1. Symmetric Groups and Ramanujan Graphs Mike Krebs, Cal State LA (joint work with A. Shaheen)

3. I begin by telling you about the motivation for this research.

4. I begin by telling you about the motivation for this research. lying to you about

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

11. Deceitful Question #1:

13. Deceitful Sub-question #1 ’ :

14. Deceitful Question #2:

16. Representations Symmetric Group of the

23. A Young diagram is a bunch of boxes stacked on top of each other, where no row is longer than the one above it.

24. A Young diagram is a bunch of boxes stacked on top of each other, where no row is longer than the one above it. Like this:

25. A Young diagram is a bunch of boxes stacked on top of each other, where no row is longer than the one above it. Young diagrams with m+n boxes are in 1-1 correspondence with (irreducible) representations of G.

26. A Young diagram is a bunch of boxes stacked on top of each other, where no row is longer than the one above it. Young diagrams with m+n boxes are in 1-1 correspondence with (irreducible) representations of G. I’m not going to tell you how to get a representation from a Young diagram.

27. A Young diagram is a bunch of boxes stacked on top of each other, where no row is longer than the one above it. Young diagrams with m+n boxes are in 1-1 correspondence with (irreducible) representations of G. But I will tell you how to get the character of the representation induced by a Young diagram.

28. The Murnaghan-Nakayama Rule

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

40. The Young diagrams with which we will be primarily concerned are those that have no more than two rows and no more than m boxes in the bottom row.

41. The Young diagrams with which we will be primarily concerned are those that have no more than two rows and no more than m boxes in the bottom row. (This is because such Young diagrams are precisely those whose associated reps appear in the induced rep of the trivial rep of the subgroup Y.)

42. Deceitful Question #3:

52. But we can evaluate this sum for all j, and answer the other deceitful questions, using the following numbers...

54. ... and all of these numbers come from spectral graph theory.

55. Background and Motivation Ramanujan Graphs Spectral Graph Theory and

56. Background and Motivation Ramanujan Graphs Spectral Graph Theory and (the real motivation, I promise)

57. The graph shown here is regular: every vertex is an endpoint for the same number of edges.

58. The graph shown here is regular: every vertex is an endpoint for the same number of edges. This number (in this example, 3) is the degree of the graph.

59. The graph shown here is connected (all in one piece).

60. This graph is also bipartite. The graph shown here is connected (all in one piece).

61. Think of a graph as a communications network. A number called the (edge) expansion constant measures how fast a message originating in some set of vertices will propogate to the entire network.

62. Think of a graph as a communications network. A number called the (edge) expansion constant measures how fast a message originating in some set of vertices will propogate to the entire network.

63. We form the adjacency matrix as follows:

64. We form the adjacency matrix as follows:

65. In spectral graph theory, one obtains information about a graph from the eigenvalues of its adjacency matrix.

66. In spectral graph theory, one obtains information about a graph from the eigenvalues of its adjacency matrix. Assume the graph is regular. Then:

67. In spectral graph theory, one obtains information about a graph from the eigenvalues of its adjacency matrix. The degree k always appears as an eigenvalue. (It’s the largest eigenvalue.) Assume the graph is regular. Then:

68. In spectral graph theory, one obtains information about a graph from the eigenvalues of its adjacency matrix. The degree k always appears as an eigenvalue. (It’s the largest eigenvalue.) If k appears with multiplicity one, then the graph is connected. Assume the graph is regular. Then:

69. In spectral graph theory, one obtains information about a graph from the eigenvalues of its adjacency matrix. If the graph is bipartite, then -k appears as an eigenvalue. Assume the graph is regular. Then:

70. In spectral graph theory, one obtains information about a graph from the eigenvalues of its adjacency matrix. Assume the graph is regular. Then: (F. Chung ’88)

71. In spectral graph theory, one obtains information about a graph from the eigenvalues of its adjacency matrix. Assume the graph is regular. Then: (Alon, Milman, Tanner)

72. The point is, graphs with small eigenvalues are good expanders.

73. The point is, graphs with small eigenvalues are good expanders. So... just how small can we get the eigenvalues to be?

74. The point is, graphs with small eigenvalues are good expanders. So... just how small can we get the eigenvalues to be?

75. The point is, graphs with small eigenvalues are good expanders. So... just how small can we get the eigenvalues to be? (Alon-Boppana, Serre)

76. A graph is Ramanujan if it satisfies:

78. (Side note: a graph is Ramanujan iff its “Ihara zeta function” satisfies the Riemann hypothesis.) A graph is Ramanujan if it satisfies:

79. In 1988, Lubotzky, Phillips and Sarnak constructed infinite families of Ramanujan graphs for k = 1 + a prime.

80. In 1988, Lubotzky, Phillips and Sarnak constructed infinite families of Ramanujan graphs for k = 1 + a prime. (They coined the term “Ramanujan graph,” as their proof makes use of the “Ramanujan conjecture,” proved by Deligne in 1974.)

81. In 1988, Lubotzky, Phillips and Sarnak constructed infinite families of Ramanujan graphs for k = 1 + a prime. (They coined the term “Ramanujan graph,” as their proof makes use of the “Ramanujan conjecture,” proved by Deligne in 1974.) Yes, this Ramanujan:

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

83. Another family of Ramanujan graphs is the set of “finite upper plane graphs.” The proof that these are Ramanujan is due to Katz and Evans.

84. Another family of Ramanujan graphs is the set of “finite upper plane graphs.” The proof that these are Ramanujan is due to Katz and Evans. As with the Lubotzky-Phillips-Sarnak graphs, the proof is quite difficult.

85. One of our goals is to find more elementary constructions of Ramanujan graphs.

86. Constructing Graphs from Symmetric Groups

88. (These are quotients of Cayley graphs.)

89. Each such graph is highly regular and hence has a collapsed adjacency matrix C.

90. Each such graph is highly regular and hence has a collapsed adjacency matrix C.

91. The eigenvalues of C coincide with the eigenvalues of A. (But with different multiplicities.) The color-coded sets of vertices are precisely the double cosets.

93. In fact, with one small change (divide each row by its right-most entry), the columns become the eigenvectors.

94. Here’s how to obtain these eigenvalues:

99. However, we don’t know about connectedness yet, since we don’t know the multiplicity of the degree as an eigenvalue. For that, we need to learn about “finite spherical functions.”

100. Spherical Functions on (G,Y)

104. But we don’t know in what order!

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

109. The first sum doesn’t help us at all---it’s always 1, no matter what r is. But the values of the second sum are always distinct for distinct values of r.

110. The first sum doesn’t help us at all---it’s always 1, no matter what r is. But the values of the second sum are always distinct for distinct values of r.

115. Truthful Answers

116. Deceitful Question #3:

117. Truthful Answer #3:

118. Deceitful Question #2:

119. No, unless m=n=j. This is equivalent to nonbipartiteness. Truthful Answer #2:

120. Deceitful Question #1:

121. Yes, unless j=0 or m=n=j. This is equivalent to connectedness. Truthful Answer #1:

122. Deceitful Sub-question #1 ’ :

123. k is exactly the diameter of the corresponding graph. So we can estimate k using the estimate of F. Chung. Truthful Sub-answer #1 ’ :

124. For example, for our infinite family of Ramanujan graphs (m=j=2), we get:

125. For example, for our infinite family of Ramanujan graphs (m=j=2), we get:

126. For example, for our infinite family of Ramanujan graphs (m=j=2), we get: