The Iwahori-Hecke Algebra, the Ramanujan Conjecture, and Expander - PowerPoint PPT Presentation

The Iwahori-Hecke Algebra, the Ramanujan Conjecture, and Expander Graphs Cristina Ballantine College of the Holy Cross ICERM April 16, 2013

Graph Theory • Graph X = ( V , E )

Graph Theory • Graph X = ( V , E ) • V = { v 1 , v 2 , . . . , v m } set of vertices

Graph Theory • Graph X = ( V , E ) • V = { v 1 , v 2 , . . . , v m } set of vertices • E set of edges (subsets of order 2 of V )

Graph Theory • Graph X = ( V , E ) • V = { v 1 , v 2 , . . . , v m } set of vertices • E set of edges (subsets of order 2 of V )

Graph Theory • Graph X = ( V , E ) • V = { v 1 , v 2 , . . . , v m } set of vertices • E set of edges (subsets of order 2 of V )

Graph Theory • Graph X = ( V , E ) • V = { v 1 , v 2 , . . . , v m } set of vertices • E set of edges (subsets of order 2 of V )

Graph Theory • Graph X = ( V , E ) • V = { v 1 , v 2 , . . . , v m } set of vertices • E set of edges (subsets of order 2 of V ) 1 4 3 2

Graph Theory • Graph X = ( V , E ) • V = { v 1 , v 2 , . . . , v m } set of vertices • E set of edges (subsets of order 2 of V ) 1 4 3 2

Graph Theory • Graph X = ( V , E ) • V = { v 1 , v 2 , . . . , v m } set of vertices • E set of edges (subsets of order 2 of V ) • adjacency matrix of X : A = a ij 1 � 1 if { v i , v j } ∈ E a ij = if { v i , v j } �∈ E 0 4 3 2

Graph Theory • Graph X = ( V , E ) • V = { v 1 , v 2 , . . . , v m } set of vertices • E set of edges (subsets of order 2 of V ) • adjacency matrix of X : A = a ij 1 � 1 if { v i , v j } ∈ E a ij = if { v i , v j } �∈ E 0   0 1 1 1 4 3 1 0 1 1     1 1 0 0   1 1 0 0 2

Primes in Graphs Graph X = ( V , E ) • Path C C = ( v 0 , v 1 , v 2 , . . . , v n ) , ( v i − 1 , v i ) ∈ E , ∀ 1 ≤ i ≤ n

Primes in Graphs Graph X = ( V , E ) • Path C C = ( v 0 , v 1 , v 2 , . . . , v n ) , ( v i − 1 , v i ) ∈ E , ∀ 1 ≤ i ≤ n

Primes in Graphs Graph X = ( V , E ) • Path C C = ( v 0 , v 1 , v 2 , . . . , v n ) , ( v i − 1 , v i ) ∈ E , ∀ 1 ≤ i ≤ n length of path: | C | = n

Primes in Graphs Graph X = ( V , E ) • Path C C = ( v 0 , v 1 , v 2 , . . . , v n ) , ( v i − 1 , v i ) ∈ E , ∀ 1 ≤ i ≤ n length of path: | C | = n • Prime [ C ] equivalence class of a

Primes in Graphs Graph X = ( V , E ) • Path C C = ( v 0 , v 1 , v 2 , . . . , v n ) , ( v i − 1 , v i ) ∈ E , ∀ 1 ≤ i ≤ n length of path: | C | = n • Prime [ C ] equivalence class of a

Primes in Graphs Graph X = ( V , E ) • Path C C = ( v 0 , v 1 , v 2 , . . . , v n ) , ( v i − 1 , v i ) ∈ E , ∀ 1 ≤ i ≤ n length of path: | C | = n • Prime [ C ] equivalence class of a closed,

Primes in Graphs Graph X = ( V , E ) • Path C C = ( v 0 , v 1 , v 2 , . . . , v n ) , ( v i − 1 , v i ) ∈ E , ∀ 1 ≤ i ≤ n length of path: | C | = n • Prime [ C ] equivalence class of a closed, backtrack-less,

