Sandpile Groups of Cubes B. Anzis & R. Prasad August 1, 2016 Sandpile Groups of Cubes August 1, 2016 1 / 27
Overview Introduction Sandpile Groups of Cubes August 1, 2016 2 / 27
Overview Introduction Definitions Sandpile Groups of Cubes August 1, 2016 2 / 27
Overview Introduction Definitions Previous Results Sandpile Groups of Cubes August 1, 2016 2 / 27
Overview Introduction Definitions Previous Results Gr¨ obner Basis Calculations Sandpile Groups of Cubes August 1, 2016 2 / 27
Overview Introduction Definitions Previous Results Gr¨ obner Basis Calculations A Bound on the Largest Cyclic Factor Size Sandpile Groups of Cubes August 1, 2016 2 / 27
Overview Introduction Definitions Previous Results Gr¨ obner Basis Calculations A Bound on the Largest Cyclic Factor Size Analogous Bounds on Other Cayley Graphs Sandpile Groups of Cubes August 1, 2016 2 / 27
Overview Introduction Definitions Previous Results Gr¨ obner Basis Calculations A Bound on the Largest Cyclic Factor Size Analogous Bounds on Other Cayley Graphs Higher Critical Groups Sandpile Groups of Cubes August 1, 2016 2 / 27
Introduction Definitions Definition The n-cube is the graph Q n with V ( Q n ) = ( Z / 2 Z ) n and an edge between two vertices v 1 , v 2 ∈ V ( Q n ) if v 1 and v 2 differ in precisely one place. Sandpile Groups of Cubes August 1, 2016 3 / 27
Introduction Definitions Definition The Laplacian of a graph G , denoted L ( G ), is the matrix � deg( v i ) if i = j L ( G ) i , j = − # { edges from v i to v j } if i � = j Example 2 − 1 − 1 0 � 1 � − 1 − 1 2 0 − 1 L ( Q 1 ) = L ( Q 2 ) = − 1 1 − 1 0 2 − 1 0 − 1 − 1 2 Sandpile Groups of Cubes August 1, 2016 4 / 27
Introduction A Final Definition Definition Let G be a graph. Since L ( G ) is an integer matrix, we may consider it as a Z -linear map L ( G ) : Z # V ( G ) → Z # V ( G ) . The torsion part of the cokernel of this map is the critical group (or sandpile group ) of G , denoted K ( G ). Sandpile Groups of Cubes August 1, 2016 5 / 27
Introduction Previous Results I Theorem [Bai] For every prime p > 2, � n � ( Z / k Z )( n k ) Syl p ( K ( Q n )) ∼ � = Syl p . k =1 Sandpile Groups of Cubes August 1, 2016 6 / 27
Introduction Previous Results I Theorem [Bai] For every prime p > 2, � n � ( Z / k Z )( n k ) Syl p ( K ( Q n )) ∼ � = Syl p . k =1 Remark To understand K ( Q n ), it then remains to understand Syl 2 ( K ( Q n )). Sandpile Groups of Cubes August 1, 2016 6 / 27
Introduction Previous Results II Lemma [Benkart, Klivans, Reiner] For every u ∈ ( Z / 2 Z ) n , let χ u ∈ Z 2 n be the vector with entry in position v ∈ ( Z / 2 Z ) n equal to ( − 1) u · v . Then χ u is an eigenvector of L ( Q n ) with eigenvalue 2 · wt( u ), where wt( u ) is the number of non-zero entries in u . Sandpile Groups of Cubes August 1, 2016 7 / 27
