Crystallizing the Schur Q -functions Maria Gillespie, University of - PowerPoint PPT Presentation

✶ ➓ Straight shapes, two letters ➓ Restrict to alphabet t 1 ✶ , 1 , 2 ✶ , 2 ✉ . Shape can have two rows: 1 1 1 2 1 1 2 2 F 1 F ′ F 1 2 2 1 1 1 1 1 F ′ 1 2 ′ 2 2 1 2 2 1 1 1 2 ′ 1 1 2 ′ 2 F ′ F 1 1 2 F 1 2 Or one row: F 1 F 1 F 1 F 1 F 1 ∅ 1 1 1 1 1 1 1 2 1 1 2 2 1 2 2 2 2 2 2 2 F ′ F ′ F ′ F ′ F ′ 1 1 1 1 1 ➓ Need two operators F 1 , F ✶ 1 and their partial inverses E 1 , E ✶ 1 . ➓ Coplacticity : Extend to skew shapes by applying outer slides. ➓ General operators: F i , F ✶ i , E i , E ✶ i act on the strip of i ✶ , i , ♣ i � 1 q ✶ , i � 1 letters, by JDT rectifying, applying the appropriate operator, and unrectifying.

Straight shapes, two letters ➓ Restrict to alphabet t 1 ✶ , 1 , 2 ✶ , 2 ✉ . Shape can have two rows: 1 1 1 2 1 1 2 2 F 1 F ′ F 1 2 2 1 1 1 1 1 F ′ 1 2 ′ 2 2 1 2 2 1 1 1 2 ′ 1 1 2 ′ 2 F ′ F 1 1 2 F 1 2 Or one row: F 1 F 1 F 1 F 1 F 1 ∅ 1 1 1 1 1 1 1 2 1 1 2 2 1 2 2 2 2 2 2 2 F ′ F ′ F ′ F ′ F ′ 1 1 1 1 1 ➓ Need two operators F 1 , F ✶ 1 and their partial inverses E 1 , E ✶ 1 . ➓ Coplacticity : Extend to skew shapes by applying outer slides. ➓ General operators: F i , F ✶ i , E i , E ✶ i act on the strip of i ✶ , i , ♣ i � 1 q ✶ , i � 1 letters, by JDT rectifying, applying the appropriate operator, and unrectifying. ➓ Highest weight iff killed by all raising operators E i , E ✶ i .

➓ ➳ ♣ ♣ qq ✏ ➓ ④ Crystal-like structure 1 1 1 2 ➓ “Crystal graph” for i ✏ 1 , 2: 1 1 2 1 1 2 ′ 1 1 1 F 1 F ✶ F 2 F ✶ 2 2 3 1 2 1 2 ′ 2 1 1 3 1 1 2 2 2 3 1 2 ′ 3 1 2 2 1 1 3 1 1 3 ′ 2 3 3 3 2 2 2 1 2 3 1 2 3 ′ 3 3 3 1 3 ′ 3 2 2 3 2 2 3 ′ 3 3 3 2 3 ′ 3 3

➓ ④ Crystal-like structure 1 1 1 2 ➓ “Crystal graph” for i ✏ 1 , 2: 1 1 2 1 1 2 ′ 1 1 1 F 1 F ✶ F 2 F ✶ 2 2 3 1 2 ➓ Characters are Schur 1 2 ′ 2 1 1 3 1 1 2 2 2 3 Q -functions: 2 ℓ ♣ wt ♣ T qq x wt T ✏ Q λ 1 2 ′ 3 ➳ 1 2 2 1 1 3 1 1 3 ′ 2 3 3 3 T in crystal 2 2 2 1 2 3 1 2 3 ′ Graph structure implies Q λ 3 3 3 is symmetric. 1 3 ′ 3 2 2 3 2 2 3 ′ 3 3 3 2 3 ′ 3 3

Crystal-like structure 1 1 1 2 ➓ “Crystal graph” for i ✏ 1 , 2: 1 1 2 1 1 2 ′ 1 1 1 F 1 F ✶ F 2 F ✶ 2 2 3 1 2 ➓ Characters are Schur 1 2 ′ 2 1 1 3 1 1 2 2 2 3 Q -functions: 2 ℓ ♣ wt ♣ T qq x wt T ✏ Q λ 1 2 ′ 3 ➳ 1 2 2 1 1 3 1 1 3 ′ 2 3 3 3 T in crystal 2 2 2 1 2 3 1 2 3 ′ Graph structure implies Q λ 3 3 3 is symmetric. ➓ Connected components for 1 3 ′ 3 2 2 3 2 2 3 ′ 3 3 3 skew shapes give LR rule for Q λ ④ µ . 2 3 ′ 3 3

