Smith Normal Form and Combinatorics

Smith Normal Form and Combinatorics p. 1 Smith Normal Form and Combinatorics Richard P. Stanley

Smith Normal Form and Combinatorics p. 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.

Smith Normal Form and Combinatorics p. 3 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.

Smith Normal Form and Combinatorics p. 4 Existence of SNF If R is a PID, such as Z or K[x] (K = field), then A has a unique SNF up to units.

Smith Normal Form and Combinatorics p. 4 Existence of SNF If R is a PID, such as Z or K[x] (K = field), then A has a unique SNF up to units. Otherwise A typically does not have a SNF but may have one in special cases.

Smith Normal Form and Combinatorics p. 5 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

Smith Normal Form and Combinatorics p. 5 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).

Smith Normal Form and Combinatorics p. 6 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

Smith Normal Form and Combinatorics p. 6 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.

Smith Normal Form and Combinatorics p. 7 An example Reduced Laplacian matrix of K 4 : 3 1 1 A = 1 3 1 1 1 3

Smith Normal Form and Combinatorics p. 7 An example Reduced Laplacian matrix of K 4 : 3 1 1 A = 1 3 1 1 1 3 Matrix-tree theorem = det(a) = 16, the number of spanning trees of K 4.

Smith Normal Form and Combinatorics p. 7 An example Reduced Laplacian matrix of K 4 : 3 1 1 A = 1 3 1 1 1 3 Matrix-tree theorem = det(a) = 16, the number of spanning trees of K 4. What about SNF?

Smith Normal Form and Combinatorics p. 8 An example (continued) 3 1 1 1 3 1 1 1 3 0 0 1 0 4 0 4 4 0 0 0 1 4 4 1 8 4 3 0 0 1 0 4 0 4 0 0 0 0 1 4 4 0 8 4 0 4 0 0 0 4 0 0 0 1

Smith Normal Form and Combinatorics p. 9 Laplacian matrices L 0 (G): reduced Laplacian matrix of the graph G Matrix-tree theorem. detl 0 (G) = κ(g), the number of spanning trees of G.

Smith Normal Form and Combinatorics p. 9 Laplacian matrices L 0 (G): reduced Laplacian matrix of the graph G Matrix-tree theorem. detl 0 (G) = κ(g), the number of spanning trees of G. Theorem. L 0 (K n ) SNF diag(1,n,n,...,n), a refinement of Cayley s theorem that κ(k n ) = n n 2.

Smith Normal Form and Combinatorics p. 10 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,...}.

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. 0 5 6 1 2 7 1 2 2 1 3 2 Smith Normal Form and Combinatorics p. 10

Smith Normal Form and Combinatorics p. 11 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.

Smith Normal Form and Combinatorics p. 12 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

Smith Normal Form and Combinatorics p. 13 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.

Smith Normal Form and Combinatorics p. 13 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 Then L 0 (G) SNF diag(e 1,...,e n 1 ). K(G) = Z/e 1 Z Z/e n 1 Z.

Smith Normal Form and Combinatorics p. 14 Second example Some matrices connected with Young diagrams

Smith Normal Form and Combinatorics p. 15 Extended Young diagrams λ: a partition (λ 1,λ 2,...), identified with its Young diagram (3,1)

Smith Normal Form and Combinatorics p. 15 Extended Young diagrams 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

Smith Normal Form and Combinatorics p. 15 Extended Young diagrams 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)

Smith Normal Form and Combinatorics p. 16 Initialization Insert 1 into each square of λ /λ. 1 1 1 1 1 1 (3,1)* = (4,4,2)

Smith Normal Form and Combinatorics p. 17 M t Let t λ. Let M t be the largest square of λ with t as the upper left-hand corner.

Smith Normal Form and Combinatorics p. 17 M t Let t λ. Let M t be the largest square of λ with t as the upper left-hand corner. t

Smith Normal Form and Combinatorics p. 18 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.

Smith Normal Form and Combinatorics p. 18 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. 1 1 1 1 1 1

Smith Normal Form and Combinatorics p. 18 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. 2 1 1 1 1 1 1

Smith Normal Form and Combinatorics p. 18 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. 2 1 2 1 1 1 1 1

Smith Normal Form and Combinatorics p. 18 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. 2 1 3 2 1 1 1 1 1

Smith Normal Form and Combinatorics p. 18 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. 5 2 1 3 2 1 1 1 1 1

Smith Normal Form and Combinatorics p. 18 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. 9 5 2 1 3 2 1 1 1 1 1

Smith Normal Form and Combinatorics p. 19 Uniqueness Easy to see: the numbers n t are well-defined and unique.

