Sandpile Groups of Cubes

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1 Sandpile Groups of Cubes B. Anzis & R. Prasad August 1, 2016 Sandpile Groups of Cubes August 1, / 27

2 Overview Introduction Sandpile Groups of Cubes August 1, / 27

3 Overview Introduction Definitions Sandpile Groups of Cubes August 1, / 27

4 Overview Introduction Definitions Previous Results Sandpile Groups of Cubes August 1, / 27

5 Overview Introduction Definitions Previous Results Gröbner Basis Calculations Sandpile Groups of Cubes August 1, / 27

6 Overview Introduction Definitions Previous Results Gröbner Basis Calculations A Bound on the Largest Cyclic Factor Size Sandpile Groups of Cubes August 1, / 27

7 Overview Introduction Definitions Previous Results Gröbner Basis Calculations A Bound on the Largest Cyclic Factor Size Analogous Bounds on Other Cayley Graphs Sandpile Groups of Cubes August 1, / 27

8 Overview Introduction Definitions Previous Results Gröbner 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, / 27

9 Introduction Definitions Definition The n-cube is the graph Q n with V (Q n ) = (Z/2Z) 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, / 27

10 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 L(Q 1 ) = ( 1 ) L(Q 2 ) = Sandpile Groups of Cubes August 1, / 27

11 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, / 27

12 Introduction Previous Results I Theorem [Bai] For every prime p > 2, ( n ) Syl p (K(Q n )) = Syl p (Z/kZ) (n k). k=1 Sandpile Groups of Cubes August 1, / 27

13 Introduction Previous Results I Theorem [Bai] For every prime p > 2, ( n ) Syl p (K(Q n )) = Syl p (Z/kZ) (n k). k=1 Remark To understand K(Q n ), it then remains to understand Syl 2 (K(Q n )). Sandpile Groups of Cubes August 1, / 27

14 Introduction Previous Results II Lemma [Benkart, Klivans, Reiner] For every u (Z/2Z) n, let χ u Z 2n be the vector with entry in position v (Z/2Z) 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, / 27

15 Introduction Previous Results II Lemma [Benkart, Klivans, Reiner] For every u (Z/2Z) n, let χ u Z 2n be the vector with entry in position v (Z/2Z) 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 2n Q 2n. When considering it as a map Z 2n Z 2n, this leaves us with the task of understanding the Z-torsion. Sandpile Groups of Cubes August 1, / 27

16 Introduction Previous Results II Lemma [Benkart, Klivans, Reiner] For every u (Z/2Z) n, let χ u Z 2n be the vector with entry in position v (Z/2Z) 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 2n Q 2n. When considering it as a map Z 2n Z 2n, 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, / 27

17 Gröbner Basis Calculations Gröbner 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 n αn of R. From now on, we implicitly assume a monomial order < on R. Sandpile Groups of Cubes August 1, / 27

18 Gröbner Basis Calculations Gröbner 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 n αn of R. From now on, we implicitly assume a monomial order < on R. Notation Let I [n]. We write x I := i I x i. Sandpile Groups of Cubes August 1, / 27

19 Gröbner Basis Calculations Gröbner 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 n αn of R. From now on, we implicitly assume a monomial order < 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 lt(f ), is the term of f greatest with respect to <. Sandpile Groups of Cubes August 1, / 27

20 Gröbner Basis Calculations Gröbner Basis Background Definition Let I R be an ideal. Then the leading term ideal of I is LT(I ) = ({lt(f ) f I }). Sandpile Groups of Cubes August 1, / 27

21 Gröbner Basis Calculations Gröbner Basis Background Definition Let I R be an ideal. Then the leading term ideal of I is LT(I ) = ({lt(f ) f I }). Definition Let I R an ideal. A Gröbner 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 lt(f ) = c 1 lt(g 1 ) + + c k lt(g k ) for some c i R. LT (I ) = (lt(g 1 ),..., lt(g k )). Sandpile Groups of Cubes August 1, / 27

22 Gröbner Basis Calculations Gröbner Basis Background Definition Let I R be an ideal. Then the leading term ideal of I is LT(I ) = ({lt(f ) f I }). Definition Let I R an ideal. A Gröbner 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 lt(f ) = c 1 lt(g 1 ) + + c k lt(g k ) for some c i R. LT (I ) = (lt(g 1 ),..., lt(g k )). Theorem When T is a PID, every ideal I R has a Gröbner basis. Sandpile Groups of Cubes August 1, / 27

