The symmetry and Schur expansion of dual stable
Grothendieck polynomials
Pavel Galashin
MIT
October 7, 2015
Joint work with Gaku Liu and Darij Grinberg
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
1 / 25
Part 1: Symmetry
Young diagrams, skew-shapes, SSYT
Skew shapes
λ = (4, 4, 3)
Pavel Galashin (MIT)
λ/µ = (4, 4, 3)/(2, 1)
Dual stable Grothendieck polynomials
October 7, 2015
3 / 25
Young diagrams, skew-shapes, SSYT
Skew shapes
λ = (4, 4, 3)
λ/µ = (4, 4, 3)/(2, 1)
Semi-standard Young tableau (SSYT)
2
Pavel Galashin (MIT)
1
3
2
2
4
6
6
7
1
4
2
2
4
6
6
Dual stable Grothendieck polynomials
October 7, 2015
3 / 25
Young diagrams, skew-shapes, SSYT
Skew shapes
λ = (4, 4, 3)
λ/µ = (4, 4, 3)/(2, 1)
Semi-standard Young tableau (SSYT)
2
Pavel Galashin (MIT)
1
3
2
2
4
6
6
7
1
4
2
2
4
6
6
Dual stable Grothendieck polynomials
October 7, 2015
3 / 25
Young diagrams, skew-shapes, SSYT
Skew shapes
λ = (4, 4, 3)
λ/µ = (4, 4, 3)/(2, 1)
Semi-standard Young tableau (SSYT)
2
Pavel Galashin (MIT)
1
3
2
2
4
6
6
1
4
2
2
4
7 6
6
Dual stable Grothendieck polynomials
October 7, 2015
3 / 25
Reverse plane partitions (RPP)
2
Pavel Galashin (MIT)
1
3
1
2
3
2
2
3
1
3
1
2
2
2
2
Dual stable Grothendieck polynomials
October 7, 2015
4 / 25
Reverse plane partitions (RPP)
2
Pavel Galashin (MIT)
1
3
1
2
3
2
2
3
1
3
1
2
2
2
2
Dual stable Grothendieck polynomials
October 7, 2015
4 / 25
Reverse plane partitions (RPP)
2
Pavel Galashin (MIT)
1
3
1
2
3
2
2
1
3
1
2
2
3 2
2
Dual stable Grothendieck polynomials
October 7, 2015
4 / 25
Reverse plane partitions (RPP)
2
1
3
1
2
3
2
2
1
3
1
2
2
3 2
2
SSYT is a special case of RPP!
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
4 / 25
Skew-Schur polynomials
Definition
If T is an SSYT then w (T ) := (#T −1 (1), #T −1 (2), . . . , #T −1 (m )),
where #T −1 (i ) = [the number of entries in T equal to i ].
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
5 / 25
Skew-Schur polynomials
Definition
If T is an SSYT then w (T ) := (#T −1 (1), #T −1 (2), . . . , #T −1 (m )),
where #T −1 (i ) = [the number of entries in T equal to i ].
Example
1
T =
2
2
2
6
6
Pavel Galashin (MIT)
3 ,
4
w (T ) = (1, 3, 1, 1, 0, 2),
Dual stable Grothendieck polynomials
x w (T ) = x11 x23 x31 x41 x50 x62 .
October 7, 2015
5 / 25
Skew-Schur polynomials
Definition
If T is an SSYT then w (T ) := (#T −1 (1), #T −1 (2), . . . , #T −1 (m )),
where #T −1 (i ) = [the number of entries in T equal to i ].
Example
1
T =
2
2
2
6
6
3 ,
4
w (T ) = (1, 3, 1, 1, 0, 2),
x w (T ) = x11 x23 x31 x41 x50 x62 .
Definition
sλ/µ (x1 , . . . , xm ) =
∑
x w (T ) .
