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Smith Normal Form andCombinatorics

Richard P. Stanley

Smith Normal Form and Combinatorics – p. 1

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Smith Normal Form and Combinatorics – p. 2

A curious connection

Little known fact: perhaps the most influentialwork of Ira is related to

Smith Normal Form and Combinatorics – p. 3

A curious connection

Little known fact: perhaps the most influentialwork of Ira is related to

QUAINT CUNEIFORM MYSTICS

Smith Normal Form and Combinatorics – p. 3

A curious connection

Little known fact: perhaps the most influentialwork of Ira is related to

QUAINT CUNEIFORM MYSTICS1 2 3 5 17 11 14 16 22 10 13 15 21 12 8 9 7 6 19 20 18 4

Smith Normal Form and Combinatorics – p. 3

A curious connection

Little known fact: perhaps the most influentialwork of Ira is related to

QUAINT CUNEIFORM MYSTICS1 2 3 5 17 11 14 16 22 10 13 15 21 12 8 9 7 6 19 20 18 4

QUASISYMMETRIC FUNCTION

Smith Normal Form and Combinatorics – p. 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(d1, d1d2, . . . d1d2 · · · dn),

where di ∈ R. We then call B a Smith normalform (SNF) of A.

Smith Normal Form and Combinatorics – p. 4

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(d1, d1d2, . . . d1d2 · · · dn),

where di ∈ R. We then call B a Smith normalform (SNF) of A.

NOTE. (1) Can extend to m× n.

(2) unit · det(A) = det(B) = dn1dn−12 · · · dn.

Thus SNF is a refinement of det.

Smith Normal Form and Combinatorics – p. 4

Existence of SNF

If R is a principal ideal ring (PIR), such as Z orK[x] (K = field), then A has a unique SNF up tounits.

Smith Normal Form and Combinatorics – p. 5

Existence of SNF

If R is a principal ideal ring (PIR), such as Z orK[x] (K = field), then A has a unique SNF up tounits.

Otherwise A “typically” does not have a SNF butmay have one in special cases.

Smith Normal Form and Combinatorics – p. 5

Row and column operations

Over a principal ideal ring, can put a matrix intoSNF 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. 6

Row and column operations

Over a principal ideal ring, can put a matrix intoSNF 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).

Smith Normal Form and Combinatorics – p. 6

Algebraic interpretation of SNF

R: a PIR

A: an n× n matrix over R with rowsv1, . . . , vn ∈ Rn

diag(e1, e2, . . . , en): SNF of A

Smith Normal Form and Combinatorics – p. 7

Algebraic interpretation of SNF

R: a PIR

A: an n× n matrix over R with rowsv1, . . . , vn ∈ Rn

diag(e1, e2, . . . , en): SNF of A

Theorem.

Rn/(v1, . . . , vn) ∼= (R/e1R)⊕ · · · ⊕ (R/enR).

Smith Normal Form and Combinatorics – p. 7

Algebraic interpretation of SNF

R: a PIR

A: an n× n matrix over R with rowsv1, . . . , vn ∈ Rn

diag(e1, e2, . . . , en): SNF of A

Theorem.

Rn/(v1, . . . , vn) ∼= (R/e1R)⊕ · · · ⊕ (R/enR).

Rn/(v1, . . . , vn): (Kastelyn) cokernel of A

Smith Normal Form and Combinatorics – p. 7

An explicit formula for SNF

R: a PIR

A: an n× n matrix over R with det(A) 6= 0

diag(e1, e2, . . . , en): SNF of A

Smith Normal Form and Combinatorics – p. 8

An explicit formula for SNF

R: a PIR

A: an n× n matrix over R with det(A) 6= 0

diag(e1, e2, . . . , en): SNF of A

Theorem. e1e2 · · · ei is the gcd of all i× i minorsof A.

minor: determinant of a square submatrix.

Special case: e1 is the gcd of all entries of A.