23 Gröbner 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, / 27

24 Gröbner 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 Gröbner basis for the ideal in Z[x 1,..., x n ]. I n := (x 2 1 1,..., x 2 n 1, n x i ) Sandpile Groups of Cubes August 1, / 27

25 Gröbner 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 Gröbner 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 Gröbner basis is very complicated. Sandpile Groups of Cubes August 1, / 27

26 Gröbner Basis Calculations Relevance to Our Situation Lemma Let J n denote the ideal (x 2 1 1,..., x 2 n 1, n x i ) in Z/2 i Z[x 1,..., x n ]. Then the factors of Z/2Z,..., 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, / 27

27 Gröbner Basis Calculations Relevance to Our Situation Lemma Let J n denote the ideal (x 2 1 1,..., x 2 n 1, n x i ) in Z/2 i Z[x 1,..., x n ]. Then the factors of Z/2Z,..., 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 Gröbner basis of J n for i = 2, and thus understand the number of Z/2Z-factors in Syl 2 K(Q n ). Sandpile Groups of Cubes August 1, / 27

28 Gröbner Basis Calculations The Case i = 2 Conjecture For every odd integer m, let W m = {(2 + ɛ 2, 4 + ɛ 4,..., m 3 + ɛ m 3, m 1, m) ɛ i {0, 1}}. Sandpile Groups of Cubes August 1, / 27

29 Gröbner Basis Calculations The Case i = 2 Conjecture For every odd integer m, let W m = {(2 + ɛ 2, 4 + ɛ 4,..., m 3 + ɛ m 3, m 1, m) ɛ i {0, 1}}. Then LT (J n ) = (x 1 ) + (x2 2,..., xn) 2 + (2x I ). m n I W m m odd Sandpile Groups of Cubes August 1, / 27

30 A Bound on the Largest Cyclic Factor Size An Observation Observation The highest cyclic factor has size equal to the highest additive order of an element in K(Q n ) = Z[x 1, x 2,..., x n ]/(x 2 1 1,..., x 2 n 1, n x 1 x 2... x n ) Sandpile Groups of Cubes August 1, / 27

31 A Bound on the Largest Cyclic Factor Size An Observation Observation The highest cyclic factor has size equal to the highest additive order of an element in K(Q n ) = Z[x 1, x 2,..., x n ]/(x 2 1 1,..., x 2 n 1, n x 1 x 2... x n ) Lemma The elements x i 1 have highest additive order in K(Q n ) for all i {1,..., n}. Sandpile Groups of Cubes August 1, / 27

32 A Bound on the Largest Cyclic Factor Size An Observation Proof Outline Show a polynomial has a multiple in I n only if it has the form f (x 1,..., x n ) = c I (x I 1) I [n] Sandpile Groups of Cubes August 1, / 27

33 A Bound on the Largest Cyclic Factor Size An Observation Proof Outline Show a polynomial has a multiple in I n only if it has the form f (x 1,..., x n ) = c I (x I 1) I [n] Show x I 1 has a multiple in I n for every I [n]. Sandpile Groups of Cubes August 1, / 27

34 A Bound on the Largest Cyclic Factor Size An Observation Proof Outline Show a polynomial has a multiple in I n only if it has the form f (x 1,..., x n ) = c I (x I 1) I [n] Show x I 1 has a multiple in I n for every I [n]. Show ord(x i 1) ord(x I 1) for any I [n]. Sandpile Groups of Cubes August 1, / 27

35 A Bound on the Largest Cyclic Factor Size The Order of x 1 1 We switch back to Q 2n : 1 1 x Sandpile Groups of Cubes August 1, / 27

36 A Bound on the Largest Cyclic Factor Size The Order of x 1 1 We want to find the smallest C such that v Z 2n satisfying C C L(Q n ) v = 0. 0 Sandpile Groups of Cubes August 1, / 27

37 A Bound on the Largest Cyclic Factor Size The Order of x 1 1 We want to find the smallest C such that v Z 2n satisfying Idea: Work in the χ u -basis! C C L(Q n ) v = 0. 0 Sandpile Groups of Cubes August 1, / 27

38 A Bound on the Largest Cyclic Factor Size The Order of x 1 1 Both terms are nice: L(Q n ) n n n n 1 Sandpile Groups of Cubes August 1, / 27

39 A Bound on the Largest Cyclic Factor Size The Order of x 1 1 We now have the χ u -coordinates of v: v ( n n n2 n ) T Sandpile Groups of Cubes August 1, / 27

40 A Bound on the Largest Cyclic Factor Size The Order of x 1 1 We now have the χ u -coordinates of v: v ( n n n2 n ) T Theorem The order of x 1 1 is 2 n LCM(1, 2,..., n) Sandpile Groups of Cubes August 1, / 27

41 A Bound on the Largest Cyclic Factor Size The Order of x 1 1 We now have the χ u -coordinates of v: v ( n n n2 n ) T Theorem The order of x 1 1 is 2 n LCM(1, 2,..., n) Corollary The size of the largest cyclic factor in Syl 2 (K(Q n )) is 2 n+log 2 n Sandpile Groups of Cubes August 1, / 27

42 Analogous Bounds on Other Cayley Graphs Other Cayley Graphs Goal Generalize the technique used for the cube graph to other Cayley graphs. Sandpile Groups of Cubes August 1, / 27

43 Analogous Bounds on Other Cayley Graphs Other Cayley Graphs Goal Generalize the technique used for the cube graph to other Cayley graphs. Key Theorem [Benkart, Klivans, Reiner] Let G be the n-th power of a directed cycle of size k. Then K(G) = Z[x 1,..., x n ]/(x k 1 1,..., x k n 1, n x i ). Sandpile Groups of Cubes August 1, / 27

44 Analogous Bounds on Other Cayley Graphs Other Cayley Graphs Goal Generalize the technique used for the cube graph to other Cayley graphs. Key Theorem [Benkart, Klivans, Reiner] Let G be the n-th power of a directed cycle of size k. Then K(G) = Z[x 1,..., x n ]/(x k 1 1,..., x k n 1, n x i ). Lemma [Benkart, Klivans, Reiner] For every u (Z/kZ) n, let χ u Z kn be the vector with entry in position v (Z/kZ) n equal to ζk u v. Then χ u is an eigenvector of L(G) with eigenvalue k wt(u), where wt(u) is the number of non-zero entries in u. Sandpile Groups of Cubes August 1, / 27

45 Analogous Bounds on Other Cayley Graphs Generalize x 1 1 Lemma As before, x i 1 has maximal order in K(G) for all i {1,..., n}. Sandpile Groups of Cubes August 1, / 27

46 Analogous Bounds on Other Cayley Graphs Generalize x 1 1 Lemma As before, x i 1 has maximal order in K(G) for all i {1,..., n}. Remark However, x i 1 does not have a nice form in the χ u -basis. So we must find another high-order term with a nice form. One such element is (k 1) x i x 2 i x k 1 i. Sandpile Groups of Cubes August 1, / 27

47 Analogous Bounds on Other Cayley Graphs Generalize x 1 1 Lemma As before, x i 1 has maximal order in K(G) for all i {1,..., n}. Remark However, x i 1 does not have a nice form in the χ u -basis. So we must find another high-order term with a nice form. One such element is (k 1) x i x 2 i x k 1 i. Lemma ( ) k ord (k 1) x i xi 2 x k 1 i = ord(x i 1). Sandpile Groups of Cubes August 1, / 27

48 Analogous Bounds on Other Cayley Graphs Form in χ u -basis The form for (k 1) x i x 2 i x k 1 i in the χ u -basis is as follows: k k n. 1 k n 0 1 k n. 1 k n.. Sandpile Groups of Cubes August 1, / 27

49 Analogous Bounds on Other Cayley Graphs Bounds for k = 3, 4 Theorem (k = 3) Let k = 3. Then the size of the largest cyclic factor of Syl 3 (K(G)) is 3 n+1+ log 3 (n). Theorem (k = 4) Let k = 4. Then the size of the largest cyclic factor of Syl 2 (K(G)) is 4 n+1+ log 4 (n). Sandpile Groups of Cubes August 1, / 27

50 Higher Critical Groups A Different Viewpoint Set C 1 (G), C 0 (G) to be formal groups of Z-linear combinations of the edges and vertices of G respectively. Sandpile Groups of Cubes August 1, / 27

