schur functions: tableaux, determinant formulas and

Schur functions: tableaux, determinant formulas andlattices paths

Olga Azenhas

CMUC

November 24 , 2008

1 / 100

Contents1 Partitions and Young diagrams2 Young tableaux3 Schur functions4 Schur functions are symmetric functions5 Bender-Knuth involution on semistandard tableaux6 The ring of symmetric functions7 Complete and and elementary homogeneous symmetric functions8 Jacobi-Trudi determinant formulas9 The plane integer lattice

10 Lattice paths11 Weighted paths12 Weight-preserving bijection between tableaux and nonintersecting

lattice paths13 Complete and elementary homogeneous symmetric functions are

weight-generating functions of lattice paths between two points14 Lindstrom-Gessel-Viennot involution

2 / 100

1. Partitions and Young diagramsFix a positive integer r ≥ 1.

λ = (λ1, . . . , λr), with λ1 ≥ · · · ≥ λr > 0 positive integers, is a partitionof length l(λ) = r .Each partition λ is identified with a Young (Ferrer) diagram Fλ

consisting of |λ| = λ1 + · · ·+ λr boxes arranged in r left adjusted rowsof lengths λ1 ≥ · · · ≥ λr > 0.

Exampleλ = (4, 3, 2), |λ| = 9, l(λ) = 3

Fλ = Fλ′ =

λ′ = (3, 3, 2, 1).

The partition λ′ conjugate of λ is such that Fλ′ is obtained from Fλ byinterchanging rows and columns.

3 / 100

1. Partitions and Young diagramsFix a positive integer r ≥ 1.λ = (λ1, . . . , λr), with λ1 ≥ · · · ≥ λr > 0 positive integers, is a partitionof length l(λ) = r .

Each partition λ is identified with a Young (Ferrer) diagram Fλ


Exampleλ = (4, 3, 2), |λ| = 9, l(λ) = 3

Fλ = Fλ′ =

λ′ = (3, 3, 2, 1).


4 / 100

1. Partitions and Young diagramsFix a positive integer r ≥ 1.λ = (λ1, . . . , λr), with λ1 ≥ · · · ≥ λr > 0 positive integers, is a partitionof length l(λ) = r .Each partition λ is identified with a Young (Ferrer) diagram Fλ


Exampleλ = (4, 3, 2), |λ| = 9, l(λ) = 3

Fλ = Fλ′ =

λ′ = (3, 3, 2, 1).


5 / 100



Exampleλ = (4, 3, 2), |λ| = 9, l(λ) = 3

Fλ =

Fλ′ =

λ′ = (3, 3, 2, 1).


6 / 100



Exampleλ = (4, 3, 2), |λ| = 9, l(λ) = 3

Fλ = Fλ′ =

λ′ = (3, 3, 2, 1).


7 / 100



Exampleλ = (4, 3, 2), |λ| = 9, l(λ) = 3

Fλ = Fλ′ =

λ′ = (3, 3, 2, 1).


8 / 100



Exampleλ = (4, 3, 2), |λ| = 9, l(λ) = 3

Fλ = Fλ′ =

λ′ = (3, 3, 2, 1).


9 / 100

2. Young TableauxGiven a partition λ with l(λ) = r , fix a positive integer n ≥ r .

An n-semistandard tableau T of shape λ is a filling of the boxes of theFerrer diagram Fλ with elements i in {1, . . . , n} which is

I weakly increasing across rows from left to rightI strictly increasing down columns

T has type α = (α1, . . . , αn) if T has αi entries equal i.

Exampleλ = (4, 3, 2), l(λ) = 3, n = 7

T =

2 3 4 64 4 65 6

7- semistandard tableau T of shape λ = (4, 3, 2), α = (0, 1, 1, 3, 1, 3, 0).

EquivalentlyT : {(i, j) ∈ Z2 : 1 ≤ i ≤ r , 1 ≤ j ≤ λi} −→ {1, . . . , n}, T(i, j) = Tij .

10 / 100

2. Young TableauxGiven a partition λ with l(λ) = r , fix a positive integer n ≥ r .An n-semistandard tableau T of shape λ is a filling of the boxes of theFerrer diagram Fλ with elements i in {1, . . . , n} which is



Exampleλ = (4, 3, 2), l(λ) = 3, n = 7

T =

2 3 4 64 4 65 6



11 / 100




Exampleλ = (4, 3, 2), l(λ) = 3, n = 7

T =

2 3 4 64 4 65 6



12 / 100




Exampleλ = (4, 3, 2), l(λ) = 3, n = 7

T =

2 3 4 64 4 65 6



13 / 100




Exampleλ = (4, 3, 2), l(λ) = 3, n = 7

T =

2 3 4 64 4 65 6

7- semistandard tableau T of shape λ = (4, 3, 2),

α = (0, 1, 1, 3, 1, 3, 0).


14 / 100




Exampleλ = (4, 3, 2), l(λ) = 3, n = 7

T =

2 3 4 64 4 65 6



15 / 100




Exampleλ = (4, 3, 2), l(λ) = 3, n = 7

T =

2 3 4 64 4 65 6



16 / 100

3. Schur functionsLet n be a fixed positive integer and x = (x1, . . . , xn) a sequence ofvariables.

The monomial weight of an n-semistandard tableau T of shape λ isthe monomial of degree |λ|, in the variables x1, . . . , xn,

XT =∏Tij

xTij

where Tij runs over all the |λ| entries of T .

Examplen = 7

T =

2 3 4 64 4 65 6 XT = x0

1 x2x3x34 x5x3

6 x07

α(T)=(0,1,1,3,1,3,0)

17 / 100

3. Schur functionsLet n be a fixed positive integer and x = (x1, . . . , xn) a sequence ofvariables.

The monomial weight of an n-semistandard tableau T of shape λ isthe monomial of degree |λ|, in the variables x1, . . . , xn,

XT =∏Tij

xTij

where Tij runs over all the |λ| entries of T .

Examplen = 7

T =

2 3 4 64 4 65 6 XT = x0

1 x2x3x34 x5x3

6 x07

α(T)=(0,1,1,3,1,3,0)

18 / 100

Schur functions continuedGiven a partition λ with l(λ) ≤ n, the Schur function sn(λ, x)associated with the partition λ is the homogeneous polynomial ofdegree |λ| on the variables x1 . . . , xn,

sn(λ, x) =∑

T

XT

where T runs over all n-semistandard tableaux of shape λ.

Examplen = 3, λ = (2, 1), |λ| = 3

1 12

1 13

1 22

1 23

1 32

1 33

2 23

2 33 .

Then

s3(λ, x1, x2, x3) = x21 x2 + x2

1 x3 + x1x22 + 2x1x2x3 + x1x2

3 + x22 x3 + x2x2

3 .

n = 2, s2(λ, x1, x2) = x21 x2 + x1x2

2 .

19 / 100


sn(λ, x) =∑

T

XT


Examplen = 3, λ = (2, 1), |λ| = 3

1 12

1 13

1 22

1 23

1 32

1 33

2 23

2 33 .

Then

s3(λ, x1, x2, x3) = x21 x2 + x2

1 x3 + x1x22 + 2x1x2x3 + x1x2

3 + x22 x3 + x2x2

3 .

n = 2, s2(λ, x1, x2) = x21 x2 + x1x2

2 .

20 / 100


sn(λ, x) =∑

T

XT


Examplen = 3, λ = (2, 1), |λ| = 3

1 12

1 13

1 22

1 23

1 32

1 33

2 23

2 33 .

Then

s3(λ, x1, x2, x3) = x21 x2 + x2

1 x3 + x1x22 + 2x1x2x3 + x1x2

3 + x22 x3 + x2x2

3 .

n = 2, s2(λ, x1, x2) = x21 x2 + x1x2

2 .21 / 100

Schur functions continued

Given a partition λ with l(λ) ≤ n, the Schur function sn(λ, x)associated with the partition λ is the homogeneous polynomial ofdegree |λ| on the variables x1 . . . , xn,

sn(λ, x) =∑

T

XT

=∑

α weak composition of |λ| of length ≤ n

Kλ, αxα,

xα = xα11 xα2

2 . . . xαnn ,

Kλα is the Kostka number, the number of SSYTs of shape λ and typeα.

22 / 100

4. Schur functions are symmetric functions

Let C[x] be the ring of polynomials in the variables x1, . . . , xn, over C.

Let Sn be the symmetric group of degree n, consisting of allpermutations of {1, . . . , n}, and π ∈ Sn. There is a natural action of πon f(x) ∈ C[x],

πf(x) = f(xπ(1), . . . , xπ(n)).

Are Schur functions fixed points of this action?

It is enough to check for adjacent transpositions (i i + 1) ∈ Sn,

(i i + 1)sn(λ, x) = sn(λ, x).

Kλ α = Kλα , with α = (i i + 1)α?

23 / 100




πf(x) = f(xπ(1), . . . , xπ(n)).



(i i + 1)sn(λ, x) = sn(λ, x).


24 / 100




πf(x) = f(xπ(1), . . . , xπ(n)).



(i i + 1)sn(λ, x) = sn(λ, x).


25 / 100




πf(x) = f(xπ(1), . . . , xπ(n)).



(i i + 1)sn(λ, x) = sn(λ, x).


26 / 100




πf(x) = f(xπ(1), . . . , xπ(n)).



(i i + 1)sn(λ, x) = sn(λ, x).


27 / 100

5. Bender-Knuth involution on semistandard tableauxBender-Knuth involution is a bijection ξ : T −→ Q , on the set ofsemistandard tableaux of shape λ, such that the numbers of i’s and(i + 1)’s are swapped when passing from T to Q with all othermultiplicities staying the same.

Exampleλ = (11, 5, 2)

1 1 1 1 2 2 2 2 3 3 42 2 3 3 33 3 ←→

ξ23

I each column of T contains either an i, i + 1 pair; exactly one of i, i + 1;or neither.

I the numbers in such pairs are called fixed and the other occurrences ofi’s or i + 1’s are free.

I if in a row one has k free i’s followed by ` free i + 1’s then replace themby ` free i’s followed by k free i + 1’s.

28 / 100


Exampleλ = (11, 5, 2)

1 1 1 1 2 2 2 2 3 3 42 2 3 3 33 3

←→ξ23




29 / 100


Exampleλ = (11, 5, 2)

1 1 1 1 2 2 2 2 3 3 42 2 3 3 33 3 ←→

ξ23




30 / 100


Exampleλ = (11, 5, 2)

1 1 1 1 2 2 2 2 3 3 42 2 3 3 33 3 ←→

ξ23

1 1 1 1 2 2 2 3 3 3 42 2 2 2 33 3

XT = x41 x6

2 x73 x1

4 XQ = x41 x7

2 x63 x1

4

CorollaryKλ β = Kλα, with β any permutation of α.The Schur function sn(λ, x) is a homogeneous symmetric function inx1, . . . , xn.

31 / 100


Exampleλ = (11, 5, 2)

1 1 1 1 2 2 2 2 3 3 42 2 3 3 33 3 ←→

ξ23

1 1 1 1 2 2 2 3 3 3 42 2 2 2 33 3

XT = x41 x6

2 x73 x1

4 XQ = x41 x7

2 x63 x1

4


32 / 100


Exampleλ = (11, 5, 2)

1 1 1 1 2 2 2 2 3 3 42 2 3 3 33 3 ←→

ξ23

1 1 1 1 2 2 2 3 3 3 42 2 2 2 33 3

XT = x41 x6

2 x73 x1

4 XQ = x41 x7

2 x63 x1

4


33 / 100

6. The ring of symmetric functions

Given λ with `(λ) ≤ n,

mλ(x1 . . . , xn) =∑α

xα,

α a permutation of λ.The ring of symmetric functions on variables the x1 . . . , xn is thevector space

Λ = Cmλ.

For each k ≥ 0, let Λk be the vector space generated by{mλ : |λ| = k }. The Schur functions sλ with |λ| = k , on the variablesx1, . . . , xn, form a basis of the vector space Λk .

The Schur functions sλ on the variables x1, . . . , xn, form an additivebasis of the ring Λ.

34 / 100

7. Complete homogeneous symmetric functions andelementary symmetric functions

row partition λ = (m), l(λ) = 1

Examplem = 3, n = 3

T = 1 2 2 Q = 1 1 3

xT = x1x2x2 xQ = x1x1x3.

h3(x1, x2, x3) = x31 +x3

2 +x33 +x2

1 x3+x21 x2+x1x2

2 +x1x23 +x1x2x3+x2

2 x3+x2x23

The mth complete homogeneous symmetric function, in the variablesx1, . . . , xn, is the sum of all monomials of degree m in the variablesx1, . . . , xn

hm(x1, . . . , xn) := sn((m), x1, . . . , xn) =∑

1≤i1≤···≤im≤n

xi1xi2 . . . xim

35 / 100



Examplem = 3, n = 3

T = 1 2 2 Q = 1 1 3


h3(x1, x2, x3) = x31 +x3

2 +x33 +x2

1 x3+x21 x2+x1x2

2 +x1x23 +x1x2x3+x2

2 x3+x2x23


hm(x1, . . . , xn) := sn((m), x1, . . . , xn) =∑

1≤i1≤···≤im≤n

xi1xi2 . . . xim

36 / 100



Examplem = 3, n = 3

T = 1 2 2 Q = 1 1 3


h3(x1, x2, x3) = x31 +x3

2 +x33 +x2

1 x3+x21 x2+x1x2

2 +x1x23 +x1x2x3+x2

2 x3+x2x23


hm(x1, . . . , xn) := sn((m), x1, . . . , xn) =∑

1≤i1≤···≤im≤n

xi1xi2 . . . xim

37 / 100

continued

column partition λ = (1, 1, . . . , 1) = (1m)

Examplem = 3, n = 4

T =

134 xT = x1x3x4.

e3(x1, x2, x3, x4) = x1x2x3 + x1x2x4 + x1x3x4 + x2x3x4

The mth elementary symmetric function, in the variables x1, . . . , xn, isthe sum of all monomials xi1 . . . xim for all strictly increasing sequences1 ≤ i1 < i2 < · · · < im ≤ n

em(x1, . . . , xn) := sn((1m), x1, . . . , xn) =∑

1≤i1<···<im≤n

xi1xi2 . . . xim

38 / 100

continued

column partition λ = (1, 1, . . . , 1) = (1m)

Examplem = 3, n = 4

T =

134 xT = x1x3x4.

e3(x1, x2, x3, x4) = x1x2x3 + x1x2x4 + x1x3x4 + x2x3x4

The mth elementary symmetric function, in the variables x1, . . . , xn, isthe sum of all monomials xi1 . . . xim for all strictly increasing sequences1 ≤ i1 < i2 < · · · < im ≤ n

em(x1, . . . , xn) := sn((1m), x1, . . . , xn) =∑

1≤i1<···<im≤n

xi1xi2 . . . xim

39 / 100

8. The (classical) Jacobi-Trudi determinant formulasLet λ be a partition with l(λ) = r ≤ n.The original definition of Schur function (and that Schur originallyused)

sn(λ, x) =|xλi+n−i

j |r×r

|xn−ij |r×r

The Schur function sn(λ, x) can be expressed as a determinant interms of complete symmetric functions and elementary symmetricfunctions

I sn(λ, x) = |hλj−j+i(x)|r×r, h-formula

=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

hλ1 (x) hλ2−1(x) hλ3−2(x) . . . hλr−1−r (x) hλr−r+1(x)hλ1+1(x) hλ2 (x) hλ3−1(x) . . . hλr−1−r+1(x) hλr−r+2(x)hλ1+2(x) hλ2+1(x) hλ3 (x) . . . hλr−1−r+2(x) hλr−r+3(x)

......

......

...hλ1+r−1(x) hλ2+r−2(x) hλ3+r−3(x) . . . hλr−1+1(x) hλr (x)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣where we set h0 = 1, hk = 0, k < 0.

I sn(λ, x) = |eλ′j−j+i(x)|λ1×λ1 , e-formula,where we set e0 = 1, ek = 0, k > n.

40 / 100

Jacobi-Trudi continuedn = 2, x = (x1, x2), λ = (2, 1)

I h-formula

|hλj−j+i(x)| =

∣∣∣∣∣∣ h2 h0

h3 h1

∣∣∣∣∣∣ = h2(x)h1(x) − h3(x).1

= (x21 + x2

2 + x1x2)(x1 + x2) − (x31 + x2

1 x2 + x1x22 + x3

2 )

= x31 + 2x1x2

2 + 2x1x22 + x3

2 − (x31 + x2

1 x2 + x1x22 + x3

2 ) = x21 x2 + x1x2

2 .

I e-formula

|eλ′j−j+i(x)| =

∣∣∣∣∣∣ e2 e0

e3 e1

∣∣∣∣∣∣ = e2(x)e1(x) − 0.1

= (x1x2)(x1 + x2) = x21 x2 + x1x2

2 .

Example

n = 2, λ = (2, 1), |λ| = 3,1 12

1 22

s2(λ,x)=x21 x2+x1x2

2=∑

2-semistandard tableaux T of shape λxT .

43 / 100


I h-formula

|hλj−j+i(x)| =

∣∣∣∣∣∣ h2 h0

h3 h1

∣∣∣∣∣∣ = h2(x)h1(x) − h3(x).1

= (x21 + x2

2 + x1x2)(x1 + x2) − (x31 + x2

1 x2 + x1x22 + x3

2 )

= x31 + 2x1x2

2 + 2x1x22 + x3

2 − (x31 + x2

1 x2 + x1x22 + x3

2 ) = x21 x2 + x1x2

2 .

I e-formula


∣∣∣∣∣∣ e2 e0

e3 e1

∣∣∣∣∣∣ = e2(x)e1(x) − 0.1

= (x1x2)(x1 + x2) = x21 x2 + x1x2

2 .

Example

n = 2, λ = (2, 1), |λ| = 3,1 12

1 22

s2(λ,x)=x21 x2+x1x2

2=∑


44 / 100


I h-formula

|hλj−j+i(x)| =

∣∣∣∣∣∣ h2 h0

h3 h1

∣∣∣∣∣∣ = h2(x)h1(x) − h3(x).1

= (x21 + x2

2 + x1x2)(x1 + x2) − (x31 + x2

1 x2 + x1x22 + x3

2 )

= x31 + 2x1x2

2 + 2x1x22 + x3

2 − (x31 + x2

1 x2 + x1x22 + x3

2 ) = x21 x2 + x1x2

2 .

I e-formula


∣∣∣∣∣∣ e2 e0

e3 e1

∣∣∣∣∣∣ = e2(x)e1(x) − 0.1

= (x1x2)(x1 + x2) = x21 x2 + x1x2

2 .

Example

n = 2, λ = (2, 1), |λ| = 3,1 12

1 22

s2(λ,x)=x21 x2+x1x2

2=∑


45 / 100


I h-formula

|hλj−j+i(x)| =

∣∣∣∣∣∣ h2 h0

h3 h1

∣∣∣∣∣∣ = h2(x)h1(x) − h3(x).1

= (x21 + x2

2 + x1x2)(x1 + x2) − (x31 + x2

1 x2 + x1x22 + x3

2 )

= x31 + 2x1x2

2 + 2x1x22 + x3

2 − (x31 + x2

1 x2 + x1x22 + x3

2 ) = x21 x2 + x1x2

2 .

I e-formula


∣∣∣∣∣∣ e2 e0

e3 e1

∣∣∣∣∣∣ = e2(x)e1(x) − 0.1

= (x1x2)(x1 + x2) = x21 x2 + x1x2

2 .

Example

n = 2, λ = (2, 1), |λ| = 3,1 12

1 22

s2(λ,x)=x21 x2+x1x2

2=∑


46 / 100

9. The plane integer lattice

The plane integer lattice is the set of integer points Z2 equipped withthe usual component wise order relation

(a, b) ≤ (c, d)⇔ a ≤ c, b ≤ d.

AB

A ≤ B

47 / 100



(a, b) ≤ (c, d)⇔ a ≤ c, b ≤ d.

AB

C

A ≤ B ≤ C

48 / 100



(a, b) ≤ (c, d)⇔ a ≤ c, b ≤ d.

AB

C

DA ≤ B ≤ C , A ≤ D

B , D are not comparable

C , D are not comparable

49 / 100



(a, b) ≤ (c, d)⇔ a ≤ c, b ≤ d.

AB

C

DA ≤ B ≤ C , A ≤ D



50 / 100



(a, b) ≤ (c, d)⇔ a ≤ c, b ≤ d.

AB

C

DA ≤ B ≤ C , A ≤ D



51 / 100



(a, b) ≤ (c, d)⇔ a ≤ c, b ≤ d.

AB

C

DA ≤ B ≤ C , A ≤ D



52 / 100



(a, b) ≤ (c, d)⇔ a ≤ c, b ≤ d.

AB

C

DE

B ∧ D = A

C ∧ D = E

53 / 100

9. The plane integer latticeThe plane integer lattice is the set of integer points Z2 equipped withthe usual component wise order relation

(a, b) ≤ (c, d)⇔ a ≤ c, b ≤ d.

AB

C

DE

F

B ∧ D = A

C ∧ D = E

C ∨ D = F

The plane of integer points Z2 is a lattice with the relation ≤.

54 / 100

9. The plane integer latticeThe plane integer lattice is the set of integer points Z2 equipped withthe usual component wise order relation

(a, b) ≤ (c, d)⇔ a ≤ c, b ≤ d.

AB

C

DE

F

B ∧ D = A

C ∧ D = E

C ∨ D = F

The plane of integer points Z2 is a lattice with the relation ≤.55 / 100

10. Lattice paths

A lattice path from u to v, with u ≤ v, is a sequence of adjacent pointsin the integer lattice, i.e. a sequence of unit horizontal (East) andvertical (North) steps in the positive direction, starting in u and endingin v.

Example

•u

•v

•u

•v

56 / 100

10. Lattice paths


Example

•u •

•v

•u

•

•v

57 / 100

10. Lattice paths


Example

•u • •

•v

•u

•

•

•v

58 / 100

10. Lattice paths


Example

•u • •

•

•v

•u

•

•

• •v

59 / 100

10. Lattice paths


Example

•u • •

•

• •

•v

•u

•

•

• • • •v

60 / 100

10. Lattice pathsA lattice path from u to v, with u ≤ v, is a sequence of adjacent pointsin the integer lattice, i.e. a sequence of unit horizontal (East) andvertical (North) steps in the positive direction, starting in u and endingin v.

Example

•u • •

•

• •

•v•

•u

•

•

•

• • • •v

Given u, v ∈ Z2, P(u, v) is the set of all lattice paths from u to v.

#P(u, v) = #P(0, v − u) =

(v1 − u1 + v2 − u2

v1 − u1

), u=(u1,u2)≤v=(v1,v2).

P(u, v) = ∅, u, v not comparable.

61 / 100


Example

•u • •

•

• •

•v•

•u

•

•

•

• • • •v


#P(u, v) = #P(0, v − u) =

(v1 − u1 + v2 − u2

v1 − u1

), u=(u1,u2)≤v=(v1,v2).

P(u, v) = ∅, u, v not comparable.

62 / 100


Example

•u • •

•

• •

•v•

•u

•

•

•

• • • •v


#P(u, v) = #P(0, v − u) =

(v1 − u1 + v2 − u2

v1 − u1

), u=(u1,u2)≤v=(v1,v2).

P(u, v) = ∅, u, v not comparable.63 / 100

(continued)Two lattice paths are nonintersecting if they do not have any (lattice)point in common.

Example

//

OO

•

• •

•

//

OO

•

•

��

__

��

•

•

xyzw

64 / 100

(continued)Two lattice paths are nonintersecting if they do not have any (lattice)point in common.

Example

//

OO

•

• •

•

//

OO

•

•

��

__

��

•

•

xyzw

65 / 100

11. Weighted paths

To each horizontal edge a in Z2 one assignes a weight w(a).

Example

� � � � � � � //

_

_

_

_

_

_

_

_

_

_

OO

2

3

5

6 6

•

•

//� � � � � � � � �??

??

?

??

??

??

?

??

??

??

??

?

??

??

??

??

?

??

??

??

??

??

?

??

??

??

??

??

??

?

??

??

??

??

??

??

??

?

??

??

??

??

??

??

??

??

?

??

??

??

??

??

??

??

?

??

??

??

??

??

??

??

??

?

_

_

_

_

_

_

OO

2

4

7

9 10

•

•

wh((i − 1, j); (i, j)) = xj we((i − 1, j); (i, j)) = xi+j

66 / 100

11. Weighted paths


Example

� � � � � � � //

_

_

_

_

_

_

_

_

_

_

OO

2

3

5

6 6

•

•

//� � � � � � � � �??

??

?

??

??

??

?

??

??

??

??

?

??

??

??

??

?

??

??

??

??

??

?

??

??

??

??

??

??

?

??

??

??

??

??

??

??

?

??

??

??

??

??

??

??

??

?

??

??

??

??

??

??

??

?

??

??

??

??

??

??

??

??

?

_

_

_

_

_

_

OO

2

4

7

9 10

•

•


67 / 100

11. Weighted paths


Example

� � � � � � � //

_

_

_

_

_

_

_

_

_

_

OO

2

3

5

6 6

•

•

//� � � � � � � � �??

??

?

??

??

??

?

??

??

??

??

?

??

??

??

??

?

??

??

??

??

??

?

??

??

??

??

??

??

?

??

??

??

??

??

??

??

?

??

??

??

??

??

??

??

??

?

??

??

??

??

??

??

??

?

??

??

??

??

??

??

??

??

?

_

_

_

_

_

_

OO

2

4

7

9 10

•

•


68 / 100

11. Weighted paths


Example

� � � � � � � //

_

_

_

_

_

_

_

_

_

_

OO

2

3

5

6 6

•

•

//� � � � � � � � �??

??

?

??

??

??

?

??

??

??

??

?

??

??

??

??

?

??

??

??

??

??

?

??

??

??

??

??

??

?

??

??

??

??

??

??

??

?

??

??

??

??

??

??

??

??

?

??

??

??

??

??

??

??

?

??

??

??

??

??

??

??

??

?

_

_

_

_

_

_

OO

2

4

7

9 10

•

•


69 / 100

The weight w(P) of a lattice path P is

w(P) =∏

a horizontal step ∈ P

w(a).

Example

� � � � � � � //

_

_

_

_

_

_

_

_

_

_

OO

2

3

5

6 6

•

•

//� � � � � � � � �??

??

?

??

??

??

?

??

??

??

??

?

??

??

??

??

?

??

??

??

??

??

?

??

??

??

??

??

??

?

??

??

??

??

??

??

??

?

??

??

??

??

??

??

??

??

?

??

??

??

??

??

??

??

?

??

??

??

??

??

??

??

??

?

_

_

_

_

_

_

OO

2

4

7

9 10

•

•

wh(P) = x2x3x5x26 we(P) = x2x4x7x9x10

70 / 100

The weight w(P) of a lattice path P is

w(P) =∏

a horizontal step ∈ P

w(a).

Example

� � � � � � � //

_

_

_

_

_

_

_

_

_

_

OO

2

3

5

6 6

•

•

//� � � � � � � � �??

??

?

??

??

??

?

??

??

??

??

?

??

??

??

??

?

??

??

??

??

??

?

??

??

??

??

??

??

?

??

??

??

??

??

??

??

?

??

??

??

??

??

??

??

??

?

??

??

??

??

??

??

??

?

??

??

??

??

??

??

??

??

?

_

_

_

_

_

_

OO

2

4

7

9 10

•

•

wh(P) = x2x3x5x26 we(P) = x2x4x7x9x10

71 / 100

12. Weight-preserving bijection between tableaux andnon intersecting lattice paths

Examplen = 6, λ = (5, 3, 2)

T =

2 3 5 6 63 4 64 5 −→

� � � � � � � //

_

_

_

_

_

_

OO

72 / 100


Examplen = 6, λ = (5, 3, 2)

T =

2 3 5 6 63 4 64 5 −→

� � � � � � � //

_

_

_

_

_

_

OO

73 / 100


Examplen = 6, λ = (5, 3, 2)

T =

2 3 5 6 63 4 64 5 −→ //

_

_

_

_

_

_

OO

2

3

5

6 6

•

•

74 / 100


Examplen = 6, λ = (5, 3, 2)

T =

2 3 5 6 63 4 64 5 −→ //

_

_

_

_

_

_

OO

2

3

5

6 6

•

•

3

4

6

•

•

75 / 100


Examplen = 6, λ = (5, 3, 2)

T =

2 3 5 6 63 4 64 5 ←→ //

_

_

_

_

_

_

OO

2

3

5

6 6

•

•

3

4

6

•

•

4

5

•

•

XT = x2x23 x2

4 x25 x3

6 P = (P1,P2,P3)wh(P) = wh(P1)wh(P2)wh(P3)

= x2x3x5x62.x3x4x6.x4x5

76 / 100


Examplen = 6, λ = (5, 3, 2)

T =

2 3 5 6 63 4 64 5 ←→ //

_

_

_

_

_

_

OO

2

3

5

6 6

•

•

3

4

6

•

•

4

5

•

•

XT = x2x23 x2

4 x25 x3


= x2x3x5x62.x3x4x6.x4x5

77 / 100


Examplen = 6, λ = (5, 3, 2)

T =

2 3 5 6 63 4 64 5 ←→ //

_

_

_

_

_

_

OO

2

3

5

6 6

•

•

3

4

6

•

•

4

5

•

•

XT = x2x23 x2

4 x25 x3


= x2x3x5x62.x3x4x6.x4x5

78 / 100

(continued)Examplen = 6, λ = (5, 3, 2)

T =

2 3 5 6 63 4 64 5 −→ //

_

_

_

_

_

_

OO

2

3

5

6 6

•

•

3

4

6

•

•

4

5

•

•

Given a n-semistandard tableau of length r ≤ n, the map

T −→ (P1,P2, . . . ,Pr),

where (P1,P2, . . . ,Pr) is an r-tuple of nonintersecting lattice paths:I i-th row of T −→ Pi from ui = (−i, 1) to vi = (λi − i, n), i = 1, . . . , r ,

whose heights of horizontal steps are the entries in the i-th row.79 / 100


T =

2 3 5 6 63 4 64 5 ←→ //

_

_

_

_

_

_

OO

� � � �

_

_

_

��

2 3 4•

•3 4 5

•

•

•

•

•

5 6

•

•

•

6

•

•

•

6

•

•

•

??

??

??

??

??

?

??

??

??

??

?

??

??

??

??

??

??

?

XT=x2x23 x2

4 x25 x3

6 Q=(Q1,Q2,Q3,Q4,Q5)

we(Q)=we(Q1)we(Q2)we(Q3)we(Q4)we(Q5)=x2x3x4.x3x5x6.x5x6.x6.x6

column i of T −→ Pi from ui = (−i + 1, i − 1) to (λ′i − i + 1, n − λ′i + i − 1)

80 / 100


T =

2 3 5 6 63 4 64 5 ←→ //

_

_

_

_

_

_

OO

� � � �

_

_

_

��

2 3 4•

•3 4 5

•

•

•

•

•

5 6

•

•

•

6

•

•

•

6

•

•

•

??

??

??

??

??

?

??

??

??

??

?

??

??

??

??

??

??

?

XT=x2x23 x2

4 x25 x3

6 Q=(Q1,Q2,Q3,Q4,Q5)

we(Q)=we(Q1)we(Q2)we(Q3)we(Q4)we(Q5)=x2x3x4.x3x5x6.x5x6.x6.x6

column i of T −→ Pi from ui = (−i + 1, i − 1) to (λ′i − i + 1, n − λ′i + i − 1)81 / 100

13. Complete and elementary homogeneous symmetricfunctions are weight generating functions of lattice pathsbetween two pointsExampleui = (−i, 1), vj = (λj − j, n)

i1 i2 . . . im︸︷︷︸λj−j+i

−→ //

OO

i1

i2 ...

•ui

•vj

P ∈ P(ui , vj)

hλj−j+i(x1, . . . , xn) =∑

1≤i1≤···≤im≤n

xi1xi2 . . . xim = GFh(P(ui , vj))

eλ′j−j+i(x1, . . . , xn) =∑

1≤i1<···<im≤nxi1xi2 . . . xim = GFe(P(ui , vj))

82 / 100

13. Complete and elementary homogeneous symmetricfunctions are weight generating functions of lattice pathsbetween two pointsExampleui = (−i, 1), vj = (λj − j, n)

i1 i2 . . . im︸︷︷︸λj−j+i

−→ //

OO

i1

i2 ...

•ui

•vj

P ∈ P(ui , vj)

hλj−j+i(x1, . . . , xn) =∑

1≤i1≤···≤im≤n

xi1xi2 . . . xim = GFh(P(ui , vj))

eλ′j−j+i(x1, . . . , xn) =∑

1≤i1<···<im≤nxi1xi2 . . . xim = GFe(P(ui , vj))

83 / 100

7. The (classical) Jacobi-Trudi determinant formulas

|hλj−j+i(x)|r×r, h-formula

=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

hλ1(x) hλ2−1(x) hλ3−2(x) . . . hλr−1−r(x) hλr−r+1(x)hλ1+1(x) hλ2(x) hλ3−1(x) . . . hλr−1−r+1(x) hλr−r+2(x)hλ1+2(x) hλ2+1(x) hλ3(x) . . . hλr−1−r+2(x) hλr−r+3(x)

......

......

...

hλ1+r−1(x) hλ2+r−2(x) hλ3+r−3(x) . . . hλr−1+1(x) hλr (x)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣where we set h0 = 1, hk = 0, k < 0.

|eλ′j−j+i(x)|λ1×λ1 , e-formula,where we set e0 = 1, ek = 0, k > n.

84 / 100

7. The (classical) Jacobi-Trudi determinant formulas

|hλj−j+i(x)|r×r, h-formula

=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

hλ1(x) hλ2−1(x) hλ3−2(x) . . . hλr−1−r(x) hλr−r+1(x)hλ1+1(x) hλ2(x) hλ3−1(x) . . . hλr−1−r+1(x) hλr−r+2(x)hλ1+2(x) hλ2+1(x) hλ3(x) . . . hλr−1−r+2(x) hλr−r+3(x)

......

......

...

hλ1+r−1(x) hλ2+r−2(x) hλ3+r−3(x) . . . hλr−1+1(x) hλr (x)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣where we set h0 = 1, hk = 0, k < 0.

|eλ′j−j+i(x)|λ1×λ1 , e-formula,where we set e0 = 1, ek = 0, k > n.

85 / 100

13. The Lindstrom-Gessel-Viennot involutionm = r , λ1

|GFP(ui , vj)|m×m =∑

σ∈Sm

sgn(σ)∏m

i=1 GF(P(ui , vσ(i))

P(u, vσ) = {P = (P1, . . . ,Pm) : Pi ∈ P(ui , vσ(i)), 1 ≤ i ≤ m}

= P(u1, vσ(1)) × · · · × P(um, vσ(m))

w(P) =m∏

i=1

w(Pi)

GF(P(u, vσ)) =∑

P∈P(u,vσ)

w(P) =m∏

i=1

GF(P(ui , vσ(i))).

=∑

σ∈Sm , P∈P(u,vσ)

sgn(σ)w(P)

86 / 100



σ∈Sm

sgn(σ)∏m



= P(u1, vσ(1)) × · · · × P(um, vσ(m))

w(P) =m∏

i=1

w(Pi)

GF(P(u, vσ)) =∑

P∈P(u,vσ)

w(P) =m∏

i=1

GF(P(ui , vσ(i))).

=∑


sgn(σ)w(P)

87 / 100



σ∈Sm

sgn(σ)∏m



= P(u1, vσ(1)) × · · · × P(um, vσ(m))

w(P) =m∏

i=1

w(Pi)

GF(P(u, vσ)) =∑

P∈P(u,vσ)

w(P) =m∏

i=1

GF(P(ui , vσ(i))).

=∑


sgn(σ)w(P)

88 / 100



σ∈Sm

sgn(σ)∏m



= P(u1, vσ(1)) × · · · × P(um, vσ(m))

w(P) =m∏

i=1

w(Pi)

GF(P(u, vσ)) =∑

P∈P(u,vσ)

w(P) =m∏

i=1

GF(P(ui , vσ(i))).

=∑


sgn(σ)w(P)

89 / 100

Example

//

OO

•v2

•u2

��

__

��

•u1

•v3

w

O�O�O�O�

/o/o/o/o/o/o/o/o/o/o/o

O�O�O�O�

•v1

•u3

LGV involution //

OO

___

��•v2

•u2

��

__

•u1

•v3

w

O�O�O�O�


O�O�O�O�

•v1

•u3

(P1,P2,P3)∈P(u,vσ) (P′1,P′2,P3)∈P(u,vπσ)

σ = (1 3) π = (2 3)(1 3)sgn(π) = −sign(σ)

The LGV-involution is a weight -preserving but sign-reversinginvolution on the set of all those m-tuples that are intersecting.

90 / 100

Example

//

OO

•v2

•u2

��

__

��

•u1

•v3

w

O�O�O�O�


O�O�O�O�

•v1

•u3

LGV involution //

OO

___

��•v2

•u2

��

__

•u1

•v3

w

O�O�O�O�


O�O�O�O�

•v1

•u3

(P1,P2,P3)∈P(u,vσ) (P′1,P′2,P3)∈P(u,vπσ)

σ = (1 3) π = (2 3)(1 3)sgn(π) = −sign(σ)

The LGV-involution is a weight -preserving but sign-reversinginvolution on the set of all those m-tuples that are intersecting.

