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Smith Normal Form and Combinatorics Richard P . Stanley Smith - - PowerPoint PPT Presentation

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

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SLIDE 1

Smith Normal Form and Combinatorics

Richard P . Stanley

Smith Normal Form and Combinatorics – p. 1

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SLIDE 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(d1, d1d2, . . . d1d2 · · · dn), where di ∈ R. We then call B a Smith normal form (SNF) of A.

Smith Normal Form and Combinatorics – p. 2

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SLIDE 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 normal form (SNF) of A.

  • NOTE. (1) Can extend to m × n.

(2) unit · det(A) = det(B) = dn

1dn−1 2

· · · dn. Thus SNF is a refinement of det.

Smith Normal Form and Combinatorics – p. 2

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SLIDE 4

Row and column operations

Can put a matrix into SNF by the following

  • perations.

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. 3

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SLIDE 5

Row and column operations

Can put a matrix into SNF by the following

  • perations.

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. 3

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SLIDE 6

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

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SLIDE 7

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. 4

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SLIDE 8

Algebraic interpretation of SNF

R: a PID A: an n × n matrix over R with rows v1, . . . , vn ∈ Rn diag(e1, e2, . . . , en): SNF of A

Smith Normal Form and Combinatorics – p. 5

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SLIDE 9

Algebraic interpretation of SNF

R: a PID A: an n × n matrix over R with rows v1, . . . , vn ∈ Rn diag(e1, e2, . . . , en): SNF of A Theorem. Rn/(v1, . . . , vn) ∼ = (R/e1R) ⊕ · · · ⊕ (R/enR).

Smith Normal Form and Combinatorics – p. 5

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SLIDE 10

Algebraic interpretation of SNF

R: a PID A: an n × n matrix over R with rows v1, . . . , 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. 5

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SLIDE 11

An explicit formula for SNF

R: a PID A: an n × n matrix over R with det(A) = 0 diag(e1, e2, . . . , en): SNF of A

Smith Normal Form and Combinatorics – p. 6

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SLIDE 12

An explicit formula for SNF

R: a PID A: an n × n matrix over R with det(A) = 0 diag(e1, e2, . . . , en): SNF of A

  • Theorem. e1e2 · · · ei is the gcd of all i × i minors
  • f A.

minor: determinant of a square submatrix. Special case: e1 is the gcd of all entries of A.

Smith Normal Form and Combinatorics – p. 6

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SLIDE 13

An example

Reduced Laplacian matrix of K4: A =    3 −1 −1 −1 3 −1 −1 −1 3   

Smith Normal Form and Combinatorics – p. 7

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SLIDE 14

An example

Reduced Laplacian matrix of K4: A =    3 −1 −1 −1 3 −1 −1 −1 3    Matrix-tree theorem = ⇒ det(A) = 16, the number of spanning trees of K4.

Smith Normal Form and Combinatorics – p. 7

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SLIDE 15

An example

Reduced Laplacian matrix of K4: A =    3 −1 −1 −1 3 −1 −1 −1 3    Matrix-tree theorem = ⇒ det(A) = 16, the number of spanning trees of K4. What about SNF?

Smith Normal Form and Combinatorics – p. 7

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SLIDE 16

An example (continued)

     

3 −1 −1 −1 3 −1 −1 −1 3

      →      

0 −1 −4 4 −1 8 −4 3

      →      

0 −1 −4 4 8 −4

     

→    0 −1 4 4 −4    →    0 0 −1 0 4 4 0    →    4 0 0 0 4 0 0 0 1   

Smith Normal Form and Combinatorics – p. 8

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SLIDE 17

Laplacian matrices

L0(G): reduced Laplacian matrix of the graph G Matrix-tree theorem. det L0(G) = κ(G), the number of spanning trees of G.

Smith Normal Form and Combinatorics – p. 9

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SLIDE 18

Laplacian matrices

L0(G): reduced Laplacian matrix of the graph G Matrix-tree theorem. det L0(G) = κ(G), the number of spanning trees of G.

  • Theorem. L0(Kn)

SNF

− → diag(1, n, n, . . . , n), a refinement of Cayley’s theorem that κ(Kn) = nn−2.