Smith Normal Form and Combinatorics – p. 8

An example

Reduced Laplacian matrix of K4:

A =

3 −1 −1

−1 3 −1

−1 −1 3

Smith Normal Form and Combinatorics – p. 9

An example

Reduced Laplacian matrix of K4:

A =

3 −1 −1

−1 3 −1

−1 −1 3

Matrix-tree theorem =⇒ det(A) = 16, thenumber of spanning trees of K4.

Smith Normal Form and Combinatorics – p. 9

An example

Reduced Laplacian matrix of K4:

A =

3 −1 −1

−1 3 −1

−1 −1 3

Matrix-tree theorem =⇒ det(A) = 16, thenumber of spanning trees of K4.

What about SNF?

Smith Normal Form and Combinatorics – p. 9

An example (continued)

3 −1 −1

−1 3 −1

−1 −1 3

→

0 0 −1

−4 4 −1

8 −4 3

→

0 0 −1

−4 4 0

8 −4 0

→

0 0 −1

0 4 0

4 −4 0

→

0 0 −1

0 4 0

4 0 0

→

1 0 0

0 4 0

0 0 4

Smith Normal Form and Combinatorics – p. 10

Laplacian matrices

L0(G): reduced Laplacian matrix of the graph G

Matrix-tree theorem. detL0(G) = κ(G), thenumber of spanning trees of G.

Smith Normal Form and Combinatorics – p. 11

Laplacian matrices

L0(G): reduced Laplacian matrix of the graph G

Matrix-tree theorem. detL0(G) = κ(G), thenumber of spanning trees of G.

Theorem. L0(Kn)SNF−→ diag(1, n, n, . . . , n), a

refinement of Cayley’s theorem thatκ(Kn) = nn−2.

Smith Normal Form and Combinatorics – p. 11

Laplacian matrices

L0(G): reduced Laplacian matrix of the graph G

Matrix-tree theorem. detL0(G) = κ(G), thenumber of spanning trees of G.

Theorem. L0(Kn)SNF−→ diag(1, n, n, . . . , n), a

refinement of Cayley’s theorem thatκ(Kn) = nn−2.

In general, SNF of L0(G) not understood.

Smith Normal Form and Combinatorics – p. 11

Chip firing

Abelian sandpile: a finite collection σ ofindistinguishable chips distributed among thevertices V of a (finite) connected graph.Equivalently,

σ : V → {0, 1, 2, . . . }.

2

Smith Normal Form and Combinatorics – p. 12

Chip firing

Abelian sandpile: a finite collection σ ofindistinguishable chips distributed among thevertices V of a (finite) connected graph.Equivalently,

σ : V → {0, 1, 2, . . . }.

toppling of a vertex v: if σ(v) ≥ deg(v), thensend a chip to each neighboring vertex.

7

1 2

1 2

31 2 2

0 65

Smith Normal Form and Combinatorics – p. 12

The sandpile group

Choose a vertex to be a sink, and ignore chipsfalling into the sink.

stable configuration: no vertex can topple

Theorem (easy). After finitely many topples astable configuration will be reached, which isindependent of the order of topples.

Smith Normal Form and Combinatorics – p. 13

The monoid of stable configurations

Define a commutative monoid M on the stableconfigurations by vertex-wise addition followedby stabilization.

ideal of M : subset J ⊆ M satisfying σJ ⊆ J forall σ ∈ M

Smith Normal Form and Combinatorics – p. 14

The monoid of stable configurations

Define a commutative monoid M on the stableconfigurations by vertex-wise addition followedby stabilization.

ideal of M : subset J ⊆ M satisfying σJ ⊆ J forall σ ∈ M

Exercise. The (unique) minimal ideal of a finitecommutative monoid is a group.

Smith Normal Form and Combinatorics – p. 14

Sandpile group

sandpile group of G: the minimal ideal K(G) ofthe monoid M

Fact. K(G) is independent of the choice of sinkup to isomorphism.

Smith Normal Form and Combinatorics – p. 15

Sandpile group

sandpile group of G: the minimal ideal K(G) ofthe monoid M

Fact. K(G) is independent of the choice of sinkup to isomorphism.