Smith Normal Form and Combinatorics p. 19 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.

Smith Normal Form and Combinatorics p. 20 λ(t) If t λ, let λ(t) consist of all squares of λ to the southeast of t.

Smith Normal Form and Combinatorics p. 20 λ(t) If t λ, let λ(t) consist of all squares of λ to the southeast of t. t λ = (4,4,3)

Smith Normal Form and Combinatorics p. 20 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)

Smith Normal Form and Combinatorics p. 21 u λ u λ = #{µ : µ λ}

Smith Normal Form and Combinatorics p. 21 u λ Example. u (2,1) = 5: u λ = #{µ : µ λ} φ

Smith Normal Form and Combinatorics p. 21 u λ Example. u (2,1) = 5: u λ = #{µ : µ λ} φ There is a determinantal formula for u λ, due essentially to MacMahon and later Kreweras (not needed here).

Smith Normal Form and Combinatorics p. 22 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 = f(λ(t)).

Smith Normal Form and Combinatorics p. 23 An example 7 3 2 1 2 1 1 1 1 1

Smith Normal Form and Combinatorics p. 23 An example 7 3 2 1 2 1 1 1 1 1 φ

Smith Normal Form and Combinatorics p. 24 Many indeterminates For each square (i,j) λ, associate an indeterminate x ij (matrix coordinates).

Smith Normal Form and Combinatorics p. 24 Many indeterminates For each square (i,j) λ, associate an indeterminate x ij (matrix coordinates). x x x x 11 12 13 x 21 22

Smith Normal Form and Combinatorics p. 25 A refinement of u λ u λ (x) = µ λ (i,j) λ/µ x ij

Smith Normal Form and Combinatorics p. 25 A refinement of u λ u λ (x) = µ λ (i,j) λ/µ x ij a b c c d e d e λ µ λ/µ (i,j) λ/µ x ij = cde

Smith Normal Form and Combinatorics p. 26 An example a d b e c abcde+bcde+bce+cde +ce+de+c+e+1 bce+ce+c +e+1 c+1 1 de+e+1 e+1 1 1 1 1 1

Smith Normal Form and Combinatorics p. 27 A t A t = (i,j) λ(t) x ij

Smith Normal Form and Combinatorics p. 27 A t A t = (i,j) λ(t) x ij t a b c d e f g h i j k l m n o

Smith Normal Form and Combinatorics p. 27 A t A t = (i,j) λ(t) x ij t a b c d e f g h i j k l m n o A t = bcdeghiklmo

Smith Normal Form and Combinatorics p. 28 The main theorem Theorem. Let t = (i,j). Then M t has SNF diag(a ij,a i 1,j 1,...,1).

Smith Normal Form and Combinatorics p. 28 The main theorem Theorem. Let t = (i,j). Then M t has SNF diag(a ij,a i 1,j 1,...,1). Proof. 1. Explicit row and column operations putting M t into SNF. 2. (C. Bessenrodt) Induction.

Smith Normal Form and Combinatorics p. 29 An example a d b e c abcde+bcde+bce+cde +ce+de+c+e+1 bce+ce+c +e+1 c+1 1 de+e+1 e+1 1 1 1 1 1

Smith Normal Form and Combinatorics p. 29 An example a d b e c abcde+bcde+bce+cde +ce+de+c+e+1 bce+ce+c +e+1 c+1 1 de+e+1 e+1 1 1 1 1 1 SNF = diag(abcde,e,1)

Smith Normal Form and Combinatorics p. 30 A special case Let λ be the staircase δ n = (n 1,n 2,...,1). Set each x ij = q.

Smith Normal Form and Combinatorics p. 30 A special case Let λ be the staircase δ n = (n 1,n 2,...,1). Set each x ij = q. u δn 1 (x) xij counts Dyck paths of length 2n by =q (scaled) area, and is thus the well-known q-analogue C n (q) of the Catalan number C n.

Smith Normal Form and Combinatorics p. 31 A q-catalan example C 3 (q) = q 3 +q 2 +2q +1

Smith Normal Form and Combinatorics p. 31 A q-catalan example C 3 (q) = q 3 +q 2 +2q +1 C 4 (q) C 3 (q) 1+q C 3 (q) 1+q 1 1+q 1 1 SNF diag(q 6,q,1)

Smith Normal Form and Combinatorics p. 31 A q-catalan example C 3 (q) = q 3 +q 2 +2q +1 C 4 (q) C 3 (q) 1+q C 3 (q) 1+q 1 1+q 1 1 SNF diag(q 6,q,1) q-catalan determinant previously known SNF is new

Smith Normal Form and Combinatorics p. 32 SNF of random matrices Huge literature on random matrices, mostly connected with eigenvalues. Very little work on SNF of random matrices over a PID.