Lattice walks of words ➓ Walk of w ✏ w 1 w 2 ☎ ☎ ☎ w n P t 1 ✶ , 1 , 2 ✶ , 2 ✉ n is a lattice walk in first quadrant from ♣ x 0 , y 0 q ✏ ♣ 0 , 0 q to ♣ x n , y n q , with w i labeling the step ♣ x i , y i q Ñ ♣ x i � 1 , y i � 1 q . Directions: 1 ✶ 1 ➑ ➑ Ý Ý Ñ Ý Ý Ñ if x i y i ✏ 0 2 ✶ 2 ➓ ➓ 1 ✶ 2 ✶ ➓ ➑ Ý Ý Ñ Ð Ý Ý if x i y i ✘ 0 1 2 ➒ ➓ ➓ Example: The walk of 1222 ✶ 11 ✶ 122 looks like: 2 2 ✶ 1 ✶ 1 2 2 1 2 1

♣♣ � � q④ ♣ ✁ ✁ q④ q ➓ ♣♣ � ✁ q④ ♣ ✁ � q④ q ➓ t ✶ ✶ ✉ ➓ ✶ ♣ q ✏ ♣ q ✏ ➓ Properties of lattice walks (G., Levinson, Purbhoo) ➓ Rectification: Endpoint ♣ x n , y n q tells much about rect ♣ w q : 2 2 ✶ 1 ✶ 1 2 2 1 1 1 1 1 2 2 2 2 2 2 1

♣♣ � ✁ q④ ♣ ✁ � q④ q ➓ t ✶ ✶ ✉ ➓ ✶ ♣ q ✏ ♣ q ✏ ➓ Properties of lattice walks (G., Levinson, Purbhoo) ➓ Rectification: Endpoint ♣ x n , y n q tells much about rect ♣ w q : ➓ Shape is ♣♣ n � x n � y n q④ 2 , ♣ n ✁ x n ✁ y n q④ 2 q . 2 2 ✶ 1 ✶ 1 2 2 1 1 1 1 1 2 2 2 2 2 2 1

t ✶ ✶ ✉ ➓ ✶ ♣ q ✏ ♣ q ✏ ➓ Properties of lattice walks (G., Levinson, Purbhoo) ➓ Rectification: Endpoint ♣ x n , y n q tells much about rect ♣ w q : ➓ Shape is ♣♣ n � x n � y n q④ 2 , ♣ n ✁ x n ✁ y n q④ 2 q . ➓ Weight is ♣♣ n � x n ✁ y n q④ 2 , ♣ n ✁ x n � y n q④ 2 q . 2 2 ✶ 1 ✶ 1 2 2 1 1 1 1 1 2 2 2 2 2 2 1

➓ Properties of lattice walks (G., Levinson, Purbhoo) ➓ Rectification: Endpoint ♣ x n , y n q tells much about rect ♣ w q : ➓ Shape is ♣♣ n � x n � y n q④ 2 , ♣ n ✁ x n ✁ y n q④ 2 q . ➓ Weight is ♣♣ n � x n ✁ y n q④ 2 , ♣ n ✁ x n � y n q④ 2 q . 2 2 ✶ 1 ✶ 1 2 2 1 1 1 1 1 2 2 2 2 2 2 1 ➓ Highest weight: A word w with letters in t 1 ✶ , 1 , 2 ✶ , 2 ✉ has E 1 ♣ w q ✏ E ✶ 1 ♣ w q ✏ ∅ iff its walk ends on the x -axis.

Properties of lattice walks (G., Levinson, Purbhoo) ➓ Rectification: Endpoint ♣ x n , y n q tells much about rect ♣ w q : ➓ Shape is ♣♣ n � x n � y n q④ 2 , ♣ n ✁ x n ✁ y n q④ 2 q . ➓ Weight is ♣♣ n � x n ✁ y n q④ 2 , ♣ n ✁ x n � y n q④ 2 q . 2 2 ✶ 1 ✶ 1 2 2 1 1 1 1 1 2 2 2 2 2 2 1 ➓ Highest weight: A word w with letters in t 1 ✶ , 1 , 2 ✶ , 2 ✉ has E 1 ♣ w q ✏ E ✶ 1 ♣ w q ✏ ∅ iff its walk ends on the x -axis. ➓ Proofs via Knuth equivalence: An elementary shifted Knuth move (Sagan, Worley) does not change the endpoint of the walk.

☎ ☎ ☎ ➓ ♣ q ☎ ☎ ☎ ➓ ♣ q ✏ ➓ The operation F 1 on words ➓ Let w P t 1 ✶ , 1 , 2 ✶ , 2 ✉ n . An F-critical substring of w is a substring of any of the types and locations below. Type Substring Starting Location Transformation 1 ♣ 1 ✶ q ✝ 2 ✶ 2 ✶ ♣ 1 ✶ q ✝ 2 1F y ✏ 0 or y ✏ 1, x ➙ 1 1 ♣ 2 q ✝ 1 ✶ 2 ✶ ♣ 2 q ✝ 1 2F x ✏ 0 or x ✏ 1, y ➙ 1 y ✏ 0 3F 1 2 1 ✶ 2 ✶ 4F x ✏ 0 1 or 2 ✶ 5F x ✏ 1, y ➙ 1 ∅

♣ q ☎ ☎ ☎ ➓ ♣ q ✏ ➓ The operation F 1 on words ➓ Let w P t 1 ✶ , 1 , 2 ✶ , 2 ✉ n . An F-critical substring of w is a substring of any of the types and locations below. Type Substring Starting Location Transformation 1 ♣ 1 ✶ q ✝ 2 ✶ 2 ✶ ♣ 1 ✶ q ✝ 2 1F y ✏ 0 or y ✏ 1, x ➙ 1 1 ♣ 2 q ✝ 1 ✶ 2 ✶ ♣ 2 q ✝ 1 2F x ✏ 0 or x ✏ 1, y ➙ 1 y ✏ 0 3F 1 2 1 ✶ 2 ✶ 4F x ✏ 0 1 or 2 ✶ 5F x ✏ 1, y ➙ 1 ∅ ➓ Final substring is the F-critical substring w i ☎ ☎ ☎ w j with largest j .

♣ q ✏ ➓ The operation F 1 on words ➓ Let w P t 1 ✶ , 1 , 2 ✶ , 2 ✉ n . An F-critical substring of w is a substring of any of the types and locations below. Type Substring Starting Location Transformation 1 ♣ 1 ✶ q ✝ 2 ✶ 2 ✶ ♣ 1 ✶ q ✝ 2 1F y ✏ 0 or y ✏ 1, x ➙ 1 1 ♣ 2 q ✝ 1 ✶ 2 ✶ ♣ 2 q ✝ 1 2F x ✏ 0 or x ✏ 1, y ➙ 1 y ✏ 0 3F 1 2 1 ✶ 2 ✶ 4F x ✏ 0 1 or 2 ✶ 5F x ✏ 1, y ➙ 1 ∅ ➓ Final substring is the F-critical substring w i ☎ ☎ ☎ w j with largest j . ➓ F 1 ♣ w q : Replace w i ☎ ☎ ☎ w j with its transformation.

The operation F 1 on words ➓ Let w P t 1 ✶ , 1 , 2 ✶ , 2 ✉ n . An F-critical substring of w is a substring of any of the types and locations below. Type Substring Starting Location Transformation 1 ♣ 1 ✶ q ✝ 2 ✶ 2 ✶ ♣ 1 ✶ q ✝ 2 1F y ✏ 0 or y ✏ 1, x ➙ 1 1 ♣ 2 q ✝ 1 ✶ 2 ✶ ♣ 2 q ✝ 1 2F x ✏ 0 or x ✏ 1, y ➙ 1 y ✏ 0 3F 1 2 1 ✶ 2 ✶ 4F x ✏ 0 1 or 2 ✶ 5F x ✏ 1, y ➙ 1 ∅ ➓ Final substring is the F-critical substring w i ☎ ☎ ☎ w j with largest j . ➓ F 1 ♣ w q : Replace w i ☎ ☎ ☎ w j with its transformation. ➓ If no F -critical substrings, F 1 ♣ w q ✏ ∅ .

Example Type Substring Starting Location Transformation 1 ♣ 1 ✶ q ✝ 2 ✶ 2 ✶ ♣ 1 ✶ q ✝ 2 1F y ✏ 0 or y ✏ 1, x ➙ 1 2F 1 ♣ 2 q ✝ 1 ✶ x ✏ 0 or x ✏ 1, y ➙ 1 2 ✶ ♣ 2 q ✝ 1 3F 1 y ✏ 0 2 1 ✶ 2 ✶ 4F x ✏ 0 1 or 2 ✶ x ✏ 1, y ➙ 1 5F ∅ The word w ✏ 1222 ✶ 11 ✶ 122 has a type 2F substring at 11 ✶ , and this is its final F -critical substring. Thus F 1 ♣ w q ✏ 1222 ✶ 2 ✶ 1122. 2 1 2 ✶ 2 2 1 2 ✶ 2 ✶ 1 ✶ 1 2 2 1 2 F 1 Ý Ñ 2 2 1 1

