arXiv subjects

Filippo Cesi

Publications and source records attributed to Filippo Cesi.

4 recordsLinked to original sources

arXiv2018

On the spectral gap of some Cayley graphs on the Weyl group $W(B_n)$

The Laplacian of a (weighted) Cayley graph on the Weyl group $W(B_n)$ is a $N\times N$ matrix with $N = 2^n n!$ equal to the order of the group. We show that for a class of (weighted) generating sets, its spectral gap (lowest nontrivial eigenvalue), is actually equal to the spectral gap of a $2n \times 2n$ matrix associated to a $2n$-dimensional permutation representation of $W_n$. This result can be viewed as an extension to $W(B_n)$ of an analogous result valid for the symmetric group, known as `Aldous' spectral gap conjecture', proven in 2010 by Caputo, Liggett and Richthammer.

math.CO↗

arXiv2013

A few remarks on the octopus inequality and Aldous' spectral gap conjecture

A conjecture by D. Aldous, which can be formulated as a statement about the first nontrivial eigenvalue of the Laplacian of certain Cayley graphs on the symmetric group generated by transpositions, has been recently proven by Caputo, Liggett and Richthammer. Their proof is a subtle combination of two ingredients: a nonlinear mapping in the group algebra of the symmetric groups which permits a proof by induction, and a quite hard estimate named the octopus inequality. In this paper we present a simpler and more transparent proof of the octopus inequality, which emerges naturally when looking at the Aldous' conjecture from an algebraic perspective. We also show that the analogous of the Aldous' conjecture, where the spectral gap is replaced by the Kazhdan constant, does not hold in general.

math.RT↗

arXiv2010

On the eigenvalues of Cayley graphs on the symmetric group generated by a complete multipartite set of transpositions

Given a finite simple graph $\cG$ with $n$ vertices, we can construct the Cayley graph on the symmetric group $S_n$ generated by the edges of $\cG$, interpreted as transpositions. We show that, if $\cG$ is complete multipartite, the eigenvalues of the Laplacian of $\Cay(\cG)$ have a simple expression in terms of the irreducible characters of transpositions, and of the Littlewood-Richardson coefficients. As a consequence we can prove that the Laplacians of $\cG$ and of $\Cay(\cG)$ have the same first nontrivial eigenvalue. This is equivalent to saying that Aldous's conjecture, asserting that the random walk and the interchange process have the same spectral gap, holds for complete multipartite graphs.

math.CO↗

arXiv2009

Cayley graphs on the symmetric group generated by initial reversals have unit spectral gap

In a recent paper Gunnells, Scott and Walden have determined the complete spectrum of the Schreier graph on the symmetric group corresponding to the Young subgroup $S_{n-2}\times S_2$ and generated by initial reversals. In particular they find that the first nonzero eigenvalue, or spectral gap, of the Laplacian is always 1, and report that "empirical evidence" suggests that this also holds for the corresponding Cayley graph. We provide a simple proof of this last assertion, based on the decomposition of the Laplacian of Cayley graphs, into a direct sum of irreducible representation matrices of the symmetric group.

math.CO↗