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Smith Normal Form andCombinatorics
Richard P. Stanley
Smith Normal Form and Combinatorics – p. 1
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GESSEL
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