Theorem. Let

L0(G)SNF−→ diag(e1, . . . , en−1).

Then

K(G) ∼= Z/e1Z⊕ · · · ⊕ Z/en−1Z.

Smith Normal Form and Combinatorics – p. 15

Second example

Some matrices connected with Youngdiagrams

Smith Normal Form and Combinatorics – p. 16

Extended Young diagrams

λ: a partition (λ1, λ2, . . . ), identified with its Youngdiagram

(3,1)

Smith Normal Form and Combinatorics – p. 17

Extended Young diagrams

λ: a partition (λ1, λ2, . . . ), identified with its Youngdiagram

(3,1)

λ∗: λ extended by a border strip along its entireboundary

Smith Normal Form and Combinatorics – p. 17

Extended Young diagrams

λ: a partition (λ1, λ2, . . . ), identified with its Youngdiagram

(3,1)

λ∗: λ extended by a border strip along its entireboundary

(3,1)* = (4,4,2)

Smith Normal Form and Combinatorics – p. 17

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. 18

Mt

Let t ∈ λ. Let Mt be the largest square of λ∗ witht as the upper left-hand corner.

Smith Normal Form and Combinatorics – p. 19

Mt

Let t ∈ λ. Let Mt be the largest square of λ∗ witht as the upper left-hand corner.

t

Smith Normal Form and Combinatorics – p. 19

Mt

Let t ∈ λ. Let Mt be the largest square of λ∗ witht as the upper left-hand corner.

t

Smith Normal Form and Combinatorics – p. 19

Determinantal algorithm

Suppose all squares to the southeast of t havebeen filled. Insert into t the number nt so thatdetMt = 1.

Smith Normal Form and Combinatorics – p. 20

Determinantal algorithm

Suppose all squares to the southeast of t havebeen filled. Insert into t the number nt so thatdetMt = 1.

1 1 1

1 1

1

Smith Normal Form and Combinatorics – p. 20

Determinantal algorithm

Suppose all squares to the southeast of t havebeen filled. Insert into t the number nt so thatdetMt = 1.

2

1 1 1

1 1

1

Smith Normal Form and Combinatorics – p. 20

Determinantal algorithm

Suppose all squares to the southeast of t havebeen filled. Insert into t the number nt so thatdetMt = 1.

2

2

1 1 1

1 1

1

Smith Normal Form and Combinatorics – p. 20

Determinantal algorithm

Suppose all squares to the southeast of t havebeen filled. Insert into t the number nt so thatdetMt = 1.

2

2

3

1 1 1

1 1

1

Smith Normal Form and Combinatorics – p. 20

Determinantal algorithm

Suppose all squares to the southeast of t havebeen filled. Insert into t the number nt so thatdetMt = 1.

3 2

25

1 1 1

1 1

1

Smith Normal Form and Combinatorics – p. 20

Determinantal algorithm

Suppose all squares to the southeast of t havebeen filled. Insert into t the number nt so thatdetMt = 1.

3

5 29

2

1 1 1

1 1

1

Smith Normal Form and Combinatorics – p. 20

Uniqueness

Easy to see: the numbers nt are well-defined andunique.

Smith Normal Form and Combinatorics – p. 21

Uniqueness

Easy to see: the numbers nt are well-defined andunique.

Why? Expand detMt by the first row. Thecoefficient of nt is 1 by induction.

Smith Normal Form and Combinatorics – p. 21

λ(t)

If t ∈ λ, let λ(t) consist of all squares of λ to thesoutheast of t.

Smith Normal Form and Combinatorics – p. 22

λ(t)

If t ∈ λ, let λ(t) consist of all squares of λ to thesoutheast of t.

tλ = (4,4,3)

Smith Normal Form and Combinatorics – p. 22

λ(t)

If t ∈ λ, let λ(t) consist of all squares of λ to thesoutheast of t.