Introduction Previous Results II Lemma [Benkart, Klivans, Reiner] For every u ∈ ( Z / 2 Z ) n , let χ u ∈ Z 2 n be the vector with entry in position v ∈ ( Z / 2 Z ) n equal to ( − 1) u · v . Then χ u is an eigenvector of L ( Q n ) with eigenvalue 2 · wt( u ), where wt( u ) is the number of non-zero entries in u . Remark Thus, we understand L ( Q n ) entirely as a map Q 2 n → Q 2 n . When considering it as a map Z 2 n → Z 2 n , this leaves us with the task of understanding the Z -torsion. Sandpile Groups of Cubes August 1, 2016 7 / 27
Introduction Previous Results II Lemma [Benkart, Klivans, Reiner] For every u ∈ ( Z / 2 Z ) n , let χ u ∈ Z 2 n be the vector with entry in position v ∈ ( Z / 2 Z ) n equal to ( − 1) u · v . Then χ u is an eigenvector of L ( Q n ) with eigenvalue 2 · wt( u ), where wt( u ) is the number of non-zero entries in u . Remark Thus, we understand L ( Q n ) entirely as a map Q 2 n → Q 2 n . When considering it as a map Z 2 n → Z 2 n , this leaves us with the task of understanding the Z -torsion. Theorem [Benkart, Klivans, Reiner] There is an isomorphism of Z -modules Z ⊕ K ( Q n ) ∼ � = Z [ x 1 , . . . , x n ] / ( x 2 1 − 1 , . . . , x 2 n − 1 , n − x i ) . Sandpile Groups of Cubes August 1, 2016 7 / 27
Gr¨ obner Basis Calculations Gr¨ obner Basis Background Definition Let R = T [ x 1 , . . . , x n ], where T is a commutative Noetherian ring. A monomial order on R is a total order < on the set of monomials x α 1 1 · · · x α n of R . From now on, we implicitly assume a monomial order < n on R . Sandpile Groups of Cubes August 1, 2016 8 / 27
Gr¨ obner Basis Calculations Gr¨ obner Basis Background Definition Let R = T [ x 1 , . . . , x n ], where T is a commutative Noetherian ring. A monomial order on R is a total order < on the set of monomials x α 1 1 · · · x α n of R . From now on, we implicitly assume a monomial order < n on R . Notation Let I ⊆ [ n ]. We write x I := � i ∈ I x i . Sandpile Groups of Cubes August 1, 2016 8 / 27
Gr¨ obner Basis Calculations Gr¨ obner Basis Background Definition Let R = T [ x 1 , . . . , x n ], where T is a commutative Noetherian ring. A monomial order on R is a total order < on the set of monomials x α 1 1 · · · x α n of R . From now on, we implicitly assume a monomial order < n on R . Notation Let I ⊆ [ n ]. We write x I := � i ∈ I x i . Definition Let f ∈ R . Then the leading term of f , denoted ℓ t( f ), is the term of f greatest with respect to < . Sandpile Groups of Cubes August 1, 2016 8 / 27
Gr¨ obner Basis Calculations Gr¨ obner Basis Background Definition Let I ⊳ R be an ideal. Then the leading term ideal of I is LT( I ) = ( { ℓ t( f ) | f ∈ I } ) . Sandpile Groups of Cubes August 1, 2016 9 / 27
Gr¨ obner Basis Calculations Gr¨ obner Basis Background Definition Let I ⊳ R be an ideal. Then the leading term ideal of I is LT( I ) = ( { ℓ t( f ) | f ∈ I } ) . Definition Let I ⊳ R an ideal. A Gr¨ obner basis of I is a generating set S = { g 1 , . . . , g k } of I satisfying either of the following two properties: For every f ∈ I , we can write ℓ t( f ) = c 1 ℓ t( g 1 ) + · · · + c k ℓ t( g k ) for some c i ∈ R . LT ( I ) = ( ℓ t( g 1 ) , . . . , ℓ t( g k )). Sandpile Groups of Cubes August 1, 2016 9 / 27
