[PPT] - Skew LittlewoodRichardson Rules from Hopf Algebras Aaron Lauve PowerPoint Presentation

SLIDE 1

Skew Littlewood–Richardson Rules from Hopf Algebras

Aaron Lauve

Texas A&M University Loyola University Chicago joint work with:

Thomas Lam

University of Michigan

Frank Sottile

Texas A&M University

FPSAC 2010, San Francisco, CA

SLIDE 2

Hopf Algebras, *@#?%!

SLIDE 3

Hopf Structure of Λ As an algebra, . . . Λ = Z[h1, h2, . . . ]

complete homogeneous symmetric functions

hn :=

i1≤i2≤···≤in

xi1xi2 · · · xin = Z[e1, e2, . . . ]

elementary symmetric functions

en :=

i1<i2<···<in

xi1xi2 · · · xin

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 3 / 20

SLIDE 4

Hopf Structure of Λ As an algebra, . . . Λ = Z[h1, h2, . . . ]

complete homogeneous symmetric functions

hn :=

i1≤i2≤···≤in

xi1xi2 · · · xin = Z[e1, e2, . . . ]

elementary symmetric functions

en :=

i1<i2<···<in

xi1xi2 · · · xin = spanZ

Schur functions sλ
(a nice basis)

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 3 / 20

SLIDE 5

Hopf Structure of Λ As a Hopf algebra, . . .

we need more maps, Λ = (Λ, ·, ∆, ε, S)

coproduct ∆ : Λ → Λ ⊗ Λ ∆(hn) =

j+k=n

hj ⊗ hk

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 4 / 20

SLIDE 6

Hopf Structure of Λ As a Hopf algebra, . . .

we need more maps, Λ = (Λ, ·, ∆, ε, S)

coproduct counit ∆ : Λ → Λ ⊗ Λ ε : Λ → Z ∆(hn) =

j+k=n

hj ⊗ hk ε(hn) = δn0

(put h0 = e0 = 1)

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 4 / 20

SLIDE 7

Hopf Structure of Λ As a Hopf algebra, . . .

we need more maps, Λ = (Λ, ·, ∆, ε, S)

coproduct counit antipode ∆ : Λ → Λ ⊗ Λ ε : Λ → Z S : Λ → Λ ∆(hn) =

j+k=n

hj ⊗ hk ε(hn) = δn0 S(hk) = (−1)kek

(put h0 = e0 = 1)

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 4 / 20

SLIDE 8

Hopf Structure of Λ As a Hopf algebra, . . .

we need more maps, Λ = (Λ, ·, ∆, ε, S)

coproduct counit antipode ∆ : Λ → Λ ⊗ Λ ε : Λ → Z S : Λ → Λ ∆(hn) =

j+k=n

hj ⊗ hk ε(hn) = δn0 S(hk) = (−1)kek

(put h0 = e0 = 1)

together with some compatibility conditions (omitted)

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 4 / 20

SLIDE 9

Schur Functions

sλ

SLIDE 10

Schur Functions A nice basis

Definition. Given a partition λ, s λ is the generating function for the

corresponding semistandard Young tableaux SSYT(λ).

Ferrers fillings satisfying a ≤ b < c .

Example:

s = 1 1 2 + 1 1 3 + 1 2 2 + 1 2 3 + 1 3 2 + · · · = x12x2 + x12x3 + x1x22 + 2x1x2x3 + · · ·

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 6 / 20

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Schur Functions A nice basis

Definition. Given a partition λ, s λ is the generating function for the

corresponding semistandard Young tableaux SSYT(λ).

Ferrers fillings satisfying a ≤ b < c .

Example:

s = 1 1 2 + 1 1 3 + 1 2 2 + 1 2 3 + 1 3 2 + · · · = x12x2 + x12x3 + x1x22 + 2x1x2x3 + · · ·

Worth noting: s

. . .

= hn and s

. . .

= en.

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 6 / 20

SLIDE 12

Schur Functions Classical problem

Problem. Understand the coefficients c ν

λ, μ in

sλ · sμ =

ν

c ν

λ, μ sν .

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 7 / 20

SLIDE 13

Schur Functions Classical problem

Problem. Understand the coefficients c ν

λ, μ in

sλ · sμ =

ν

c ν

λ, μ sν .

Special case (Pieri rule). sλ · hj =

λ +j

−→h λ+

s λ+

sum over all ways (λ+) to add j boxes in a horizontal strip to the diagram λ.

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 7 / 20

SLIDE 14

Schur Functions Classical problem

Problem. Understand the coefficients c ν

λ, μ in

sλ · sμ =

ν

c ν

λ, μ sν .

Special case (Pieri rule). sλ · hj =

λ +j

−→h λ+

s λ+

sum over all ways (λ+) to add j boxes in a horizontal strip to the diagram λ.

Example (j = 3):

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 7 / 20

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Schur Functions Nice answer!

Problem. Understand the coefficients c ν

λ, μ in sλ · sμ =

ν c ν

λ, μ sν .

Theorem (Littlewood–Richardson rule) Fix T ∈ SSYT(ν). Then

c ν

λ, μ = #

(R, S): R ∈ SSYT(λ), S ∈ SSYT(μ), R ∗ S = T
.

play jeu-de-taquin)

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 8 / 20

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Schur Functions Nice answer!

Problem. Understand the coefficients c ν

λ, μ in sλ · sμ =

ν c ν

λ, μ sν .

Theorem (Littlewood–Richardson rule) Fix T ∈ SSYT(ν). Then

c ν

λ, μ = #

(R, S): R ∈ SSYT(λ), S ∈ SSYT(μ), R ∗ S = T
.

play jeu-de-taquin)

Example: Pick T = 1 1 2 . Guess R = 1 2 and S = 1 . 1 1 2∗

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 8 / 20

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Schur Functions Nice answer!

Problem. Understand the coefficients c ν

λ, μ in sλ · sμ =

ν c ν

λ, μ sν .

Theorem (Littlewood–Richardson rule) Fix T ∈ SSYT(ν). Then

c ν

λ, μ = #

(R, S): R ∈ SSYT(λ), S ∈ SSYT(μ), R ∗ S = T
.

play jeu-de-taquin)

Example: Pick T = 1 1 2 . Guess R = 1 2 and S = 1 . 1 1 2 −→ 1 1 2

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 8 / 20

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Schur Functions Nice answer!

Problem. Understand the coefficients c ν

λ, μ in sλ · sμ =

ν c ν

λ, μ sν .

Theorem (Littlewood–Richardson rule) Fix T ∈ SSYT(ν). Then

c ν

λ, μ = #

(R, S): R ∈ SSYT(λ), S ∈ SSYT(μ), R ∗ S = T
.

play jeu-de-taquin)

Example: Pick T = 1 1 2 . Guess R = 1 2 and S = 1 . 1 1 2 −→ 1 1 2 −→ 1 1 2

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 8 / 20

SLIDE 19

Schur Functions Nice answer!

Problem. Understand the coefficients c ν

λ, μ in sλ · sμ =

ν c ν

λ, μ sν .

Theorem (Littlewood–Richardson rule) Fix T ∈ SSYT(ν). Then

c ν

λ, μ = #

(R, S): R ∈ SSYT(λ), S ∈ SSYT(μ), R ∗ S = T
.

play jeu-de-taquin)

Example: Pick T = 1 1 2 . Guess R = 1 2 and S = 1 . 1 1 2 −→ 1 1 2 −→ 1 1 2 −→ 1 1 2

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 8 / 20

SLIDE 20

Schur Functions Nice answer!

Problem. Understand the coefficients c ν

λ, μ in sλ · sμ =

ν c ν

λ, μ sν .

Theorem (Littlewood–Richardson rule) Fix T ∈ SSYT(ν). Then

c ν

λ, μ = #

(R, S): R ∈ SSYT(λ), S ∈ SSYT(μ), R ∗ S = T
.

play jeu-de-taquin)

Example: Pick T = 1 1 2 . Guess R = 1 2 and S = 1 . 1 1 2 −→ 1 1 2 −→ 1 1 2 −→ 1 1 2

c 21

2,1 ≥ 1

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 8 / 20

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Schur Functions More nice facts

More Facts.

∆(sν) =

λ, μ

c ν

λ, μ s λ ⊗ sμ

Same coefficients as for product! (Λ is a self-dual Hopf algebra.)

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 9 / 20

SLIDE 22

Schur Functions More nice facts

More Facts.

∆(sν) =

λ, μ

c ν

λ, μ s λ ⊗ sμ

=

μ

sν/μ ⊗ sμ

Same coefficients as for product! (Λ is a self-dual Hopf algebra.) “Skew ν by μ” ≡ “collect terms (−) ⊗ sμ in the coproduct.”

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 9 / 20

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Schur Functions More nice facts

More Facts.

∆(sν) =

λ, μ

c ν

λ, μ s λ ⊗ sμ

S(sλ) = (−1)

|λ| sλ′

=

μ

sν/μ ⊗ sμ

Same coefficients as for product! (Λ is a self-dual Hopf algebra.) “Skew ν by μ” ≡ “collect terms (−) ⊗ sμ in the coproduct.” E.g., S

s
= −s

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 9 / 20

SLIDE 24

Skew Schur Functions

sλ/μ | μ ⊆ λ

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Skew Schur Functions Assaf-McNamara problem

Problem. Understand the coefficients in

sλ/μ · sσ/τ =

d

.

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 11 / 20

SLIDE 26

Skew Schur Functions Assaf-McNamara problem

Problem. Understand the coefficients in

sλ/μ · sσ/τ =

d

.

Natural to take to be Schur functions. Assaf-McNamara take to be skew Schur functions.

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 11 / 20

SLIDE 27

Skew Schur Functions Assaf-McNamara problem

Problem. Understand the coefficients in

sλ/μ · sσ/τ =

d

.

Natural to take to be Schur functions. Assaf-McNamara take to be skew Schur functions.

in the spirit of Pieri rule. . . λ μ

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 11 / 20

SLIDE 28

Skew Schur Functions Assaf-McNamara problem

Problem. Understand the coefficients in

sλ/μ · sσ/τ =

d

.

Natural to take to be Schur functions. Assaf-McNamara take to be skew Schur functions.

in the spirit of Pieri rule. . . sum over ways to grow (λ → λ+) and shrink (μ → μ−) the skew-shape λ/μ λ μ λ+ μ−

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 11 / 20

SLIDE 29

Skew Schur Functions Assaf-McNamara problem

Problem. Understand the coefficients in sλ/μ · sσ/τ =
d

s . Special case (Assaf–McNamara rule). Given μ ⊆ λ and n ≥ 0,

sλ/μ · hn =

j+k=n
λ +j

−→h λ+ μ −k −→v μ−

(−1)k sλ+/μ− .

Proof: a sign-reversing involution.

add (+j) a horizontal strip to λ remove (−k) a vertical strip from μ

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 12 / 20

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Skew Schur Functions Assaf-McNamara problem

Problem. Understand the coefficients in sλ/μ · sσ/τ =
d

s . Theorem (Lam–L–Sottile) Fix T ∈ SSYT(σ). Then

sλ/μ · sσ/τ =

(−1)|R−| sλ+/μ− ,

the sum taken over R− ∈ SSYT((μ/μ−)′), R+ ∈ SSYT(λ+/λ), and S ∈ SSYT(τ) such that R− ∗ R+ ∗ S = T. Proof: a Hopf formula.

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 13 / 20

SLIDE 31

A Hopf Formula

SLIDE 32

Dual Pairs of Hopf Algebras The harpoon action

Fix H and its dual H∗ under a bilinear pairing 〈·, ·〉: H ⊗ H∗ → . For h ∈ H and a ∈ H∗, put ∆(h) =

(h) h(1) ⊗ h(2) .
Definition. Construct left action (): coproduct + evaluation.

a h :=

(h)
h(2), a
h(1) .

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 15 / 20

SLIDE 33

Dual Pairs of Hopf Algebras The harpoon action

Fix H and its dual H∗ under a bilinear pairing 〈·, ·〉: H ⊗ H∗ → . For h ∈ H and a ∈ H∗, put ∆(h) =

(h) h(1) ⊗ h(2)

and ∆(a) =

(a) a(1) ⊗ a(2) .
Definition. Construct left actions (): coproduct + evaluation.

a h :=

(h)
h(2), a
h(1)

and h a :=

(a)
h, a(2)
a(1) .

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 15 / 20

SLIDE 34

Dual Pairs of Hopf Algebras The harpoon action

Fix H and its dual H∗ under a bilinear pairing 〈·, ·〉: H ⊗ H∗ → . For h ∈ H and a ∈ H∗, put ∆(h) =

(h) h(1) ⊗ h(2)

and ∆(a) =

(a) a(1) ⊗ a(2) .
Definition. Construct left actions (): coproduct + evaluation.

a h :=

(h)
h(2), a
h(1)

and h a :=

(a)
h, a(2)
a(1) .

Examples: Take H = Λ

( ≃ Λ∗ under Hall inner product )

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 15 / 20

SLIDE 35

Dual Pairs of Hopf Algebras The harpoon action

Fix H and its dual H∗ under a bilinear pairing 〈·, ·〉: H ⊗ H∗ → . For h ∈ H and a ∈ H∗, put ∆(h) =

(h) h(1) ⊗ h(2)

and ∆(a) =

(a) a(1) ⊗ a(2) .
Definition. Construct left actions (): coproduct + evaluation.

a h :=

(h)
h(2), a
h(1)

and h a :=

(a)
h, a(2)
a(1) .

Examples: Take H = Λ

( ≃ Λ∗ under Hall inner product )

s∗

μ sλ =

id ⊗ 〈·, s∗

μ 〉 κ

sλ/κ ⊗ sκ = sλ/μ ek s∗

μ = s∗ μ/

. . .

=

μ −k

−→v μ−

s∗

μ−

a dual Pieri rule

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 15 / 20

SLIDE 36

Dual Pairs of Hopf Algebras A Hopf formula

Lemma For all g, h ∈ H, for all a ∈ H∗, we have (a g) · h =

(h)
S(h(2)) a
(g · h(1)) .

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 16 / 20

SLIDE 37

Dual Pairs of Hopf Algebras A Hopf formula

Lemma For all g, h ∈ H, for all a ∈ H∗, we have (a g) · h =

(h)
S(h(2)) a
(g · h(1)) .

Example: Take a = s∗

μ , g = sλ, and h = hn.

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 16 / 20

SLIDE 38

Dual Pairs of Hopf Algebras A Hopf formula

Lemma For all g, h ∈ H, for all a ∈ H∗, we have (a g) · h =

(h)
S(h(2)) a
(g · h(1)) .

Example: Take a = s∗

μ , g = sλ, and h = hn.

LHS:

(s∗

μ sλ) · hn = sλ/μ · hn

the left-hand side in the Assaf–McNamara Pieri rule

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 16 / 20

SLIDE 39

Dual Pairs of Hopf Algebras A Hopf formula

Lemma For all g, h ∈ H, for all a ∈ H∗, we have (a g) · h =

(h)
S(h(2)) a
(g · h(1)) .

Example: Take a = s∗

μ , g = sλ, and h = hn.

LHS:

(s∗

μ sλ) · hn = sλ/μ · hn

the left-hand side in the Assaf–McNamara Pieri rule

RHS:

j+k=n
S(hk) s∗

μ

(sλ · hj) (unravelling coproduct)

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 16 / 20

SLIDE 40

Dual Pairs of Hopf Algebras A Hopf formula

RHS =

j+k=n
(−1)kek s∗

μ

(sλ · hj)

(antipode)

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 17 / 20

SLIDE 41

Dual Pairs of Hopf Algebras A Hopf formula

RHS =

j+k=n
(−1)kek s∗

μ

(sλ · hj)

(antipode)

RHS =

j+k=n
(−1)kek s∗

μ

λ +j

−→h λ+

sλ+

(Pieri rule)

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 17 / 20

SLIDE 42

Dual Pairs of Hopf Algebras A Hopf formula

RHS =

j+k=n
(−1)kek s∗

μ

(sλ · hj)

(antipode)

RHS =

j+k=n
(−1)kek s∗

μ

λ +j

−→h λ+

sλ+

(Pieri rule)

RHS =

j+k=n

(−1)k

μ −k

−→v μ−

sμ−

λ +j

−→h λ+

sλ+

(dual Pieri rule)

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 17 / 20

SLIDE 43

Dual Pairs of Hopf Algebras A Hopf formula

RHS =

j+k=n
(−1)kek s∗

μ

(sλ · hj)

(antipode)

RHS =

j+k=n
(−1)kek s∗

μ

λ +j

−→h λ+

sλ+

(Pieri rule)

RHS =

j+k=n

(−1)k

μ −k

−→v μ−

sμ−

λ +j

−→h λ+

sλ+

(dual Pieri rule)

RHS =

j+k=n
λ +j

−→h λ+ μ −k −→v μ−

(−1)k sλ+/μ−

the right-hand side in the Assaf–McNamara Pieri rule

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 17 / 20

SLIDE 44

Dual Pairs of Hopf Algebras More generally. . .

Fix basis indexing set P Fix bases H = span

Lλ
λ∈P and H∗ = span
Rλ
λ∈P

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 18 / 20

SLIDE 45

Dual Pairs of Hopf Algebras More generally. . .

Fix basis indexing set P Fix bases H = span

Lλ
λ∈P and H∗ = span
Rλ
λ∈P

Define skewing as harpooning: Lλ/μ := Rμ Lλ Find structure coefficients: Lλ · Lμ =

νc ν

λ,μ Lν

and Rλ · Rμ =

νd ν

λ,μ Rν

need not be the same!

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 18 / 20

SLIDE 46

Dual Pairs of Hopf Algebras More generally. . .

Fix basis indexing set P Fix bases H = span

Lλ
λ∈P and H∗ = span
Rλ
λ∈P

Define skewing as harpooning: Lλ/μ := Rμ Lλ Find structure coefficients: Lλ · Lμ =

νc ν

λ,μ Lν

and Rλ · Rμ =

νd ν

λ,μ Rν

need not be the same!

Theorem IF: S(Lλ) = e(λ)LT(λ) for functions e : P → and T : P → P, THEN: Lλ/μ · Lσ/τ =

μ−,λ+,π,κ,ρ

e(ρ)d κ

π,ρ d σ κ,τ c μ μ−,T(ρ) c λ+ λ,π Lλ+/μ−

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 18 / 20

SLIDE 47

Dual Pairs of Hopf Algebras More generally. . . We have found combinatorial interpretations for skew product coefficients in several settings arising in algebraic combinatorics:

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 19 / 20

SLIDE 48

Dual Pairs of Hopf Algebras More generally. . . We have found combinatorial interpretations for skew product coefficients in several settings arising in algebraic combinatorics:

Schur functions Schur P-functions and Q-functions Noncommutative ribbon Schur functions (≈ Gessel quasi-symmetric functions) k-Schur functions

(Pieri rule only)

Schubert classes for affine GrSp(2n,C)

(Pieri rule only)

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 19 / 20

SLIDE 49

T H E E N D

Thank You!

References

Assaf, McNamara, A Pieri rule for skew shapes, FPSAC 2010. Lam, Lauve, Sottile, Skew Littlewood–Richardson rules from Hopf algebras, FPSAC 2010. Macdonald, Symmetric functions and Hall polynomials, 2 ed., 1995. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series, v.82., 1993.

Aaron Lauve (TAMU, LUC) Skew Littlewood–Richardson Rules 4 August 2010 20 / 20