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

Crystallizing the Schur Q-functions

Maria Gillespie, University of California, Davis Jake Levinson, University of Washington Kevin Purbhoo, University of Waterloo AMS Fall Western Sectional Nov 4, 2017

Shifted tableaux

➓ Shifted partitions: Partitions with distinct parts; ith row

shifted to the right i steps.

λ = (6, 4, 2, 1)

➓

④

➓ ✶ ➔

➔

✶ ➔

➔

✶ ➔

➔ ☎ ☎ ☎

➓ ✶ ✶ ✶ ✶

Shifted tableaux

➓ Shifted partitions: Partitions with distinct parts; ith row

shifted to the right i steps.

λ = (6, 4, 2, 1) µ = (3, 1)

➓ Skew shape: λ④µ ➓ ✶ ➔

➔

✶ ➔

➔

✶ ➔

➔ ☎ ☎ ☎

➓ ✶ ✶ ✶ ✶

Shifted tableaux

➓ Shifted partitions: Partitions with distinct parts; ith row

shifted to the right i steps.

3 1′ 1 2′ 1′ 2 1 1

➓ Skew shape: λ④µ ➓ Semistandard tableaux: 1✶ ➔ 1 ➔ 2✶ ➔ 2 ➔ 3✶ ➔ 3 ➔ ☎ ☎ ☎ is

alphabet, entries weakly increasing down and right. Primed letters can only repeat in columns and unprimed only in rows.

➓ ✶ ✶ ✶ ✶

Shifted tableaux

➓ Shifted partitions: Partitions with distinct parts; ith row

shifted to the right i steps.

3 1′ 1 2′ 1′ 2 1 1

➓ Skew shape: λ④µ ➓ Semistandard tableaux: 1✶ ➔ 1 ➔ 2✶ ➔ 2 ➔ 3✶ ➔ 3 ➔ ☎ ☎ ☎ is

alphabet, entries weakly increasing down and right. Primed letters can only repeat in columns and unprimed only in rows.

➓ Canonical form: First i or i✶ is always unprimed in reading

• rder (read rows from bottom to top, 3111✶21✶12✶).

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

3 1′ 1 2′ 1′ 2 1 1

➓

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

3 1′ 1 2′ 1′ 2 1 1

➓

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

3 1′ 1 2′ 1′ 2 1 1

➓

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

3 1′ 1 2′ 1′ 1 2 1

➓

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

1′ 1 2′ 1′ 1 2 1 3

➓

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

1 1′ 1 2′ 1 2 1 3

➓

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

1 1′ 1 2′ 1 1 2 3

➓

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

1 1′ 1 2′ 1 1 2 3

➓

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

1 1′ 1 2′ 1 1 2 3

➓

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

1 1′ 1 1 2′ 1 2 3

➓

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

1 1′ 1 1 2′ 1 2 3

➓

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

1′ 3 1 1 2′ 1 1 2

➓

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

1′ 3 1 1 1 2′ 1 2

➓

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

1′ 3 1 1 1 2′ 1 2

➓

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

1 3 1 1 1 2′ 1 2

➓

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

1 1 3 1 1 1 2′ 2

➓

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

1 1 3 1 1 1 2′ 2

➓

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

1 1 1 1 1 2′ 2 3

➓

Shifted tableaux

➓ Jeu de Taquin sliding: Primed letters lose their primes when

sliding into the diagonal.

1 1 1 1 1 2′ 2 3

➓ Highest weight: Rectifies to shifted tableau with all i’s in ith

row: 1 1 1 1 1 2 2 3

Standardization

➓ Standardization order: Break ties by reading order for

unprimed entries, reverse reading order for primed entries

3 1′ 1 2′ 1′ 2 1 1 8 1 5 6 2 7 3 4

➓

♣ q ✏ ♣ q

✶

♣ q

➓ ♣ q ✏

☎ ☎ ☎

Standardization

➓ Standardization order: Break ties by reading order for

unprimed entries, reverse reading order for primed entries

3 1′ 1 2′ 1′ 2 1 1 8 1 5 6 2 7 3 4

➓ Weight: wt♣Tq ✏ ♣m1, m2, . . .q where mi is the total number

• f i and i✶ entries in T. Above, weight is ♣5, 2, 1q.

➓ ♣ q ✏

☎ ☎ ☎

Standardization

➓ Standardization order: Break ties by reading order for

unprimed entries, reverse reading order for primed entries

3 1′ 1 2′ 1′ 2 1 1 8 1 5 6 2 7 3 4

➓ Weight: wt♣Tq ✏ ♣m1, m2, . . .q where mi is the total number

• f i and i✶ entries in T. Above, weight is ♣5, 2, 1q.