=

( ) = (3,2)tλ

(4,4,3)λ t

Smith Normal Form and Combinatorics – p. 22

uλ

uλ = #{µ : µ ⊆ λ}

Smith Normal Form and Combinatorics – p. 23

uλ

uλ = #{µ : µ ⊆ λ}

Example. u(2,1) = 5:

φ

Smith Normal Form and Combinatorics – p. 23

uλ

uλ = #{µ : µ ⊆ λ}

Example. u(2,1) = 5:

φ

There is a determinantal formula for uλ, dueessentially to MacMahon and later Kreweras(not needed here).

Smith Normal Form and Combinatorics – p. 23

Carlitz-Scoville-Roselle theorem

Berlekamp (1963) first asked for nt (mod 2)in connection with a coding theory problem.

Carlitz-Roselle-Scoville (1971):combinatorial interpretation of nt (over Z).

Smith Normal Form and Combinatorics – p. 24

Carlitz-Scoville-Roselle theorem

Berlekamp (1963) first asked for nt (mod 2)in connection with a coding theory problem.

Carlitz-Roselle-Scoville (1971):combinatorial interpretation of nt (over Z).

Theorem. nt = uλ(t)

Smith Normal Form and Combinatorics – p. 24

Carlitz-Scoville-Roselle theorem

Berlekamp (1963) first asked for nt (mod 2)in connection with a coding theory problem.

Carlitz-Roselle-Scoville (1971):combinatorial interpretation of nt (over Z).

Theorem. nt = uλ(t)

Proofs. 1. Induction (row and columnoperations).

2. Nonintersecting lattice paths.

Smith Normal Form and Combinatorics – p. 24

An example

37 2 1

1 1 12

1 1

Smith Normal Form and Combinatorics – p. 25

An example

37 2 1

1 1 12

1 1

φ

Smith Normal Form and Combinatorics – p. 25

Many indeterminates

For each square (i, j) ∈ λ, associate anindeterminate xij (matrix coordinates).

Smith Normal Form and Combinatorics – p. 26

Many indeterminates

For each square (i, j) ∈ λ, associate anindeterminate xij (matrix coordinates).

x

x x x

x

11 12 13

21 22

Smith Normal Form and Combinatorics – p. 26

A refinement of uλ

uλ(x) =∑

µ⊆λ

∏

(i,j)∈λ/µ

xij

Smith Normal Form and Combinatorics – p. 27

A refinement of uλ

uλ(x) =∑

µ⊆λ

∏

(i,j)∈λ/µ

xij

d e

c

λ/µ

cba

d e

λ µ

∏

(i,j)∈λ/µ

xij = cde

Smith Normal Form and Combinatorics – p. 27

An example

ed

a cb

abcde+bcde+bce+cde +ce+de+c+e+1

bce+ce+c +e+1 c+1

de+e+1 e+1 1 1

1

111

Smith Normal Form and Combinatorics – p. 28

At

At =∏

(i,j)∈λ(t)

xij

Smith Normal Form and Combinatorics – p. 29

At

At =∏

(i,j)∈λ(t)

xij

t

o

a cb d e

f g h i

j k ml

n

Smith Normal Form and Combinatorics – p. 29

At

At =∏

(i,j)∈λ(t)

xij

t

o

a c d e

f g h i

j k ml

b

n

At = bcdeghiklmoSmith Normal Form and Combinatorics – p. 29

The main theorem

Theorem. Let t = (i, j). Then Mt has SNF

diag(1, . . . , Ai−2,j−2, Ai−1,j−1, Aij).

Smith Normal Form and Combinatorics – p. 30

The main theorem

Theorem. Let t = (i, j). Then Mt has SNF

diag(1, . . . , Ai−2,j−2, Ai−1,j−1, Aij).

Proof. 1. Explicit row and column operationsputting Mt into SNF.

2. (C. Bessenrodt) Induction.