Primes in Graphs Graph X = ( V , E ) • Path C C = ( v 0 , v 1 , v 2 , . . . , v n ) , ( v i − 1 , v i ) ∈ E , ∀ 1 ≤ i ≤ n length of path: | C | = n • Prime [ C ] equivalence class of a closed, backtrack-less, tail-less,

Primes in Graphs Graph X = ( V , E ) • Path C C = ( v 0 , v 1 , v 2 , . . . , v n ) , ( v i − 1 , v i ) ∈ E , ∀ 1 ≤ i ≤ n length of path: | C | = n • Prime [ C ] equivalence class of a closed, backtrack-less, tail-less, primitive path C

Zeta Function for Graphs � 1 − u | C | � − 1 � Z X ( u ) = , u ∈ C [ C ] prime

Zeta Function for Graphs � 1 − u | C | � − 1 � Z X ( u ) = , u ∈ C [ C ] prime Theorem (Ihara 1966) If X = ( V , E ) is a finite, connected, ( q + 1) -regular multigraph (q odd), then Z X ( u ) − 1 = (1 − u 2 ) r − 1 det( I − uA + qu 2 I ) , where r = | E | − | V | + 1 and I is the | V | × | V | identity matrix.

Riemann Hypothesis for Graphs Definition A finite, connected, ( q + 1)-regular graph X satisfies the Riemann Hypothesis if for Re ( s ) ∈ (0 , 1) � − 1 = 0 implies Re ( s ) = 1 � Z X ( q − s ) 2

Riemann Hypothesis for Graphs Definition A finite, connected, ( q + 1)-regular graph X satisfies the Riemann Hypothesis if for Re ( s ) ∈ (0 , 1) � − 1 = 0 implies Re ( s ) = 1 � Z X ( q − s ) 2 • Which graphs satisfy the Riemann Hypothesis?

Expander Graphs • If ( X n , k ) is a k -regular graph on n vertices, the eigenvalues of A are k ≥ λ 1 ≥ λ 2 ≥ . . . ≥ λ n − 1 ≥ − k

Expander Graphs • If ( X n , k ) is a k -regular graph on n vertices, the eigenvalues of A are k ≥ λ 1 ≥ λ 2 ≥ . . . ≥ λ n − 1 ≥ − k • λ ( X ) = max {| λ ( X ) | : λ ∈ Spec( X ) , | λ | � = k }

Expander Graphs • If ( X n , k ) is a k -regular graph on n vertices, the eigenvalues of A are k ≥ λ 1 ≥ λ 2 ≥ . . . ≥ λ n − 1 ≥ − k • λ ( X ) = max {| λ ( X ) | : λ ∈ Spec( X ) , | λ | � = k } • ∂ ( F ) = {{ u , v } ∈ E | u ∈ F , v ∈ V \ F } (boundary of F )

Expander Graphs • If ( X n , k ) is a k -regular graph on n vertices, the eigenvalues of A are k ≥ λ 1 ≥ λ 2 ≥ . . . ≥ λ n − 1 ≥ − k • λ ( X ) = max {| λ ( X ) | : λ ∈ Spec( X ) , | λ | � = k } • ∂ ( F ) = {{ u , v } ∈ E | u ∈ F , v ∈ V \ F } (boundary of F ) Definition An ( n , k , c )-expander: X n , k = ( V , E ) with expansion coefficient � | ∂ ( F ) | � : F ⊆ V , 0 < | F | ≤ n c = inf | F | 2

Expander Graphs • If ( X n , k ) is a k -regular graph on n vertices, the eigenvalues of A are k ≥ λ 1 ≥ λ 2 ≥ . . . ≥ λ n − 1 ≥ − k • λ ( X ) = max {| λ ( X ) | : λ ∈ Spec( X ) , | λ | � = k } • ∂ ( F ) = {{ u , v } ∈ E | u ∈ F , v ∈ V \ F } (boundary of F ) Definition An ( n , k , c )-expander: X n , k = ( V , E ) with expansion coefficient � | ∂ ( F ) | � : F ⊆ V , 0 < | F | ≤ n c = inf | F | 2

Expander Graphs • If ( X n , k ) is a k -regular graph on n vertices, the eigenvalues of A are k ≥ λ 1 ≥ λ 2 ≥ . . . ≥ λ n − 1 ≥ − k • λ ( X ) = max {| λ ( X ) | : λ ∈ Spec( X ) , | λ | � = k } • ∂ ( F ) = {{ u , v } ∈ E | u ∈ F , v ∈ V \ F } (boundary of F ) Definition An ( n , k , c )-expander: X n , k = ( V , E ) with expansion coefficient � | ∂ ( F ) | � : F ⊆ V , 0 < | F | ≤ n c = inf | F | 2 Want large expansion coefficient.