The operation E 1 on words ➓ Let w P t 1 ✶ , 1 , 2 ✶ , 2 ✉ n . An E-critical substring of w is a substring of any of the types and locations below. Type Substring Starting Location Transformation 2 ✶ ♣ 2 q ✝ 1 1 ♣ 2 q ✝ 1 ✶ 1E x ✏ 0 or x ✏ 1, y ➙ 1 2 ✶ ♣ 1 ✶ q ✝ 2 1 ♣ 1 ✶ q ✝ 2 ✶ 2E y ✏ 0 or y ✏ 1, x ➙ 1 2 ✶ y ✏ 0 1 ✶ 3E 4E 2 x ✏ 0 1 1 or 2 ✶ 5E y ✏ 1, x ➙ 1 ∅ ➓ Final substring is the E-critical substring w i ☎ ☎ ☎ w j with largest i , breaking ties by largest j . ➓ E 1 ♣ w q defined by applying the appropriate transformation to the final E -critical substring of w . ➓ If there are no E -critical substrings we define E 1 ♣ w q ✏ ∅ .

Properties of E 1 and F 1 (G., Levinson, Purbhoo.) Theorem. The operators E 1 and F 1 are: ➓ Defined on tableaux: Applying E 1 or F 1 to the reading word of a skew shifted semistandard tableau preserves semistandardness of the entries. ➓ Agree with diagram on straight shapes: 1 1 1 2 F 1 1 1 2 2 F ′ F 1 2 2 1 1 1 1 1 F ′ 1 2 ′ 2 2 1 2 2 1 1 1 2 ′ 1 1 2 ′ 2 F 1 F ′ 1 2 F 1 2 ➓ Coplactic: E 1 and F 1 commute with all sequences of inner or outer JDT slides. (Difficult!) ➓ Partial inverses: E 1 ♣ T q ✏ T ✶ if and only if F 1 ♣ T ✶ q ✏ T .

✶ ✶ ➓ ➓ ✶ ♣ q ✏ ✶ ✶ ♣ ✶ q ✏ ➓ ➓ Primed operators E ✶ 1 and F ✶ 1 1 ♣ w q is defined by changing the last 2 ✶ in w to a 1 if this ➓ E ✶ does not change the standardization. Otherwise E ✶ 1 ♣ w q ✏ ∅ . 1 ♣ w q is defined by changing the last 1 in w to a 2 ✶ if this ➓ F ✶ does not change the standardization. Otherwise F ✶ 1 ♣ w q ✏ ∅ . ➓ Two maximal F ✶ 1 chains: F ✶ F ✶ 12211 ✶ Ñ 1222 ✶ 1 ✶ 1 1 Ý Ý Ñ ∅ F ✶ F ✶ F ✶ F ✶ F ✶ F ✶ 1111 ✶ 1 ✶ Ñ 1121 ✶ 1 ✶ 1 Ñ 1221 ✶ 1 ✶ 1 Ñ 22211 ✶ 1 Ñ 2222 ✶ 1 1 Ñ 2222 ✶ 2 ✶ 1 1 Ý Ý Ý Ý Ý Ý Ñ ∅

Primed operators E ✶ 1 and F ✶ 1 1 ♣ w q is defined by changing the last 2 ✶ in w to a 1 if this ➓ E ✶ does not change the standardization. Otherwise E ✶ 1 ♣ w q ✏ ∅ . 1 ♣ w q is defined by changing the last 1 in w to a 2 ✶ if this ➓ F ✶ does not change the standardization. Otherwise F ✶ 1 ♣ w q ✏ ∅ . ➓ Two maximal F ✶ 1 chains: F ✶ F ✶ 12211 ✶ Ñ 1222 ✶ 1 ✶ 1 1 Ý Ý Ñ ∅ F ✶ F ✶ F ✶ F ✶ F ✶ F ✶ 1111 ✶ 1 ✶ Ñ 1121 ✶ 1 ✶ 1 Ñ 1221 ✶ 1 ✶ 1 Ñ 22211 ✶ 1 Ñ 2222 ✶ 1 1 Ñ 2222 ✶ 2 ✶ 1 1 Ý Ý Ý Ý Ý Ý Ñ ∅ ➓ Theorem. The operations E ✶ 1 and F ✶ 1 are: ➓ Coplactic and well-defined on skew shifted tableaux. 1 ♣ T q ✏ T ✶ then F ✶ ➓ Partial inverses: if E ✶ 1 ♣ T ✶ q ✏ T . ➓ Have chains of length 2 unless the rectification shape has one row; in the latter case they coincide with E 1 and F 1 .

