symmetric groups and ramanujan graphs mike krebs, cal state la (joint work with a. shaheen)
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Symmetric Groups
and
Ramanujan Graphs
Mike Krebs, Cal State LA
(joint work with A. Shaheen)
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I begin by
telling you about
the motivation for this research.
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I begin by
telling you about
the motivation for this research.
lying to you about
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To fix notation: (1,2)(2,3)=(1,3,2).
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Deceitful Question #1:
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Deceitful Question #1:
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Deceitful Question #1:
Deceitful Sub-question #1’:
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Deceitful Question #2:
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Deceitful Question #2:
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Representations
Symmetric Group
of the
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A Young diagram is a bunch of boxesstacked on top of each other, whereno row is longer than the one above it.
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A Young diagram is a bunch of boxesstacked on top of each other, whereno row is longer than the one above it.
Like this:
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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.
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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.
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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.
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The Murnaghan-Nakayama Rule
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The Murnaghan-Nakayama Rule
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The Murnaghan-Nakayama Rule
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The Murnaghan-Nakayama Rule
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The Murnaghan-Nakayama Rule
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The Murnaghan-Nakayama Rule
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The Murnaghan-Nakayama Rule
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The Murnaghan-Nakayama Rule
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The Murnaghan-Nakayama Rule
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The Murnaghan-Nakayama Rule
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The Murnaghan-Nakayama Rule
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Nota bene: a Young diagram with justone row yields the trivial representation,which has degree one.
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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.
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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.)
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Deceitful Question #3:
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But we can evaluate this sum for all j, andanswer the other deceitful questions, usingthe following numbers . . .
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. . . and all of these numbers comefrom spectral graph theory.
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Background and Motivation
Ramanujan Graphs
Spectral Graph Theoryand
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Background and Motivation
Ramanujan Graphs
Spectral Graph Theoryand
(the real motivation, I promise)
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The graph shownhere is regular:every vertex is anendpoint for thesame number ofedges.
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The graph shownhere is regular:every vertex is anendpoint for thesame number ofedges.
This number (in this example, 3) is thedegree of the graph.
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The graph shownhere is connected(all in one piece).
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This graph is also bipartite.
The graph shownhere is connected(all in one piece).
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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.
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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.
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We form theadjacency matrixas follows:
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We form theadjacency matrixas follows:
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In spectral graph theory, one obtainsinformation about a graph from theeigenvalues of its adjacency matrix.
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In spectral graph theory, one obtainsinformation about a graph from theeigenvalues of its adjacency matrix.
Assume the graph is regular. Then:
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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:
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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:
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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:
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In spectral graph theory, one obtainsinformation about a graph from theeigenvalues of its adjacency matrix.
Assume the graph is regular. Then:
(F. Chung ’88)
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In spectral graph theory, one obtainsinformation about a graph from theeigenvalues of its adjacency matrix.
Assume the graph is regular. Then:
(Alon, Milman, Tanner)
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The point is, graphs with smalleigenvalues are good expanders.
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The point is, graphs with smalleigenvalues are good expanders.
So . . . just how small can we getthe eigenvalues to be?
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The point is, graphs with smalleigenvalues are good expanders.
So . . . just how small can we getthe eigenvalues to be?
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The point is, graphs with smalleigenvalues are good expanders.
So . . . just how small can we getthe eigenvalues to be?
(Alon-Boppana, Serre)
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A graph is Ramanujan if it satisfies:
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A graph is Ramanujan if it satisfies:
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(Side note: a graph is Ramanujan iff its “Iharazeta function” satisfies the Riemann hypothesis.)
A graph is Ramanujan if it satisfies:
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In 1988, Lubotzky, Phillips and Sarnakconstructed infinite families of Ramanujangraphs for k = 1 + a prime.
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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.)
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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:
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Another family of Ramanujan graphsis the set of “finite upper plane graphs.”
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Another family of Ramanujan graphsis the set of “finite upper plane graphs.”
The proof that these are Ramanujanis due to Katz and Evans.
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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.
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One of our goals is to find moreelementary constructions of Ramanujangraphs.
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Constructing Graphs
from Symmetric Groups
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(These are quotients of Cayley graphs.)
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Each such graph ishighly regular andhence has a collapsedadjacency matrix C.
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Each such graph ishighly regular andhence has a collapsedadjacency matrix C.
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The eigenvalues of C coincide with theeigenvalues of A. (But with differentmultiplicities.)
The color-coded sets of vertices areprecisely the double cosets.
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In fact, with one small change (divideeach row by its right-most entry), thecolumns become the eigenvectors.
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Here’s how to obtain these eigenvalues:
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Here’s how to obtain these eigenvalues:
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Here’s how to obtain these eigenvalues:
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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.”
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Spherical Functions
on (G,Y)
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But we don’t know in what order!
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The first sum doesn’t help us at all---it’salways 1, no matter what r is.
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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.
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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.
![Page 111: Symmetric Groups and Ramanujan Graphs Mike Krebs, Cal State LA (joint work with A. Shaheen)](https://reader036.vdocuments.site/reader036/viewer/2022062714/56649d2a5503460f949fe7e8/html5/thumbnails/111.jpg)
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Truthful Answers
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Deceitful Question #3:
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Truthful Answer #3:
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Deceitful Question #2:
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Deceitful Question #2:
No, unless m=n=j.This is equivalent to nonbipartiteness.
Truthful Answer #2:
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Deceitful Question #1:
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Deceitful Question #1:
Yes, unless j=0 or m=n=j.This is equivalent to connectedness.
Truthful Answer #1:
![Page 122: Symmetric Groups and Ramanujan Graphs Mike Krebs, Cal State LA (joint work with A. Shaheen)](https://reader036.vdocuments.site/reader036/viewer/2022062714/56649d2a5503460f949fe7e8/html5/thumbnails/122.jpg)
Deceitful Sub-question #1’:
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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’:
![Page 124: Symmetric Groups and Ramanujan Graphs Mike Krebs, Cal State LA (joint work with A. Shaheen)](https://reader036.vdocuments.site/reader036/viewer/2022062714/56649d2a5503460f949fe7e8/html5/thumbnails/124.jpg)
For example, for our infinite family ofRamanujan graphs (m=j=2), we get:
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For example, for our infinite family ofRamanujan graphs (m=j=2), we get:
![Page 126: Symmetric Groups and Ramanujan Graphs Mike Krebs, Cal State LA (joint work with A. Shaheen)](https://reader036.vdocuments.site/reader036/viewer/2022062714/56649d2a5503460f949fe7e8/html5/thumbnails/126.jpg)
For example, for our infinite family ofRamanujan graphs (m=j=2), we get: