Smith Normal Form and Combinatorics

Transcription

1 Smith Normal Form and Combinatorics Richard P. Stanley June 8, 2017

2 Smith normal form A: n n matrix over commutative ring R (with 1) Suppose there exist P,Q GL(n,R) such that PAQ := B = diag(d 1,d 1 d 2,...,d 1 d 2 d n ), where d i R. We then call B a Smith normal form (SNF) of A.

3 Smith normal form A: n n matrix over commutative ring R (with 1) Suppose there exist P,Q GL(n,R) such that PAQ := B = diag(d 1,d 1 d 2,...,d 1 d 2 d n ), where d i R. We then call B a Smith normal form (SNF) of A. Note. (1) Can extend to m n. (2) unit det(a) = det(b) = d n 1 dn 1 2 d n. Thus SNF is a refinement of det.

4 Who is Smith? Henry John Stephen Smith

5 Who is Smith? Henry John Stephen Smith born 2 November 1826 in Dublin, Ireland

6 Who is Smith? Henry John Stephen Smith born 2 November 1826 in Dublin, Ireland educated at Oxford University (England)

7 Who is Smith? Henry John Stephen Smith born 2 November 1826 in Dublin, Ireland educated at Oxford University (England) remained at Oxford throughout his career

8 Who is Smith? Henry John Stephen Smith born 2 November 1826 in Dublin, Ireland educated at Oxford University (England) remained at Oxford throughout his career twice president of London Mathematical Society

9 Who is Smith? Henry John Stephen Smith born 2 November 1826 in Dublin, Ireland educated at Oxford University (England) remained at Oxford throughout his career twice president of London Mathematical Society 1861: SNF paper in Phil. Trans. R. Soc. London

10 Who is Smith? Henry John Stephen Smith born 2 November 1826 in Dublin, Ireland educated at Oxford University (England) remained at Oxford throughout his career twice president of London Mathematical Society 1861: SNF paper in Phil. Trans. R. Soc. London 1868: Steiner Prize of Royal Academy of Sciences of Berlin

11 Who is Smith? Henry John Stephen Smith born 2 November 1826 in Dublin, Ireland educated at Oxford University (England) remained at Oxford throughout his career twice president of London Mathematical Society 1861: SNF paper in Phil. Trans. R. Soc. London 1868: Steiner Prize of Royal Academy of Sciences of Berlin died 9 February 1883 (age 56)

12 Who is Smith? Henry John Stephen Smith born 2 November 1826 in Dublin, Ireland educated at Oxford University (England) remained at Oxford throughout his career twice president of London Mathematical Society 1861: SNF paper in Phil. Trans. R. Soc. London 1868: Steiner Prize of Royal Academy of Sciences of Berlin died 9 February 1883 (age 56) April 1883: shared Grand prix des sciences mathématiques with Minkowski

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14 Row and column operations Can put a matrix into SNF by the following operations. Add a multiple of a row to another row. Add a multiple of a column to another column. Multiply a row or column by a unit in R.

15 Row and column operations Can put a matrix into SNF by the following operations. Add a multiple of a row to another row. Add a multiple of a column to another column. Multiply a row or column by a unit in R. Over a field, SNF is row reduced echelon form (with all unit entries equal to 1).

16 Existence of SNF PIR: principal ideal ring, e.g., Z, K[x], Z/mZ. Theorem (Smith, for Z). If R is a PIR then A has a unique SNF up to units.

17 Existence of SNF PIR: principal ideal ring, e.g., Z, K[x], Z/mZ. Theorem (Smith, for Z). If R is a PIR then A has a unique SNF up to units. Otherwise A typically does not have a SNF but may have one in special cases.

18 Algebraic interpretation of SNF R: a PID A: an n n matrix over R with rows v 1,...,v n R n diag(e 1,e 2,...,e n ): SNF of A

19 Algebraic interpretation of SNF R: a PID A: an n n matrix over R with rows v 1,...,v n R n diag(e 1,e 2,...,e n ): SNF of A Theorem. R n /(v 1,...,v n ) = (R/e 1 R) (R/e n R).

20 Algebraic interpretation of SNF R: a PID A: an n n matrix over R with rows v 1,...,v n R n diag(e 1,e 2,...,e n ): SNF of A Theorem. R n /(v 1,...,v n ) = (R/e 1 R) (R/e n R). R n /(v 1,...,v n ): (Kasteleyn) cokernel of A

21 An explicit formula for SNF R: a PID A: an n n matrix over R with det(a) 0 diag(e 1,e 2,...,e n ): SNF of A

22 An explicit formula for SNF R: a PID A: an n n matrix over R with det(a) 0 diag(e 1,e 2,...,e n ): SNF of A Theorem. e 1 e 2 e i is the gcd of all i i minors of A. minor: determinant of a square submatrix

23 An explicit formula for SNF R: a PID A: an n n matrix over R with det(a) 0 diag(e 1,e 2,...,e n ): SNF of A Theorem. e 1 e 2 e i is the gcd of all i i minors of A. minor: determinant of a square submatrix Special case: e 1 is the gcd of all entries of A.

24 Laplacian matrices L(G): Laplacian matrix of the graph G rows and columns indexed by vertices of G { #(edges uv), u v L(G) uv = deg(u), u = v.

25 Laplacian matrices L(G): Laplacian matrix of the graph G rows and columns indexed by vertices of G { #(edges uv), u v L(G) uv = deg(u), u = v. reduced Laplacian matrix L 0 (G): for some vertex v, remove from L(G) the row and column indexed by v

26 Matrix-tree theorem Matrix-tree theorem. detl 0 (G) = κ(g), the number of spanning trees of G.

27 Matrix-tree theorem Matrix-tree theorem. detl 0 (G) = κ(g), the number of spanning trees of G. In general, SNF of L 0 (G) not understood.

28 Matrix-tree theorem Matrix-tree theorem. detl 0 (G) = κ(g), the number of spanning trees of G. In general, SNF of L 0 (G) not understood. L 0 (G) snf diag(e 1,...,e n 1 ) L(G) snf diag(e 1,...,e n 1,0)

29 Matrix-tree theorem Matrix-tree theorem. detl 0 (G) = κ(g), the number of spanning trees of G. In general, SNF of L 0 (G) not understood. L 0 (G) snf diag(e 1,...,e n 1 ) L(G) snf diag(e 1,...,e n 1,0) Applications to sandpile models, chip firing, etc.

30 An example Reduced Laplacian matrix of K 4 : A =

31 An example Reduced Laplacian matrix of K 4 : A = Matrix-tree theorem = det(a) = 16, the number of spanning trees of K 4.

32 An example Reduced Laplacian matrix of K 4 : A = Matrix-tree theorem = det(a) = 16, the number of spanning trees of K 4. What about SNF?

33 An example (continued)

34 Reduced Laplacian matrix of K n L 0 (K n ) = ni n 1 J n 1 detl 0 (K n ) = n n 2

35 Reduced Laplacian matrix of K n L 0 (K n ) = ni n 1 J n 1 detl 0 (K n ) = n n 2 Theorem. L 0 (K n ) SNF diag(1,n,n,...,n), a refinement of Cayley s theorem that κ(k n ) = n n 2.

36 Proof that L 0 (K n ) SNF diag(1, n, n,...,n) Trick: 2 2 submatrices (up to row and column permutations): [ n n 1 ], [ n ], [ with determinants n(n 2), n, and 0. Hence e 1 e 2 = n. Since ei = n n 2 and e i e i+1, we get the SNF diag(1,n,n,...,n). ],

37 Chip firing Abelian sandpile: a finite collection σ of indistinguishable chips distributed among the vertices V of a (finite) connected graph. Equivalently, σ: V {0,1,2,...}.

38 Chip firing Abelian sandpile: a finite collection σ of indistinguishable chips distributed among the vertices V of a (finite) connected graph. Equivalently, σ: V {0,1,2,...}. toppling of a vertex v: if σ(v) deg(v), then send a chip to each neighboring vertex

39 The sandpile group Choose a vertex to be a sink, and ignore chips falling into the sink. stable configuration: no vertex can topple Theorem (easy). After finitely many topples a stable configuration will be reached, which is independent of the order of topples.

40 The monoid of stable configurations Define a commutative monoid M on the stable configurations by vertex-wise addition followed by stabilization. ideal of M: subset J M satisfying σj J for all σ M

41 The monoid of stable configurations Define a commutative monoid M on the stable configurations by vertex-wise addition followed by stabilization. ideal of M: subset J M satisfying σj J for all σ M Exercise. The (unique) minimal ideal of a finite commutative monoid is a group.

42 Sandpile group sandpile group of G: the minimal ideal K(G) of the monoid M Fact. K(G) is independent of the choice of sink up to isomorphism.

43 Sandpile group sandpile group of G: the minimal ideal K(G) of the monoid M Fact. K(G) is independent of the choice of sink up to isomorphism. Theorem. Let L 0 (G) SNF diag(e 1,...,e n 1 ). Then K(G) = Z/e 1 Z Z/e n 1 Z.

44 The n-cube C n : graph of the n-cube

45 The n-cube C n : graph of the n-cube

46 The 4-cube

47 The 4-cube

48 The 4-cube C 4

49 An open problem n κ(c n ) = 2 2n n 1 i (n i) i=1

50 An open problem κ(c n ) = 2 2n n 1 n i (n i) Easy to prove by linear algebra; combinatorial (but not bijective) proof by O. Bernardi, i=1

51 An open problem κ(c n ) = 2 2n n 1 n i (n i) Easy to prove by linear algebra; combinatorial (but not bijective) proof by O. Bernardi, p-sylow subgroup of K(C n ) known for all odd primes p (Hua Bai, 2003) i=1

52 An open problem κ(c n ) = 2 2n n 1 n i (n i) Easy to prove by linear algebra; combinatorial (but not bijective) proof by O. Bernardi, p-sylow subgroup of K(C n ) known for all odd primes p (Hua Bai, 2003) i=1 2-Sylow subgroup of K(C n ) is unknown.

53 SNF of random matrices Huge literature on random matrices, mostly connected with eigenvalues. Very little work on SNF of random matrices over a PID.

54 Is the question interesting? Mat k (n): all n n Z-matrices with entries in [ k,k] (uniform distribution) p k (n, d): probability that if M Mat k (n) and SNF(M) = (e 1,...,e n ), then e 1 = d.

55 Is the question interesting? Mat k (n): all n n Z-matrices with entries in [ k,k] (uniform distribution) p k (n, d): probability that if M Mat k (n) and SNF(M) = (e 1,...,e n ), then e 1 = d. Recall: e 1 = gcd of 1 1 minors (entries) of M

56 Is the question interesting? Mat k (n): all n n Z-matrices with entries in [ k,k] (uniform distribution) p k (n, d): probability that if M Mat k (n) and SNF(M) = (e 1,...,e n ), then e 1 = d. Recall: e 1 = gcd of 1 1 minors (entries) of M Theorem. lim k p k (n,d) = 1 d n2 ζ(n 2 )

57 Specifying some e i with Yinghui Wang

58 Specifying some e i with Yinghui Wang ( )

59 Specifying some e i with Yinghui Wang ( ) Two general results. Let α 1,...,α n 1 P, α i α i+1. µ k (n): probability that the SNF of a random A Mat k (n) satisfies e i = α i for 1 α i n 1. µ(n) = lim k µ k(n). Then µ(n) exists, and 0 < µ(n) < 1.

60 Second result Let α n P. ν k (n): probability that the SNF of a random A Mat k (n) satisfies e n = α n. Then lim ν k(n) = 0. k

61 Sample result µ k (n): probability that the SNF of a random A Mat k (n) satisfies e 1 = 2, e 2 = 6. µ(n) = lim k µ k(n).

62 Conclusion µ(n) = 2 n2 1 n(n 1) i=(n 1) 2 2 i + n 2 1 i=n(n 1) (n 1)2 (1 3 (n 1)2 )(1 3 n ) 2 n(n 1) n p i + p>3 i=(n 1) 2 i=n(n 1)+1 2 i p i.

63 Cyclic cokernel κ(n): probability that an n n Z-matrix has SNF diag(e 1,e 2,...,e n ) with e 1 = e 2 = = e n 1 = 1

64 Cyclic cokernel κ(n): probability that an n n Z-matrix has SNF diag(e 1,e 2,...,e n ) with e 1 = e 2 = = e n 1 = 1 (1+ 1p 2 + 1p p ) n Theorem. κ(n) = p ζ(2)ζ(3)

65 Cyclic cokernel κ(n): probability that an n n Z-matrix has SNF diag(e 1,e 2,...,e n ) with e 1 = e 2 = = e n 1 = 1 (1+ 1p 2 + 1p p ) n Theorem. κ(n) = p ζ(2)ζ(3) Corollary. lim κ(n) = 1 n ζ(6) j 4 ζ(j)

66 Small number of generators g: number of generators of cokernel (number of entries of SNF 1) as n previous slide: Prob(g = 1) =

67 Small number of generators g: number of generators of cokernel (number of entries of SNF 1) as n previous slide: Prob(g = 1) = Prob(g 2) =

68 Small number of generators g: number of generators of cokernel (number of entries of SNF 1) as n previous slide: Prob(g = 1) = Prob(g 2) = Prob(g 3) =

69 Small number of generators g: number of generators of cokernel (number of entries of SNF 1) as n previous slide: Prob(g = 1) = Prob(g 2) = Prob(g 3) = Theorem. Prob(g l) = 1 ( )2 (l+1)2 (1+O(2 l ))

70 Small number of generators g: number of generators of cokernel (number of entries of SNF 1) as n previous slide: Prob(g = 1) = Prob(g 2) = Prob(g 3) = Theorem. Prob(g l) = 1 ( )2 (l+1)2 (1+O(2 l ))

71 = 1 (1 12 ) j j 1

72 Example of SNF computation λ: a partition (λ 1,λ 2,...), identified with its Young diagram (3,1)

73 Example of SNF computation λ: a partition (λ 1,λ 2,...), identified with its Young diagram (3,1) λ : λ extended by a border strip along its entire boundary

74 Example of SNF computation λ: a partition (λ 1,λ 2,...), identified with its Young diagram (3,1) λ : λ extended by a border strip along its entire boundary (3,1)* = (4,4,2)

75 Initialization Insert 1 into each square of λ /λ (3,1)* = (4,4,2)

76 M t Let t λ. Let M t be the largest square of λ with t as the upper left-hand corner.

77 M t Let t λ. Let M t be the largest square of λ with t as the upper left-hand corner. t

78 M t Let t λ. Let M t be the largest square of λ with t as the upper left-hand corner. t

79 Determinantal algorithm Suppose all squares to the southeast of t have been filled. Insert into t the number n t so that detm t = 1.

80 Determinantal algorithm Suppose all squares to the southeast of t have been filled. Insert into t the number n t so that detm t =

81 Determinantal algorithm Suppose all squares to the southeast of t have been filled. Insert into t the number n t so that detm t =

82 Determinantal algorithm Suppose all squares to the southeast of t have been filled. Insert into t the number n t so that detm t =

83 Determinantal algorithm Suppose all squares to the southeast of t have been filled. Insert into t the number n t so that detm t =

84 Determinantal algorithm Suppose all squares to the southeast of t have been filled. Insert into t the number n t so that detm t =

85 Determinantal algorithm Suppose all squares to the southeast of t have been filled. Insert into t the number n t so that detm t =

86 Uniqueness Easy to see: the numbers n t are well-defined and unique.

87 Uniqueness Easy to see: the numbers n t are well-defined and unique. Why? Expand detm t by the first row. The coefficient of n t is 1 by induction.

88 λ(t) If t λ, let λ(t) consist of all squares of λ to the southeast of t.

89 λ(t) If t λ, let λ(t) consist of all squares of λ to the southeast of t. t λ = (4,4,3)

90 λ(t) If t λ, let λ(t) consist of all squares of λ to the southeast of t. t λ = (4,4,3) λ( t ) = (3,2)

91 u λ u λ = #{µ : µ λ}

92 u λ u λ = #{µ : µ λ} Example. u (2,1) = 5: φ

93 u λ u λ = #{µ : µ λ} Example. u (2,1) = 5: φ There is a determinantal formula for u λ, due essentially to MacMahon and later Kreweras (not needed here).

94 Carlitz-Scoville-Roselle theorem Berlekamp (1963) first asked for n t (mod 2) in connection with a coding theory problem. Carlitz-Roselle-Scoville (1971): combinatorial interpretation of n t (over Z).

95 Carlitz-Scoville-Roselle theorem Berlekamp (1963) first asked for n t (mod 2) in connection with a coding theory problem. Carlitz-Roselle-Scoville (1971): combinatorial interpretation of n t (over Z). Theorem. n t = u λ(t)

96 Carlitz-Scoville-Roselle theorem Berlekamp (1963) first asked for n t (mod 2) in connection with a coding theory problem. Carlitz-Roselle-Scoville (1971): combinatorial interpretation of n t (over Z). Theorem. n t = u λ(t) Proofs. 1. Induction (row and column operations). 2. Nonintersecting lattice paths.

97 An example

98 An example φ

99 A q-analogue Weight each µ λ by q λ/µ.

100 A q-analogue Weight each µ λ by q λ/µ. λ = 64431, µ = 42211, q λ/µ = q 8

101 u λ (q) u λ (q) = µ λ q λ/µ u (2,1) (q) = 1+2q +q 2 +q 3 :

102 Diagonal hooks d i (λ) = λ i +λ i 2i +1 d 1 = 9, d 2 = 4, d 3 = 1

103 Main result (with C. Bessenrodt) Theorem. M t has an SNF over Z[q]. Write d i = d i (λ t ). If M t is a (k +1) (k +1) matrix then M t has SNF diag(1,q d k,q d k 1+d k,...,q d 1+d 2 + +d k ).