T is a SSYT
of shape λ/µ
with entries ≤ m
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
5 / 25
Example
Example
Let m = 2, λ = (3, 2), µ = (1).
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
6 / 25
Example
Example
Let m = 2, λ = (3, 2), µ = (1).
Pavel Galashin (MIT)
1
1
1
1
2
2
2
2
Dual stable Grothendieck polynomials
October 7, 2015
6 / 25
Example
Example
Let m = 2, λ = (3, 2), µ = (1).
1
1
Pavel Galashin (MIT)
2
1
1
1
2
2
1
2
Dual stable Grothendieck polynomials
2
1
1
2
2
2
October 7, 2015
6 / 25
Example
Example
Let m = 2, λ = (3, 2), µ = (1).
1
1
w (T ) =
Pavel Galashin (MIT)
2
(3, 1)
1
1
1
2
(2, 2)
2
1
2
Dual stable Grothendieck polynomials
2
(2, 2)
1
1
2
2
2
(1, 3)
October 7, 2015
6 / 25
Example
Example
Let m = 2, λ = (3, 2), µ = (1).
1
1
w (T ) =
sλ/µ (x1 , x2 ) =
Pavel Galashin (MIT)
2
(3, 1)
x13 x2
1
1
1
2
2
(2, 2)
+x12 x22
1
2
Dual stable Grothendieck polynomials
1
2
(2, 2)
+x12 x22
1
2
2
2
(1, 3)
+x1 x23 .
October 7, 2015
6 / 25
Problem
Example
Let m = 2, λ = (3, 2), µ = (1).
1
“w (R ) =”
Pavel Galashin (MIT)
1
1
(3, 0)
1
2
1
(2, 1)
1
1
2
(2, 1)
Dual stable Grothendieck polynomials
1
2
2
(1, 2)
2
2
2
(0, 3)
October 7, 2015
7 / 25
Problem
Example
Let m = 2, λ = (3, 2), µ = (1).
1
“w (R ) =”
Pavel Galashin (MIT)
1
1 2
1 1
1 2
2
1
(3, 0)
1
(2, 1)
2
(2, 1)
2
(1, 2)
2
(0, 3)
Dual stable Grothendieck polynomials
2
October 7, 2015
7 / 25
Dual stable Grothendieck polynomials
Definition
If R is an RPP then w (R ) := (w1 (R ), w2 (R ), . . . , wm (R )), where
wi (R ) = [the number of columns in R containing i ].
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
8 / 25
Dual stable Grothendieck polynomials
Definition
If R is an RPP then w (R ) := (w1 (R ), w2 (R ), . . . , wm (R )), where
wi (R ) = [the number of columns in R containing i ].
Definition
gλ/µ (x1 , . . . , xm ) =
∑
x w (R ) .
R is a RPP
of shape λ/µ
with entries ≤ m
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
8 / 25
Dual stable Grothendieck polynomials
Definition
If R is an RPP then w (R ) := (w1 (R ), w2 (R ), . . . , wm (R )), where
wi (R ) = [the number of columns in R containing i ].
Definition
∑
gλ/µ (x1 , . . . , xm ) =
x w (R ) .