Smith Normal Form and Combinatorics p. 33 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.

Work of Yinghui Wang Smith Normal Form and Combinatorics p. 34

Work of Yinghui Wang ( ) Smith Normal Form and Combinatorics p. 35

Smith Normal Form and Combinatorics p. 36 Work of Yinghui Wang ( ) 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).

Smith Normal Form and Combinatorics p. 37 Conclusion µ(n) = 2 n2 1 n(n 1) i=(n 1) 2 2 i + n 2 1 i=n(n 1)+1 3 (1 3 2 3 (n 1)2 (n 1)2 )(1 3 n ) 2 n(n 1) n 2 1 1 p i + p>3 i=(n 1) 2 i=n(n 1)+1 2 i p i.

Smith Normal Form and Combinatorics p. 38 A note on the proof uses a 2014 result of C. Feng, R. W. Nóbrega, F. R. Kschischang, and D. Silva, Communication over finite-chain-ring matrix channels: number of m n matrices over Z/p s Z with specified SNF

Smith Normal Form and Combinatorics p. 39 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.

Smith Normal Form and Combinatorics p. 39 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 + 1p + + 1p ) 2 3 n Theorem. κ(n) = p ζ(2)ζ(3)

Smith Normal Form and Combinatorics p. 40 Third example In collaboration with Tommy Wuxing Cai.

Smith Normal Form and Combinatorics p. 40 Third example In collaboration with.

Smith Normal Form and Combinatorics p. 40 Third example In collaboration with. Par(n): set of all partitions of n E.g., Par(4) = {4, 31, 22, 211, 1111}.

Smith Normal Form and Combinatorics p. 40 Third example In collaboration with. Par(n): set of all partitions of n E.g., Par(4) = {4, 31, 22, 211, 1111}. V n : real vector space with basis Par(n)

Smith Normal Form and Combinatorics p. 41 U Define U = U n : V n V n+1 by U(λ) = µ µ, where µ Par(n+1) and µ i λ i i. Example. U(42211) = 52211+43211+42221+422111

Smith Normal Form and Combinatorics p. 42 D Dually, define D = D n : V n V n 1 by D(λ) = ν ν, where ν Par(n 1) and ν i λ i i. Example. D(42211) = 32211+42111+4221

Smith Normal Form and Combinatorics p. 43 Symmetric functions NOTE. Identify V n with the space Λ n Q of all homogeneous symmetric functions of degree n over Q, and identify λ V n with the Schur function s λ. Then U(f) = p 1 f, D(f) = p 1 f.

Smith Normal Form and Combinatorics p. 44 Commutation relation Basic commutation relation: DU U D = I Allows computation of eigenvalues of DU: V n V n. Or note that the eigenvectors of p 1 p 1 are the p λ s, λ n.

Smith Normal Form and Combinatorics p. 45 Eigenvalues of DU Let p(n) = #Par(n) = dimv n. Theorem. Let 1 i n+1, i n. Then i is an eigenvalue of D n+1 U n with multiplicity p(n+1 i) p(n i). Hence detd n+1 U n = n+1 i=1 i p(n+1 i) p(n i).

Smith Normal Form and Combinatorics p. 46 Conjecture of A. R. Miller, 2005 Conjecture (first form). Let e 1,...,e p(n) be the eigenvalues of D n+1 U n. Then [D n+1 U n ] has the same SNF as diag(e 1,...,e p(n) ). Conjecture (second form). The diagonal entries of the SNF of [D n+1 U n ] are: (n+1)(n 1)!, with multiplicity 1 (n k)! with multiplicity p(k +1) 2p(k)+p(k 1), 3 k n 2 1, with multiplicity p(n) p(n 1)+p(n 2).

Smith Normal Form and Combinatorics p. 47 Not a trivial result NOTE. {p λ } λ n is not an integral basis.

Smith Normal Form and Combinatorics p. 48 Another form m 1 (λ): number of 1 s in λ M 1 (n): multiset of all numbers m 1 (λ)+1, λ Par(n) Let SNF of [D n+1 U n ] be diag(f 1,f 2,...,f p(n) ). Conjecture (third form). f 1 is the product of the distinct entries of M 1 (n); f 2 is the product of the remaining distinct entries of M 1 (n), etc.