Smith Normal Form and Combinatorics – p. 9

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SLIDE 19

Laplacian matrices

L0(G): reduced Laplacian matrix of the graph G Matrix-tree theorem. det L0(G) = κ(G), the number of spanning trees of G.

  • Theorem. L0(Kn)

SNF

− → diag(1, n, n, . . . , n), a refinement of Cayley’s theorem that κ(Kn) = nn−2. In general, SNF of L0(G) not understood.

Smith Normal Form and Combinatorics – p. 9

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SLIDE 20

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, . . . }.

Smith Normal Form and Combinatorics – p. 10

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SLIDE 21

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.

7 1 2 1 2 3 1 2 2 6 5

Smith Normal Form and Combinatorics – p. 10

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SLIDE 22

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. 11

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SLIDE 23

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. 12

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SLIDE 24

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.

Smith Normal Form and Combinatorics – p. 12

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SLIDE 25

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

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SLIDE 26

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

L0(G)

SNF

− → diag(e1, . . . , en−1). Then K(G) ∼ = Z/e1Z ⊕ · · · ⊕ Z/en−1Z.

Smith Normal Form and Combinatorics – p. 13

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SLIDE 27

Second example

Some matrices connected with Young diagrams

Smith Normal Form and Combinatorics – p. 14

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SLIDE 28

Extended Young diagrams

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

(3,1)

Smith Normal Form and Combinatorics – p. 15

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SLIDE 29

Extended Young diagrams

λ: a partition (λ1, λ2, . . . ), 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

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SLIDE 30

Extended Young diagrams

λ: a partition (λ1, λ2, . . . ), 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. 15

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SLIDE 31

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. 16

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SLIDE 32

Mt

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

Smith Normal Form and Combinatorics – p. 17

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SLIDE 33

Mt

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

t

Smith Normal Form and Combinatorics – p. 17

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SLIDE 34

Mt

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

t

Smith Normal Form and Combinatorics – p. 17

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SLIDE 35

Determinantal algorithm

Suppose all squares to the southeast of t have been filled. Insert into t the number nt so that det Mt = 1.

Smith Normal Form and Combinatorics – p. 18

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SLIDE 36

Determinantal algorithm

Suppose all squares to the southeast of t have been filled. Insert into t the number nt so that det Mt = 1.

1 1 1 1 1 1

Smith Normal Form and Combinatorics – p. 18

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SLIDE 37

Determinantal algorithm

Suppose all squares to the southeast of t have been filled. Insert into t the number nt so that det Mt = 1.

2 1 1 1 1 1 1

Smith Normal Form and Combinatorics – p. 18

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SLIDE 38

Determinantal algorithm

Suppose all squares to the southeast of t have been filled. Insert into t the number nt so that det Mt = 1.

2 2 1 1 1 1 1 1

Smith Normal Form and Combinatorics – p. 18

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SLIDE 39

Determinantal algorithm

Suppose all squares to the southeast of t have been filled. Insert into t the number nt so that det Mt = 1.

2 2 3 1 1 1 1 1 1

Smith Normal Form and Combinatorics – p. 18

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SLIDE 40

Determinantal algorithm

Suppose all squares to the southeast of t have been filled. Insert into t the number nt so that det Mt = 1.

3 2 2 5 1 1 1 1 1 1

Smith Normal Form and Combinatorics – p. 18

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SLIDE 41

Determinantal algorithm

Suppose all squares to the southeast of t have been filled. Insert into t the number nt so that det Mt = 1.

3 5 2 9 2 1 1 1 1 1 1

Smith Normal Form and Combinatorics – p. 18

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SLIDE 42

Uniqueness

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

Smith Normal Form and Combinatorics – p. 19

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SLIDE 43

Uniqueness

Easy to see: the numbers nt are well-defined and unique. Why? Expand det Mt by the first row. The coefficient of nt is 1 by induction.