R is a RPP
of shape λ/µ
with entries ≤ m
Example
1
w (R ) =
Pavel Galashin (MIT)
1
1
(2, 0)
1
2
1
(1, 1)
1
1
2
(2, 1)
Dual stable Grothendieck polynomials
1
2
2
(1, 2)
2
2
2
(0, 2)
October 7, 2015
8 / 25
Properties of gλ/µ
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
9 / 25
Properties of gλ/µ
“represent the classes in K-homology of the ideal sheaves of the
boundaries of Schubert varieties” (see [Lam, Pylyavskyy (2007)]);
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
9 / 25
Properties of gλ/µ
“represent the classes in K-homology of the ideal sheaves of the
boundaries of Schubert varieties” (see [Lam, Pylyavskyy (2007)]);
SSYT(λ/µ, ≤ m ) ⊂ RPP(λ/µ, ≤ m ) and the top-degree
homogeneous component of gλ/µ is sλ/µ ;
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
9 / 25
Properties of gλ/µ
“represent the classes in K-homology of the ideal sheaves of the
boundaries of Schubert varieties” (see [Lam, Pylyavskyy (2007)]);
SSYT(λ/µ, ≤ m ) ⊂ RPP(λ/µ, ≤ m ) and the top-degree
homogeneous component of gλ/µ is sλ/µ ;
gλ/µ are symmetric (see [Lam, Pylyavskyy (2007)]);
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
9 / 25
Properties of gλ/µ
“represent the classes in K-homology of the ideal sheaves of the
boundaries of Schubert varieties” (see [Lam, Pylyavskyy (2007)]);
SSYT(λ/µ, ≤ m ) ⊂ RPP(λ/µ, ≤ m ) and the top-degree
homogeneous component of gλ/µ is sλ/µ ;
gλ/µ are symmetric (see [Lam, Pylyavskyy (2007)]);
there exist involutions Bi : RPP(λ/µ, ≤ m ) → RPP(λ/µ, ≤ m ) such
that
w (Bi (R )) = si w (R );
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
9 / 25
Properties of gλ/µ
“represent the classes in K-homology of the ideal sheaves of the
boundaries of Schubert varieties” (see [Lam, Pylyavskyy (2007)]);
SSYT(λ/µ, ≤ m ) ⊂ RPP(λ/µ, ≤ m ) and the top-degree
homogeneous component of gλ/µ is sλ/µ ;
gλ/µ are symmetric (see [Lam, Pylyavskyy (2007)]);
there exist involutions Bi : RPP(λ/µ, ≤ m ) → RPP(λ/µ, ≤ m ) such
that
w (Bi (R )) = si w (R );
Bi restricted to SSYT(λ/µ, ≤ m ) are classical Bender-Knuth
involutions.
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
9 / 25
sλ/µ and gλ/µ are symmetric!
Bender-Knuth involutions
Want to construct Bi : RPP(λ/µ, ≤ m ) → RPP(λ/µ, ≤ m ). Note that it
is enough to consider the case i = 1, m = 2:
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
10 / 25
sλ/µ and gλ/µ are symmetric!
Bender-Knuth involutions
Want to construct Bi : RPP(λ/µ, ≤ m ) → RPP(λ/µ, ≤ m ). Note that it
is enough to consider the case i = 1, m = 2:
Reduction to the case m = 2
Let i = 5.
1
5
5
2
6
7
1
3
3
7
8
6
8
9
1
1
5
6
5
5
6
6
9
6
7
7
8
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
10 / 25
sλ/µ and gλ/µ are symmetric!
Bender-Knuth involutions
Want to construct Bi : RPP(λ/µ, ≤ m ) → RPP(λ/µ, ≤ m ). Note that it
is enough to consider the case i = 1, m = 2:
Reduction to the case m = 2
Let i = 5.
1
5 5
2
6
7
3
7
8
5 6 6
8
9
1
1
1
5 5 6 6
6
7
7
3
9
8
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
10 / 25
sλ/µ and gλ/µ are symmetric!
Bender-Knuth involutions
Want to construct Bi : RPP(λ/µ, ≤ m ) → RPP(λ/µ, ≤ m ). Note that it
is enough to consider the case i = 1, m = 2:
Reduction to the case m = 2
Let i = 5.