✶ ♣ ✁ q ➓ ✶ ✶ ♣✁ q Properties ➓ E ✶ 1 , E 1 , F ✶ 1 , F 1 all commute with each other.

Properties ➓ E ✶ 1 , E 1 , F ✶ 1 , F 1 all commute with each other. ➓ E 1 and E ✶ 1 move the endpoint of the walk by ♣ 1 , ✁ 1 q , F 1 and F ✶ 1 by ♣✁ 1 , 1 q . Example of repeated F 1 followed by one F ✶ 1 :

Properties ➓ E ✶ 1 , E 1 , F ✶ 1 , F 1 all commute with each other. ➓ E 1 and E ✶ 1 move the endpoint of the walk by ♣ 1 , ✁ 1 q , F 1 and F ✶ 1 by ♣✁ 1 , 1 q . Example of repeated F 1 followed by one F ✶ 1 : 2 1 1 ✶ 1 ✶ 2 1 1 1 1

Properties ➓ E ✶ 1 , E 1 , F ✶ 1 , F 1 all commute with each other. ➓ E 1 and E ✶ 1 move the endpoint of the walk by ♣ 1 , ✁ 1 q , F 1 and F ✶ 1 by ♣✁ 1 , 1 q . Example of repeated F 1 followed by one F ✶ 1 : 2 1 1 ✶ 1 ✶ 2 1 2 1 1

Properties ➓ E ✶ 1 , E 1 , F ✶ 1 , F 1 all commute with each other. ➓ E 1 and E ✶ 1 move the endpoint of the walk by ♣ 1 , ✁ 1 q , F 1 and F ✶ 1 by ♣✁ 1 , 1 q . Example of repeated F 1 followed by one F ✶ 1 : 2 1 2 1 ✶ 1 ✶ 2 2 1 1

Properties ➓ E ✶ 1 , E 1 , F ✶ 1 , F 1 all commute with each other. ➓ E 1 and E ✶ 1 move the endpoint of the walk by ♣ 1 , ✁ 1 q , F 1 and F ✶ 1 by ♣✁ 1 , 1 q . Example of repeated F 1 followed by one F ✶ 1 : 2 2 ✶ 1 ✶ 1 2 2 1 2 1

Properties ➓ E ✶ 1 , E 1 , F ✶ 1 , F 1 all commute with each other. ➓ E 1 and E ✶ 1 move the endpoint of the walk by ♣ 1 , ✁ 1 q , F 1 and F ✶ 1 by ♣✁ 1 , 1 q . Example of repeated F 1 followed by one F ✶ 1 : 2 1 2 ✶ 2 1 2 ✶ 2 2 1

Properties ➓ E ✶ 1 , E 1 , F ✶ 1 , F 1 all commute with each other. ➓ E 1 and E ✶ 1 move the endpoint of the walk by ♣ 1 , ✁ 1 q , F 1 and F ✶ 1 by ♣✁ 1 , 1 q . Example of repeated F 1 followed by one F ✶ 1 : 2 2 2 ✶ 1 2 ✶ 2 ✶ 2 2 1

➓ � ①♣ q ♣ q② ✏ ➦ � ✁ ♣ q ➓ ✂ ➓ Application: Type B Schubert curves ➓ Orthogonal Grassmannian OG ♣ 2 n � 1 , n q : the type B analog of Gr ♣ n , k q

♣ q ➓ ✂ ➓ Application: Type B Schubert curves ➓ Orthogonal Grassmannian OG ♣ 2 n � 1 , n q : the type B analog of Gr ♣ n , k q ➓ Can be defined as the variety of n -dimensional isotropic (self-orthogonal) subspaces V of C 2 n � 1 with respect to the symmetric inner form ①♣ a i q , ♣ b j q② ✏ ➦ a i b 2 n � 1 ✁ i .

➓ Application: Type B Schubert curves ➓ Orthogonal Grassmannian OG ♣ 2 n � 1 , n q : the type B analog of Gr ♣ n , k q ➓ Can be defined as the variety of n -dimensional isotropic (self-orthogonal) subspaces V of C 2 n � 1 with respect to the symmetric inner form ①♣ a i q , ♣ b j q② ✏ ➦ a i b 2 n � 1 ✁ i . ➓ Schubert varieties Ω λ ♣ F q defined for shifted partitions λ in the n ✂ n staircase.