Smith Normal Form and Combinatorics p. 49 An example: n = 6 Par(6) = {6,51,42,33,411,321,222,3111, 2211, 21111, 111111} M 1 (6) = {1,2,1,1,3,2,1,4,3,5,7} (f 1,...,f 11 ) = (7 5 4 3 2 1, 3 2 1, 1,1,1,1,1,1,1,1,1) = (840,6,1,1,1,1,1,1,1,1,1)

Smith Normal Form and Combinatorics p. 50 Yet another form Conjecture (fourth form). The matrix [D n+1 U n +xi] has an SNF over Z[x]. Note that Z[x] is not a PID.

Smith Normal Form and Combinatorics p. 51 Resolution of conjecture Theorem. The conjecture of Miller is true.

Smith Normal Form and Combinatorics p. 51 Resolution of conjecture Theorem. The conjecture of Miller is true. Proof (first step). Rather than use the basis {s λ } λ Par(n) (Schur functions) for Λ n Q, use the basis {h λ } λ Par(n) (complete symmetric functions). Since the two bases differ by a matrix in SL(p(n),Z), the SNF s stay the same.

Smith Normal Form and Combinatorics p. 52 Conclusion of proof (second step) Row and column operations.

Smith Normal Form and Combinatorics p. 52 Conclusion of proof (second step) Row and column operations. Not very insightful.

Smith Normal Form and Combinatorics p. 53 An unsolved conjecture m j (λ): number of j s in λ M j (n): multiset of all numbers j(m j (λ)+1), λ Par(n) p j : power sum symmetric function x j i Let SNF of the operator f j p j p j f with respect to the basis {s λ } be diag(g 1,g 2,...,g p(n) ). Conjecture.g 1 is the product of the distinct entries of M j (n); g 2 is the product of the remaining distinct entries of M j (n), etc.

Smith Normal Form and Combinatorics p. 54 Jacobi-Trudi specialization Jacobi-Trudi identity: s λ = det[h λi i+j], where s λ is a Schur function and h i is a complete symmetric function.

Smith Normal Form and Combinatorics p. 54 Jacobi-Trudi specialization Jacobi-Trudi identity: s λ = det[h λi i+j], where s λ is a Schur function and h i is a complete symmetric function. We consider the specialization x 1 = x 2 = = x n = 1, other x i = 0. Then ( ) n+i 1 h i. i

Smith Normal Form and Combinatorics p. 55 Specialized Schur function s λ u λ n+c(u) h(u). c(u): content of the square u 0 1 2 3 4 1 0 1 2 2 1 3 2 0 1

Smith Normal Form and Combinatorics p. 56 Diagonal hooks D 1,...,D m 0 1 2 3 4 1 0 1 2 2 1 3 2 0 1 λ = (5,4,4,2)

Smith Normal Form and Combinatorics p. 56 Diagonal hooks D 1,...,D m 0 1 2 3 4 1 0 1 2 2 1 3 2 0 1 D 1

Smith Normal Form and Combinatorics p. 56 Diagonal hooks D 1,...,D m 0 1 2 3 4 1 0 1 2 2 1 3 2 0 1 D 2

Smith Normal Form and Combinatorics p. 56 Diagonal hooks D 1,...,D m 0 1 2 3 4 1 0 1 2 2 1 3 2 0 1 D 3

Smith Normal Form and Combinatorics p. 57 SNF result Let SNF R = Q[n] [( )] n+λi i+j 1 λ i i+j = diag(e 1,...,e m ). Then e i = n+c(u). h(u) u D m i+1

Smith Normal Form and Combinatorics p. 58 Idea of proof f i = n+c(u) h(u) u D m i+1 Then f 1 f 2 ] f i is the value of the lower-left i i minor. (Special argument for 0 minors.)

Smith Normal Form and Combinatorics p. 58 Idea of proof f i = n+c(u) h(u) u D m i+1 Then f 1 f 2 ] f i is the value of the lower-left i i minor. (Special argument for 0 minors.) Every i i minor is a specialized skew Schur function s µ/ν. Let s α correspond to the lower left i i minor.

Smith Normal Form and Combinatorics p. 59 Conclusion of proof Let s µ/ν = ρ c µ νρs ρ. By Littlewood-Richardson rule, c µ νρ 0 α ρ.

Smith Normal Form and Combinatorics p. 59 Conclusion of proof Let s µ/ν = ρ c µ νρs ρ. By Littlewood-Richardson rule, c µ νρ 0 α ρ. Hence f i = gcd(i i minors) = e i e i 1.

The last slide Smith Normal Form and Combinatorics p. 60

The last slide Smith Normal Form and Combinatorics p. 61