SEALow Team 2 Presents: More Miles for Your (Sand)Piles Jiyang Gao, - PowerPoint PPT Presentation

SEALow Team 2 Presents: More Miles for Your (Sand)Piles Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald August 3, 2018 Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald Sandpile Groups August 3, 2018 1 / 41

Overview Definitions and Previous Results 1 Results on the Number of Even Invariant factors 2 Reducing the Sandpile Group and Results for Small Cases 3 Largest Cyclic Factors 4 Future Areas of Investigation 5 Future Work and Acknowledgements 6 Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald Sandpile Groups August 3, 2018 2 / 41

Definitions and Previous Results Section 1 Definitions and Previous Results Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald Sandpile Groups August 3, 2018 3 / 41

Definitions and Previous Results Definitions Definition Given F r 2 , M , where M = { v 1 , . . . , v n } is a set of generators, we define the Cayley graph G ( F r 2 , M ) with V ( G ) = F r 2 and u , w ∈ V ( G ) share an edge if u − w = v i for some generator. Multiple edges are allowed. Example Let M = { e 1 , . . . , e n } . Then G ( F r 2 , M ) = Q n , is called the hypercube graph. If M = { v ∈ F r 2 − { 0 }} , then G ( F r 2 , M ) = K 2 r is called the complete graph on 2 r vertices. See board for image of Q 2 and K 4 with generators labelled Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald Sandpile Groups August 3, 2018 4 / 41

Definitions and Previous Results Definitions Definition The Laplacian of a nondirected graph G , denoted L ( G ) has entries � deg ( v i ) i = j L ( G ) i , j = − # edges from v i to v j i � = j Definition Given a connected graph G with | V ( G ) | = w , L ( G ) is an integer w × w matrix, so we can view it as map of Z -modules Z w → Z w . The kernel is span ( 1 ) , so coker L ( G ) ∼ = Z ⊕ K ( G ) where K ( G ) is a finite abelian group. We call K ( G ) the sandpile group of G . Example It is well known that K ( K n ) ∼ = ( Z / n Z ) n − 2 . So we can determine at least one case of Cayley graphs. Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald Sandpile Groups August 3, 2018 5 / 41

Definitions and Previous Results Previous Results for F r 2 Lemma Let M = { v 1 , . . . , v n } be the set of generators. For every u ∈ F r 2 , let n � � (− 1 ) u · v e v (− 1 ) u · v i f u = λ u , M = n − v ∈ F r i = 1 2 Then { f u } is an eigenbasis of R 2 r each with eigenvalues { λ u , M } , which is � always even. Moreover, e v = 1 2 (− 1 ) u · v f v . v ∈ F r 2 r Theorem (Ducey-Jalil) Let G be a Cayley graph of F r 2 . For all p � = 2 , � 2 r � Syl p ( K ( G )) ∼ � = Syl p Z /λ u , M Z k = 1 Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald Sandpile Groups August 3, 2018 6 / 41

Definitions and Previous Results More Previous Results Remark L ( G ) is diagonalizable over Z [ 1 2 ] , and we can describe the Sylow-p structure for all p � = 2 in terms of the eigenvalues. What about p = 2 ? Is the Sylow- 2 group uniquely determined by the eigenvalues? Theorem There is an isomorphism of abelian groups   n � � x ( v i ) j Z ⊕ K ( G ) ∼  x 2 1 − 1 , . . . , x 2 = Z [ x 1 , . . . , x r ] / r − 1 , n − j  i = 1 j Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald Sandpile Groups August 3, 2018 7 / 41

Definitions and Previous Results Previous results for p = 2 Theorem (Bai) For G = Q n , the number of Sylow- 2 cyclic factors is 2 n − 1 − 1 . Additionally, the number of ( Z / 2 Z ) ’s in K ( G ) is 2 n − 2 − 2 ⌊ ( n − 2 ) / 2 ⌋ . Theorem (Anzis-Prasad) The size of the largest factor in Syl 2 ( K ( Q n )) is ≤ 2 n + ⌊ log 2 n ⌋ . We will generalize Bai’s first result and Anzis-Prasad, but not Bai’s second result. Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald Sandpile Groups August 3, 2018 8 / 41

Results on the Number of Even Invariant factors Section 2 Results on the Number of Even Invariant Factors Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald Sandpile Groups August 3, 2018 9 / 41

Results on the Number of Even Invariant factors Invariant factors Definition We define d ( M ) to be the number of Sylow-2 cyclic factors in K ( G ) . Proposition (Parity Invariance) Let our matrix of generators M have multiplicities ( a v 1 , . . . a v 2 r − 1 ) for each nonzero vector in F r 2 . Then d ( M ) only depends on the parity of the multiplicities of generators. Example  1 0 0   1 0 0 1 1   and M ′ =  then If M = 0 1 0 0 1 0 1 1   0 0 1 0 0 1 1 1 K ( G ( F 3 2 , M )) = ( Z / 2 Z ) ⊕ ( Z / 8 Z ) ⊕ ( Z / 24 Z ) K ( G ( F 3 2 , M ′ )) = ( Z / 6 Z ) ⊕ ( Z / 24 Z ) ⊕ ( Z / 120 Z ) Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald Sandpile Groups August 3, 2018 10 / 41