➓ Monomial weight: xwt♣Tq ✏ xm1 1 xm2 2

☎ ☎ ☎ . Above, x5

1x2 2x3.

Shifted Littlewood-Richardson Rule

➓ Schur Q-functions: Let ℓ♣wtTq be the number of nonzero

entries in wtT. Qλ④µ♣x1, x2, . . .q ✏ ➳

TPShST♣λ④µq

2ℓ♣wtTqxwtT

➓ ④ ✏

➳ ④

➓

Shifted Littlewood-Richardson Rule

➓ Schur Q-functions: Let ℓ♣wtTq be the number of nonzero

entries in wtT. Qλ④µ♣x1, x2, . . .q ✏ ➳

TPShST♣λ④µq

2ℓ♣wtTqxwtT

➓ Shifted Littlewood-Richardson rule:

Qλ④µ ✏ ➳ f λ

µνQν

where f λ

µν is the number of highest weight canonical shifted

semistandard tableaux of shape λ④µ and weight ν.

➓

Shifted Littlewood-Richardson Rule

➓ Schur Q-functions: Let ℓ♣wtTq be the number of nonzero

entries in wtT. Qλ④µ♣x1, x2, . . .q ✏ ➳

TPShST♣λ④µq

2ℓ♣wtTqxwtT

➓ Shifted Littlewood-Richardson rule:

Qλ④µ ✏ ➳ f λ

µνQν

where f λ

µν is the number of highest weight canonical shifted

semistandard tableaux of shape λ④µ and weight ν.

➓ Question: Can we detect these highest weight skew shifted

tableaux with crystal-like raising operators?

Shifted Littlewood-Richardson Rule

➓ Schur Q-functions: Let ℓ♣wtTq be the number of nonzero

entries in wtT. Qλ④µ♣x1, x2, . . .q ✏ ➳

TPShST♣λ④µq

2ℓ♣wtTqxwtT

➓ Shifted Littlewood-Richardson rule:

Qλ④µ ✏ ➳ f λ

µνQν

where f λ

µν is the number of highest weight canonical shifted

semistandard tableaux of shape λ④µ and weight ν.

➓ Question: Can we detect these highest weight skew shifted

tableaux with crystal-like raising operators? (Main result: yes!)

Straight shapes, two letters

➓ Restrict to alphabet t1✶, 1, 2✶, 2✉. Shape can have two rows:

2 1 1 1 1 2 1 1 1 2 2 1 1 2 2 2 1 1 1 2′ 2 1 1 2′ 2 2 1 2′ 2 2

F1 F1 F1 F1 F ′

1

F ′

1

F ′

1

Or one row:

1 1 1 1 1 1 1 2 1 1 2 2 1 2 2 2 2 2 2 2 F1 F1 F1 F1 F1 ∅ F ′

1

F ′

1

F ′

1

F ′

1

F ′

1

➓ ✶ ✶ ➓ ➓ ✶ ✶ ✶

♣ q✶

• ➓

✶

Straight shapes, two letters

➓ Restrict to alphabet t1✶, 1, 2✶, 2✉. Shape can have two rows:

2 1 1 1 1 2 1 1 1 2 2 1 1 2 2 2 1 1 1 2′ 2 1 1 2′ 2 2 1 2′ 2 2

F1 F1 F1 F1 F ′

1

F ′

1

F ′

1

Or one row:

1 1 1 1 1 1 1 2 1 1 2 2 1 2 2 2 2 2 2 2 F1 F1 F1 F1 F1 ∅ F ′

1

F ′

1

F ′

1

F ′

1

F ′

1

➓ Need two operators F1, F ✶ 1 and their partial inverses E1, E ✶ 1. ➓ ➓ ✶ ✶ ✶

♣ q✶

• ➓

✶

Straight shapes, two letters

➓ Restrict to alphabet t1✶, 1, 2✶, 2✉. Shape can have two rows:

2 1 1 1 1 2 1 1 1 2 2 1 1 2 2 2 1 1 1 2′ 2 1 1 2′ 2 2 1 2′ 2 2

F1 F1 F1 F1 F ′

1

F ′

1

F ′

1

Or one row:

1 1 1 1 1 1 1 2 1 1 2 2 1 2 2 2 2 2 2 2 F1 F1 F1 F1 F1 ∅ F ′

1

F ′

1

F ′

1

F ′

1

F ′

1

➓ Need two operators F1, F ✶ 1 and their partial inverses E1, E ✶ 1. ➓ Coplacticity: Extend to skew shapes by applying outer slides. ➓ ✶ ✶ ✶