51 Higher Critical Groups A Different Viewpoint Set C 1 (G), C 0 (G) to be formal groups of Z-linear combinations of the edges and vertices of G respectively. There is a chain complex 0 C 1 (G) E C 0 (G) ɛ Z 0 where E is the incidence matrix of G and ɛ( n i v i ) = n i is the augmentation map. Sandpile Groups of Cubes August 1, / 27

52 Higher Critical Groups A Different Viewpoint Set C 1 (G), C 0 (G) to be formal groups of Z-linear combinations of the edges and vertices of G respectively. There is a chain complex 0 C 1 (G) E C 0 (G) ɛ Z 0 where E is the incidence matrix of G and ɛ( n i v i ) = n i is the augmentation map. Lemma L(G) = EE T and K(G) = ker(ɛ)/im(l(g)) = ker(ɛ)/im(ee T ) Sandpile Groups of Cubes August 1, / 27

53 Higher Critical Groups Extension to Cell Complexes Fix a cell complex X. There is a cellular chain complex... C i (X ) i C i 1 (X )... C 1 (X ) 1 C 0 (X ) ɛ Z 0 Sandpile Groups of Cubes August 1, / 27

54 Higher Critical Groups Extension to Cell Complexes Fix a cell complex X. There is a cellular chain complex... C i (X ) i C i 1 (X )... C 1 (X ) 1 C 0 (X ) ɛ Z 0 Definition The i-th critical group of X is K i (X ) = ker( i )/Im( i+1 T i+1 ) Sandpile Groups of Cubes August 1, / 27

55 Higher Critical Groups Extension to Cell Complexes Fix a cell complex X. There is a cellular chain complex... C i (X ) i C i 1 (X )... C 1 (X ) 1 C 0 (X ) ɛ Z 0 Definition The i-th critical group of X is K i (X ) = ker( i )/Im( i+1 T i+1 ) Related to cellular spannng trees, higher-dimensional dynamical systems on X. Sandpile Groups of Cubes August 1, / 27

56 Higher Critical Groups Initial Results We have an extension of Bai s Theorem: Theorem For any prime p > 2, Syl p (K i (Q n )) Syl p n (Z/jZ) (n j)( j 1 i ) j=i+1 Sandpile Groups of Cubes August 1, / 27

57 Higher Critical Groups Initial Results We have an extension of Bai s Theorem: Theorem For any prime p > 2, Proof Outline Syl p (K i (Q n )) Syl p n (Z/jZ) (n j)( j 1 i ) j=i+1 Can show i+1 T i+1 + T i i = L(Q n i ) (n i). Sandpile Groups of Cubes August 1, / 27

58 Higher Critical Groups Initial Results We have an extension of Bai s Theorem: Theorem For any prime p > 2, Proof Outline Syl p (K i (Q n )) Syl p n (Z/jZ) (n j)( j 1 i ) j=i+1 Can show i+1 T i+1 + T i i = L(Q n i ) (n i). i+1 T i+1 and T i i are diagonalizable and commute, so they have the same eigenvectors. Sandpile Groups of Cubes August 1, / 27

59 Further Directions Further Directions Further Directions A lower bound on the top cyclic factor: Examine minors of L(Q n )? Top cyclic factor bounds on K s1 K s2... K sn. Extend the top cyclic factor bound to higher critical groups. Sandpile Groups of Cubes August 1, / 27

60 Conclusion Acknowledgments Acknowledgments We would like to acknowledge support from NSF RTG grant DMS as well as the UMN Twin Cities 2016 Math REU. Special thanks to Vic Reiner for his mentorship and advice, as well as to Will Grodzicki for helpful comments. Sandpile Groups of Cubes August 1, / 27

61 Conclusion Acknowledgments Acknowledgments We would like to acknowledge support from NSF RTG grant DMS as well as the UMN Twin Cities 2016 Math REU. Special thanks to Vic Reiner for his mentorship and advice, as well as to Will Grodzicki for helpful comments. Questions? Sandpile Groups of Cubes August 1, / 27

On the critical group of the n-cube

Linear Algebra and its Applications 369 003 51 61 wwwelseviercom/locate/laa On the critical group of the n-cube Hua Bai School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Received