m$, we present a family of $PI$-eigenfunctions of $S_n$ with eigenvalue $n-m-1$. We establish a connection of these functions with the standard basis of a Specht module. In the case of largest non-principal eigenvalue $n-2$ we prove that any eigenfunction of $S_n$ can be reconstructed by its values on the second neighbourhood of a vertex.