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Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual...

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What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions Gaku Liu Joint work with Pavel Galashin and Darij Grinberg MIT FPSAC 2016
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Page 1: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Refined dual stable Grothendieck polynomials andgeneralized Bender-Knuth involutions

Gaku Liu

Joint work with Pavel Galashin and Darij Grinberg

MIT

FPSAC 2016

Page 2: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

History

Grothendieck polynomials and their variations are K -theoryanalogues of Schubert and Schur polynomials.

Grothendieck polynomials (Lascoux-Schutzenberger ’82):polynomial representatives of structure sheaves of Schubertvarieties in the K -theory of flag manifolds

stable Grothendieck polynomials (Fomin-Kirillov ’96):symmetric power series representatives of structure sheaves ofSchubert varieties in the K -theory of the Grassmannian

dual stable Grothendieck polynomials (Lam-Pylyavskyy ’07):symmetric functions which are the continuous dual basis tothe stable Grothendieck polynomials with respect to the Hallinner product

Page 3: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

History

Grothendieck polynomials and their variations are K -theoryanalogues of Schubert and Schur polynomials.

Grothendieck polynomials (Lascoux-Schutzenberger ’82):polynomial representatives of structure sheaves of Schubertvarieties in the K -theory of flag manifolds

stable Grothendieck polynomials (Fomin-Kirillov ’96):symmetric power series representatives of structure sheaves ofSchubert varieties in the K -theory of the Grassmannian

dual stable Grothendieck polynomials (Lam-Pylyavskyy ’07):symmetric functions which are the continuous dual basis tothe stable Grothendieck polynomials with respect to the Hallinner product

Page 4: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

History

Grothendieck polynomials and their variations are K -theoryanalogues of Schubert and Schur polynomials.

Grothendieck polynomials (Lascoux-Schutzenberger ’82):polynomial representatives of structure sheaves of Schubertvarieties in the K -theory of flag manifolds

stable Grothendieck polynomials (Fomin-Kirillov ’96):symmetric power series representatives of structure sheaves ofSchubert varieties in the K -theory of the Grassmannian

dual stable Grothendieck polynomials (Lam-Pylyavskyy ’07):symmetric functions which are the continuous dual basis tothe stable Grothendieck polynomials with respect to the Hallinner product

Page 5: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

History

Grothendieck polynomials and their variations are K -theoryanalogues of Schubert and Schur polynomials.

Grothendieck polynomials (Lascoux-Schutzenberger ’82):polynomial representatives of structure sheaves of Schubertvarieties in the K -theory of flag manifolds

stable Grothendieck polynomials (Fomin-Kirillov ’96):symmetric power series representatives of structure sheaves ofSchubert varieties in the K -theory of the Grassmannian

dual stable Grothendieck polynomials (Lam-Pylyavskyy ’07):symmetric functions which are the continuous dual basis tothe stable Grothendieck polynomials with respect to the Hallinner product

Page 6: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Reverse plane partitions

A reverse plane partition (rpp) is a filling of a skew diagram λ/µwith positive integers such that entries are weakly increasing alongrows and columns.

1 1 3

1 1

2 2

1 3 4

2 3

Page 7: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Irredundant content

We define the irredundant content of an rpp T to be the sequencec(T ) = (c1, c2, c3, . . . ) where ci is the number of columns of Twhich contain an i .

1 1 3

1 1

2 2

1 3 4

2 3

c(T ) = (3, 3, 2, 1, 0, 0, . . . )

Page 8: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Dual stable Grothendieck polynomials

For each skew shape λ/µ, define

gλ/µ =∑

T is an rppof shape λ/µ

xc(T )

where x (c1,c2,c3,... ) = xc11 xc2

2 xc33 · · · .

The gλ/µ are called dual stable Grothendieck polynomials.

Page 9: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Dual stable Grothendieck polynomials

For each skew shape λ/µ, define

gλ/µ =∑

T is an rppof shape λ/µ

xc(T )

where x (c1,c2,c3,... ) = xc11 xc2

2 xc33 · · · .

The gλ/µ are called dual stable Grothendieck polynomials.

Page 10: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Dual stable Grothendiecks are symmetric

Theorem (Lam-Pylyavskyy ’07)

For every λ/µ, the power series gλ/µ is symmetric in the xi .

Their proof uses Fomin-Greene operators—fundamentallycombinatorial, but the combinatorics are mysterious.

Our result: A bijective proof of this theorem.

Bijection is a generalization of the Bender-Knuth involutionsfor semistandard tableaux.

Page 11: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Dual stable Grothendiecks are symmetric

Theorem (Lam-Pylyavskyy ’07)

For every λ/µ, the power series gλ/µ is symmetric in the xi .

Their proof uses Fomin-Greene operators—fundamentallycombinatorial, but the combinatorics are mysterious.

Our result: A bijective proof of this theorem.

Bijection is a generalization of the Bender-Knuth involutionsfor semistandard tableaux.

Page 12: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Dual stable Grothendiecks are symmetric

Theorem (Lam-Pylyavskyy ’07)

For every λ/µ, the power series gλ/µ is symmetric in the xi .

Their proof uses Fomin-Greene operators—fundamentallycombinatorial, but the combinatorics are mysterious.

Our result: A bijective proof of this theorem.

Bijection is a generalization of the Bender-Knuth involutionsfor semistandard tableaux.

Page 13: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Dual stable Grothendiecks are symmetric

Theorem (Lam-Pylyavskyy ’07)

For every λ/µ, the power series gλ/µ is symmetric in the xi .

Their proof uses Fomin-Greene operators—fundamentallycombinatorial, but the combinatorics are mysterious.

Our result: A bijective proof of this theorem.

Bijection is a generalization of the Bender-Knuth involutionsfor semistandard tableaux.

Page 14: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Schur functions

A semistandard Young tableau (SSYT) is a filling of a skewdiagram λ/µ with positive integers such that entries are weaklyincreasing along rows and strictly increasing down columns.

For each skew shape λ/µ, define the Schur function

sλ/µ =∑

T is a SSYTof shape λ/µ

xc(T ).

The Bender-Knuth involutions are a way to prove the sλ/µ aresymmetric.

Page 15: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Schur functions

A semistandard Young tableau (SSYT) is a filling of a skewdiagram λ/µ with positive integers such that entries are weaklyincreasing along rows and strictly increasing down columns.

For each skew shape λ/µ, define the Schur function

sλ/µ =∑

T is a SSYTof shape λ/µ

xc(T ).

The Bender-Knuth involutions are a way to prove the sλ/µ aresymmetric.

Page 16: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Schur functions

A semistandard Young tableau (SSYT) is a filling of a skewdiagram λ/µ with positive integers such that entries are weaklyincreasing along rows and strictly increasing down columns.

For each skew shape λ/µ, define the Schur function

sλ/µ =∑

T is a SSYTof shape λ/µ

xc(T ).

The Bender-Knuth involutions are a way to prove the sλ/µ aresymmetric.

Page 17: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Bender-Knuth involutions

Suffices to show that sλ/µ is symmetric in the variables xi and xi+1

for all i .

Let SSYT(λ/µ) be the set of all SSYT’s of shape λ/µ.

For each i , we define an involution Bi : SSYT(λ/µ)→ SSYT(λ/µ)such that c(BiT ) = sic(T ), where si is the permutation (i i + 1).

Page 18: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Bender-Knuth involutions

Suffices to show that sλ/µ is symmetric in the variables xi and xi+1

for all i .

Let SSYT(λ/µ) be the set of all SSYT’s of shape λ/µ.

For each i , we define an involution Bi : SSYT(λ/µ)→ SSYT(λ/µ)such that c(BiT ) = sic(T ), where si is the permutation (i i + 1).

Page 19: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Bender-Knuth involutions

Suffices to show that sλ/µ is symmetric in the variables xi and xi+1

for all i .

Let SSYT(λ/µ) be the set of all SSYT’s of shape λ/µ.

For each i , we define an involution Bi : SSYT(λ/µ)→ SSYT(λ/µ)such that c(BiT ) = sic(T ), where si is the permutation (i i + 1).

Page 20: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

1 . . .

1 1 1 1 1 1 1 2 2 2 2

. . . 2 2

1 . . .

1 1 1 1 1 2 2 2 2 2 2

. . . 2 2

Page 21: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Generalized Bender-Knuth involutions

To prove gλ/µ is symmetric, suffices to show it is symmetric in thevariables xi and xi+1 for all i .

Let RPP(λ/µ) be the set of all RPP’s of shape λ/µ.

For each i , we define an involution Bi : RPP(λ/µ)→ RPP(λ/µ)such that c(BiT ) = sic(T ), where si is the permutation (i i + 1).

Page 22: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Generalized Bender-Knuth involutions

To prove gλ/µ is symmetric, suffices to show it is symmetric in thevariables xi and xi+1 for all i .

Let RPP(λ/µ) be the set of all RPP’s of shape λ/µ.

For each i , we define an involution Bi : RPP(λ/µ)→ RPP(λ/µ)such that c(BiT ) = sic(T ), where si is the permutation (i i + 1).

Page 23: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Generalized Bender-Knuth involutions

To prove gλ/µ is symmetric, suffices to show it is symmetric in thevariables xi and xi+1 for all i .

Let RPP(λ/µ) be the set of all RPP’s of shape λ/µ.

For each i , we define an involution Bi : RPP(λ/µ)→ RPP(λ/µ)such that c(BiT ) = sic(T ), where si is the permutation (i i + 1).

Page 24: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Three types of columns

1 2

12

Restricting an rpp to cells with entries 1 or2, we have three types of columns:

1-pure: Contains 1’s and no 2’s.

mixed: Contains both 1’s and 2’s.

2-pure: Contains 2’s and no 1’s.

Page 25: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Three types of columns

1 2

12

Restricting an rpp to cells with entries 1 or2, we have three types of columns:

1-pure: Contains 1’s and no 2’s.

mixed: Contains both 1’s and 2’s.

2-pure: Contains 2’s and no 1’s.

Page 26: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Three types of columns

1 2

12

Restricting an rpp to cells with entries 1 or2, we have three types of columns:

1-pure: Contains 1’s and no 2’s.

mixed: Contains both 1’s and 2’s.

2-pure: Contains 2’s and no 1’s.

Page 27: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Three types of columns

1 2

12

Restricting an rpp to cells with entries 1 or2, we have three types of columns:

1-pure: Contains 1’s and no 2’s.

mixed: Contains both 1’s and 2’s.

2-pure: Contains 2’s and no 1’s.

Page 28: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Defintion of B1

Let T ∈ RPP(λ/µ). Construct B1(T ) from T as follows.

1 Change all 1-pure columns to 2-pure columns and all 2-purecolumns to 1-pure columns (of the same size).

Page 29: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Defintion of B1

Let T ∈ RPP(λ/µ). Construct B1(T ) from T as follows.

1 Change all 1-pure columns to 2-pure columns and all 2-purecolumns to 1-pure columns (of the same size).

1 2

12

−→

1 1

22

Page 30: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Defintion of B1

Let T ∈ RPP(λ/µ). Construct B1(T ) from T as follows.

1 Change all 1-pure columns to 2-pure columns and all 2-purecolumns to 1-pure columns (of the same size).

2 “Resolve descents” one at a time until none remain.

Page 31: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Defintion of B1

Let T ∈ RPP(λ/µ). Construct B1(T ) from T as follows.

1 Change all 1-pure columns to 2-pure columns and all 2-purecolumns to 1-pure columns (of the same size).

2 “Resolve descents” one at a time until none remain.

A “descent” is a pair of adjacent columns which contain a 2immediately to the left of a 1.

Page 32: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Resolving descents: Example

1 1

2

−→

1

12

Page 33: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Resolving descents: Example

1 1

2

−→

1

12

Page 34: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

1 1

2

−→

1

12

1

22

−→ 1 2

2

12

−→ 21

Page 35: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Defintion of B1

Let T ∈ RPP(λ/µ). Construct B1(T ) from T as follows.

1 Change all 1-pure columns to 2-pure columns and all 2-purecolumns to 1-pure columns (of the same length).

2 “Resolve descents” one at a time until none remain.

How do we know that this process will terminate?

Page 36: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Defintion of B1

Let T ∈ RPP(λ/µ). Construct B1(T ) from T as follows.

1 Change all 1-pure columns to 2-pure columns and all 2-purecolumns to 1-pure columns (of the same length).

2 “Resolve descents” one at a time until none remain.

How do we know that this process will terminate?

Page 37: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

1 1

2

−→

1

12

1

22

−→ 1 2

2

12

−→ 21

Page 38: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Defintion of B1

Let T ∈ RPP(λ/µ). Construct B1(T ) from T as follows.

1 Change all 1-pure columns to 2-pure columns and all 2-purecolumns to 1-pure columns (of the same length).

2 “Resolve descents” one at a time until none remain.How do we know that this process will terminate?

Look at positions of 1-pure and 2-pure columns.

How do we know the end result is unique?

Page 39: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Defintion of B1

Let T ∈ RPP(λ/µ). Construct B1(T ) from T as follows.

1 Change all 1-pure columns to 2-pure columns and all 2-purecolumns to 1-pure columns (of the same length).

2 “Resolve descents” one at a time until none remain.How do we know that this process will terminate?

Look at positions of 1-pure and 2-pure columns.

How do we know the end result is unique?

Page 40: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

A lemma

Let S be the set of all intermediate tableaux that can be achievedduring the above algorithm.

For T , T ′ ∈ S , write Tu−→ T ′ if T ′ is obtained from T by

resolving a descent in columns u, u + 1.

Write T∗−→ T ′ if T ′ can be obtained from T through a sequence

of descent resolutions.

Lemma

If T , Tu, and Tv ∈ S such that Tu−→ Tu and T

v−→ Tv , then thereexists T ′ ∈ S such that Tu

∗−→ T ′ and Tv∗−→ T ′.

Page 41: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

A lemma

Let S be the set of all intermediate tableaux that can be achievedduring the above algorithm.

For T , T ′ ∈ S , write Tu−→ T ′ if T ′ is obtained from T by

resolving a descent in columns u, u + 1.

Write T∗−→ T ′ if T ′ can be obtained from T through a sequence

of descent resolutions.

Lemma

If T , Tu, and Tv ∈ S such that Tu−→ Tu and T

v−→ Tv , then thereexists T ′ ∈ S such that Tu

∗−→ T ′ and Tv∗−→ T ′.

Page 42: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

A lemma

Let S be the set of all intermediate tableaux that can be achievedduring the above algorithm.

For T , T ′ ∈ S , write Tu−→ T ′ if T ′ is obtained from T by

resolving a descent in columns u, u + 1.

Write T∗−→ T ′ if T ′ can be obtained from T through a sequence

of descent resolutions.

Lemma

If T , Tu, and Tv ∈ S such that Tu−→ Tu and T

v−→ Tv , then thereexists T ′ ∈ S such that Tu

∗−→ T ′ and Tv∗−→ T ′.

Page 43: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

A lemma

Let S be the set of all intermediate tableaux that can be achievedduring the above algorithm.

For T , T ′ ∈ S , write Tu−→ T ′ if T ′ is obtained from T by

resolving a descent in columns u, u + 1.

Write T∗−→ T ′ if T ′ can be obtained from T through a sequence

of descent resolutions.

Lemma

If T , Tu, and Tv ∈ S such that Tu−→ Tu and T

v−→ Tv , then thereexists T ′ ∈ S such that Tu

∗−→ T ′ and Tv∗−→ T ′.

Page 44: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Proof of lemma

Lemma

If T , Tu, and Tv ∈ S such that Tu−→ Tu and T

v−→ Tv , then thereexists T ′ ∈ S such that Tu

∗−→ T ′ and Tv∗−→ T ′.

Proof: If |u − v | ≥ 2, then the result is easy.

Assume u = v − 1. Columns u, u + 1, u + 2 must look like:

1 1

22

Page 45: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Proof of lemma

Lemma

If T , Tu, and Tv ∈ S such that Tu−→ Tu and T

v−→ Tv , then thereexists T ′ ∈ S such that Tu

∗−→ T ′ and Tv∗−→ T ′.

Proof: If |u − v | ≥ 2, then the result is easy.

Assume u = v − 1. Columns u, u + 1, u + 2 must look like:

1 1

22

Page 46: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Proof of lemma

Lemma

If T , Tu, and Tv ∈ S such that Tu−→ Tu and T

v−→ Tv , then thereexists T ′ ∈ S such that Tu

∗−→ T ′ and Tv∗−→ T ′.

Proof: If |u − v | ≥ 2, then the result is easy.

Assume u = v − 1. Columns u, u + 1, u + 2 must look like:

1 1

22

Page 47: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

1 1

22

u−→

11 2

2

u+1−→

21 1

2

u−→

1 2

12

1 1

22

u+1−→

11

22

u−→

12

12

u+1−→

1 2

12

Page 48: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Resolving descents: End result is unique

Proposition

For each T ∈ S , there is a unique T ′ ∈ RPP(λ/µ) such that

T∗−→ T ′.

Proof: Let ` : S → N be a function such that if T1u−→ T2, then

`(T1) < `(T2).

We use backward induction on `(T ). Suppose T /∈ RPP(λ/µ).Suppose T

u−→ Tu and Tv−→ Tv .

By induction, there are unique T ′u, T ′v ∈ RPP(λ/µ) such that

Tu∗−→ T ′u, Tv

∗−→ T ′v .

By the Lemma, we must have T ′u = T ′v .

Since this holds for any u, v , the Proposition is proved.

Page 49: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Resolving descents: End result is unique

Proposition

For each T ∈ S , there is a unique T ′ ∈ RPP(λ/µ) such that

T∗−→ T ′.

Proof: Let ` : S → N be a function such that if T1u−→ T2, then

`(T1) < `(T2).

We use backward induction on `(T ). Suppose T /∈ RPP(λ/µ).Suppose T

u−→ Tu and Tv−→ Tv .

By induction, there are unique T ′u, T ′v ∈ RPP(λ/µ) such that

Tu∗−→ T ′u, Tv

∗−→ T ′v .

By the Lemma, we must have T ′u = T ′v .

Since this holds for any u, v , the Proposition is proved.

Page 50: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Resolving descents: End result is unique

Proposition

For each T ∈ S , there is a unique T ′ ∈ RPP(λ/µ) such that

T∗−→ T ′.

Proof: Let ` : S → N be a function such that if T1u−→ T2, then

`(T1) < `(T2).

We use backward induction on `(T ). Suppose T /∈ RPP(λ/µ).Suppose T

u−→ Tu and Tv−→ Tv .

By induction, there are unique T ′u, T ′v ∈ RPP(λ/µ) such that

Tu∗−→ T ′u, Tv

∗−→ T ′v .

By the Lemma, we must have T ′u = T ′v .

Since this holds for any u, v , the Proposition is proved.

Page 51: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Resolving descents: End result is unique

Proposition

For each T ∈ S , there is a unique T ′ ∈ RPP(λ/µ) such that

T∗−→ T ′.

Proof: Let ` : S → N be a function such that if T1u−→ T2, then

`(T1) < `(T2).

We use backward induction on `(T ). Suppose T /∈ RPP(λ/µ).Suppose T

u−→ Tu and Tv−→ Tv .

By induction, there are unique T ′u, T ′v ∈ RPP(λ/µ) such that

Tu∗−→ T ′u, Tv

∗−→ T ′v .

By the Lemma, we must have T ′u = T ′v .

Since this holds for any u, v , the Proposition is proved.

Page 52: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Resolving descents: End result is unique

Proposition

For each T ∈ S , there is a unique T ′ ∈ RPP(λ/µ) such that

T∗−→ T ′.

Proof: Let ` : S → N be a function such that if T1u−→ T2, then

`(T1) < `(T2).

We use backward induction on `(T ). Suppose T /∈ RPP(λ/µ).Suppose T

u−→ Tu and Tv−→ Tv .

By induction, there are unique T ′u, T ′v ∈ RPP(λ/µ) such that

Tu∗−→ T ′u, Tv

∗−→ T ′v .

By the Lemma, we must have T ′u = T ′v .

Since this holds for any u, v , the Proposition is proved.

Page 53: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Resolving descents: End result is unique

Proposition

For each T ∈ S , there is a unique T ′ ∈ RPP(λ/µ) such that

T∗−→ T ′.

Proof: Let ` : S → N be a function such that if T1u−→ T2, then

`(T1) < `(T2).

We use backward induction on `(T ). Suppose T /∈ RPP(λ/µ).Suppose T

u−→ Tu and Tv−→ Tv .

By induction, there are unique T ′u, T ′v ∈ RPP(λ/µ) such that

Tu∗−→ T ′u, Tv

∗−→ T ′v .

By the Lemma, we must have T ′u = T ′v .

Since this holds for any u, v , the Proposition is proved.

Page 54: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Newman’s Lemma

Note about the above proof: We are implicitly basing ourargument on Newman’s lemma (or the diamond lemma): Aterminating rewriting system is confluent if it locally confluent.

Page 55: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Defintion of B1

Let T ∈ RPP(λ/µ). Construct B1(T ) from T as follows.

1 Change all 1-pure columns to 2-pure columns and all 2-purecolumns to 1-pure columns (of the same length).

2 “Resolve descents” one at a time until none remain.How do we know that this process will terminate?

Look at positions of 1-pure and 2-pure columns.

How do we know the end result is unique?

We do.

Page 56: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Defintion of B1

Let T ∈ RPP(λ/µ). Construct B1(T ) from T as follows.

1 Change all 1-pure columns to 2-pure columns and all 2-purecolumns to 1-pure columns (of the same length).

2 “Resolve descents” one at a time until none remain.How do we know that this process will terminate?

Look at positions of 1-pure and 2-pure columns.

How do we know the end result is unique?

We do.

Page 57: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Defintion of B1

Let T ∈ RPP(λ/µ). Construct B1(T ) from T as follows.

1 Change all 1-pure columns to 2-pure columns and all 2-purecolumns to 1-pure columns (of the same length).

2 “Resolve descents” one at a time until none remain.How do we know that this process will terminate?

Look at positions of 1-pure and 2-pure columns.

How do we know the end result is unique?

We do.

Easy to check that B1 : RPP(λ/µ)→ RPP(λ/µ) is an involution.

Page 58: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Defintion of B1

Let T ∈ RPP(λ/µ). Construct B1(T ) from T as follows.

1 Change all 1-pure columns to 2-pure columns and all 2-purecolumns to 1-pure columns (of the same length).

2 “Resolve descents” one at a time until none remain.How do we know that this process will terminate?

Look at positions of 1-pure and 2-pure columns.

How do we know the end result is unique?

We do.

Easy to check that B1 : RPP(λ/µ)→ RPP(λ/µ) is an involution.

Thus, gλ/µ is symmetric.

Page 59: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

1 . . .

1 1 1 1 1 1 1 2 2 2 2

. . . 2 2

↓1 . . .

1 1 2 2 2 2 2 1 1 1 2

. . . 2 2

↓1 . . .

1 1 1 1 1 2 2 2 2 2 2

. . . 2 2

Page 60: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Generalized Bender-Knuth involutions

The Bi are the unique extensions of the Bender-Knuth involutions(to rpp) that satisfies a certain “locality” condition (see the lastsection of our paper).

The Bi also give some additional structure to RPP(λ/µ) beyondthe above symmetry: they preserve some of the behavior betweenadjacent rows of an rpp.

Page 61: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Generalized Bender-Knuth involutions

The Bi are the unique extensions of the Bender-Knuth involutions(to rpp) that satisfies a certain “locality” condition (see the lastsection of our paper).

The Bi also give some additional structure to RPP(λ/µ) beyondthe above symmetry: they preserve some of the behavior betweenadjacent rows of an rpp.

Page 62: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

The statistic ceq

For T ∈ RPP(λ/µ), define ceq(T ) = (q1, q2, q3, . . . ) where qi isthe number of vertically adjacent pairs of cells in rows i , i + 1 of Twith equal entries.

1 1 3

1 1

2 2

1 3 4

2 3

ceq(T ) = (2, 0, 0, 1, 0, 0, . . . )

Page 63: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Refined dual stable Grothendieck polynomials

For each skew shape λ/µ, define

gλ/µ =∑

T∈RPP(λ/µ)

tceq(T )xc(T )

where t(q1,q2,q3,... ) = tq11 tq2

2 tq33 · · · .

If t = 1, then gλ/µ = gλ/µ.If t = 0, then gλ/µ = sλ/µ.

From the previous proof, gλ/µ is symmetric in x .

Page 64: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Refined dual stable Grothendieck polynomials

For each skew shape λ/µ, define

gλ/µ =∑

T∈RPP(λ/µ)

tceq(T )xc(T )

where t(q1,q2,q3,... ) = tq11 tq2

2 tq33 · · · .

If t = 1, then gλ/µ = gλ/µ.If t = 0, then gλ/µ = sλ/µ.

From the previous proof, gλ/µ is symmetric in x .

Page 65: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

Refined dual stable Grothendieck polynomials

For each skew shape λ/µ, define

gλ/µ =∑

T∈RPP(λ/µ)

tceq(T )xc(T )

where t(q1,q2,q3,... ) = tq11 tq2

2 tq33 · · · .

If t = 1, then gλ/µ = gλ/µ.If t = 0, then gλ/µ = sλ/µ.

From the previous proof, gλ/µ is symmetric in x .

Page 66: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

An example and a conjecture

Example: If λ/µ is a single column with n cells, then

gλ/µ = en(t1, t2, . . . , tn−1, x1, x2, . . . ).

Conjecture (Grinberg):

gλ′/µ′ = det(eλi−µj−i+j(tµj+1, . . . , tλi−1, x1, x2, . . . )

)`(λ)

i ,j=1

Page 67: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

An example and a conjecture

Example: If λ/µ is a single column with n cells, then

gλ/µ = en(t1, t2, . . . , tn−1, x1, x2, . . . ).

Conjecture (Grinberg):

gλ′/µ′ = det(eλi−µj−i+j(tµj+1, . . . , tλi−1, x1, x2, . . . )

)`(λ)

i ,j=1

Page 68: Refined dual stable Grothendieck polynomials and ...galashin/slides/Gaku.pdf · What is a dual stable Grothendieck polynomial?Generalized BK involutionsRe ned dual stable Grothendieck

Thank you!

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What is a dual stable Grothendieck polynomial? Generalized BK involutions Refined dual stable Grothendieck polynomials

References

Sergey Fomin, Curtis Greene, Noncommutative Schurfunctions and their applications, Discrete Mathematics 306(2006) 1080–1096.

Pavel Galashin, Darij Grinberg, Gaku Liu, Refined dual stablepolynomials and generalized Bender-Knuth involutions,October 15, 2015, arXiv:1509.03803v2

Thomas Lam, Pavlo Pylyavskyy, Combinatorial Hopf algebrasand K -homology of Grassmanians, arxiv:0705.2189v1.

Alain Lascoux, Marcel-Paul Schutzenberger, Structure deHopf de l’anneau de cohomologie et de l’anneau deGrothendieck d’une variete de drapeaux, C. R. Acad. Sci.Paris Sr. I Math 295 (1982), 11, 629–633.


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