Application: Type B Schubert curves ➓ Orthogonal Grassmannian OG ♣ 2 n � 1 , n q : the type B analog of Gr ♣ n , k q ➓ Can be defined as the variety of n -dimensional isotropic (self-orthogonal) subspaces V of C 2 n � 1 with respect to the symmetric inner form ①♣ a i q , ♣ b j q② ✏ ➦ a i b 2 n � 1 ✁ i . ➓ Schubert varieties Ω λ ♣ F q defined for shifted partitions λ in the n ✂ n staircase. ➓ Schubert curves: certain 1-dimensional intersections of Schubert varieties

❜ ✏ ❜ ❜ ✏ ✽ ❜ ➓ ✶ ➓ Schubert curves in the Orthogonal Grassmannian ➓ Real Schubert curves have a natural smooth covering of RP 1 , monodromy operator given by a certain operation on highest weight skew tableau with a marked inner corner: 1 ′ 1 1 × 1 ′ 2 ′ 1 2 3

✶ ➓ Schubert curves in the Orthogonal Grassmannian ➓ Real Schubert curves have a natural smooth covering of RP 1 , monodromy operator given by a certain operation on highest weight skew tableau with a marked inner corner: 1 ′ 1 1 × 1 ′ 2 ′ 1 2 3 ➓ Monodromy operator: 1. Rectify , with ❜ ✏ 0 2. Slide the ❜ to an outer corner with an outer JDT slide 3. Unrectify to the original shape, with ❜ ✏ ✽ 4. Slide the ❜ back to an inner corner

Schubert curves in the Orthogonal Grassmannian ➓ Real Schubert curves have a natural smooth covering of RP 1 , monodromy operator given by a certain operation on highest weight skew tableau with a marked inner corner: 1 ′ 1 1 × 1 ′ 2 ′ 1 2 3 ➓ Monodromy operator: 1. Rectify , with ❜ ✏ 0 2. Slide the ❜ to an outer corner with an outer JDT slide 3. Unrectify to the original shape, with ❜ ✏ ✽ 4. Slide the ❜ back to an inner corner ➓ Operators F i , F ✶ i give us a new easier rule that avoids rectification!

Thank You!

✶ ❜ ➓ � ❜ Local rule (G., J. Levinson and K. Purbhoo) ➓ Local rule for steps 1 ✁ 3, without rectifying: ➓ Phase 1. Switch ❜ with the nearest 1 ✶ after it in reading order, if one exists, and then with the nearest 1 before it in reading order. Do the same for 2 ✶ and 2, and so on until there is no i ✶ after it for some i . 1 ′ 1 1 × 1 ′ 2 ′ 1 2 3

✶ ❜ ➓ � ❜ Local rule (G., J. Levinson and K. Purbhoo) ➓ Local rule for steps 1 ✁ 3, without rectifying: ➓ Phase 1. Switch ❜ with the nearest 1 ✶ after it in reading order, if one exists, and then with the nearest 1 before it in reading order. Do the same for 2 ✶ and 2, and so on until there is no i ✶ after it for some i . 1 ′ 1 1 1 ′ × 2 ′ 1 2 3

✶ ❜ ➓ � ❜ Local rule (G., J. Levinson and K. Purbhoo) ➓ Local rule for steps 1 ✁ 3, without rectifying: ➓ Phase 1. Switch ❜ with the nearest 1 ✶ after it in reading order, if one exists, and then with the nearest 1 before it in reading order. Do the same for 2 ✶ and 2, and so on until there is no i ✶ after it for some i . 1 ′ 1 1 1 ′ 1 2 ′ × 2 3

✶ ❜ ➓ � ❜ Local rule (G., J. Levinson and K. Purbhoo) ➓ Local rule for steps 1 ✁ 3, without rectifying: ➓ Phase 1. Switch ❜ with the nearest 1 ✶ after it in reading order, if one exists, and then with the nearest 1 before it in reading order. Do the same for 2 ✶ and 2, and so on until there is no i ✶ after it for some i . 1 ′ 1 1 1 ′ 1 × 2 ′ 2 3

✶ ❜ ➓ � ❜ Local rule (G., J. Levinson and K. Purbhoo) ➓ Local rule for steps 1 ✁ 3, without rectifying: ➓ Phase 1. Switch ❜ with the nearest 1 ✶ after it in reading order, if one exists, and then with the nearest 1 before it in reading order. Do the same for 2 ✶ and 2, and so on until there is no i ✶ after it for some i . At this point go to Phase 2. 1 ′ 1 1 1 ′ 1 2 2 ′ × 3

Local rule (G., J. Levinson and K. Purbhoo) ➓ Local rule for steps 1 ✁ 3, without rectifying: ➓ Phase 1. Switch ❜ with the nearest 1 ✶ after it in reading order, if one exists, and then with the nearest 1 before it in reading order. Do the same for 2 ✶ and 2, and so on until there is no i ✶ after it for some i . At this point go to Phase 2. ➓ Phase 2. Replace the ❜ with i ✶ and apply F i , F i � 1 , . . . in that order until only one entry is changing. Then replace that entry with ❜ . 1 ′ 1 1 1 ′ 1 2 2 ′ × 3

Local rule (G., J. Levinson and K. Purbhoo) ➓ Local rule for steps 1 ✁ 3, without rectifying: ➓ Phase 1. Switch ❜ with the nearest 1 ✶ after it in reading order, if one exists, and then with the nearest 1 before it in reading order. Do the same for 2 ✶ and 2, and so on until there is no i ✶ after it for some i . At this point go to Phase 2. ➓ Phase 2. Replace the ❜ with i ✶ and apply F i , F i � 1 , . . . in that order until only one entry is changing. Then replace that entry with ❜ . 1 ′ 1 1 1 1 2 2 3 ′ 3

Local rule (G., J. Levinson and K. Purbhoo) ➓ Local rule for steps 1 ✁ 3, without rectifying: ➓ Phase 1. Switch ❜ with the nearest 1 ✶ after it in reading order, if one exists, and then with the nearest 1 before it in reading order. Do the same for 2 ✶ and 2, and so on until there is no i ✶ after it for some i . At this point go to Phase 2. ➓ Phase 2. Replace the ❜ with i ✶ and apply F i , F i � 1 , . . . in that order until only one entry is changing. Then replace that entry with ❜ . 1 ′ 1 1 1 1 2 2 3 4

Local rule (G., J. Levinson and K. Purbhoo) ➓ Local rule for steps 1 ✁ 3, without rectifying: ➓ Phase 1. Switch ❜ with the nearest 1 ✶ after it in reading order, if one exists, and then with the nearest 1 before it in reading order. Do the same for 2 ✶ and 2, and so on until there is no i ✶ after it for some i . At this point go to Phase 2. ➓ Phase 2. Replace the ❜ with i ✶ and apply F i , F i � 1 , . . . in that order until only one entry is changing. Then replace that entry with ❜ . 1 ′ 1 1 1 1 2 2 3 5

Local rule (G., J. Levinson and K. Purbhoo) ➓ Local rule for steps 1 ✁ 3, without rectifying: ➓ Phase 1. Switch ❜ with the nearest 1 ✶ after it in reading order, if one exists, and then with the nearest 1 before it in reading order. Do the same for 2 ✶ and 2, and so on until there is no i ✶ after it for some i . At this point go to Phase 2. ➓ Phase 2. Replace the ❜ with i ✶ and apply F i , F i � 1 , . . . in that order until only one entry is changing. Then replace that entry with ❜ . 1 ′ 1 1 1 1 2 2 3 6

Local rule (G., J. Levinson and K. Purbhoo) ➓ Local rule for steps 1 ✁ 3, without rectifying: ➓ Phase 1. Switch ❜ with the nearest 1 ✶ after it in reading order, if one exists, and then with the nearest 1 before it in reading order. Do the same for 2 ✶ and 2, and so on until there is no i ✶ after it for some i . At this point go to Phase 2. ➓ Phase 2. Replace the ❜ with i ✶ and apply F i , F i � 1 , . . . in that order until only one entry is changing. Then replace that entry with ❜ . 1 ′ 1 1 1 1 2 2 3 ×

Larger Phase 2 example 1 1 2 × 3 1 1 2 3 2 4

Larger Phase 2 example 1 1 2 1 ′ 3 1 1 2 3 2 4

Larger Phase 2 example 1 1 2 1 3 1 2 ′ 2 3 2 4

Larger Phase 2 example 1 1 2 1 3 1 2 3 ′ 3 2 4

Larger Phase 2 example 1 1 2 1 3 1 2 3 4 ′ 2 4

Larger Phase 2 example 1 1 2 1 3 1 2 3 4 2 5

Larger Phase 2 example 1 1 2 1 3 1 2 3 4 2 6

Larger Phase 2 example 1 1 2 1 3 1 2 3 4 2 7

Larger Phase 2 example 1 1 2 1 3 1 2 3 4 2 ×

♣ q ➓ ✂ ♣ ✁ q ➓ ➓ ⑤ ⑤ � ⑤ ⑤ � ⑤ ⑤ ✏ ♣ ✁ q ✁ ✏ ♣ q ✏ ♣ q ❳ ♣ q ❳ ♣ ✽ q ✁ ✁ q ♣ q ÞÑ ♣ ☎ ☎ ☎ Application: Schubert curves ➓ Grassmannian : Gr ♣ n , k q is the variety of k -dimensional subspaces of C n .

➓ ➓ ⑤ ⑤ � ⑤ ⑤ � ⑤ ⑤ ✏ ♣ ✁ q ✁ ✏ ♣ q ✏ ♣ q ❳ ♣ q ❳ ♣ ✽ q ✁ ✁ q ♣ q ÞÑ ♣ ☎ ☎ ☎ Application: Schubert curves ➓ Grassmannian : Gr ♣ n , k q is the variety of k -dimensional subspaces of C n . ➓ Schubert varieties: Certain subvarieties Ω λ ♣ F q where λ fits in a k ✂ ♣ n ✁ k q rectangle and F is a complete flag.

➓ ⑤ ⑤ � ⑤ ⑤ � ⑤ ⑤ ✏ ♣ ✁ q ✁ ✏ ♣ q ✏ ♣ q ❳ ♣ q ❳ ♣ ✽ q ✁ ✁ q ♣ q ÞÑ ♣ ☎ ☎ ☎ Application: Schubert curves ➓ Grassmannian : Gr ♣ n , k q is the variety of k -dimensional subspaces of C n . ➓ Schubert varieties: Certain subvarieties Ω λ ♣ F q where λ fits in a k ✂ ♣ n ✁ k q rectangle and F is a complete flag. ➓ Schubert curve: A one-dimensional intersection of Schubert varieties.

Application: Schubert curves ➓ Grassmannian : Gr ♣ n , k q is the variety of k -dimensional subspaces of C n . ➓ Schubert varieties: Certain subvarieties Ω λ ♣ F q where λ fits in a k ✂ ♣ n ✁ k q rectangle and F is a complete flag. ➓ Schubert curve: A one-dimensional intersection of Schubert varieties. ➓ Special Schubert curves: Three partitions α, β, γ with ⑤ α ⑤ � ⑤ β ⑤ � ⑤ γ ⑤ ✏ k ♣ n ✁ k q ✁ 1. Define S ✏ S ♣ α, β, γ q ✏ Ω α ♣ F 0 q ❳ Ω β ♣ F 1 q ❳ Ω γ ♣ F ✽ q where the flag F t is the maximally tangent flag at t of the rational normal curve in P n ✁ 1 : ♣ 1 : t q ÞÑ ♣ 1 : t : t 2 : ☎ ☎ ☎ : t n ✁ 1 q

♣ q ➓ ✁ � q Ý Ý Ñ ♣ ♣ q Ð Ý Ý ✏ ✆ ➓ ♣ q Real geometry of S

♣ q ➓ ✁ � q Ý Ý Ñ ♣ ♣ q Ð Ý Ý ✏ ✆ ➓ ♣ q Real geometry of S Theorem(s). (Levinson, Speyer.) There is a degree- N map f : S Ñ P 1 that makes S ♣ R q a smooth covering of the circle RP 1 , with finite fibers of size N ✏ c α, ,β,γ .

♣ q ➓ ✁ � q Ý Ý Ñ ♣ ♣ q Ð Ý Ý ✏ ✆ ➓ ♣ q Real geometry of S ➓ (Fiber over 0) Ø Tableaux of shape γ c ④ α with one inner corner ✂ and the rest a highest weight tableau of weight β . ➓ (Fiber over ✽ ) Ø Tableaux of shape γ c ④ α with one outer corner ✂ and the rest a highest weight tableau of weight β .

Real geometry of S ➓ The arcs of S ♣ R q covering R ✁ and R � respectively induce the shuffling and evacuation-shuffling bijections sh and esh : esh Ý Ý Ñ LR ♣ α, , β, γ q LR ♣ α, β, , γ q Ð Ý Ý sh ➓ Monodromy operator: ω ✏ sh ✆ esh . Cycles of ω correspond to connected components of S ♣ R q .

Real geometry of S ➓ The arcs of S ♣ R q covering R ✁ and R � respectively induce the shuffling and evacuation-shuffling bijections sh and esh : esh Ý Ý Ñ LR ♣ α, , β, γ q LR ♣ α, β, , γ q Ð Ý Ý sh ➓ Monodromy operator: ω ✏ sh ✆ esh . Cycles of ω correspond to connected components of S ♣ R q .