Smith Normal Form and Combinatorics – p. 19

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SLIDE 44

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

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SLIDE 45

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

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SLIDE 46

λ(t)

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

= ( ) = (3,2) t λ (4,4,3) λ t

Smith Normal Form and Combinatorics – p. 20

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SLIDE 47

uλ

uλ = #{µ : µ ⊆ λ}

Smith Normal Form and Combinatorics – p. 21

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SLIDE 48

uλ

uλ = #{µ : µ ⊆ λ}

  • Example. u(2,1) = 5:

φ

Smith Normal Form and Combinatorics – p. 21

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SLIDE 49

uλ

uλ = #{µ : µ ⊆ λ}

  • Example. u(2,1) = 5:

φ

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

Smith Normal Form and Combinatorics – p. 21

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SLIDE 50

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. 22

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SLIDE 51

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 = f(λ(t)).

Smith Normal Form and Combinatorics – p. 22

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SLIDE 52

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 = f(λ(t)).
  • Proofs. 1. Induction (row and column
  • perations).
  • 2. Nonintersecting lattice paths.

Smith Normal Form and Combinatorics – p. 22

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SLIDE 53

An example

3 7 2 1 1 1 1 2 1 1

Smith Normal Form and Combinatorics – p. 23

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SLIDE 54

An example

3 7 2 1 1 1 1 2 1 1

φ

Smith Normal Form and Combinatorics – p. 23

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SLIDE 55

Many indeterminates

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

Smith Normal Form and Combinatorics – p. 24

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SLIDE 56

Many indeterminates

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

x x x x x

11 12 13 21 22

Smith Normal Form and Combinatorics – p. 24

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SLIDE 57

A refinement of uλ

uλ(x) =

  • µ⊆λ
  • (i,j)∈λ/µ

xij

Smith Normal Form and Combinatorics – p. 25

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SLIDE 58

A refinement of uλ

uλ(x) =

  • µ⊆λ
  • (i,j)∈λ/µ

xij

d e c λ/µ c b a d e λ µ

  • (i,j)∈λ/µ

xij = cde

Smith Normal Form and Combinatorics – p. 25

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SLIDE 59

An example

e d a c b abcde+bcde+bce+cde +ce+de+c+e+1 bce+ce+c +e+1 c+1 de+e+1 e+1 1 1 1 1 1 1

Smith Normal Form and Combinatorics – p. 26

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SLIDE 60

At

At =

  • (i,j)∈λ(t)

xij

Smith Normal Form and Combinatorics – p. 27

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SLIDE 61

At

At =

  • (i,j)∈λ(t)

xij

t

  • a

c b d e f g h i j k m l n

Smith Normal Form and Combinatorics – p. 27

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SLIDE 62

At

At =

  • (i,j)∈λ(t)

xij

t

  • a

c d e f g h i j k m l b n

At = bcdeghiklmo

Smith Normal Form and Combinatorics – p. 27

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SLIDE 63

The main theorem

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

diag(Aij, Ai−1,j−1, . . . , 1).

Smith Normal Form and Combinatorics – p. 28

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SLIDE 64

The main theorem

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

diag(Aij, Ai−1,j−1, . . . , 1).

  • Proof. 1. Explicit row and column operations

putting Mt into SNF .

  • 2. (C. Bessenrodt) Induction.

Smith Normal Form and Combinatorics – p. 28

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SLIDE 65

An example

e d a c b abcde+bcde+bce+cde +ce+de+c+e+1 bce+ce+c +e+1 c+1 de+e+1 e+1 1 1 1 1 1 1

Smith Normal Form and Combinatorics – p. 29

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SLIDE 66

An example

e d a c b abcde+bcde+bce+cde +ce+de+c+e+1 bce+ce+c +e+1 c+1 de+e+1 e+1 1 1 1 1 1 1

SNF = diag(abcde, e, 1)

Smith Normal Form and Combinatorics – p. 29

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SLIDE 67

A special case

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

Smith Normal Form and Combinatorics – p. 30

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SLIDE 68

A special case

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

Smith Normal Form and Combinatorics – p. 30

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SLIDE 69

A special case

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

  • xij=q counts Dyck paths of length 2n by

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

Smith Normal Form and Combinatorics – p. 30

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SLIDE 70

A q-Catalan example

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

Smith Normal Form and Combinatorics – p. 31

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SLIDE 71

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(q6, q, 1)

Smith Normal Form and Combinatorics – p. 31

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SLIDE 72

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(q6, q, 1) q-Catalan determinant previously known SNF is new

Smith Normal Form and Combinatorics – p. 31

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SLIDE 73

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. 32

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SLIDE 74

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) and SNF(M) = (e1, . . . , en), then e1 = d.