1
5 5
2
6
7
3
7
8
5 6 6
8
9
1
1
1
5 5 6 6
6
7
7
8
Pavel Galashin (MIT)
3
9
5
−→
5
6
5
5
5
6
6
6
6
6
Dual stable Grothendieck polynomials
October 7, 2015
10 / 25
Three types of columns
1
2
2
1
Pavel Galashin (MIT)
1
2
2
1
Dual stable Grothendieck polynomials
October 7, 2015
11 / 25
Three types of columns
1
2
2
1
1
2
2
1
Definition
Let R ∈ RPP(λ/µ, 2). A column of R is called
mixed, if it contains a 1 and a 2;
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
11 / 25
Three types of columns
1
2
2
1
1
2
2
1
Definition
Let R ∈ RPP(λ/µ, 2). A column of R is called
mixed, if it contains a 1 and a 2;
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
11 / 25
Three types of columns
1
2
2
1
1
2
2
1
Definition
Let R ∈ RPP(λ/µ, 2). A column of R is called
mixed, if it contains a 1 and a 2;
1-pure, if it contains a 1 and not a 2;
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
11 / 25
Three types of columns
1
2
2
1
1
2
2
1
Definition
Let R ∈ RPP(λ/µ, 2). A column of R is called
mixed, if it contains a 1 and a 2;
1-pure, if it contains a 1 and not a 2;
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
11 / 25
Three types of columns
1
2
2
1
1
2
2
1
Definition
Let R ∈ RPP(λ/µ, 2). A column of R is called
mixed, if it contains a 1 and a 2;
1-pure, if it contains a 1 and not a 2;
2-pure, if it contains a 2 and not a 1;
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
11 / 25
Three types of columns
1
2
2
1
1
2
2
1
Definition
Let R ∈ RPP(λ/µ, 2). A column of R is called
mixed, if it contains a 1 and a 2;
1-pure, if it contains a 1 and not a 2;
2-pure, if it contains a 2 and not a 1;
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
11 / 25
Construction of Bender-Knuth involutions
Flip map
1
2
2
1
1
2
2
1
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
12 / 25
Construction of Bender-Knuth involutions
Flip map
1
1
2
2
1
1
2
1
Pavel Galashin (MIT)
2
1
−→
1
2
1
2
Dual stable Grothendieck polynomials
2
2
October 7, 2015
12 / 25
Descent-resolution step
A lot of descents
1
1
1
2
1
2
2
2
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
13 / 25
Descent-resolution step
A lot of descents
1
(M1) mixed vs. 1-pure;
1
1
2
1
2
2
2
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
13 / 25
Descent-resolution step
A lot of descents
1
(M1) mixed vs. 1-pure;
1
1
2
1
2
2
2
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
13 / 25
Descent-resolution step
A lot of descents
1
(M1) mixed vs. 1-pure;
1
1
2
1
2
2
(2M) 2-pure vs. mixed;
2
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
13 / 25
Descent-resolution step
A lot of descents
1
(M1) mixed vs. 1-pure;
1
1
2
1
2
2
(2M) 2-pure vs. mixed;
2
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
13 / 25
Descent-resolution step
A lot of descents
1
(M1) mixed vs. 1-pure;
1
1
2
1
2
2
Pavel Galashin (MIT)
2
(2M) 2-pure vs. mixed;
(21) 2-pure vs. 1-pure.
Dual stable Grothendieck polynomials
October 7, 2015
13 / 25
Descent-resolution step
A lot of descents
1
(M1) mixed vs. 1-pure;
1
1
2
1
2
2
Pavel Galashin (MIT)
2
(2M) 2-pure vs. mixed;
(21) 2-pure vs. 1-pure.
Dual stable Grothendieck polynomials
October 7, 2015
13 / 25
Descent-resolution step
A lot of descents
1
(M1) mixed vs. 1-pure;
1
1
2
1
2
2
Pavel Galashin (MIT)
2
(2M) 2-pure vs. mixed;
(21) 2-pure vs. 1-pure.
(MM) mixed vs. mixed
Dual stable Grothendieck polynomials
October 7, 2015
13 / 25
Descent-resolution step
A lot of descents
1
(M1) mixed vs. 1-pure;
1
1
2
1
2
2
Pavel Galashin (MIT)
2
(2M) 2-pure vs. mixed;
(21) 2-pure vs. 1-pure.
