Symmetric Groups

and

Ramanujan Graphs

Mike Krebs, Cal State LA

(joint work with A. Shaheen)

I begin by

telling you about

the motivation for this research.

I begin by

telling you about

the motivation for this research.

lying to you about

To fix notation: (1,2)(2,3)=(1,3,2).

Deceitful Question #1:

Deceitful Question #1:

Deceitful Question #1:

Deceitful Sub-question #1’:

Deceitful Question #2:

Deceitful Question #2:

Representations

Symmetric Group

of the

A Young diagram is a bunch of boxesstacked on top of each other, whereno row is longer than the one above it.

A Young diagram is a bunch of boxesstacked on top of each other, whereno row is longer than the one above it.

Like this:

A Young diagram is a bunch of boxesstacked on top of each other, whereno row is longer than the one above it.

Young diagrams with m+n boxesare in 1-1 correspondence with(irreducible) representations of G.

A Young diagram is a bunch of boxesstacked on top of each other, whereno row is longer than the one above it.

Young diagrams with m+n boxesare in 1-1 correspondence with(irreducible) representations of G.

I’m not going to tell you how to geta representation from a Youngdiagram.

A Young diagram is a bunch of boxesstacked on top of each other, whereno row is longer than the one above it.

Young diagrams with m+n boxesare in 1-1 correspondence with(irreducible) representations of G.

But I will tell you how to get the characterof the representation induced by a Youngdiagram.

The Murnaghan-Nakayama Rule

The Murnaghan-Nakayama Rule

The Murnaghan-Nakayama Rule

The Murnaghan-Nakayama Rule

The Murnaghan-Nakayama Rule

The Murnaghan-Nakayama Rule

The Murnaghan-Nakayama Rule

The Murnaghan-Nakayama Rule

The Murnaghan-Nakayama Rule

The Murnaghan-Nakayama Rule

The Murnaghan-Nakayama Rule

Nota bene: a Young diagram with justone row yields the trivial representation,which has degree one.

The Young diagrams with which wewill be primarily concerned are thosethat have no more than two rows andno more than m boxes in the bottom row.

The Young diagrams with which wewill be primarily concerned are thosethat have no more than two rows andno more than m boxes in the bottom row.

(This is because such Young diagramsare precisely those whose associatedreps appear in the induced rep of thetrivial rep of the subgroup Y.)

Deceitful Question #3:

But we can evaluate this sum for all j, andanswer the other deceitful questions, usingthe following numbers . . .

. . . and all of these numbers comefrom spectral graph theory.

Background and Motivation

Ramanujan Graphs

Spectral Graph Theoryand

Background and Motivation

Ramanujan Graphs

Spectral Graph Theoryand

(the real motivation, I promise)

The graph shownhere is regular:every vertex is anendpoint for thesame number ofedges.

The graph shownhere is regular:every vertex is anendpoint for thesame number ofedges.

This number (in this example, 3) is thedegree of the graph.

The graph shownhere is connected(all in one piece).

This graph is also bipartite.

The graph shownhere is connected(all in one piece).

Think of a graph as a communicationsnetwork.  A number called the (edge)expansion constant measures how fast amessage originating in some set of verticeswill propogate to the entire network.

Think of a graph as a communicationsnetwork.  A number called the (edge)expansion constant measures how fast amessage originating in some set of verticeswill propogate to the entire network.

We form theadjacency matrixas follows:

We form theadjacency matrixas follows:

In spectral graph theory, one obtainsinformation about a graph from theeigenvalues of its adjacency matrix.

In spectral graph theory, one obtainsinformation about a graph from theeigenvalues of its adjacency matrix.

Assume the graph is regular. Then:

In spectral graph theory, one obtainsinformation about a graph from theeigenvalues of its adjacency matrix.

The degree k always appears as aneigenvalue. (It’s the largest eigenvalue.)

Assume the graph is regular. Then:

In spectral graph theory, one obtainsinformation about a graph from theeigenvalues of its adjacency matrix.

The degree k always appears as aneigenvalue. (It’s the largest eigenvalue.)

If k appears with multiplicityone, then the graph is connected.

Assume the graph is regular. Then:

In spectral graph theory, one obtainsinformation about a graph from theeigenvalues of its adjacency matrix.

If the graph is bipartite, then-k appears as an eigenvalue.