Smith Normal Form and Combinatorics – p. 30

An example

ed

a cb

abcde+bcde+bce+cde +ce+de+c+e+1

bce+ce+c +e+1 c+1

de+e+1 e+1 1 1

1

111

Smith Normal Form and Combinatorics – p. 31

An example

ed

a cb

abcde+bcde+bce+cde +ce+de+c+e+1

bce+ce+c +e+1 c+1

de+e+1 e+1 1 1

1

111

SNF = diag(1, e, abcde)

Smith Normal Form and Combinatorics – p. 31

A special case

Let λ be the staircase δn = (n− 1, n− 2, . . . , 1).Set each xij = q.

Smith Normal Form and Combinatorics – p. 32

A special case

Let λ be the staircase δn = (n− 1, n− 2, . . . , 1).Set each xij = q.

Smith Normal Form and Combinatorics – p. 32

A special case

Let λ be the staircase δn = (n− 1, n− 2, . . . , 1).Set each xij = q.

uδn−1(x)

∣

∣

xij=qcounts Dyck paths of length 2n by

(scaled) area, and is thus the well-knownq-analogue Cn(q) of the Catalan number Cn.

Smith Normal Form and Combinatorics – p. 32

A q-Catalan example

C3(q) = q3 + q2 + 2q + 1

Smith Normal Form and Combinatorics – p. 33

A q-Catalan example

C3(q) = q3 + q2 + 2q + 1

∣

∣

∣

∣

∣

∣

∣

C4(q) C3(q) 1 + q

C3(q) 1 + q 1

1 + q 1 1

∣

∣

∣

∣

∣

∣

∣

SNF∼ diag(1, q, q6)

Smith Normal Form and Combinatorics – p. 33

A q-Catalan example

C3(q) = q3 + q2 + 2q + 1

∣

∣

∣

∣

∣

∣

∣

C4(q) C3(q) 1 + q

C3(q) 1 + q 1

1 + q 1 1

∣

∣

∣

∣

∣

∣

∣

SNF∼ diag(1, q, q6)

q-Catalan determinant previously known

SNF is new

Smith Normal Form and Combinatorics – p. 33

SNF of random matrices

Huge literature on random matrices, mostlyconnected with eigenvalues.

Very little work on SNF of random matrices overa PIR.

Smith Normal Form and Combinatorics – p. 34

Is the question interesting?

Matk(n): all n× n Z-matrices with entries in[−k, k] (uniform distribution)

pk(n, d): probability that if M ∈ Matk(n) andSNF(M) = (e1, . . . , en), then e1 = d.

Smith Normal Form and Combinatorics – p. 35

Is the question interesting?

Matk(n): all n× n Z-matrices with entries in[−k, k] (uniform distribution)

pk(n, d): probability that if M ∈ Matk(n) andSNF(M) = (e1, . . . , en), then e1 = d.

Recall: e1 = gcd of 1× 1 minors (entries) of M

Smith Normal Form and Combinatorics – p. 35

Is the question interesting?

Matk(n): all n× n Z-matrices with entries in[−k, k] (uniform distribution)

pk(n, d): probability that if M ∈ Matk(n) andSNF(M) = (e1, . . . , en), then e1 = d.

Recall: e1 = gcd of 1× 1 minors (entries) of M

Theorem. limk→∞ pk(n, d) = 1/dn2

ζ(n2)

Smith Normal Form and Combinatorics – p. 35

Work of Yinghui Wang

Smith Normal Form and Combinatorics – p. 36

Work of Yinghui Wang ( )

Smith Normal Form and Combinatorics – p. 37

Work of Yinghui Wang ( )

Sample result. µk(n): probability that the SNFof a random A ∈ Matk(n) satisfies e1 = 2, e2 = 6.

µ(n) = limk→∞

µk(n).

Smith Normal Form and Combinatorics – p. 38

Conclusion

µ(n) = 2−n2

1−

n(n−1)∑

i=(n−1)2

2−i +n2−1∑

i=n(n−1)+1

2−i

·3

2· 3−(n−1)2(1− 3(n−1)2)(1− 3−n)2

·∏

p>3

1−

n(n−1)∑

i=(n−1)2

p−i +n2−1∑

i=n(n−1)+1

p−i

.