Results on the Number of Even Invariant factors Computing d ( M ) Definition A Cayley graph G ( F r 2 , M ) with M = { v 1 , . . . , v n } is called generic if � n i = 1 v i � = � 0. For example, Q n is generic for all n ≥ 1. Theorem 2 , M ) is generic, then d ( M ) = 2 r − 1 − 1 . If G ( F r Proof Sketch. Consider ( Z ⊕ K ( G )) ⊗ ( Z / 2 Z ) . d ( M ) is equal to the dimension of K ( G ) ⊗ ( Z / 2 Z ) as a vector space. Theorem’s condition gives us a nonzero degree 1 term of the form u i which allows us to construct an explicit ( Z / 2 Z )− mod ∼ ( Z / 2 Z ) 2 r − 1 . isomorphism ( Z ⊕ K ( G )) ⊗ ( Z / 2 Z ) = Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald Sandpile Groups August 3, 2018 11 / 41

Results on the Number of Even Invariant factors Conjectures Conjecture For a collection of generators, M, yielding a connected Cayley graph on 2 , d ( M ) ≥ 2 r − 1 − 1 with equality occurring iff M is generic. F r Conjecture d ( M ) is odd unless all of the eigenvalues have the same power of 2 , in which case d ( M ) = 2 n − 2 . Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald Sandpile Groups August 3, 2018 12 / 41

Reducing the Sandpile Group and Results for Small Cases Section 3 Reducing the Sandpile Group and Results for Small Cases Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald Sandpile Groups August 3, 2018 13 / 41

Reducing the Sandpile Group and Results for Small Cases Reducing multiplicities in M Given generators for V = F r 2 , we can express these generators in terms of their multiplicities � a = ( a v 1 , . . . , a v 2 r − 1 ) , where the multiplicity, a v i , denotes the number of times the vector v i occurs. Here, we will use the binary naming convention for vectors, so v 3 = ( 1 , 1 , 0 ) . Lemma Let G 1 = G ( F r 2 , M 1 ) and G 2 = G ( F r 2 , M 2 ) such that � a 2 = λ � a 2 for λ ∈ N and let { α i } be the invariant factors in the Smith Normal Form of L ( G 1 ) . Then 2 r 2 r � � K ( G 1 ) = Z /α i Z = ⇒ K ( G 2 ) = Z / ( λα i ) Z i = 1 i = 1 Proof. a 2 = λ � a 2 = ⇒ L ( G 2 ) = ( λ Id ) · L ( G 1 ) . Now consider SNF of L ( G 2 ) . (Note: � reduces analysis to gcd ( � a ) = 1 case.) Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald Sandpile Groups August 3, 2018 14 / 41

Reducing the Sandpile Group and Results for Small Cases Example Consider the two matrices of generators: � 1 � 0 1 M 1 = 0 1 1 ⇒ K ( G ( F 2 = 2 , M 1 )) = ( Z / 1 Z ) ⊕ ( Z / 4 Z ) ⊕ ( Z / 4 Z ) � 1 � 1 0 0 1 1 M 2 = 0 0 1 1 1 1 ⇒ K ( G ( F 2 = 2 , M 2 )) = ( Z / 2 Z ) ⊕ ( Z / 8 Z ) ⊕ ( Z / 8 Z ) Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald Sandpile Groups August 3, 2018 15 / 41

Reducing the Sandpile Group and Results for Small Cases Invariance Under GL Action Theorem Given a matrix of generators on F r 2   | | . . . | M = v 1 v 2 . . . v n   | | . . . | and an element g ∈ GL r ( F 2 ) , define M ′ := g · M, then G ( F r 2 , M ) and G ( F r 2 , M ′ ) have the same sandpile group. Proof. An element of GL r permutes the nonzero vertices of the graphs and the edges in a consistent manner. This induces a graph isomorphism, and thus a sandpile group isomorphism. Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald Sandpile Groups August 3, 2018 16 / 41

Reducing the Sandpile Group and Results for Small Cases Example Assume a v 1 = a v 2 = a v 4 = a v 8 = 2. Let { i 1 , . . . , i k } denote a v i 1 = · · · = a v ik = 2 and a v j = 1 for all j �∈ { 1 , 2 , 4 , 8 } ∪ { i 1 , . . . , i k } : { 6 , 10 , 12 } , { 5 , 9 , 12 } , { 3 , 5 , 6 } , { 3 , 9 , 10 } , { 10 , 12 , 14 } , { 9 , 12 , 13 } , { 5 , 6 , 7 } , { 3 , 10 , 11 } , { 6 , 12 , 14 } , { 5 , 9 , 13 } , { 5 , 12 , 13 } , { 3 , 6 , 7 } , { 3 , 9 , 11 } , { 6 , 10 , 14 } , { 3 , 5 , 7 } , { 9 , 10 , 11 } All of the 16 previous cases yield K ( G ) = ( Z / 3 Z ) ⊕ ( Z / 6 Z ) ⊕ ( Z / 48 Z ) ⊕ ( Z / 48 Z ) ⊕ ( Z / 528 Z ) ⊕ ( Z / 6864 Z ) ⊕ ( Z / 6864 Z ) Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald Sandpile Groups August 3, 2018 17 / 41

Reducing the Sandpile Group and Results for Small Cases Size of 2-Sylow component By Kirchoff’s Matrix Tree Theorem i , i = λ 2 · · · λ m | K ( G ) | = det L ( G ) m where λ 1 = 0 is only 0 eigenvalue by convention. Here, m = 2 r , so   � | Syl 2 ( K ( G )) | = 1 2 r Pow 2 λ u , M   u ∈ F r 2 − { 0 } where for n = 2 k · b with k -maximal, we define Pow 2 ( n ) := 2 k and v 2 ( n ) := k . Jiyang Gao, Jared Marx-Kuo, Amal Mattoo, Vaughan McDonald Sandpile Groups August 3, 2018 18 / 41