Assume the graph is regular. Then:

In spectral graph theory, one obtainsinformation about a graph from theeigenvalues of its adjacency matrix.

Assume the graph is regular. Then:

(F. Chung ’88)

In spectral graph theory, one obtainsinformation about a graph from theeigenvalues of its adjacency matrix.

Assume the graph is regular. Then:

(Alon, Milman, Tanner)

The point is, graphs with smalleigenvalues are good expanders.

The point is, graphs with smalleigenvalues are good expanders.

So . . . just how small can we getthe eigenvalues to be?

The point is, graphs with smalleigenvalues are good expanders.

So . . . just how small can we getthe eigenvalues to be?

The point is, graphs with smalleigenvalues are good expanders.

So . . . just how small can we getthe eigenvalues to be?

(Alon-Boppana, Serre)

A graph is Ramanujan if it satisfies:

A graph is Ramanujan if it satisfies:

(Side note: a graph is Ramanujan iff its “Iharazeta function” satisfies the Riemann hypothesis.)

A graph is Ramanujan if it satisfies:

In 1988, Lubotzky, Phillips and Sarnakconstructed infinite families of Ramanujangraphs for k = 1 + a prime.

In 1988, Lubotzky, Phillips and Sarnakconstructed infinite families of Ramanujangraphs for k = 1 + a prime.

(They coined the term “Ramanujangraph,” as their proof makes use ofthe “Ramanujan conjecture,” provedby Deligne in 1974.)

In 1988, Lubotzky, Phillips and Sarnakconstructed infinite families of Ramanujangraphs for k = 1 + a prime.

(They coined the term “Ramanujangraph,” as their proof makes use ofthe “Ramanujan conjecture,” provedby Deligne in 1974.)

Yes, this Ramanujan:

Another family of Ramanujan graphsis the set of “finite upper plane graphs.”

Another family of Ramanujan graphsis the set of “finite upper plane graphs.”

The proof that these are Ramanujanis due to Katz and Evans.

Another family of Ramanujan graphsis the set of “finite upper plane graphs.”

The proof that these are Ramanujanis due to Katz and Evans.

As with the Lubotzky-Phillips-Sarnakgraphs, the proof is quite difficult.

One of our goals is to find moreelementary constructions of Ramanujangraphs.

Constructing Graphs

from Symmetric Groups

(These are quotients of Cayley graphs.)

Each such graph ishighly regular andhence has a collapsedadjacency matrix C.

Each such graph ishighly regular andhence has a collapsedadjacency matrix C.

The eigenvalues of C coincide with theeigenvalues of A. (But with differentmultiplicities.)

The color-coded sets of vertices areprecisely the double cosets.

In fact, with one small change (divideeach row by its right-most entry), thecolumns become the eigenvectors.

Here’s how to obtain these eigenvalues:

Here’s how to obtain these eigenvalues:

Here’s how to obtain these eigenvalues:

However, we don’t know aboutconnectedness yet, since we don’t know themultiplicity of the degree as an eigenvalue.

For that, we need to learn about “finitespherical functions.”

Spherical Functions

on (G,Y)

But we don’t know in what order!

The first sum doesn’t help us at all---it’salways 1, no matter what r is.

The first sum doesn’t help us at all---it’salways 1, no matter what r is.

But the values of the second sum arealways distinct for distinct values of r.

The first sum doesn’t help us at all---it’salways 1, no matter what r is.

But the values of the second sum arealways distinct for distinct values of r.

Truthful Answers

Deceitful Question #3:

Truthful Answer #3:

Deceitful Question #2:

Deceitful Question #2:

No, unless m=n=j.This is equivalent to nonbipartiteness.

Truthful Answer #2:

Deceitful Question #1:

Deceitful Question #1:

Yes, unless j=0 or m=n=j.This is equivalent to connectedness.

Truthful Answer #1:

Deceitful Sub-question #1’:

Deceitful Sub-question #1’:

k is exactly the diameter of thecorresponding graph. So we canestimate k using the estimate of F. Chung.

Truthful Sub-answer #1’:

For example, for our infinite family ofRamanujan graphs (m=j=2), we get:

For example, for our infinite family ofRamanujan graphs (m=j=2), we get:

For example, for our infinite family ofRamanujan graphs (m=j=2), we get: