Graph of $S_n$ with respect to transposition - MathOverflow

Graph of $S_n$ with respect to transposition

2012-12-29T20:53:54Z

Consider the graph $G_n$, with $V(G_n) = S_n$ (the set of permutations of a set of size $n$) and having an edge $\sigma\sigma'$ iif $\sigma'$ can be obtained from $\sigma$ by applying a transposition.

This $G_n$, defined that way, has a "name"?

It seems pretty easy, (and this is the main question) but I'm not sure how $G_n$ "looks like". For instante:

$G_1$ is $K_1$
$G_2$ is $K_2$
$G_3$ is $K_{3,3}$
$G_4$ is ??
$\dots$
$G_n$ is ??

Answer by Michael Joyce for Graph of $S_n$ with respect to transposition

2012-12-29T21:43:14Z

This is the undirected version of the Bruhat graph. To make the graph directed, direct an edge $\sigma \rightarrow \sigma'$ if $\ell(\sigma') > \ell(\sigma)$, where $\ell(\sigma)$ denotes the length of $\sigma$ defined to be the number of inversions of $\sigma$. A related graph is the Hasse diagram of the Bruhat order, which is the subgraph of the Bruhat graph where only the edges $\sigma \rightarrow \sigma'$ with $\ell(\sigma') = \ell(\sigma) + 1$ are kept. A basic fact is that for any edge $\sigma \rightarrow \sigma'$ in the Bruhat graph with $\ell(\sigma') - \ell(\sigma) > 1$, there is a path of edges in the Hasse diagram starting at $\sigma$ and ending at $\sigma'$.

A standard reference for this material is Chapter 2 of Bjorner and Brenti's Combinatorics of Coxeter Groups.

Answer by Dima Pasechnik for Graph of $S_n$ with respect to transposition

2012-12-30T05:23:02Z

1) One can say something about the $k$-neighbourhoods of a vertex $v$ (i.e. sets of vertices at distance $k$ from $v$). For $v=()$, the identity element of $S_n$, each $k$-neighbourhood is a union of conjugacy classes of $S_n$.

E.g. for $k=1$ you get $n \choose 2$ vertices, corresponding to involutions of type $2^1$, a.k.a. transpositions $(ab)$, and there are no edges between them. That is, there are no triangles in your graph. By the way this immediately tells you that $K_{888}$ for $G_4$ is very far off.

For $k=2$ you get two types of vertices, namely, the ones corresponding to involutions of type $2^2$, i.e. $(ab)(cd)$, and the ones corresponding to $3$-cycles $(abc)$. This will tell you that for any two vertices at distance 2 there is unique 4-cycle which contains them. Indeed, you can get $(ab)(cd)$ using two transpositions either as $(ab)\cdot (cd)$ or as $(cd)\cdot (ab)$, and you can get $(abc)$ either as $(ac)\cdot (bc)$ or as $(bc)\cdot (ab)$.

2) Another interesting and sometimes useful fact is that the eigenvalues of the adjacency matrix $A$ of $G_n$ can be computed from the values of the irreducible characters of $S_n$; namely, $A$ can be viewed as an element $\sum\limits_{\pi \text{ a transposition}}\pi$ in the center of the group algebra $\mathbb{C}[S_n]$, and the center is generated by such conjugacy class sums; this allows you to simultaneously diagnonalise them, etc.