Grothendieck expansions of symmetric polynomials Jason Bandlow - - PowerPoint PPT Presentation

Grothendieck expansions of symmetric polynomials

Jason Bandlow (joint work with Jennifer Morse)

University of Pennsylvania

August 3rd, 2010 – FPSAC San Francisco State University

Outline

Symmetric functions Tableaux–Schur expansions Grothendieck functions and their dual basis Main theorem Examples Sketch of proof

The monomial basis

The monomial symmetric functions are indexed by partitions λ = (λ1, λ2, . . . , λk) λi ≥ λi+1 mλ =

• α

xα α a rearrangement of the parts of λ and infinitely many 0’s

Example

m2,1 =(x2

1x2 + x1x2 2) + (x1x2 3 + x2 1x3) + . . .

+ (x2

2x3 + x2x2 3) + . . .

The complete homogeneous basis

The (complete) homogeneous symmetric functions are defined by hi =

• λ⊢i

mλ hλ = hλ1hλ2 . . . hλk

Example

h3 = m3 + m2,1 + m1,1,1 h4,2,1 = h4h2h1

The Hall inner product

Defined by hλ, mµ =

• 1

if λ = µ

• therwise

The Hall inner product

Defined by hλ, mµ =

• 1

if λ = µ

• therwise

Proposition

If {fλ}, {f ∗

λ } and {gλ}, {g∗ λ} are two pairs of dual bases with

fλ =

• µ

Mλ,µgµ then g∗

µ =

• λ

Mλ,µf ∗

λ

Semistandard Young tableaux

A left-and-bottom justified, partition-shaped array of numbers, weakly increasing across rows and strictly increasing up columns.

Example

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

Semistandard Young tableaux

A left-and-bottom justified, partition-shaped array of numbers, weakly increasing across rows and strictly increasing up columns.

Example

7 7 5 6 6 8 3 3 4 7 1 1 2 2 2 Shape: (5, 4, 4, 2) Evaluation: (2, 3, 2, 1, 1, 2, 3, 1) Reading word: 775668334711222

Knuth equivalence

An equivalence relation on words generated by yxz ≡ yzx if x < y ≤ z xzy ≡ zxy if x ≤ y < z

Key Fact

Every word is Knuth equivalent to the reading word of exactly one tableau.

Example

rw

• 3 4

1 2 3

• = 34123 ≡ 31423 ≡ 31243 ≡ 13243 ≡ 13423

The Schur basis

Definition

The Schur functions are given by sλ =

• T∈SSYT(λ)

xev(T)

The Schur basis

Definition

The Schur functions are given by sλ =

• T∈SSYT(λ)

xev(T)

Example

s2,1 = x2

1x2+

x1x2

2+

2x1x2x3+ · · · 2 1 1 2 1 2 3 1 2 2 1 3 · · ·

The Schur basis

Definition

The Schur functions are given by sλ =

• T∈SSYT(λ)

xev(T)

Example

s2,1 = x2

1x2+

x1x2

2+

2x1x2x3+ · · · 2 1 1 2 1 2 3 1 2 2 1 3 · · · Fact: Schur functions are a self-dual basis of the symmetric functions.

The Schur basis

Using the fact that Schur functions are symmetric, we can rewrite the definition as sλ =

• µ

Kλ,µmµ where Kλ,µ is the number of semistandard tableaux of shape λ and evaluation µ.

The Schur basis

Using the fact that Schur functions are symmetric, we can rewrite the definition as sλ =

• µ

Kλ,µmµ where Kλ,µ is the number of semistandard tableaux of shape λ and evaluation µ. Using the proposition about dual bases, we get hµ =

• λ

Kλ,µsλ

The Schur basis

Using the fact that Schur functions are symmetric, we can rewrite the definition as sλ =

• µ

Kλ,µmµ where Kλ,µ is the number of semistandard tableaux of shape λ and evaluation µ. Using the proposition about dual bases, we get hµ =

• λ

Kλ,µsλ which can be rewritten as hµ =

• T∈Tµ

ssh(T) where Tµ is the set of all semistandard tableaux of evaluation µ.

Tableaux–Schur expansions

Elements of a family {fα} of symmetric functions have tableaux–Schur expansions if there exist sets Tα of semistandard tableaux and weight functions wtα such that fα =

• T∈Tα

wtα(T)ssh(T)

Tableaux–Schur expansions

Elements of a family {fα} of symmetric functions have tableaux–Schur expansions if there exist sets Tα of semistandard tableaux and weight functions wtα such that fα =

• T∈Tα

wtα(T)ssh(T) Goal: find appropriate sets and modifications of wt so that fα =

• S∈Sα

wtα(S)Gsh(S) fα =

• R∈Rα

wtα(R)gsh(R) where G and g are, respectively, the Grothendieck and dual-Grothendieck functions.

Tableaux–Schur expansions

Examples

Tableaux–Schur expansions

Examples

◮ hµ = T∈Tµ ssh(T)

Tableaux–Schur expansions

Examples

◮ hµ = T∈Tµ ssh(T) ◮ Hµ[X; t] = T∈Tµ tch(T)ssh(T)

Tableaux–Schur expansions

Examples

◮ hµ = T∈Tµ ssh(T) ◮ Hµ[X; t] = T∈Tµ tch(T)ssh(T) ◮ sλsµ = T∈Tλ,µ ssh(T)

Tableaux–Schur expansions

Examples

◮ hµ = T∈Tµ ssh(T) ◮ Hµ[X; t] = T∈Tµ tch(T)ssh(T) ◮ sλsµ = T∈Tλ,µ ssh(T) ◮ Fσ = T∈Tσ ssh(T)

Tableaux–Schur expansions

Examples

◮ hµ = T∈Tµ ssh(T) ◮ Hµ[X; t] = T∈Tµ tch(T)ssh(T) ◮ sλsµ = T∈Tλ,µ ssh(T) ◮ Fσ = T∈Tσ ssh(T) ◮ A(k) µ [X; t] = T∈Tk,µ tch(T)ssh(T)

Tableaux–Schur expansions

Examples

◮ hµ = T∈Tµ ssh(T) ◮ Hµ[X; t] = T∈Tµ tch(T)ssh(T) ◮ sλsµ = T∈Tλ,µ ssh(T) ◮ Fσ = T∈Tσ ssh(T) ◮ A(k) µ [X; t] = T∈Tk,µ tch(T)ssh(T) ◮ Hµ[X; 1, t] = T∈T(1n) tchµ(T)ssh(T)

Tableaux–Schur expansions

Examples

◮ hµ = T∈Tµ ssh(T) ◮ Hµ[X; t] = T∈Tµ tch(T)ssh(T) ◮ sλsµ = T∈Tλ,µ ssh(T) ◮ Fσ = T∈Tσ ssh(T) ◮ A(k) µ [X; t] = T∈Tk,µ tch(T)ssh(T) ◮ Hµ[X; 1, t] = T∈T(1n) tchµ(T)ssh(T) ◮ Hµ[X; q, t] ?

=

T∈T(1n) qaµ(T)tbµ(T)ssh(T)

Grothendieck polynomials

◮ Introduced by Lascoux-Sch¨

utzenberger (1982)

◮ Represent K-theory classes of structure sheaves of Schubert

varieties in GLn

◮ Given by elements of Z[[x1, . . . , xn]] ◮ Analogous to Schubert polynomials ◮ Sign-alternating by degree, equal to Schubert polynomial in

bottom degree

Stable Grassmannian Grothendieck functions

Stable:

◮ Due to Fomin-Kirillov ◮ Limit as number of variables → ∞

Stable Grassmannian Grothendieck functions

Stable:

◮ Due to Fomin-Kirillov ◮ Limit as number of variables → ∞

Grassmannian:

◮ Indexed by partitions ◮ Power series of symmetric functions ◮ Sign-alternating by degree in the Schur basis ◮ Analog of Schur functions

Grassmannian Grothendieck functions

Theorem (Buch)

Gλ =

• S∈SVT(λ)

ε(S)xev(S) where SVT(λ) is the set of set-valued tableaux of shape λ.

Grassmannian Grothendieck functions

Theorem (Buch)

Gλ =

• S∈SVT(λ)

ε(S)xev(S) where SVT(λ) is the set of set-valued tableaux of shape λ.

Example

G1 = s1 −s1,1 +s1,1,1 . . . 1 12 123 · · · 23 234 · · · . . . . . .

Set-valued tableaux

Example

234 45 1 12 3

Set-valued tableaux

Example

234 45 1 12 3 shape(S) = (3, 2) evaluation(S) = (2, 2, 2, 2, 1) ε(S) = (−1)|ev(S)|+|sh(S)| = 1

Set-valued tableaux

Example

234 45 1 12 3 shape(S) = (3, 2) evaluation(S) = (2, 2, 2, 2, 1) ε(S) = (−1)|ev(S)|+|sh(S)| = 1 From Gλ =

• S∈SVT(λ)

ε(S)xev(S) we see Gλ = sλ ± higher degree terms

Set-valued tableaux

We introduce the reading word of a set-valued tableau, defined by:

◮ Proceed row by row, from top to bottom ◮ In each row, first ignore the smallest element of each cell.

Then read the remaining elements from right to left, and from largest to smallest within each cell.

◮ Read the smallest elements of each cell from left to right.

Example

234 45 1 12 3 has reading word: 543242113.

Dual Grothendieck functions

We denote the dual basis to the Grothendieck by g

Theorem (Lam, Pylyavskyy)

gλ =

• R∈RPP(λ)

xev(R) where RPP(λ) is the set of reverse plane partitions of shape λ.

Dual Grothendieck functions

We denote the dual basis to the Grothendieck by g

Theorem (Lam, Pylyavskyy)

gλ =

• R∈RPP(λ)

xev(R) where RPP(λ) is the set of reverse plane partitions of shape λ.

Example

g2,1 = s2,1 +s2 2 1 1 1 1 1 2 1 3 1 1 2 . . . . . .

Reverse plane partitions

Example

2 3 2 2 4 1 2 2 1 1 1 1 shape(R) = (4, 3, 3, 2) evaluation(R) = (4, 3, 1, 1)

Reverse plane partitions

Example

2 3 2 2 4 1 2 2 1 1 1 1 shape(R) = (4, 3, 3, 2) evaluation(R) = (4, 3, 1, 1) From gλ =

• R∈RPP(λ)

xev(R) we see gλ = sλ ± lower degree terms

Dual Grothendieck functions

We define the reading word of a reverse plane partition to be a subsequence of the usual reading word, where we only take the bottommost occurence of every letter in each column.

Example

2 3 2 2 4 1 2 2 1 1 1 1 has reading word 324221111.

Generalizing tableaux–Schur expansions

Given any set Tα of semistandard tableaux, we define corresponding sets Sα (respectively Rα) of set-valued tableaux (reverse plane partitions) defined by S ∈ Sα ⇐ ⇒ rw(S) ≡ rw(T) for some T ∈ Tα R ∈ Rα ⇐ ⇒ rw(R) ≡ rw(T) for some T ∈ Tα

Generalizing tableaux–Schur expansions

Given any set Tα of semistandard tableaux, we define corresponding sets Sα (respectively Rα) of set-valued tableaux (reverse plane partitions) defined by S ∈ Sα ⇐ ⇒ rw(S) ≡ rw(T) for some T ∈ Tα R ∈ Rα ⇐ ⇒ rw(R) ≡ rw(T) for some T ∈ Tα Given a statistic wtα on Tα, we can extend it to reading words by wtα(w) = wtα(T) if w ≡ rw(T).

Examples

If T = 4 3 4 1 2 ∈ Tα, then 4 3 4 1 2 4 3 1 24 34 4 1 2 4 4 1 23 4 1 234 43412 43412 43412 44312 44312 are in Sα

Examples

If T = 4 3 4 1 2 ∈ Tα, then 4 3 4 1 2 4 3 1 24 34 4 1 2 4 4 1 23 4 1 234 43412 43412 43412 44312 44312 are in Sα and 4 3 4 1 2 4 4 1 3 1 2 4 4 3 4 1 2 4 4 4 1 3 1 2 · · · 43412 44312 43412 44312 are in Rα. All of these will have the same weight.

Statement of main theorem

Theorem (B-Morse)

Let fα be a family of symmetric functions with fα =

• T∈Tα

wtα(T)ssh(T). Then fα =

• S∈Sα

ε(S)wtα(S)gsh(S) and fα =

• R∈Rα

wtα(R)Gsh(R).

Complete homogeneous functions

We have a tableaux-Schur expansion of the homogeneous functions by hµ =

• T∈Tµ

ssh(T) where Tµ is the set of all tableaux of evaluation µ. Thus we have hµ =

• S∈Sµ

ε(S)gsh(S) =

• R∈Rµ

Gsh(R) where Sµ is the set of all set-valued tableaux of evaluation µ and Rµ is the set of all reverse plane partitions of evaluation µ.

Example

h3,2 = s3,2 +s4,1 +s5 2 2 1 1 1 2 1 1 1 2 1 1 1 2 2

Example

h3,2 = s3,2 +s4,1 +s5 2 2 1 1 1 2 1 1 1 2 1 1 1 2 2 = g3,2 +g4,1 +g5 −g3,1 −g4 2 1 1 12 1 1 12 2

Example

h3,2 = s3,2 +s4,1 +s5 2 2 1 1 1 2 1 1 1 2 1 1 1 2 2 = g3,2 +g4,1 +g5 −g3,1 −g4 2 1 1 12 1 1 12 2 = G3,2 +G4,1 +G5 +2G3,2,1 +2G4,1,1 + · · · 2 2 2 1 1 1 2 1 2 1 1 1 2 2 1 1 1 2 2 1 1 1 1 2

Hall-Littlewood functions

The Hall-Littlewood functions form a basis for the symmetric functions over the field Q(t). Interpretations as

◮ Deformation of Weyl character formula ◮ Graded Sn-character of certain cohomology rings ◮ Representation theory of groups of matrices over finite fields

among others. We will use the version typically denoted by Hµ[X; t] or Q′

µ(x; t) in

the literature.

Hall-Littlewood functions

The Hall-Littlewood functions form a basis for the symmetric functions over the field Q(t). Interpretations as

◮ Deformation of Weyl character formula ◮ Graded Sn-character of certain cohomology rings ◮ Representation theory of groups of matrices over finite fields

among others. We will use the version typically denoted by Hµ[X; t] or Q′

µ(x; t) in

the literature.

Example

H1,1,1[X; t] = s1,1,1 + (t2 + t)s2,1 + t3s3

Hall-Littlewood functions

Theorem (Lascoux-Sch¨ utzenberger)

Hµ[X; t] =

• T∈Tµ

tch(T)ssh(T) where charge is a non-negative integer statistic defined on words and constant on Knuth equivalence classes.

The charge statistic

Example

3 4 2 2 3 1 1 1 2 3 4 2 2 3 1 1 1 2

The charge statistic

Example

3 4 2 2 3 1 1 1 2 3 4 2 2 3 1 1 1 2

The charge statistic

Example

3 4 2 2 3 1 1 1 2 3 4 2 2 3 1 1 1 2

The charge statistic

Example

3 4 2 2 3 1 1 1 2 3 4 2 2 3 1 1 1 2

The charge statistic

Example

3 4 2 2 3 1 1 1 2 1 3 4 2 2 3 1 1 1 2

The charge statistic

Example

3 4 2 2 3 1 1 1 2 1 3 4 2 2 3 1 1 1 2

The charge statistic

Example

3 4 2 2 3 1 1 1 2 1 3 4 2 2 3 1 1 1 2

The charge statistic

Example

3 4 2 2 3 1 1 1 2 1 1 3 4 2 2 3 1 1 1 2

The charge statistic

Example

3 4 2 2 3 1 1 1 2 1 1 3 4 2 2 3 1 1 1 2

The charge statistic

Example

3 4 2 2 3 1 1 1 2 1 1 1 3 4 2 2 3 1 1 1 2 charge(T) = 3

Hall-Littlewood functions

We have Hµ[X; t] =

• T∈Tµ

tch(T)ssh(T) where Tµ is again the set of all tableaux of evaluation µ. Hence Hµ[X; t] =

• S∈Sµ

ε(S)tch(S)gsh(S) =

• R∈Rµ

tch(R)Gsh(R) where Sµ (resp. Rµ) is the set of set-valued tableaux (resp. reverse plane partitions) of evaluation µ and the charge of S or R is the charge of the reading word.

Example

H1,1,1[X; t] = s1,1,1 + (t + t2)s2,1 + t3s3 3 2 1 2 1 3 3 1 2 1 2 3

Example

H1,1,1[X; t] = s1,1,1 + (t + t2)s2,1 + t3s3 =g1,1,1 + (t + t2)g2,1 + t3g3 − 2g1,1 − (t + t2)g2 + g1 3 2 1 2 1 3 3 1 2 1 2 3 23 1 3 12 12 3 1 23 123

Littlewood-Richardson tableaux

One version of the Littlewood-Richardson rule states that sλsµ =

• T∈Tλ,µ

ssh(T) where Tλ,µ is the set of tableaux:

◮ with evaluation (λ1, · · · , λℓ(λ), µ1, · · · , µℓ(µ)) ◮ which have the reverse lattice property with respect to the

letters 1, · · · , ℓ(λ), and

◮ which have the reverse lattice property with respect to the

letters ℓ(λ) + 1, · · · , ℓ(λ) + ℓ(µ)

Littlewood-Richardson tableaux

Example

T(2,1),(2,1) is 2 4 1 1 3 3 4 2 1 1 3 3 2 3 4 1 1 3 4 2 3 1 1 3 3 2 4 1 1 3 4 3 2 1 1 3 3 4 2 3 1 1 4 3 2 3 1 1

Littlewood-Richardson tableaux

Corollary

sλsµ =

• S∈Sλ,µ

ε(S)gsh(S) =

• R∈Rλ,µ

Gsh(R) where Sλ,µ (resp. Rλ,µ) is the set of all set-valued tableaux (resp. reverse plane partitions)

◮ with evaluation (λ1, · · · , λℓ(λ), µ1, · · · , µℓ(µ)) ◮ which have the reverse lattice property with respect to the

letters 1, · · · , ℓ(λ), and

◮ which have the reverse lattice property with respect to the

letters ℓ(λ) + 1, · · · , ℓ(λ) + ℓ(µ)

Grothendieck and Schur functions

Corollary

sλ =

• S∈Sλ,∅

ε(S)gsh(S) sλ =

• R∈Rλ,∅

Gsh(S) Duality gives expansions of G and g into Schur functions. A different form of these expansions was given by Lenart, in terms

• f combinatorial objects now called elegant fillings.

Sketch of proof

Given fα =

• T∈Tα

wt(T)ssh(T) =

• T∈Tα

wt(T)

• T ′∈SSYT(sh(T))

xev(T ′)

Sketch of proof

Given fα =

• T∈Tα

wt(T)ssh(T) =

• T∈Tα

wt(T)

• T ′∈SSYT(sh(T))

xev(T ′) we want to show fα =

• S∈Sα

ε(S)wt(S)gsh(S) =

• S∈Sα

ε(S)wt(S)

• R∈RPP(sh(S))

xev(R)

The involution

Example

4 23 4 1 23 4 2 1 3 1 3 4 4 23 4 1 23 4 2 1 3 1 3 4

The involution

Example

4 23 4 1 23 4 2 1 3 1 3 4 4 23 4 1 23 4 2 1 3 1 3 4

The involution

Example

4 23 4 1 23 4 2 1 3 1 3 4 4 3 2 4 1 23 4 2 1 3 1 3 4

The involution

Example

4 23 4 1 23 4 2 1 3 1 3 4 4 3 2 4 1 23 4

• 2

1 3 1 3 4

The involution

Example

4 23 4 1 23 4 2 1 3 1 3 4 4 3 2 4 1 23 4 2

• 1 3

1 3 4

The involution

Example

4 23 4 1 23 4 2 1 3 1 3 4 4 3 2 4 1 23 4 2 1

• 3

1 3 4

The involution

Example

4 23 4 1 23 4 2 1 3 1 3 4 4 3 2 4 1 23 4 2 1 1 3 1 3 4

The involution

Example

4 23 4 1 23 4 2 1 3 1 3 4 4 3 2 4 1 23 4 2 1 1 3 1 3 4

The involution

Example

4 23 4 1 23 4 2 1 3 1 3 4 4 3 2 4 1 23 4 2 1

• 3

1 3 4

The involution

Example

4 23 4 1 23 4 2 1 3 1 3 4 4 3 2 4 1 23 4 2

• 1 3

1 3 4

The involution

Example

4 23 4 1 23 4 2 1 3 1 3 4 4 3 2 4 1 23 4

• 2

1 3 1 3 4

The involution

Example

4 23 4 1 23 4 2 1 3 1 3 4 4 3 2 4 1 23 4 2 1 3 1 3 4

The involution

Example

4 23 4 1 23 4 2 1 3 1 3 4 4 3 2 4 1 23 4 2 1 3 1 3 4

The involution

Example

4 23 4 1 23 4 2 1 3 1 3 4 4 23 4 1 23 4 2 1 3 1 3 4

The involution

Example

4 23 4 1 23 4 2 1 3 1 3 4 4 23 4 1 23 4 2 1 3 1 3 4

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