104 Main result (with C. Bessenrodt) Theorem. M t has an SNF over Z[q]. Write d i = d i (λ t ). If M t is a (k +1) (k +1) matrix then M t has SNF diag(1,q d k,q d k 1+d k,...,q d 1+d 2 + +d k ). Corollary. detm t = q id i.

105 Main result (with C. Bessenrodt) Theorem. M t has an SNF over Z[q]. Write d i = d i (λ t ). If M t is a (k +1) (k +1) matrix then M t has SNF diag(1,q d k,q d k 1+d k,...,q d 1+d 2 + +d k ). Corollary. detm t = q id i. Note. There is a multivariate generalization.

106 An example t λ = 6431, d 1 = 9, d 2 = 4, d 3 = 1

107 An example t λ = 6431, d 1 = 9, d 2 = 4, d 3 = 1 SNF of M t : (1,q,q 5,q 14 )

108 A special case Let λ be the staircase δ n = (n 1,n 2,...,1).

109 A special case Let λ be the staircase δ n = (n 1,n 2,...,1). u δn 1 (q) counts Dyck paths of length 2n by (scaled) area, and is thus the well-known q-analogue C n (q) of the Catalan number C n.

110 A q-catalan example C 3 (q) = q 3 +q 2 +2q +1

111 A q-catalan example C 4 (q) C 3 (q) 1+q C 3 (q) 1+q 1 1+q 1 1 since d 1 (3,2,1) = 1, d 2 (3,2,1) = 5. C 3 (q) = q 3 +q 2 +2q +1 SNF diag(1,q,q 6 )

112 A q-catalan example C 4 (q) C 3 (q) 1+q C 3 (q) 1+q 1 1+q 1 1 since d 1 (3,2,1) = 1, d 2 (3,2,1) = 5. C 3 (q) = q 3 +q 2 +2q +1 q-catalan determinant previously known SNF is new SNF diag(1,q,q 6 )

113 Ramanujan F(q, x) := n 0 C n (q)x n = x qx 1 q2 x 1

114 Ramanujan F(q, x) := n 0 C n (q)x n = x qx 1 q2 x 1 e 2π/5 F(e 2π, e 2π ) =

115 An open problem l(w): length (number of inversions) of w = a 1 a n S n, i.e., l(w) = #{(i,j) : i < j, w i > w j }. V(n): the n! n! matrix with rows and columns indexed by w S n, and V(n) uv = q l(uv 1).

116 n = 3 det 1 q q q 2 q 2 q 3 q 1 q 2 q q 3 q 2 q q 2 1 q 3 q q 2 q 2 q q 3 1 q 2 q q 2 q 3 q q 2 1 q q 3 q 2 q 2 q q 1 = (1 q 2 ) 6 (1 q 6 )

117 n = 3 det 1 q q q 2 q 2 q 3 q 1 q 2 q q 3 q 2 q q 2 1 q 3 q q 2 q 2 q q 3 1 q 2 q q 2 q 3 q q 2 1 q q 3 q 2 q 2 q q 1 = (1 q 2 ) 6 (1 q 6 ) V(3) snf diag(1,1 q 2,1 q 2,1 q 2,(1 q 2 ) 2,(1 q 2 )(1 q 6 ))

118 n = 3 det 1 q q q 2 q 2 q 3 q 1 q 2 q q 3 q 2 q q 2 1 q 3 q q 2 q 2 q q 3 1 q 2 q q 2 q 3 q q 2 1 q q 3 q 2 q 2 q q 1 = (1 q 2 ) 6 (1 q 6 ) V(3) snf diag(1,1 q 2,1 q 2,1 q 2,(1 q 2 ) 2,(1 q 2 )(1 q 6 )) special case of q-varchenko matrix of a real hyperplane arrangment

119 Zagier s theorem Theorem (D. Zagier, 1992) n detv(n) = (1 q j(j 1)) ( n j)(j 2)!(n j+1)! j=2

120 Zagier s theorem Theorem (D. Zagier, 1992) detv(n) = n (1 q j(j 1)) ( n j)(j 2)!(n j+1)! j=2 SNF is open. Partial result: Theorem (Denham-Hanlon, 1997) Let V(n) snf diag(e 1,e 2,...,e n! ). The number of e i s exactly divisible by (q 1) j (or by (q 2 1) j ) is the number c(n,n j) of w S n with n j cycles (signless Stirling number of the first kind).

121 The last slide

122 The last slide

123 The last slide