Smith Normal Form and Combinatorics – p. 33

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SLIDE 75

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) and SNF(M) = (e1, . . . , en), then e1 = d. Recall: e1 = gcd of 1 × 1 minors (entries) of M

Smith Normal Form and Combinatorics – p. 33

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SLIDE 76

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) and SNF(M) = (e1, . . . , en), then e1 = d. Recall: e1 = gcd of 1 × 1 minors (entries) of M

  • Theorem. limk→∞ pk(n, d) = 1/dn2ζ(n2)

Smith Normal Form and Combinatorics – p. 33

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SLIDE 77

Work of Yinghui Wang

Smith Normal Form and Combinatorics – p. 34

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SLIDE 78

Work of Yinghui Wang ( )

Smith Normal Form and Combinatorics – p. 35

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SLIDE 79

Work of Yinghui Wang ( )

Sample result. µk(n): probability that the SNF

  • f a random A ∈ Matk(n) satisfies e1 = 2, e2 = 6.

µ(n) = lim

k→∞ µk(n).

Smith Normal Form and Combinatorics – p. 36

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SLIDE 80

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. 37

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SLIDE 81

A note on the proof

uses a 2014 result of C. Feng, R. W. Nóbrega, F.

  • R. Kschischang, and D. Silva, Communication
  • ver finite-chain-ring matrix channels: number of

m × n matrices over Z/psZ with specified SNF

Smith Normal Form and Combinatorics – p. 38

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SLIDE 82

A note on the proof

uses a 2014 result of C. Feng, R. W. Nóbrega, F.

  • R. Kschischang, and D. Silva, Communication
  • ver finite-chain-ring matrix channels: number of

m × n matrices over Z/psZ with specified SNF

  • Note. Z/psZ is not a PID, but SNF still exists

because its ideals form a finite chain.

Smith Normal Form and Combinatorics – p. 38

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SLIDE 83

Cyclic cokernel

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

Smith Normal Form and Combinatorics – p. 39

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SLIDE 84

Cyclic cokernel

κ(n): probability that an n × n Z-matrix has SNF diag(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. 39

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SLIDE 85

Cyclic cokernel

κ(n): probability that an n × n Z-matrix has SNF diag(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. lim

n→∞ κ(n) =

1 ζ(6)

j≥4 ζ(j)

≈ 0.846936 · · · .

Smith Normal Form and Combinatorics – p. 39

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SLIDE 86

Third example

In collaboration with Tommy Wuxing Cai.

Smith Normal Form and Combinatorics – p. 40

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SLIDE 87

Third example

In collaboration with .

Smith Normal Form and Combinatorics – p. 40

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SLIDE 88

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

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SLIDE 89

Third example

In collaboration with . Par(n): set of all partitions of n E.g., Par(4) = {4, 31, 22, 211, 1111}. Vn: real vector space with basis Par(n)

Smith Normal Form and Combinatorics – p. 40

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SLIDE 90

U

Define U = Un : Vn → Vn+1 by U(λ) =

  • µ

µ, where µ ∈ Par(n + 1) and µi ≥ λi ∀i. Example. U(42211) = 52211 + 43211 + 42221 + 422111

Smith Normal Form and Combinatorics – p. 41

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SLIDE 91

D

Dually, define D = Dn: Vn → Vn−1 by D(λ) =

  • ν

ν, where ν ∈ Par(n − 1) and νi ≤ λi ∀i.

  • Example. D(42211) = 32211 + 42111 + 4221

Smith Normal Form and Combinatorics – p. 42

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SLIDE 92

Symmetric functions

  • NOTE. Identify Vn with the space Λn

Q of all

homogeneous symmetric functions of degree n

  • ver Q, and identify λ ∈ Vn with the Schur

function sλ. Then U(f) = p1f, D(f) = ∂ ∂p1 f.