91 / 100

By the LGV involution all the intersecting m-tuples of paths will canceland only the nonintersecting will survive. The associated permutationfor such m-tuples is the identity. These m -paths correspond exactlyto the n-semistandard tableaux of shape λ


σ∈Sm

sgn(σ)∏m


=∑

σ∈Sm , P∈P(u,vσ)sgn(σ)w(P)

=∑

nonintersecting m-tuple P ∈ P(u, v)w(P)

=∑

n-semistandard tableau T of shape λxT

= sn(λ, x)

92 / 100



σ∈Sm

sgn(σ)∏m


=∑


=∑


=∑


= sn(λ, x)

93 / 100



σ∈Sm

sgn(σ)∏m


=∑


=∑


=∑


= sn(λ, x)

94 / 100

Exampleu1=(−1,1), v1=(1,2), u2=(−2,1), v2=(−1,2)

(P1,P2)∈P(u1,v1)×P(u2,v2)

//

OO

___

��

• •

• •

//

OO

__

��

__

• •

• •

x21 x2 + x1x2

2

//

OO

��

___

• •

• •

//

OO

__

��

__

• •

• •

//

OO

___

��

• •

• •

//

OO

��

___

• •

• •

(x1 + x2)(x21 + x1x2 + x2

2 ) = x21 x2 + x1x2

2 +x31 +x2

1 x2+x1x22 +x3

2

(P1,P2)∈P(u1,v2)×P(u2,v1)

//

OO

��

• •

• •

//

OO

��

• •

• •

//

OO

��

• •

• •

//

OO

��

• •

• •

x31 +x2

1 x2+x1x22 +x3

2

95 / 100

Exampleu1=(−1,1), v1=(1,2), u2=(−2,1), v2=(−1,2)

(P1,P2)∈P(u1,v1)×P(u2,v2)

//

OO

___

��

• •

• •

//

OO

__

��

__

• •

• •

x21 x2 + x1x2

2

//

OO

��

___

• •

• •

//

OO

__

��

__

• •

• •

//

OO

___

��

• •

• •

//

OO

��

___

• •

• •

(x1 + x2)(x21 + x1x2 + x2

2 ) = x21 x2 + x1x2

2 +x31 +x2

1 x2+x1x22 +x3

2

(P1,P2)∈P(u1,v2)×P(u2,v1)

//

OO

��

• •

• •

//

OO

��

• •

• •

//

OO

��

• •

• •

//

OO

��

• •

• •

x31 +x2

1 x2+x1x22 +x3

2

96 / 100

Exampleu1=(−1,1), v1=(1,2), u2=(−2,1), v2=(−1,2)

(P1,P2)∈P(u1,v1)×P(u2,v2)

//

OO

___

��

• •

• •

//

OO

__

��

__

• •

• •

x21 x2 + x1x2

2

//

OO

��

___

• •

• •

//

OO

__

��

__

• •

• •

//

OO

___

��

• •

• •

//

OO

��

___

• •

• •

(x1 + x2)(x21 + x1x2 + x2

2 ) = x21 x2 + x1x2

2 +x31 +x2

1 x2+x1x22 +x3

2

(P1,P2)∈P(u1,v2)×P(u2,v1)

//

OO

��

• •

• •

//

OO

��

• •

• •

//

OO

��

• •

• •

//

OO

��

• •

• •

x31 +x2

1 x2+x1x22 +x3

297 / 100

Jacobi-Trudi continued

n = 2, x = (x1, x2), λ = (2, 1)I h-formula

|hλj−j+i(x)| =

∣∣∣∣∣∣ h2 h0

h3 h1

∣∣∣∣∣∣ = h2(x)h1(x) − h3(x).1

= (x21 + x2

2 + x1x2)(x1 + x2) − (x31 + x2

1 x2 + x1x22 + x3

2 )

= x31 + 2x1x2

2 + 2x1x22 + x3

2 − (x31 + x2

1 x2 + x1x22 + x3

2 ) = x21 x2 + x1x2

2 .

I e-formula


∣∣∣∣∣∣ e2 e0

e3 e1

∣∣∣∣∣∣ = e2(x)e1(x) − 0.1

= (x1x2)(x1 + x2) = x21 x2 + x1x2

2 .

98 / 100

Jacobi-Trudi continued

n = 2, x = (x1, x2), λ = (2, 1)I h-formula

|hλj−j+i(x)| =

∣∣∣∣∣∣ h2 h0

h3 h1

∣∣∣∣∣∣ = h2(x)h1(x) − h3(x).1

= (x21 + x2

2 + x1x2)(x1 + x2) − (x31 + x2

1 x2 + x1x22 + x3

2 )

= x31 + 2x1x2

2 + 2x1x22 + x3

2 − (x31 + x2

1 x2 + x1x22 + x3

2 ) = x21 x2 + x1x2

2 .

I e-formula


∣∣∣∣∣∣ e2 e0

e3 e1

∣∣∣∣∣∣ = e2(x)e1(x) − 0.1

= (x1x2)(x1 + x2) = x21 x2 + x1x2

2 .

99 / 100

ReferencesE. Bender, D. Knuth,Enumeration of plane partitions, J. Combin. Theory, Serie A,13 (1972), 40-54.

C. Jacobi, De functionibus alternantibus earumque divisione per productum ediferentlis elementorum conflatum, J. Reine Angew. Math. 1841B. Lindstrom, On the vector representation of induced matroids, Bull. LondonMath. Soc. 5 (1973), 85-90.

M. Gessel, X. Viennot, Determinants, paths, and plane partitions, preprint 1988.

R. Proctor, Equivalence of the combinatorial and the classical definitions of Schurfunctions, J. Combin. Theory Ser. A 51 (1989), no. 1, 130–135.

W. Fulton, Young tableaux: With Applications to Representation Theory andGeometry, Cambridge University Press, 1999.

M. Fulmek, C. Krattenthaler, Lattice paths proofs for determinantal formulas forsymplectic and orthogonal characters,J. Combin. Theory Ser. A 77 (1997) 3–50.

B. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms,and Symmetric Functions, Springer 2001.

I. Schur, Uber eine Klasse von Matrizen die sich einer gegeben Matrix lassen,Inaugural dissertation, Berlin, 1901.R. Stanley, Enumerative Combinatorics, Cambridge University Press, vol 2, 2001.100 / 100

schur functions: tableaux, determinant formulas and

Documents