(MM) mixed vs. mixed – never happens!
Dual stable Grothendieck polynomials
October 7, 2015
13 / 25
Descent-resolution steps
(M1)
1
1
2
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
14 / 25
Descent-resolution steps
(M1)
1
1
1
→
1
2
Pavel Galashin (MIT)
2
Dual stable Grothendieck polynomials
October 7, 2015
14 / 25
Descent-resolution steps
(M1)
(2M)
1
1
1
→
1
2
Pavel Galashin (MIT)
1
2
2
2
Dual stable Grothendieck polynomials
October 7, 2015
14 / 25
Descent-resolution steps
(M1)
(2M)
1
1
1
→
1
2
Pavel Galashin (MIT)
1
→
2
2
2
1
2
2
Dual stable Grothendieck polynomials
October 7, 2015
14 / 25
Descent-resolution steps
(M1)
(2M)
1
1
1
→
1
2
Pavel Galashin (MIT)
1
→
2
2
(21)
2
1
2
1
2
2
Dual stable Grothendieck polynomials
October 7, 2015
14 / 25
Descent-resolution steps
(M1)
(2M)
1
1
1
→
1
2
Pavel Galashin (MIT)
1
→
2
2
(21)
2
1
2
1
2
→
2
1
2
Dual stable Grothendieck polynomials
October 7, 2015
14 / 25
Properties
The descent-resolution process
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
15 / 25
Properties
The descent-resolution process
ends after a finite number of steps;
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
15 / 25
Properties
The descent-resolution process
ends after a finite number of steps;
the result does not depend on the order!
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
15 / 25
Properties
The descent-resolution process
ends after a finite number of steps;
the result does not depend on the order!
Corollary
B1 is an involution on RPP(λ/µ, 2) that switches the number of 1-pure
columns with the number of 2-pure columns.
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
15 / 25
Why is Bi an involution?
(M1)
(2M)
1
1
1
→
1
2
Pavel Galashin (MIT)
1
→
2
2
(21)
2
1
2
1
2
→
2
1
2
Dual stable Grothendieck polynomials
October 7, 2015
16 / 25
Why is Bi an involution?
(M1)
(2M)
1
1
1
→
1
1
→
2
2
2
↑
↓
flip
2
↑
1
1
2
←
2
2
(2M)
Pavel Galashin (MIT)
1
2
1
2
→
2
1
2
↓
flip
↑
↓
flip
1
←
1
2
(21)
2
1
1
2
1
←
1
2
2
(M1)
(21)
Dual stable Grothendieck polynomials
October 7, 2015
16 / 25
Part 2: Schur expansion
Knuth equivalence
Definition
Two words are Knuth equivalent if they can be obtained from each other by
moves
yzx ↔ yxz, if x < y ≤ z;
xzy ↔ zxy , if x ≤ y < z.
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
18 / 25
Knuth equivalence
Definition
Two words are Knuth equivalent if they can be obtained from each other by
moves
yzx ↔ yxz, if x < y ≤ z;
xzy ↔ zxy , if x ≤ y < z.
Definition
Reading word: concatenate rows from bottom to top.
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
18 / 25
Knuth equivalence
Definition
Two words are Knuth equivalent if they can be obtained from each other by
moves
yzx ↔ yxz, if x < y ≤ z;
xzy ↔ zxy , if x ≤ y < z.
Definition
Reading word: concatenate rows from bottom to top.
1 3
T=
2
2
4
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
18 / 25
Knuth equivalence
Definition
Two words are Knuth equivalent if they can be obtained from each other by
moves
yzx ↔ yxz, if x < y ≤ z;
xzy ↔ zxy , if x ≤ y < z.
Definition
Reading word: concatenate rows from bottom to top.
1 3
T=
rw(T ) = (2, 2, 4, 1, 3).