Good expanders/Ramanujan Graphs Proposition (Cheeger’s inequality) Connected X n , k is an ( n , k , c ) -expander with c ≥ k − λ ( X n , k ) . 2

Good expanders/Ramanujan Graphs Proposition (Cheeger’s inequality) Connected X n , k is an ( n , k , c ) -expander with c ≥ k − λ ( X n , k ) . 2 Theorem (Alon-Boppana) √ n →∞ λ ( X n , k ) ≥ 2 k − 1 lim inf

Good expanders/Ramanujan Graphs Proposition (Cheeger’s inequality) Connected X n , k is an ( n , k , c ) -expander with c ≥ k − λ ( X n , k ) . 2 Theorem (Alon-Boppana) √ n →∞ λ ( X n , k ) ≥ 2 k − 1 lim inf Definition (Lubotzky, Phillips, Sarnak) √ A k -regular graph X is a Ramanujan graph if | λ ( X ) | ≤ 2 k − 1.

Good expanders/Ramanujan Graphs Proposition (Cheeger’s inequality) Connected X n , k is an ( n , k , c ) -expander with c ≥ k − λ ( X n , k ) . 2 Theorem (Alon-Boppana) √ n →∞ λ ( X n , k ) ≥ 2 k − 1 lim inf Definition (Lubotzky, Phillips, Sarnak) √ A k -regular graph X is a Ramanujan graph if | λ ( X ) | ≤ 2 k − 1. Corollary (to Ihara’s Theorem) X satisfies the Riemann hypothesis ⇐ ⇒ X is a Ramanujan graph

Constructing Ramanujan Graphs • infinite family of Ramanujan graphs X n , k , k fixed, n → ∞

Constructing Ramanujan Graphs • infinite family of Ramanujan graphs X n , k , k fixed, n → ∞ • for large n hard to estimate eigenvalues of A

Constructing Ramanujan Graphs • infinite family of Ramanujan graphs X n , k , k fixed, n → ∞ • for large n hard to estimate eigenvalues of A • attach a graph to a group

Constructing Ramanujan Graphs • infinite family of Ramanujan graphs X n , k , k fixed, n → ∞ • for large n hard to estimate eigenvalues of A • attach a graph to a group • group structure makes estimation of eigenvalues possible

Graphs from Groups • graph X = ( V , E ): Bruhat-Tits building of G = GL 2 ( Q p )

Graphs from Groups • graph X = ( V , E ): Bruhat-Tits building of G = GL 2 ( Q p ) • one vertex for each conjugate of K = GL 2 ( Z p ); V ↔ G / K

Graphs from Groups • graph X = ( V , E ): Bruhat-Tits building of G = GL 2 ( Q p ) • one vertex for each conjugate of K = GL 2 ( Z p ); V ↔ G / K • ( p + 1)-regular tree × affine line

Graphs from Groups • graph X = ( V , E ): Bruhat-Tits building of G = GL 2 ( Q p ) • one vertex for each conjugate of K = GL 2 ( Z p ); V ↔ G / K • ( p + 1)-regular tree × affine line • G acts simplicially on X

Graphs from Groups • graph X = ( V , E ): Bruhat-Tits building of G = GL 2 ( Q p ) • one vertex for each conjugate of K = GL 2 ( Z p ); V ↔ G / K • ( p + 1)-regular tree × affine line • G acts simplicially on X • Hecke algebra: H ( G , K ) = { f : G → C | f ( kgk ′ ) = f ( g ) } C � G φ ( x ) ψ ( x − 1 g ) dx ( φ ∗ ψ )( g ) :=