Smith Normal Form and Combinatorics – p. 39

A note on the proof

uses a 2014 result of C. Feng, R. W. Nóbrega, F.R. Kschischang, and D. Silva, Communicationover finite-chain-ring matrix channels: number ofm× n matrices over Z/psZ with specified SNF

Smith Normal Form and Combinatorics – p. 40

A note on the proof

uses a 2014 result of C. Feng, R. W. Nóbrega, F.R. Kschischang, and D. Silva, Communicationover finite-chain-ring matrix channels: number ofm× n matrices over Z/psZ with specified SNF

Note. Z/psZ is not a PID but is a PIR.

Smith Normal Form and Combinatorics – p. 40

Cyclic cokernel

κ(n): probability that an n× n Z-matrix has SNFdiag(e1, e2, . . . , en) with e1 = e2 = · · · = en−1 = 1.

Smith Normal Form and Combinatorics – p. 41

Cyclic cokernel

κ(n): probability that an n× n Z-matrix has SNFdiag(e1, e2, . . . , en) with e1 = e2 = · · · = en−1 = 1.

Theorem. κ(n) =

∏

p

(

1 +1

p2+

1

p3+ · · ·+

1

pn

)

ζ(2)ζ(3) · · ·

Smith Normal Form and Combinatorics – p. 41

Cyclic cokernel

κ(n): probability that an n× n Z-matrix has SNFdiag(e1, e2, . . . , en) with e1 = e2 = · · · = en−1 = 1.

Theorem. κ(n) =

∏

p

(

1 +1

p2+

1

p3+ · · ·+

1

pn

)

ζ(2)ζ(3) · · ·

Corollary. limn→∞

κ(n) =1

ζ(6)∏

j≥4 ζ(j)

≈ 0.846936 · · · .

Smith Normal Form and Combinatorics – p. 41

Small number of generators

g: number of generators of cokernel (number ofentries of SNF 6= 1) as n → ∞

previous slide: Prob(g = 1) = 0.846936 · · ·

Smith Normal Form and Combinatorics – p. 42

Small number of generators

g: number of generators of cokernel (number ofentries of SNF 6= 1) as n → ∞

previous slide: Prob(g = 1) = 0.846936 · · ·

Prob(g ≤ 2) = 0.99462688 · · ·

Smith Normal Form and Combinatorics – p. 42

Small number of generators

g: number of generators of cokernel (number ofentries of SNF 6= 1) as n → ∞

previous slide: Prob(g = 1) = 0.846936 · · ·

Prob(g ≤ 2) = 0.99462688 · · ·

Prob(g ≤ 3) = 0.99995329 · · ·

Smith Normal Form and Combinatorics – p. 42

Small number of generators

g: number of generators of cokernel (number ofentries of SNF 6= 1) as n → ∞

previous slide: Prob(g = 1) = 0.846936 · · ·

Prob(g ≤ 2) = 0.99462688 · · ·

Prob(g ≤ 3) = 0.99995329 · · ·

Theorem. Prob(g ≤ ℓ) =

1− (3.46275 · · · )2−(ℓ+1)2(1 +O(2−ℓ))

Smith Normal Form and Combinatorics – p. 42

Jacobi-Trudi specialization

Jacobi-Trudi identity:

sλ = det[hλi−i+j],

where sλ is a Schur function and hi is acomplete symmetric function.

Smith Normal Form and Combinatorics – p. 43

Jacobi-Trudi specialization

Jacobi-Trudi identity:

sλ = det[hλi−i+j],

where sλ is a Schur function and hi is acomplete symmetric function.

We consider the specializationx1 = x2 = · · · = xn = 1, other xi = 0. Then

hi →

(

n+ i− 1

i

)

.