2
2
4
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
18 / 25
Knuth equivalence
Definition
Two words are Knuth equivalent if they can be obtained from each other by
moves
yzx ↔ yxz, if x < y ≤ z;
xzy ↔ zxy , if x ≤ y < z.
Definition
Reading word: concatenate rows from bottom to top.
1 3
T=
rw(T ) = (2, 2, 4, 1, 3).
2
2
4
Proposition
Every word is Knuth equivalent to exactly one word which is a reading
word of a SSYT of straight shape (µ = ()).
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
18 / 25
Crystal operators
Definition
Ei : [m ]r → [m ]r ∪ {0} is defined as follows. For u ∈ [m ]r ,
Assume i = 3.
u=
1,4,1,3,5,4,4,3,3,1,3,1,4,5,3,1,3,3,4,1,4
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
19 / 25
Crystal operators
Definition
Ei : [m ]r → [m ]r ∪ {0} is defined as follows. For u ∈ [m ]r ,
ignore all letters of u except for i and i + 1;
Assume i = 3.
u=
1,4,1,3,5,4,4,3,3,1,3,1,4,5,3,1,3,3,4,1,4
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Dual stable Grothendieck polynomials
October 7, 2015
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Crystal operators
Definition
Ei : [m ]r → [m ]r ∪ {0} is defined as follows. For u ∈ [m ]r ,
ignore all letters of u except for i and i + 1;
Assume i = 3.
u=
1,4,1,3,5,4,4,3,3,1,3,1,4,5,3,1,3,3,4,1,4
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
19 / 25
Crystal operators
Definition
Ei : [m ]r → [m ]r ∪ {0} is defined as follows. For u ∈ [m ]r ,
ignore all letters of u except for i and i + 1;
label each i by ) and each i + 1 by (;
Assume i = 3.
u=
1,4,1,3,5,4,4,3,3,1,3,1,4,5,3,1,3,3,4,1,4
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
19 / 25
Crystal operators
Definition
Ei : [m ]r → [m ]r ∪ {0} is defined as follows. For u ∈ [m ]r ,
ignore all letters of u except for i and i + 1;
label each i by ) and each i + 1 by (;
Assume i = 3.
u=
1,4,1,3,5,4,4,3,3,1,3,1,4,5,3,1,3,3,4,1,4
1,(,1,),5,(,(,),),1,),1,(,5,),1,),),(,1,(
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
19 / 25
Crystal operators
Definition
Ei : [m ]r → [m ]r ∪ {0} is defined as follows. For u ∈ [m ]r ,
ignore all letters of u except for i and i + 1;
label each i by ) and each i + 1 by (;
ignore all pairs of matching parentheses;
Assume i = 3.
u=
1,4,1,3,5,4,4,3,3,1,3,1,4,5,3,1,3,3,4,1,4
1,(,1,),5,(,(,),),1,),1,(,5,),1,),),(,1,(
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
19 / 25
Crystal operators
Definition
Ei : [m ]r → [m ]r ∪ {0} is defined as follows. For u ∈ [m ]r ,
ignore all letters of u except for i and i + 1;
label each i by ) and each i + 1 by (;
ignore all pairs of matching parentheses;
Assume i = 3.
u=
1,4,1,3,5,4,4,3,3,1,3,1,4,5,3,1,3,3,4,1,4
1,(,1,),5,(,(,),),1,),1,(,5,),1,),),(,1,(
1,(,1,),5,(,(,),),1,),1,(,5,),1,),),(,1,(
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
19 / 25
Crystal operators
Definition
Ei : [m ]r → [m ]r ∪ {0} is defined as follows. For u ∈ [m ]r ,
ignore all letters of u except for i and i + 1;
label each i by ) and each i + 1 by (;
ignore all pairs of matching parentheses;
replace the rightmost unmatched ) by (.
Assume i = 3.
u=
1,4,1,3,5,4,4,3,3,1,3,1,4,5,3,1,3,3,4,1,4
1,(,1,),5,(,(,),),1,),1,(,5,),1,),),(,1,(
1,(,1,),5,(,(,),),1,),1,(,5,),1,),),(,1,(
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
19 / 25
Crystal operators
Definition
Ei : [m ]r → [m ]r ∪ {0} is defined as follows. For u ∈ [m ]r ,
ignore all letters of u except for i and i + 1;
label each i by ) and each i + 1 by (;
ignore all pairs of matching parentheses;
replace the rightmost unmatched ) by (.
Assume i = 3.
u=
1,4,1,3,5,4,4,3,3,1,3,1,4,5,3,1,3,3,4,1,4
1,(,1,),5,(,(,),),1,),1,(,5,),1,),),(,1,(
1,(,1,),5,(,(,),),1,),1,(,5,),1,),),(,1,(
1,(,1,),5,(,(,),),1,),1,(,5,),1,),(,(,1,(
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
19 / 25
Crystal operators
Definition
Ei : [m ]r → [m ]r ∪ {0} is defined as follows. For u ∈ [m ]r ,
ignore all letters of u except for i and i + 1;
label each i by ) and each i + 1 by (;
ignore all pairs of matching parentheses;
replace the rightmost unmatched ) by (.
Assume i = 3.
u=
1,4,1,3,5,4,4,3,3,1,3,1,4,5,3,1,3,3,4,1,4
1,(,1,),5,(,(,),),1,),1,(,5,),1,),),(,1,(
1,(,1,),5,(,(,),),1,),1,(,5,),1,),),(,1,(
1,(,1,),5,(,(,),),1,),1,(,5,),1,),(,(,1,(
Ei (u ) = 1,4,1,3,5,4,4,3,3,1,3,1,4,5,3,1,3,4,4,1,4
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
19 / 25
Crystal operators on SSYT
(a
picture
from
[Kashiwara (1995)])
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
20 / 25
Littlewood-Richardson rule
Crystal operators on words commute with Knuth equivalence relations;
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
21 / 25
Littlewood-Richardson rule
Crystal operators on words commute with Knuth equivalence relations;
For any straight shape λ and any m, the crystal graph on SSYT(λ, m )
is connected;
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
21 / 25
Littlewood-Richardson rule
Crystal operators on words commute with Knuth equivalence relations;
For any straight shape λ and any m, the crystal graph on SSYT(λ, m )
is connected;
Only one tableau Tλ ∈ SSYT(λ, m ) satisfies Ei−1 (Tλ ) = ∅ for all
i < m:
1 1 1 1
Pavel Galashin (MIT)
2
2
2
3
3
3
2
Dual stable Grothendieck polynomials
October 7, 2015
21 / 25
Littlewood-Richardson rule
Crystal operators on words commute with Knuth equivalence relations;
For any straight shape λ and any m, the crystal graph on SSYT(λ, m )
is connected;
Only one tableau Tλ ∈ SSYT(λ, m ) satisfies Ei−1 (Tλ ) = ∅ for all
i < m:
1 1 1 1
2
2
2
3
3
3
2
We have w (Tλ ) = λ.
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
21 / 25
Littlewood-Richardson rule
Crystal operators on words commute with Knuth equivalence relations;
For any straight shape λ and any m, the crystal graph on SSYT(λ, m )
is connected;
Only one tableau Tλ ∈ SSYT(λ, m ) satisfies Ei−1 (Tλ ) = ∅ for all
i < m:
1 1 1 1
2
2
2
3
3
3
2
We have w (Tλ ) = λ.
Corollary
If W ⊂ [m ]r is closed under the action of Ei , then
∑
u ∈W
Pavel Galashin (MIT)
x w (u ) =
∑
sw (u ) (x1 , . . . , xm ).
u ∈W :Ei−1 (u )=∅ ∀i
Dual stable Grothendieck polynomials
October 7, 2015
21 / 25
Reading words for RPP
Definition
1
R= 1
2
2
3
1
1
3
3
4
4
1
1
1
3
5
5
Pavel Galashin (MIT)
2
4
4
4
Dual stable Grothendieck polynomials
October 7, 2015
22 / 25
Reading words for RPP
Definition
1
R= 1
2
2
3
1
1
3
3
4
4
1
1
1
3
5
5
Pavel Galashin (MIT)
2
4
4
4
2
1
→
1
1
3
4
3
2
3
5
4
Dual stable Grothendieck polynomials
October 7, 2015
22 / 25
Reading words for RPP
Definition
1
R= 1
2
2
3
1
1
3
3
4
4
1
1
1
3
5
5
Pavel Galashin (MIT)
2
4
4
4
2
1
→
1
1
3
4
rw(R ) = 34253134112.
3
2
3
5
4
Dual stable Grothendieck polynomials
October 7, 2015
22 / 25
Crystal action on RPP
Definition
For R ∈ RPP(λ/µ, m ) define ceq(R ) = (ceq1 (R ), ceq2 (R ), . . . ) where
ceqi (R ) := [number of equalities between rows i and i + 1].
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
23 / 25
Crystal action on RPP
Definition
For R ∈ RPP(λ/µ, m ) define ceq(R ) = (ceq1 (R ), ceq2 (R ), . . . ) where
ceqi (R ) := [number of equalities between rows i and i + 1].
Proposition
If R ∈ RPP(λ/µ, m ) then there exists a unique Q ∈ RPP(λ/µ, m ) with
rw(Q ) = Ei (rw(R ));
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
23 / 25
Crystal action on RPP
Definition
For R ∈ RPP(λ/µ, m ) define ceq(R ) = (ceq1 (R ), ceq2 (R ), . . . ) where
ceqi (R ) := [number of equalities between rows i and i + 1].
Proposition
If R ∈ RPP(λ/µ, m ) then there exists a unique Q ∈ RPP(λ/µ, m ) with
rw(Q ) = Ei (rw(R ));
ceq(Q ) = ceq(R ).
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
23 / 25
Crystal action on RPP
Definition
For R ∈ RPP(λ/µ, m ) define ceq(R ) = (ceq1 (R ), ceq2 (R ), . . . ) where
ceqi (R ) := [number of equalities between rows i and i + 1].
Proposition
If R ∈ RPP(λ/µ, m ) then there exists a unique Q ∈ RPP(λ/µ, m ) with
rw(Q ) = Ei (rw(R ));
ceq(Q ) = ceq(R ).
Proposition
The ceq-statistics is preserved by flips and descent resolution steps.
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
23 / 25
Schur expansion of gλ/µ
Theorem
gλ/µ (x1 , . . . , xm ) =
∑
sw (R ) (x1 , . . . , xm ).
R is a RPP
of shape λ/µ
with entries ≤ m
such that Ei−1 (rw(R )) = ∅ for all i < m
Pavel Galashin (MIT)
Dual stable Grothendieck polynomials
October 7, 2015
24 / 25
Thank you!
Lam, T., Pylyavskyy, P., Combinatorial Hopf algebras and K -homology
of Grassmanians, International Mathematics Research Notices, Vol.
2007, doi:10.1093/imrn/rnm125.
Galashin, P. (2014). A Littlewood-Richardson Rule for Dual Stable
Grothendieck Polynomials. arXiv preprint arXiv:1501.00051.
Galashin, P., Grinberg, D., and Liu, G. (2015). Refined dual stable
Grothendieck polynomials and generalized Bender-Knuth involutions.
arXiv preprint arXiv:1509.03803.
Kashiwara, M. (1995). On crystal bases. Representations of groups
(Banff, AB, 1994), 16, 155-197.
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Dual stable Grothendieck polynomials
October 7, 2015
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