A Pieri rule for sk ew shapes Peter McNamara Bucknell University - - PowerPoint PPT Presentation

▶
a pieri rule for sk ew shapes
SMART_READER_LITE
LIVE PREVIEW

A Pieri rule for sk ew shapes Peter McNamara Bucknell University - - PowerPoint PPT Presentation

Jan 10, 2023 248 likes •1.24k views

A Pieri rule for sk ew shapes Peter McNamara Bucknell University Joint work with: Sami Assaf MIT 4 August 2010 Full paper available from www.facstaff.bucknell.edu/pm040/ A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 1 A

slide-1
SLIDE 1

A Pieri rule for sk ew shapes

Peter McNamara Bucknell University Joint work with: Sami Assaf MIT 4 August 2010 Full paper available from www.facstaff.bucknell.edu/pm040/

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 1

slide-2
SLIDE 2

A Pieri rule for sk ew shapes

Peter McNamara Bucknell University Joint work with: Sami Assaf MIT 4 August 2010 Full paper available from www.facstaff.bucknell.edu/pm040/

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 1

slide-3
SLIDE 3

Outline

◮ Background on skew Schur functions and Pieri rule ◮ Main result ◮ Some highlights of the combinatorial proof ◮ 3 further-development nuggets

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 2

slide-4
SLIDE 4

Schur functions

Cauchy, 1815

◮ Partition

λ = (λ1, λ2, . . . , λℓ)

◮ Young diagram

Example: λ = (4, 4, 3, 1)

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 3

slide-5
SLIDE 5

Schur functions

Cauchy, 1815

◮ Partition

λ = (λ1, λ2, . . . , λℓ)

◮ Young diagram

Example: λ = (4, 4, 3, 1)

◮ Semistandard Young

tableau (SSYT)

6 3 4 9 1 4 5 7 6 4 3 4

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 3

slide-6
SLIDE 6

Schur functions

Cauchy, 1815

◮ Partition

λ = (λ1, λ2, . . . , λℓ)

◮ Young diagram

Example: λ = (4, 4, 3, 1)

◮ Semistandard Young

tableau (SSYT)

6 3 4 9 1 4 5 7 6 4 3 4

The Schur function sλ in the variables x = (x1, x2, . . .) is then defined by sλ =

  • SSYT T

x#1’s in T

1

x#2’s in T

2

· · · . Example: s4431 = x1x2

3x4 4x5x2 6x7x9 + · · · .

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 3

slide-7
SLIDE 7

Skew Schur functions

Cauchy, 1815

◮ Partition

λ = (λ1, λ2, . . . , λℓ)

◮ µ fits inside λ ◮ Young diagram

Example: λ/µ = (4, 4, 3, 1)/(3, 1)

◮ Semistandard Young

tableau (SSYT)

4 9 5 7 6 4 4 6

The skew Schur function sλ/µ in the variables x = (x1, x2, . . .) is then defined by sλ/µ =

  • SSYT T

x#1’s in T

1

x#2’s in T

2

· · · . Example: s4431/31 = x3

4x5x2 6x7x9 + · · · .

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 3

slide-8
SLIDE 8

Skew Schur functions

Example: s3(x1, x2, . . .)

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 4

slide-9
SLIDE 9

Skew Schur functions

Example:

k j i

s3(x1, x2, . . .) =

  • i≤j≤k

xixjxk

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 4

slide-10
SLIDE 10

Skew Schur functions

Example:

k j i

s3(x1, x2, . . .) =

  • i≤j≤k

xixjxk Question: Why do we care about skew Schur functions?

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 4

slide-11
SLIDE 11

Skew Schur functions

Example:

k j i

s3(x1, x2, . . .) =

  • i≤j≤k

xixjxk Question: Why do we care about skew Schur functions?

◮ Fact: Skew Schur functions are symmetric in x1, x2, . . ..

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 4

slide-12
SLIDE 12

Skew Schur functions

Example:

k j i

s3(x1, x2, . . .) =

  • i≤j≤k

xixjxk Question: Why do we care about skew Schur functions?

◮ Fact: Skew Schur functions are symmetric in x1, x2, . . .. ◮ Fact: The Schur functions form a basis for the algebra of

symmetric functions.

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 4

slide-13
SLIDE 13

Skew Schur functions

Example:

k j i

s3(x1, x2, . . .) =

  • i≤j≤k

xixjxk Question: Why do we care about skew Schur functions?

◮ Fact: Skew Schur functions are symmetric in x1, x2, . . .. ◮ Fact: The Schur functions form a basis for the algebra of

symmetric functions.

◮ Strong connections with representation theory of Sn and

GL(n, C), Schubert Calculus, eigenvalues of Hermitian matrices, ....

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 4

slide-14
SLIDE 14

The Pieri rule

The (classical) Pieri rule expands sλsn in terms of {sµ}.

=

+ + + +

=

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 5

slide-15
SLIDE 15

The Pieri rule

The (classical) Pieri rule expands sλsn in terms of {sµ}. Theorem [Pieri, 1893]: For a partition λ and positive integer n, sλsn =

  • λ+/λ n−hor. strip

sλ+, where the sum is over all λ+ such that λ+/λ is a horizontal strip with n boxes.

=

+ + + +

=

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 5

slide-16
SLIDE 16

The Pieri rule

The (classical) Pieri rule expands sλsn in terms of {sµ}. Theorem [Pieri, 1893]: For a partition λ and positive integer n, sλsn =

  • λ+/λ n−hor. strip

sλ+, where the sum is over all λ+ such that λ+/λ is a horizontal strip with n boxes. Example: s322s2 = s3222 + s3321 + s4221 + s432 + s522.

+ + + +

= =

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 5

slide-17
SLIDE 17

The Pieri rule

The (classical) Pieri rule expands sλsn in terms of {sµ}. Theorem [Pieri, 1893]: For a partition λ and positive integer n, sλsn =

  • λ+/λ n−hor. strip

sλ+, where the sum is over all λ+ such that λ+/λ is a horizontal strip with n boxes. Example: s322s2 = s3222 + s3321 + s4221 + s432 + s522.

+ + + +

= =

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 5

slide-18
SLIDE 18

The Pieri rule

The (classical) Pieri rule expands sλsn in terms of {sµ}. Theorem [Pieri, 1893]: For a partition λ and positive integer n, sλsn =

  • λ+/λ n−hor. strip

sλ+, where the sum is over all λ+ such that λ+/λ is a horizontal strip with n boxes. Example: s(322)∗(2) = s322s2 = s3222 + s3321 + s4221 + s432 + s522.

+ + + +

= =

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 5

slide-19
SLIDE 19

Pieri-type rules in other settings

◮ k-Schur functions [Lapointe–Morse] ◮ Schubert polynomials [Lascoux–Schützenberger,

Lenart–Sottile, Manivel, Sottile, Winkel]

◮ LLT polynomials [Lam] ◮ Schubert classes in the affine Grassmannian

[Lam–Lapointe–Morse–Shimozono]

◮ Hall-Littlewood polynomials [Morris] ◮ Jack polynomials [Lassalle, Stanley] ◮ Macdonald polynomials [Koornwinder, Macdonald] ◮ Quasisymmetric Schur functions [Haglund, Luoto, Mason,

van Willigenburg]

◮ Grothendieck polynomials [Lenart–Sottile] ◮ Factorial Grothendieck polynomials

[McNamara (no relation!)]

◮ ....

Notably absent: skew Schur functions

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 6

slide-20
SLIDE 20

The skew Pieri rule

The skew Pieri rule expands sλ/µsn in terms of {sτ/σ}.

+ +

=

+ + +

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 7

slide-21
SLIDE 21

The skew Pieri rule

The skew Pieri rule expands sλ/µsn in terms of {sτ/σ}. Theorem [Assaf–McN.]: For a skew shape λ/µ and positive integer n, sλ/µsn =

n

  • k=0

(−1)k

  • λ+/λ (n−k)-hor. strip

µ/µ− k-vert. strip

sλ+/µ− , where λ+/λ is a horizontal strip with n − k boxes and µ/µ− is a vertical strip with k boxes.

+ +

=

+ + +

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 7

slide-22
SLIDE 22

The skew Pieri rule

The skew Pieri rule expands sλ/µsn in terms of {sτ/σ}. Theorem [Assaf–McN.]: For a skew shape λ/µ and positive integer n, sλ/µsn =

n

  • k=0

(−1)k

  • λ+/λ (n−k)-hor. strip

µ/µ− k-vert. strip

sλ+/µ− , where λ+/λ is a horizontal strip with n − k boxes and µ/µ− is a vertical strip with k boxes. Example:

+ +

=

+ + +

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 7

slide-23
SLIDE 23

The skew Pieri rule

The skew Pieri rule expands sλ/µsn in terms of {sτ/σ}. Theorem [Assaf–McN.]: For a skew shape λ/µ and positive integer n, sλ/µsn =

n

  • k=0

(−1)k

  • λ+/λ (n−k)-hor. strip

µ/µ− k-vert. strip

sλ+/µ− , where λ+/λ is a horizontal strip with n − k boxes and µ/µ− is a vertical strip with k boxes. Example:

+ +

=

+ + +

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 7

slide-24
SLIDE 24

The skew Pieri rule

The skew Pieri rule expands sλ/µsn in terms of {sτ/σ}. Theorem [Assaf–McN.]: For a skew shape λ/µ and positive integer n, sλ/µsn =

n

  • k=0

(−1)k

  • λ+/λ (n−k)-hor. strip

µ/µ− k-vert. strip

sλ+/µ− , where λ+/λ is a horizontal strip with n − k boxes and µ/µ− is a vertical strip with k boxes. Example:

+

=

+ + + +

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 7

slide-25
SLIDE 25

The combinatorial proof

sλ/µsn =

n

  • k=0

(−1)k

  • λ+/λ (n−k)-hor. strip

µ/µ− k-vert. strip

sλ+/µ− ,

+

=

+ + + +

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 8

slide-26
SLIDE 26

The combinatorial proof

sλ/µsn =

n

  • k=0

(−1)k

  • λ+/λ (n−k)-hor. strip

µ/µ− k-vert. strip

sλ+/µ− ,

+

=

+ + + +

Technique: a sign-reversing involution on SSYT that:

◮ Preserves entries appearing in each SSYT; ◮ Has fixed points with k = 0 in bijection with SSYT of shape

(λ/µ) ∗ (n);

◮ (Remaining SSYT with k even) ←

→ (SSYT with k odd).

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 8

slide-27
SLIDE 27

Robinson-Schensted-Knuth insertion

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 9

slide-28
SLIDE 28

Robinson-Schensted-Knuth insertion

7 4 9 2 7 7 4 5 8 1 1 7 4 3 7 7 2 7 7 7 5 4 8 9 4 2 1 1 4 3 2

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 9

slide-29
SLIDE 29

Robinson-Schensted-Knuth insertion

2 7 7 7 5 4 8 9 4 2 1 1 4 9 2 7 7 4 5 8 1 1 7 4 3 2 4 3 7 7 7

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 9

slide-30
SLIDE 30

Robinson-Schensted-Knuth insertion

2 7 7 7 5 4 8 9 4 2 1 1 4 9 2 7 7 4 5 8 1 1 7 4 3 2 4 3 7 7 7

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 9

slide-31
SLIDE 31

Robinson-Schensted-Knuth insertion

3 7 7 7 5 4 8 9 4 2 1 1 4 9 2 7 7 4 5 8 1 1 7 4 3 2 4 2 7 7 7

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 9

slide-32
SLIDE 32

Robinson-Schensted-Knuth insertion

3 7 7 7 5 4 8 9 4 2 1 1 4 9 2 7 7 4 5 8 1 1 7 4 3 2 4 2 7 7 7

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 9

slide-33
SLIDE 33

Robinson-Schensted-Knuth insertion

4 7 7 7 5 4 8 9 4 2 1 1 4 9 2 7 7 4 5 8 1 1 7 4 3 2 3 2 7 7 7

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 9

slide-34
SLIDE 34

Robinson-Schensted-Knuth insertion

4 7 7 7 5 4 8 9 4 2 1 1 4 9 2 7 7 4 5 8 1 1 7 4 3 2 3 2 7 7 7

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 9

slide-35
SLIDE 35

Robinson-Schensted-Knuth insertion

7 7 7 7 5 4 8 9 4 2 1 1 4 9 2 7 7 4 5 8 1 1 7 4 3 2 4 3 2 7

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 9

slide-36
SLIDE 36

Robinson-Schensted-Knuth insertion

7 7 7 7 5 4 8 9 4 2 1 1 4 9 2 7 7 4 5 8 1 1 7 4 3 2 4 3 2 7 7

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 9

slide-37
SLIDE 37

Robinson-Schensted-Knuth insertion

7 7 7 7 5 4 8 9 4 2 1 1 4 9 2 7 7 4 5 8 1 1 7 4 3 2 4 3 2 7 7

Facts:

◮ The result is an SSYT. ◮ This process is reversible.

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 9

slide-38
SLIDE 38

Robinson-Schensted-Knuth insertion

7 7 7 7 5 4 8 9 4 2 1 1 4 9 2 7 7 4 5 8 1 1 7 4 3 2 4 3 2 7 7

Facts:

◮ The result is an SSYT. ◮ This process is reversible. ◮ RSK insertion can be used to give a combinatorial proof of

the classicial Pieri rule.

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 9

slide-39
SLIDE 39

Sagan & Stanley’s generalization to skew shapes

External insertion (just like RSK):

7 7 7 7 5 4 8 9 4 4 9 7 7 4 5 8 7 4 3 4 3 2 7 7 2 4 9 2 7 7 4 5 8 1 1 7 4 3 7 7 2 7 7 7 5 2 8 9 4 3 7 2

1

3 2

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 10

slide-40
SLIDE 40

Sagan & Stanley’s generalization to skew shapes

External insertion (just like RSK):

2 4 9 2 7 7 4 5 8 1 1 7 4 3 7 7 2 7 7 5 1 8 4 3 3 4 3 7 6 4 9 2 7 7 4 5 8 1 1 7 4 3 7 7 2 7 7 7 5 2 8 9 4 3 7 2

1

3 2

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 10

slide-41
SLIDE 41

Sagan & Stanley’s generalization to skew shapes

External insertion (just like RSK):

6 4 9 2 7 7 4 5 8 1 1 7 4 3 7 7 2 7 7 5 8 4 3 3 4 3 1 7 2 4 9 2 7 7 4 5 8 1 1 7 4 3 7 7 2 7 7 7 5 2 8 9 4 3 7 2

1

3 2

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 10

slide-42
SLIDE 42

Sagan & Stanley’s generalization to skew shapes

External insertion (just like RSK):

2 7 7 5 1 8 4 1 4 7 3 5 3 3 3 3 4 7 3 7 4 7 7 8 2 6 6 4 9 2 7 7 4 5 8 1 1 7 4 3 7 7 2 7 7 7 5 2 8 9 4 3 7 2

1

3 2

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 10

slide-43
SLIDE 43

Sagan & Stanley’s generalization to skew shapes

External insertion (just like RSK):

2 7 7 5 1 8 4 1 4 7 3 5 3 3 3 3 4 7 3 7 4 7 7 8 2 6 6

Internal insertion:

4 9 2 7 7 4 5 8 1 1 7 4 3 7 7 2 7 7 7 5 2 8 9 4 3 7 2

1

3 2

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 10

slide-44
SLIDE 44

Sagan & Stanley’s generalization to skew shapes

External insertion (just like RSK):

2 7 7 5 1 8 4 1 4 7 3 5 3 3 3 3 4 7 3 7 4 7 7 8 2 6 6

Internal insertion:

4 9 2 7 7 4 5 8 1 1 7 4 3 7 7 2 7 7 7 5 2 8 9 4 3 2

1

2 3 7

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 10

slide-45
SLIDE 45

Sagan & Stanley’s generalization to skew shapes

External insertion (just like RSK):

2 7 7 5 1 8 4 1 4 7 3 5 3 3 3 3 4 7 3 7 4 7 7 8 2 6 6

Internal insertion:

4 9 2 7 7 4 5 8 1 1 7 4 3 7 7 2 7 7 7 5 2 8 9 4 3 2

1

2 3 7

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 10

slide-46
SLIDE 46

Sagan & Stanley’s generalization to skew shapes

External insertion (just like RSK):

2 7 7 5 1 8 4 1 4 7 3 5 3 3 3 3 4 7 3 7 4 7 7 8 2 6 6

Internal insertion:

7 7 7 5 2 8 9 4 3 2

1

2 3 7 2 9 7 7 4 5 8 7 2 3 3 7

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 10

slide-47
SLIDE 47

Sagan & Stanley’s generalization to skew shapes

External insertion (just like RSK):

2 7 7 5 1 8 4 1 4 7 3 5 3 3 3 3 4 7 3 7 4 7 7 8 2 6 6

Internal insertion:

7 7 7 5 2 8 9 4 3 2

1

2 3 7 2 9 7 7 4 5 8 7 2 3 3 7

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 10

slide-48
SLIDE 48

Sagan & Stanley’s generalization to skew shapes

External insertion (just like RSK):

2 7 7 5 1 8 4 1 4 7 3 5 3 3 3 3 4 7 3 7 4 7 7 8 2 6 6

Internal insertion:

7 7 7 5 2 8 9 4 3 2

1

2 3 7 2 9 7 7 4 5 8 7 2 3 3 7

Would be great: if internal insertion gave the necessary bijection for skew Pieri rule.

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 10

slide-49
SLIDE 49

Sagan & Stanley’s generalization to skew shapes

External insertion (just like RSK):

2 7 7 5 1 8 4 1 4 7 3 5 3 3 3 3 4 7 3 7 4 7 7 8 2 6 6

Internal insertion:

7 7 7 5 2 8 9 4 3 2

1

2 3 7 2 9 7 7 4 5 8 7 2 3 3 7

Would be great: if internal insertion gave the necessary bijection for skew Pieri rule. In general, it’s not that easy....

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 10

slide-50
SLIDE 50

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

7 2 2 2 3 9 5 2 1 1 2 3 7 6 5 2 2 6 5 2 3 9 5 2 1 3 2 1 2 2 2 3 9 5 2 1 1 2 3 7 6 5 slide 2 2 6 5 2 3 9 5 2 1 upward 2 3 7 1

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-51
SLIDE 51

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

5 2 2 2 3 9 2 3 2 2 6 5 1 2 3 9 5 7 5 2 1 2 1 1 2 3 7 6 2 2 2 3 9 5 2 1 1 2 3 7 6 5 slide 2 2 6 5 2 3 9 5 2 1 upward 2 3 7 1

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-52
SLIDE 52

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

1 2 2 2 3 9 2 2 2 6 5 1 2 3 9 5 7 6 5 2 1 5 3 2 1 7 3 2 2 2 2 3 9 5 2 1 1 2 3 7 6 5 slide 2 2 6 5 2 3 9 5 2 1 upward 2 3 7 1

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-53
SLIDE 53

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

5 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 6 5 2 1 7 3 2 1 2 2 2 3 9 5 2 1 1 2 3 7 6 5 slide 2 2 6 5 2 3 9 5 2 1 upward 2 3 7 1

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-54
SLIDE 54

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

1 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 6 2 3 7 5 1 5 2 2 2 2 3 9 5 2 1 1 2 3 7 6 5 slide 2 2 6 5 2 3 9 5 2 1 upward 2 3 7 1

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-55
SLIDE 55

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

1 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 6 2 3 7 5 1 5 2 2 2 2 3 9 5 2 1 1 2 3 7 6 5 slide 2 2 6 5 2 3 9 5 2 1 upward 2 3 7 1

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-56
SLIDE 56

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

1 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 5 6 2 3 7 2 5 2 2 2 3 9 5 2 1 1 2 3 7 6 5 slide 2 2 6 5 2 3 9 5 2 1 upward 2 3 7 1

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-57
SLIDE 57

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

1 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 5 6 2 3 7 2 5 2 2 2 3 9 5 2 1 1 2 3 7 6 5 slide 2 2 6 5 2 3 9 5 2 1 upward 2 3 7 1

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-58
SLIDE 58

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

1 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 5 6 2 3 7 2 5 2 2 2 3 9 5 2 1 1 2 3 7 6 5 slide 2 2 6 5 2 3 9 5 2 1 upward 2 3 7 1

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-59
SLIDE 59

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

5 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 6 2 2 1 7 3 5 2 2 2 3 9 5 2 1 1 2 3 7 6 5 slide 2 2 6 5 2 3 9 5 2 1 upward 2 3 7 1

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-60
SLIDE 60

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

5 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 6 2 2 1 7 3 5 2 2 2 3 9 5 2 1 1 2 3 7 6 5 slide 2 2 6 5 2 3 9 5 2 1 upward 2 3 7 1

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-61
SLIDE 61

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

5 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 2 2 1 7 3 5 6 2 2 2 3 9 5 2 1 1 2 3 7 6 5 slide 2 2 6 5 2 3 9 5 2 1 upward 2 3 7 1

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-62
SLIDE 62

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

5 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 2 2 1 7 3 5 6

Example 2: but stop if you’re left of an upward path. Then perform the internal insertion, and then insert the overflow.

1 2 2 2 3 9 5 2 1 1 2 3 7 6 5 2 2 6 5 2 3 9 5 2 1 2 3 7

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-63
SLIDE 63

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

5 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 2 2 1 7 3 5 6

Example 2: but stop if you’re left of an upward path. Then perform the internal insertion, and then insert the overflow.

5 2 2 2 3 9 5 2 1 2 3 7 1 2 2 2 3 9 5 1 2 1 2 3 7 5 6 6

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-64
SLIDE 64

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

5 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 2 2 1 7 3 5 6

Example 2: but stop if you’re left of an upward path. Then perform the internal insertion, and then insert the overflow.

6 2 2 2 3 9 5 2 1 2 3 7 1 2 2 2 3 9 5 1 2 1 2 3 7 5 6 5

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-65
SLIDE 65

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

5 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 2 2 1 7 3 5 6

Example 2: but stop if you’re left of an upward path. Then perform the internal insertion, and then insert the overflow.

2 2 2 3 9 5 2 3 7 1 2 2 2 3 9 1 2 7 5 6 5 6 5 3 2 1 1 2

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-66
SLIDE 66

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

5 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 2 2 1 7 3 5 6

Example 2: but stop if you’re left of an upward path. Then perform the internal insertion, and then insert the overflow.

5 2 2 3 9 5 2 3 7 1 2 2 2 3 9 1 2 7 5 6 6 1 2 2 5 1 2 3

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-67
SLIDE 67

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

5 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 2 2 1 7 3 5 6

Example 2: but stop if you’re left of an upward path. Then perform the internal insertion, and then insert the overflow.

2 2 9 5 2 3 7 1 2 2 1 2 7 5 6 6 1 2 2 5 1 2 3 5 9 3 2 3

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-68
SLIDE 68

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

5 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 2 2 1 7 3 5 6

Example 2: but stop if you’re left of an upward path. Then perform the internal insertion, and then insert the overflow.

! 2 9 5 2 3 7 1 2 2 1 2 7 5 6 6 1 2 2 5 1 2 3 5 9 3 2 3 2

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-69
SLIDE 69

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

5 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 2 2 1 7 3 5 6

Example 2: but stop if you’re left of an upward path. Then perform the internal insertion, and then insert the overflow.

6 2 9 5 2 3 7 1 2 2 1 2 7 5 6 1 2 2 5 1 3 9 2 2 3 5 3 2

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-70
SLIDE 70

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

5 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 2 2 1 7 3 5 6

Example 2: but stop if you’re left of an upward path. Then perform the internal insertion, and then insert the overflow.

5 2 9 5 2 3 7 1 2 2 7 5 6 1 2 2 5 1 3 9 2 2 3 3 2 6 1 2

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-71
SLIDE 71

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

5 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 2 2 1 7 3 5 6

Example 2: but stop if you’re left of an upward path. Then perform the internal insertion, and then insert the overflow.

1 2 9 5 2 3 7 1 2 2 7 5 6 1 2 2 5 1 3 9 2 2 3 3 2 6 5 2

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-72
SLIDE 72

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

5 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 2 2 1 7 3 5 6

Example 2: but stop if you’re left of an upward path. Then perform the internal insertion, and then insert the overflow.

6 2 9 5 2 3 7 1 2 2 5 6 1 2 2 5 1 3 9 2 2 3 5 2 1 2 3 7

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-73
SLIDE 73

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

5 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 2 2 1 7 3 5 6

Example 2: but stop if you’re left of an upward path. Then perform the internal insertion, and then insert the overflow.

6 2 9 5 2 3 7 1 2 2 5 6 1 2 2 3 9 2 2 3 5 2 1 7 3 2 1 5

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-74
SLIDE 74

The sign-reversing involution on SSYT of form λ+/µ−

Example 1: reverse insert until you perform a reverse internal

  • insertion. Then externally insert the overflow.

5 2 2 2 3 9 2 3 2 2 1 2 6 5 1 2 3 9 5 7 1 2 2 1 7 3 5 6

Example 2: but stop if you’re left of an upward path. Then perform the internal insertion, and then insert the overflow.

6 2 9 5 2 3 7 1 2 2 5 6 1 2 2 3 9 2 2 3 5 2 1 7 3 2 1 5

Bijection between these two types that is sign-reversing.

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 11

slide-75
SLIDE 75

The sign-reversing involution on SSYT of form λ+/µ−

Example 3: Fixed points. These should be in bijection with SSYT of shape (λ/µ) ∗ (n). 3 2 4 1 2 3 3 7 3 1 5

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 12

slide-76
SLIDE 76

The sign-reversing involution on SSYT of form λ+/µ−

Example 3: Fixed points. These should be in bijection with SSYT of shape (λ/µ) ∗ (n). 3 2 1 3 1 3 2 1 3 1 3 5 4 3 2 3 7 5 4 3 2 7

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 12

slide-77
SLIDE 77

The sign-reversing involution on SSYT of form λ+/µ−

Example 3: Fixed points. These should be in bijection with SSYT of shape (λ/µ) ∗ (n). 3 2 1 3 1 2 1 3 1 3 5 4 3 2 3 7 5 4 3 2 7 3

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 12

slide-78
SLIDE 78

The sign-reversing involution on SSYT of form λ+/µ−

Example 3: Fixed points. These should be in bijection with SSYT of shape (λ/µ) ∗ (n). 7 2 1 3 1 2 1 3 1 3 5 4 3 2 5 4 3 2 3 7 3 3

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 12

slide-79
SLIDE 79

The sign-reversing involution on SSYT of form λ+/µ−

Example 3: Fixed points. These should be in bijection with SSYT of shape (λ/µ) ∗ (n). 7 2 1 3 1 2 1 3 1 3 5 4 3 2 5 4 3 2 7 3 3 3

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 12

slide-80
SLIDE 80

The sign-reversing involution on SSYT of form λ+/µ−

Example 3: Fixed points. These should be in bijection with SSYT of shape (λ/µ) ∗ (n). 5 2 1 3 1 2 1 3 1 3 7 3 3 3 7 4 3 2 5 4 3 2

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 12

slide-81
SLIDE 81

The sign-reversing involution on SSYT of form λ+/µ−

Example 3: Fixed points. These should be in bijection with SSYT of shape (λ/µ) ∗ (n). 3 2 1 3 1 3 7 3 5 4 3 2 2 1 3 1 2 3 4 5 3 7

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 12

slide-82
SLIDE 82

The sign-reversing involution on SSYT of form λ+/µ−

Example 3: Fixed points. These should be in bijection with SSYT of shape (λ/µ) ∗ (n). 3 2 1 3 1 3 7 3 5 4 3 2 2 1 3 1 2 3 4 5 3 7 Conclusion: sλ/µsn =

n

  • k=0

(−1)k

  • λ+/λ (n−k)-hor. strip

µ/µ− k-vert. strip

sλ+/µ− ,

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 12

slide-83
SLIDE 83

The sign-reversing involution on SSYT of form λ+/µ−

Example 3: Fixed points. These should be in bijection with SSYT of shape (λ/µ) ∗ (n). 3 2 1 3 1 3 7 3 5 4 3 2 2 1 3 1 2 3 4 5 3 7 Conclusion: sλ/µsn =

n

  • k=0

(−1)k

  • λ+/λ (n−k)-hor. strip

µ/µ− k-vert. strip

sλ+/µ− , Proof 2: Algebraic proof given by Thomas Lam (as appendix in full paper).

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 12

slide-84
SLIDE 84

Development 1: Rectification one row at a time

A possible application of the skew Pieri rule:

+

=

+ + + +

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 13

slide-85
SLIDE 85

Development 1: Rectification one row at a time

A possible application of the skew Pieri rule:

+

=

+ + + +

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 13

slide-86
SLIDE 86

Development 1: Rectification one row at a time

A possible application of the skew Pieri rule:

+

=

+ + + +

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 13

slide-87
SLIDE 87

Development 1: Rectification one row at a time

A possible application of the skew Pieri rule:

+

=

+ + + +

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 13

slide-88
SLIDE 88

Development 1: Rectification one row at a time

A possible application of the skew Pieri rule:

+

=

+ + + +

Exactly the same proof works.

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 13

slide-89
SLIDE 89

Development 1: Rectification one row at a time

A possible application of the skew Pieri rule:

+

=

+ + + +

Exactly the same proof works. Allows you to rectify a skew shape (i.e. expand sλ/µ in terms of {sν}) one row at a time.

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 13

slide-90
SLIDE 90

Development 2: Skew Littlewood–Richardson rule

Conjecture [Assaf, McN.]: An expansion of sλ/µsσ/τ in terms of {sλ+/µ−} that generalizes the skew Pieri rule. Exact statement is coming up in Aaron’s talk (in terms of jeu-de-taquin). Proof [Lam–Lauve–Sottile]: using Hopf algebras. Open problem: find a combinatorial proof.

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 14

slide-91
SLIDE 91

Development 3: 2 combinatorial proofs for the $ of 1

Theorem: For λ and a positive integer n, sλ pn =

  • λ+

(−1)ht(λ+/λ)sλ+ where λ+/λ is a ribbon with n boxes Example: + +

=

p3 +

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 15

slide-92
SLIDE 92

Development 3: 2 combinatorial proofs for the $ of 1

Theorem: For λ/µ and a positive integer n, sλ/µpn =

  • λ+

(−1)ht(λ+/λ)sλ+/µ where λ+/λ is a ribbon with n boxes Example: +

=

p3 + +

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 15

slide-93
SLIDE 93

Development 3: 2 combinatorial proofs for the $ of 1

Theorem: For λ/µ and a positive integer n, sλ/µpn =

  • λ+

(−1)ht(λ+/λ)sλ+/µ −

  • µ−

(−1)ht(µ/µ−)sλ/µ− where λ+/λ is a ribbon with n boxes and so is µ/µ−. Example: + +

=

p3 +

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 15

slide-94
SLIDE 94

Development 3: 2 combinatorial proofs for the $ of 1

Theorem: For λ/µ and a positive integer n, sλ/µpn =

  • λ+

(−1)ht(λ+/λ)sλ+/µ −

  • µ−

(−1)ht(µ/µ−)sλ/µ− where λ+/λ is a ribbon with n boxes and so is µ/µ−. Example: + +

=

p3 + Question for you: is this a new result? Proof 1: Algebraic. Special case of LLS Hopf Formula Lemma (or à la Lam’s skew Pieri proof, but easier). Proof 2 [McN.?]: Combinatorial, except that it uses skew Littlewood-Richardson rule.

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 15

slide-95
SLIDE 95

Development 3: 2 combinatorial proofs for the $ of 1

Theorem: For λ/µ and a positive integer n, sλ/µpn =

  • λ+

(−1)ht(λ+/λ)sλ+/µ −

  • µ−

(−1)ht(µ/µ−)sλ/µ− where λ+/λ is a ribbon with n boxes and so is µ/µ−. Example: + +

=

p3 + Question for you: is this a new result? Proof 1: Algebraic. Special case of LLS Hopf Formula Lemma (or à la Lam’s skew Pieri proof, but easier). Proof 2 [McN.?]: Combinatorial, except that it uses skew Littlewood-Richardson rule. Easier(?) open problem: or find a combinatorial proof that doesn’t need the skew LR-rule.

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 15

slide-96
SLIDE 96

The End

Full paper available on the arXiv: Sami H. Assaf and Peter R.W. McNamara. A Pieri rule for skew shapes, JCT-A, to appear, arXiv:0908.0345

T h a n k · Y o u

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 16

slide-97
SLIDE 97

The End

Full paper available on the arXiv: Sami H. Assaf and Peter R.W. McNamara. A Pieri rule for skew shapes, JCT-A, to appear, arXiv:0908.0345

T h a n k · Y o u

+ + + + + + − − − − − + + +

A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 16