Graph of $S_n$ with respect to transposition - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T19:24:11Z http://mathoverflow.net/feeds/question/117562 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117562/graph-of-s-n-with-respect-to-transposition Graph of $S_n$ with respect to transposition GraphTransposition 2012-12-29T20:53:54Z 2012-12-30T05:23:02Z <p>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. </p> <p>This $G_n$, defined that way, has a "name"?</p> <p>It seems pretty easy, (and this is the <strong>main question</strong>) but I'm not sure how $G_n$ "looks like". For instante:</p> <ul> <li>$G_1$ is $K_1$</li> <li>$G_2$ is $K_2$</li> <li>$G_3$ is $K_{3,3}$</li> <li>$G_4$ is <strong>??</strong></li> <li>$\dots$</li> <li>$G_n$ is <strong>??</strong></li> </ul> http://mathoverflow.net/questions/117562/graph-of-s-n-with-respect-to-transposition/117565#117565 Answer by Michael Joyce for Graph of $S_n$ with respect to transposition Michael Joyce 2012-12-29T21:43:14Z 2012-12-30T03:36:31Z <p>This is the undirected version of the <em>Bruhat graph</em>. To make the graph directed, direct an edge $\sigma \rightarrow \sigma'$ if $\ell(\sigma') > \ell(\sigma)$, where $\ell(\sigma)$ denotes the <em>length</em> 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 $\sigma'$.</p> <p>A standard reference for this material is Chapter 2 of Bjorner and Brenti's <em>Combinatorics of Coxeter Groups</em>.</p> http://mathoverflow.net/questions/117562/graph-of-s-n-with-respect-to-transposition/117590#117590 Answer by Dima Pasechnik for Graph of $S_n$ with respect to transposition Dima Pasechnik 2012-12-30T05:23:02Z 2012-12-30T05:23:02Z <p>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 $S_n$.</p> <p>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.</p> <p>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 $(bc)\cdot (ab)$.</p> <p>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.</p>