Smith Normal Form and Combinatorics – p. 43

Specialized Schur function

sλ →∏

u∈λ

n+ c(u)

h(u).

c(u): content of the square u

−1

0 1 2 3 4

0 1 2

0 1

−1

−2

−3 −2

Smith Normal Form and Combinatorics – p. 44

Diagonal hooks D1, . . . , Dm

λ = (5,4,4,2)

0 1 2 3 4

0 1 2

0 1

−1

−2

−3 −2

−1

Smith Normal Form and Combinatorics – p. 45

Diagonal hooks D1, . . . , Dm

D1

0 1 2 3 4

1 2

0 1

−1

−2

−3 −2

−1

0

Smith Normal Form and Combinatorics – p. 45

Diagonal hooks D1, . . . , Dm

D2

0 1 2 3 4

1 2

0 1−2

−3 −2

−1

−1 0

Smith Normal Form and Combinatorics – p. 45

Diagonal hooks D1, . . . , Dm

D3

0 1 2 3 4

1 2

0 1

−1

−2

−3 −2

−1

0

Smith Normal Form and Combinatorics – p. 45

SNF result

R = Q[n]

Let

SNF

[(

n+ λi − i+ j − 1

λi − i+ j

)]

= diag(e1, . . . , em).

Then

ei =∏

u∈Dm−i+1

n+ c(u)

h(u).

Smith Normal Form and Combinatorics – p. 46

Idea of proof

We will use the fact that if

SNF(A) = diag(e1, e2, . . . , en),

then e1e2 · · · ei is the gcd of the i× i minors of A.

Smith Normal Form and Combinatorics – p. 47

Idea of proof (cont.)

fi =∏

u∈Dm−i+1

n+ c(u)

h(u)

Then f1f2 · · · fi is the value of the lower-left i× iminor. (Special argument for 0 minors.)

Smith Normal Form and Combinatorics – p. 48

Idea of proof (cont.)

fi =∏

u∈Dm−i+1

n+ c(u)

h(u)

Then f1f2 · · · fi is the value of the lower-left i× iminor. (Special argument for 0 minors.)

Every i× i minor is a specialized skew Schurfunction sµ/ν. Let sα correspond to the lower left

i× i minor.

Smith Normal Form and Combinatorics – p. 48

An example

s5442 =

h5 h6 h7 h9

h3 h4 h5 h6

h2 h3 h4 h5

0 1 h1 h2

Smith Normal Form and Combinatorics – p. 49

An example

s5442 =

∣

∣

∣

∣

∣

∣

∣

∣

∣

h5 h6 h7 h9

h3 h4 h5 h6

h2 h3 h4 h5

0 1 h1 h2

∣

∣

∣

∣

∣

∣

∣

∣

∣

s331 =

∣

∣

∣

∣

∣

∣

∣

h3 h4 h5

h2 h3 h4

0 1 h1

∣

∣

∣

∣

∣

∣

∣

Smith Normal Form and Combinatorics – p. 49

Conclusion of proof

Let

sµ/ν =∑

ρ

cµνρsρ.

By Littlewood-Richardson rule,

cµνρ 6= 0 ⇒ α ⊆ ρ.

Smith Normal Form and Combinatorics – p. 50

Conclusion of proof

Let

sµ/ν =∑

ρ

cµνρsρ.

By Littlewood-Richardson rule,

cµνρ 6= 0 ⇒ α ⊆ ρ.

Hence

f1 · · · fi = gcd(i× i minors) = e1 · · · ei.

Smith Normal Form and Combinatorics – p. 50

A generalization?

What about the specialization xi = qi−1,1 ≤ i ≤ n, other xi = 0?

hi →

(

n+ i− 1

i

)

q

Smith Normal Form and Combinatorics – p. 51

A generalization?

What about the specialization xi = qi−1,1 ≤ i ≤ n, other xi = 0?

hi →

(

n+ i− 1

i

)

q

Now it seems the ring should be Q[q]. Looksdifficult.

Smith Normal Form and Combinatorics – p. 51

The last slide

Smith Normal Form and Combinatorics – p. 52

The last slide

Smith Normal Form and Combinatorics – p. 53

The last slide

Smith Normal Form and Combinatorics – p. 53