Smith Normal Form and Combinatorics – p. 43

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SLIDE 93

Commutation relation

Basic commutation relation: DU − UD = I Allows computation of eigenvalues of DU : Vn → Vn. Or note that the eigenvectors of

∂ ∂p1p1 are the

pλ’s, λ ⊢ n.

Smith Normal Form and Combinatorics – p. 44

slide-94
SLIDE 94

Eigenvalues of DU

Let p(n) = #Par(n) = dim Vn.

  • Theorem. Let 1 ≤ i ≤ n + 1, i = n. Then i is an

eigenvalue of Dn+1Un with multiplicity p(n + 1 − i) − p(n − i). Hence det Dn+1Un =

n+1

  • i=1

ip(n+1−i)−p(n−i).

Smith Normal Form and Combinatorics – p. 45

slide-95
SLIDE 95

Eigenvalues of DU

Let p(n) = #Par(n) = dim Vn.

  • Theorem. Let 1 ≤ i ≤ n + 1, i = n. Then i is an

eigenvalue of Dn+1Un with multiplicity p(n + 1 − i) − p(n − i). Hence det Dn+1Un =

n+1

  • i=1

ip(n+1−i)−p(n−i). What about SNF of the matrix [Dn+1Un] (with respect to the basis Par(n))?

Smith Normal Form and Combinatorics – p. 45

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SLIDE 96

Conjecture of A. R. Miller, 2005

Conjecture (first form). Let e1, . . . , ep(n) be the eigenvalues of Dn+1Un. Then [Dn+1Un] has the same SNF as diag(e1, . . . , ep(n)).

Smith Normal Form and Combinatorics – p. 46

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SLIDE 97

Conjecture of A. R. Miller, 2005

Conjecture (first form). Let e1, . . . , ep(n) be the eigenvalues of Dn+1Un. Then [Dn+1Un] has the same SNF as diag(e1, . . . , ep(n)). Conjecture (second form). The diagonal entries

  • f the SNF of [Dn+1Un] 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. 46

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SLIDE 98

Not a trivial result

  • NOTE. {pλ}λ⊢n is not an integral basis.

Smith Normal Form and Combinatorics – p. 47

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SLIDE 99

Another form

m1(λ): number of 1’s in λ M1(n): multiset of all numbers m1(λ) + 1, λ ∈ Par(n) Let SNF of [Dn+1Un] be diag(f1, f2, . . . , fp(n)). Conjecture (third form). f1 is the product of the distinct entries of M1(n); f2 is the product of the remaining distinct entries of M1(n), etc.

Smith Normal Form and Combinatorics – p. 48

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SLIDE 100

An example: n = 6

Par(6) = {6, 51, 42, 33, 411, 321, 222, 3111, 2211, 21111, 111111} M1(6) = {1, 2, 1, 1, 3, 2, 1, 4, 3, 5, 7} (f1, . . . , f11) = (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. 49

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SLIDE 101

Yet another form

Conjecture (fourth form). The matrix [Dn+1Un + xI] has an SNF over Z[x]. Note that Z[x] is not a PID.

Smith Normal Form and Combinatorics – p. 50

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SLIDE 102

Resolution of conjecture

  • Theorem. The conjecture of Miller is true.

Smith Normal Form and Combinatorics – p. 51

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SLIDE 103

Resolution of conjecture

  • Theorem. The conjecture of Miller is true.

Proof (first step). Rather than use the basis {sλ}λ∈Par(n) (Schur functions) for Λn

Q, use the

basis {hλ}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. 51

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SLIDE 104

Conclusion of proof

(second step) Row and column operations.

Smith Normal Form and Combinatorics – p. 52

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SLIDE 105

Conclusion of proof

(second step) Row and column operations. Not very insightful.

Smith Normal Form and Combinatorics – p. 52

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SLIDE 106

Conclusion of proof

(second step) Row and column operations. Not very insightful.

Smith Normal Form and Combinatorics – p. 52

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SLIDE 107

An unsolved conjecture

mj(λ): number of j’s in λ Mj(n): multiset of all numbers j(mj(λ) + 1), λ ∈ Par(n) pj: power sum symmetric function xj

i

Let SNF of the operator f → j ∂

∂pjpjf with respect

to the basis {sλ} be diag(g1, g2, . . . , gp(n)).

Smith Normal Form and Combinatorics – p. 53

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SLIDE 108

An unsolved conjecture

mj(λ): number of j’s in λ Mj(n): multiset of all numbers j(mj(λ) + 1), λ ∈ Par(n) pj: power sum symmetric function xj

i

Let SNF of the operator f → j ∂

∂pjpjf with respect

to the basis {sλ} be diag(g1, g2, . . . , gp(n)). Conjecture.g1 is the product of the distinct entries of Mj(n); g2 is the product of the remaining distinct entries of Mj(n), etc.

Smith Normal Form and Combinatorics – p. 53

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SLIDE 109

Jacobi-Trudi specialization

Jacobi-Trudi identity: sλ = det[hλi−i+j], where sλ is a Schur function and hi is a complete symmetric function.

Smith Normal Form and Combinatorics – p. 54

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SLIDE 110

Jacobi-Trudi specialization

Jacobi-Trudi identity: sλ = det[hλi−i+j], where sλ is a Schur function and hi is a complete symmetric function. We consider the specialization x1 = x2 = · · · = xn = 1, other xi = 0. Then hi → n + i − 1 i

  • .

Smith Normal Form and Combinatorics – p. 54

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SLIDE 111

Specialized Schur function

sλ →

  • u∈λ

n + c(u) h(u) . c(u): content of the square u

−1 1 2 3 4 1 2 1 −1 −2 −3 −2

Smith Normal Form and Combinatorics – p. 55

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SLIDE 112

Diagonal hooks D1, . . . , Dm

λ = (5,4,4,2) 1 2 3 4 1 2 1 −1 −2 −3 −2 −1

Smith Normal Form and Combinatorics – p. 56

slide-113
SLIDE 113

Diagonal hooks D1, . . . , Dm

D

1

1 2 3 4 1 2 1 −1 −2 −3 −2 −1

Smith Normal Form and Combinatorics – p. 56

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SLIDE 114

Diagonal hooks D1, . . . , Dm

D2 1 2 3 4 1 2 1 −2 −3 −2 −1 −1

Smith Normal Form and Combinatorics – p. 56

slide-115
SLIDE 115

Diagonal hooks D1, . . . , Dm

D3 1 2 3 4 1 2 1 −1 −2 −3 −2 −1

Smith Normal Form and Combinatorics – p. 56

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SLIDE 116

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. 57

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SLIDE 117

Idea of proof

fi =

  • u∈Dm−i+1

n + c(u) h(u) Then f1f2] · · · fi is the value of the lower-left i × i

  • minor. (Special argument for 0 minors.)

Smith Normal Form and Combinatorics – p. 58

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SLIDE 118

Idea of proof

fi =

  • u∈Dm−i+1

n + c(u) h(u) Then f1f2] · · · fi 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 sµ/ν. Let sα correspond to the lower left i × i minor.

Smith Normal Form and Combinatorics – p. 58

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SLIDE 119

Conclusion of proof

Let sµ/ν =

  • ρ

cµ

νρsρ.

By Littlewood-Richardson rule, cµ

νρ = 0 ⇐ α ⊆ ρ.

Smith Normal Form and Combinatorics – p. 59

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SLIDE 120

Conclusion of proof

Let sµ/ν =

  • ρ

cµ

νρsρ.

By Littlewood-Richardson rule, cµ

νρ = 0 ⇐ α ⊆ ρ.

Hence fi = gcd(i × i minors) = ei ei−1 .

Smith Normal Form and Combinatorics – p. 59

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SLIDE 121

The last slide

Smith Normal Form and Combinatorics – p. 60

slide-122
SLIDE 122

The last slide

Smith Normal Form and Combinatorics – p. 61

slide-123
SLIDE 123

The last slide

Smith Normal Form and Combinatorics – p. 61