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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
The symmetry and Schur expansion of dual stable Grothendieck - - PowerPoint PPT Presentation
The symmetry and Schur expansion of dual stable Grothendieck - - PowerPoint PPT Presentation
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
SLIDE 2
SLIDE 3
Young diagrams, skew-shapes, SSYT
Skew shapes
λ = (4, 4, 3) λ/µ = (4, 4, 3)/(2, 1)
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 3 / 25
SLIDE 4
Young diagrams, skew-shapes, SSYT
Skew shapes
λ = (4, 4, 3) λ/µ = (4, 4, 3)/(2, 1)
Semi-standard Young tableau (SSYT)
1 3 2 2 4 2 6 6 1 4 2 2 4 7 6 6
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 3 / 25
SLIDE 5
Young diagrams, skew-shapes, SSYT
Skew shapes
λ = (4, 4, 3) λ/µ = (4, 4, 3)/(2, 1)
Semi-standard Young tableau (SSYT)
1 3 2 2 4 2 6 6 1 4 2 2 4 7 6 6
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 3 / 25
SLIDE 6
Young diagrams, skew-shapes, SSYT
Skew shapes
λ = (4, 4, 3) λ/µ = (4, 4, 3)/(2, 1)
Semi-standard Young tableau (SSYT)
1 3 2 2 4 2 6 6 1 4 2 2 4 7 6 6
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 3 / 25
SLIDE 7
Reverse plane partitions (RPP)
1 3 1 2 3 2 2 2 1 3 1 2 2 3 2 2
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 4 / 25
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Reverse plane partitions (RPP)
1 3 1 2 3 2 2 2 1 3 1 2 2 3 2 2
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 4 / 25
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Reverse plane partitions (RPP)
1 3 1 2 3 2 2 2 1 3 1 2 2 3 2 2
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 4 / 25
SLIDE 10
Reverse plane partitions (RPP)
1 3 1 2 3 2 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
SLIDE 11
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
SLIDE 12
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
T = 1 3 2 2 4 2 6 6 , w(T) = (1, 3, 1, 1, 0, 2), xw(T) = x1
1x3 2x1 3x1 4x0 5x2 6.
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 5 / 25
SLIDE 13
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
T = 1 3 2 2 4 2 6 6 , w(T) = (1, 3, 1, 1, 0, 2), xw(T) = x1
1x3 2x1 3x1 4x0 5x2 6.
Definition
sλ/µ(x1, . . . , xm) =
∑
T is a SSYT
- f shape λ/µ
with entries ≤ m
xw(T).
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 5 / 25
SLIDE 14
Example
Example
Let m = 2, λ = (3, 2), µ = (1).
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 6 / 25
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Example
Example
Let m = 2, λ = (3, 2), µ = (1). 1 2 1 2 1 2 1 2
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 6 / 25
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Example
Example
Let m = 2, λ = (3, 2), µ = (1). 1 1 1 2 1 2 1 2 1 1 2 2 1 2 2 2
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 6 / 25
SLIDE 17
Example
Example
Let m = 2, λ = (3, 2), µ = (1). 1 1 1 2 1 2 1 2 1 1 2 2 1 2 2 2 w(T) = (3, 1) (2, 2) (2, 2) (1, 3)
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 6 / 25
SLIDE 18
Example
Example
Let m = 2, λ = (3, 2), µ = (1). 1 1 1 2 1 2 1 2 1 1 2 2 1 2 2 2 w(T) = (3, 1) (2, 2) (2, 2) (1, 3) sλ/µ(x1, x2) = x3
1x2
+x2
1x2 2
+x2
1x2 2
+x1x3
2.
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 6 / 25
SLIDE 19
Problem
Example
Let m = 2, λ = (3, 2), µ = (1). 1 1 1 1 2 1 1 1 2 1 2 2 2 2 2 “w(R) =” (3, 0) (2, 1) (2, 1) (1, 2) (0, 3)
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 7 / 25
SLIDE 20
Problem
Example
Let m = 2, λ = (3, 2), µ = (1). 1 1 1 1 2 1 1 1 2 1 2 2 2 2 2 “w(R) =” (3, 0) (2, 1) (2, 1) (1, 2) (0, 3)
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 7 / 25
SLIDE 21
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
SLIDE 22
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) =
∑
R is a RPP
- f shape λ/µ
with entries ≤ m
xw(R).
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 8 / 25
SLIDE 23
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) =
∑
R is a RPP
- f shape λ/µ
with entries ≤ m
xw(R).
Example
1 1 1 1 2 1 1 1 2 1 2 2 2 2 2 w(R) = (2, 0) (1, 1) (2, 1) (1, 2) (0, 2)
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 8 / 25
SLIDE 24
Properties of gλ/µ
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 9 / 25
SLIDE 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
SLIDE 26
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
SLIDE 27
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
SLIDE 28
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)) = siw(R);
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 9 / 25
SLIDE 29
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)) = siw(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
SLIDE 30
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
SLIDE 31
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 1 1 5 6 6 8 9 5 5 6 6 9 6 7 7 8
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 10 / 25
SLIDE 32
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 1 1 5 6 6 8 9 5 5 6 6 9 6 7 7 8
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 10 / 25
SLIDE 33
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 1 1 5 6 6 8 9 5 5 6 6 9 6 7 7 8 − → 5 5 6 5 6 6 5 5 6 6 6
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 10 / 25
SLIDE 34
Three types of columns
1 2 2 2 1 1 1 2
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 11 / 25
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Three types of columns
1 2 2 2 1 1 1 2
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
SLIDE 36
Three types of columns
1 2 2 2 1 1 1 2
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
SLIDE 37
Three types of columns
1 2 2 2 1 1 1 2
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
SLIDE 38
Three types of columns
1 2 2 2 1 1 1 2
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
SLIDE 39
Three types of columns
1 2 2 2 1 1 1 2
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
SLIDE 40
Three types of columns
1 2 2 2 1 1 1 2
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
SLIDE 41
Construction of Bender-Knuth involutions
Flip map
1 2 2 2 1 1 1 2
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 12 / 25
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Construction of Bender-Knuth involutions
Flip map
1 2 2 2 1 1 1 2 − → 1 1 1 2 2 1 2 2
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 12 / 25
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Descent-resolution step
A lot of descents
1 1 1 2 2 1 2 2
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 13 / 25
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Descent-resolution step
A lot of descents
1 1 1 2 2 1 2 2 (M1) mixed vs. 1-pure;
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 13 / 25
SLIDE 45
Descent-resolution step
A lot of descents
1 1 1 2 2 1 2 2 (M1) mixed vs. 1-pure;
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 13 / 25
SLIDE 46
Descent-resolution step
A lot of descents
1 1 1 2 2 1 2 2 (M1) mixed vs. 1-pure; (2M) 2-pure vs. mixed;
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 13 / 25
SLIDE 47
Descent-resolution step
A lot of descents
1 1 1 2 2 1 2 2 (M1) mixed vs. 1-pure; (2M) 2-pure vs. mixed;
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 13 / 25
SLIDE 48
Descent-resolution step
A lot of descents
1 1 1 2 2 1 2 2 (M1) mixed vs. 1-pure; (2M) 2-pure vs. mixed; (21) 2-pure vs. 1-pure.
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 13 / 25
SLIDE 49
Descent-resolution step
A lot of descents
1 1 1 2 2 1 2 2 (M1) mixed vs. 1-pure; (2M) 2-pure vs. mixed; (21) 2-pure vs. 1-pure.
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 13 / 25
SLIDE 50
Descent-resolution step
A lot of descents
1 1 1 2 2 1 2 2 (M1) mixed vs. 1-pure; (2M) 2-pure vs. mixed; (21) 2-pure vs. 1-pure. (MM) mixed vs. mixed
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 13 / 25
SLIDE 51
Descent-resolution step
A lot of descents
1 1 1 2 2 1 2 2 (M1) mixed vs. 1-pure; (2M) 2-pure vs. mixed; (21) 2-pure vs. 1-pure. (MM) mixed vs. mixed – never happens!
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 13 / 25
SLIDE 52
Descent-resolution steps
(M1) 1 1 2
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 14 / 25
SLIDE 53
Descent-resolution steps
(M1) 1 1 2 → 1 1 2
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 14 / 25
SLIDE 54
Descent-resolution steps
(M1) (2M) 1 1 2 → 1 1 2 1 2 2
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 14 / 25
SLIDE 55
Descent-resolution steps
(M1) (2M) 1 1 2 → 1 1 2 1 2 2 → 1 2 2
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 14 / 25
SLIDE 56
Descent-resolution steps
(M1) (2M) (21) 1 1 2 → 1 1 2 1 2 2 → 1 2 2 1 2
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 14 / 25
SLIDE 57
Descent-resolution steps
(M1) (2M) (21) 1 1 2 → 1 1 2 1 2 2 → 1 2 2 1 2 → 2 1
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 14 / 25
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Properties
The descent-resolution process
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 15 / 25
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Properties
The descent-resolution process
ends after a finite number of steps;
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 15 / 25
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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
SLIDE 61
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
SLIDE 62
Why is Bi an involution?
(M1) (2M) (21) 1 1 2 → 1 1 2 1 2 2 → 1 2 2 1 2 → 2 1
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 16 / 25
SLIDE 63
Why is Bi an involution?
(M1) (2M) (21) 1 1 2 → 1 1 2 1 2 2 → 1 2 2 1 2 → 2 1 ↑ flip ↓ ↑ flip ↓ ↑ flip ↓ 1 2 2 ← 1 2 2 1 1 2 ← 1 1 2 2 1 ← 1 2 (2M) (M1) (21)
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 16 / 25
SLIDE 64
Part 2: Schur expansion
SLIDE 65
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
SLIDE 66
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
SLIDE 67
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. T= 1 3 2 2 4
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 18 / 25
SLIDE 68
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. T= 1 3 2 2 4 rw(T) = (2, 2, 4, 1, 3).
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 18 / 25
SLIDE 69
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. T= 1 3 2 2 4 rw(T) = (2, 2, 4, 1, 3).
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
SLIDE 70
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
SLIDE 71
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
SLIDE 72
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
SLIDE 73
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
SLIDE 74
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
SLIDE 75
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
SLIDE 76
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
SLIDE 77
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
SLIDE 78
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
SLIDE 79
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
SLIDE 80
Crystal operators on SSYT
(a picture from [Kashiwara (1995)])
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SLIDE 81
Littlewood-Richardson rule
Crystal operators on words commute with Knuth equivalence relations;
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SLIDE 82
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
SLIDE 83
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 E −1
i
(Tλ) = ∅ for all i < m: 1 1 1 1 2 2 2 2 3 3 3
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 21 / 25
SLIDE 84
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 E −1
i
(Tλ) = ∅ for all i < m: 1 1 1 1 2 2 2 2 3 3 3 We have w(Tλ) = λ.
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 21 / 25
SLIDE 85
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 E −1
i
(Tλ) = ∅ for all i < m: 1 1 1 1 2 2 2 2 3 3 3 We have w(Tλ) = λ.
Corollary
If W ⊂ [m]r is closed under the action of Ei, then
∑
u∈W
xw(u) =
∑
u∈W :E −1
i
(u)=∅ ∀i
sw(u)(x1, . . . , xm).
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 21 / 25
SLIDE 86
Reading words for RPP
Definition
R = 1 2 1 1 4 1 1 1 4 1 3 3 4 2 3 5 2 4 5 3 4
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 22 / 25
SLIDE 87
Reading words for RPP
Definition
R = 1 2 1 1 4 1 1 1 4 1 3 3 4 2 3 5 2 4 5 3 4 → 2 1 1 1 3 4 3 2 5 3 4
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 22 / 25
SLIDE 88
Reading words for RPP
Definition
R = 1 2 1 1 4 1 1 1 4 1 3 3 4 2 3 5 2 4 5 3 4 → 2 1 1 1 3 4 3 2 5 3 4 rw(R) = 34253134112.
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 22 / 25
SLIDE 89
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
SLIDE 90
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
SLIDE 91
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
SLIDE 92
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
SLIDE 93
Schur expansion of gλ/µ
Theorem
gλ/µ(x1, . . . , xm) =
∑
R is a RPP
- f shape λ/µ
with entries ≤ m such that E −1
i
(rw(R)) = ∅ for all i < m
sw(R)(x1, . . . , xm).
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 24 / 25
SLIDE 94
Thank you!
SLIDE 95
Lam, T., Pylyavskyy, P., Combinatorial Hopf algebras and K-homology
- f 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.
Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 25 / 25