Gr¨ obner Basis Calculations Gr¨ obner Basis Background Definition Let I ⊳ R be an ideal. Then the leading term ideal of I is LT( I ) = ( { ℓ t( f ) | f ∈ I } ) . Definition Let I ⊳ R an ideal. A Gr¨ obner basis of I is a generating set S = { g 1 , . . . , g k } of I satisfying either of the following two properties: For every f ∈ I , we can write ℓ t( f ) = c 1 ℓ t( g 1 ) + · · · + c k ℓ t( g k ) for some c i ∈ R . LT ( I ) = ( ℓ t( g 1 ) , . . . , ℓ t( g k )). Theorem When T is a PID, every ideal I ⊳ R has a Gr¨ obner basis. Sandpile Groups of Cubes August 1, 2016 9 / 27
Gr¨ obner Basis Calculations Relevance to Our Situation Theorem Let I ⊳ R be an ideal. Then, as T -modules, R / I ∼ = R / LT( I ) . Sandpile Groups of Cubes August 1, 2016 10 / 27
Gr¨ obner Basis Calculations Relevance to Our Situation Theorem Let I ⊳ R be an ideal. Then, as T -modules, R / I ∼ = R / LT( I ) . Remark By the isomorphism mentioned previously, to understand K ( Q n ) it suffices to understand a Gr¨ obner basis for the ideal I n := ( x 2 1 − 1 , . . . , x 2 � n − 1 , n − x i ) in Z [ x 1 , . . . , x n ]. Sandpile Groups of Cubes August 1, 2016 10 / 27
Gr¨ obner Basis Calculations Relevance to Our Situation Theorem Let I ⊳ R be an ideal. Then, as T -modules, R / I ∼ = R / LT( I ) . Remark By the isomorphism mentioned previously, to understand K ( Q n ) it suffices to understand a Gr¨ obner basis for the ideal I n := ( x 2 1 − 1 , . . . , x 2 � n − 1 , n − x i ) in Z [ x 1 , . . . , x n ]. However, the Gr¨ obner basis is very complicated. Sandpile Groups of Cubes August 1, 2016 10 / 27
Gr¨ obner Basis Calculations Relevance to Our Situation Lemma n − 1 , n − � x i ) in Z / 2 i Z [ x 1 , . . . , x n ]. Let J n denote the ideal ( x 2 1 − 1 , . . . , x 2 Then the factors of Z / 2 Z , . . . , Z / 2 i − 1 Z in Z [ x 1 , . . . , x n ] / I n and Z / 2 i Z [ x 1 , . . . , x n ] / J n are the same. Sandpile Groups of Cubes August 1, 2016 11 / 27
Gr¨ obner Basis Calculations Relevance to Our Situation Lemma n − 1 , n − � x i ) in Z / 2 i Z [ x 1 , . . . , x n ]. Let J n denote the ideal ( x 2 1 − 1 , . . . , x 2 Then the factors of Z / 2 Z , . . . , Z / 2 i − 1 Z in Z [ x 1 , . . . , x n ] / I n and Z / 2 i Z [ x 1 , . . . , x n ] / J n are the same. Goal Understand a Gr¨ obner basis of J n for i = 2, and thus understand the number of Z / 2 Z -factors in Syl 2 K ( Q n ). Sandpile Groups of Cubes August 1, 2016 11 / 27
Gr¨ obner Basis Calculations The Case i = 2 Conjecture For every odd integer m , let W m = { (2 + ǫ 2 , 4 + ǫ 4 , . . . , m − 3 + ǫ m − 3 , m − 1 , m ) | ǫ i ∈ { 0 , 1 }} . Sandpile Groups of Cubes August 1, 2016 12 / 27
Gr¨ obner Basis Calculations The Case i = 2 Conjecture For every odd integer m , let W m = { (2 + ǫ 2 , 4 + ǫ 4 , . . . , m − 3 + ǫ m − 3 , m − 1 , m ) | ǫ i ∈ { 0 , 1 }} . Then LT ( J n ) = ( x 1 ) + ( x 2 2 , . . . , x 2 � � n ) + (2 x I ) . m ≤ n I ∈ W m m odd Sandpile Groups of Cubes August 1, 2016 12 / 27
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