♣ q✶

• ➓

✶

Straight shapes, two letters

➓ Restrict to alphabet t1✶, 1, 2✶, 2✉. Shape can have two rows:

1 2 1′ 1 1 1 2 1′ 1 2 2 2 1 1 2 1 2 1′ 1 2′ 1 2 1′ 2′ 2 2 2 1 2′ 2

F1 F1 F1 F1 F ′

1

F ′

1

F ′

1

Or one row:

1 1′ 1 1 1 1′ 1 2 1 1′ 2 2 2 1 2 2 2 2′ 2 2 F1 F1 F1 F1 F1 ∅ F ′

1

F ′

1

F ′

1

F ′

1

F ′

1

➓ Need two operators F1, F ✶ 1 and their partial inverses E1, E ✶ 1. ➓ Coplacticity: Extend to skew shapes by applying outer slides. ➓ ✶ ✶ ✶

♣ q✶

• ➓

✶

Straight shapes, two letters

➓ Restrict to alphabet t1✶, 1, 2✶, 2✉. Shape can have two rows:

2 1 1 1 1 2 1 1 1 2 2 1 1 2 2 2 1 1 1 2′ 2 1 1 2′ 2 2 1 2′ 2 2

F1 F1 F1 F1 F ′

1

F ′

1

F ′

1

Or one row:

1 1 1 1 1 1 1 2 1 1 2 2 1 2 2 2 2 2 2 2 F1 F1 F1 F1 F1 ∅ F ′

1

F ′

1

F ′

1

F ′

1

F ′

1

➓ Need two operators F1, F ✶ 1 and their partial inverses E1, E ✶ 1. ➓ Coplacticity: Extend to skew shapes by applying outer slides. ➓ General operators: Fi, F ✶ i , Ei, E ✶ i act on the strip of

i✶, i, ♣i 1q✶, i 1 letters, by JDT rectifying, applying the appropriate operator, and unrectifying.

➓ ✶

Straight shapes, two letters

➓ Restrict to alphabet t1✶, 1, 2✶, 2✉. Shape can have two rows:

2 1 1 1 1 2 1 1 1 2 2 1 1 2 2 2 1 1 1 2′ 2 1 1 2′ 2 2 1 2′ 2 2

F1 F1 F1 F1 F ′

1

F ′

1

F ′

1

Or one row:

1 1 1 1 1 1 1 2 1 1 2 2 1 2 2 2 2 2 2 2 F1 F1 F1 F1 F1 ∅ F ′

1

F ′

1

F ′

1

F ′

1

F ′

1

➓ Need two operators F1, F ✶ 1 and their partial inverses E1, E ✶ 1. ➓ Coplacticity: Extend to skew shapes by applying outer slides. ➓ General operators: Fi, F ✶ i , Ei, E ✶ i act on the strip of

i✶, i, ♣i 1q✶, i 1 letters, by JDT rectifying, applying the appropriate operator, and unrectifying.

➓ Highest weight iff killed by all raising operators Ei, E ✶ i .

Crystal-like structure

➓ “Crystal graph” for i ✏ 1, 2:

F1 F ✶

1

F2 F ✶

2 ➓

➳

♣ ♣ qq

✏

➓ ④

2 1 1 1 2 1 1 2 2 1 1 2′ 3 1 1 1 2 1 2′ 2 2 1 1 3 3 1 1 2 2 1 2′ 3 3 1 2 2 3 1 1 3 3 1 1 3′ 3 2 2 2 3 1 2 3 3 1 2 3′ 3 2 2 3 3 2 2 3′ 3 1 3′ 3 3 2 3′ 3

Crystal-like structure

➓ “Crystal graph” for i ✏ 1, 2:

F1 F ✶

1

F2 F ✶

2 ➓ Characters are Schur

Q-functions: ➳

T in crystal

2ℓ♣wt♣TqqxwtT ✏ Qλ Graph structure implies Qλ is symmetric.

➓ ④

2 1 1 1 2 1 1 2 2 1 1 2′ 3 1 1 1 2 1 2′ 2 2 1 1 3 3 1 1 2 2 1 2′ 3 3 1 2 2 3 1 1 3 3 1 1 3′ 3 2 2 2 3 1 2 3 3 1 2 3′ 3 2 2 3 3 2 2 3′ 3 1 3′ 3 3 2 3′ 3

Crystal-like structure

➓ “Crystal graph” for i ✏ 1, 2:

F1 F ✶

1

F2 F ✶

2 ➓ Characters are Schur

Q-functions: ➳

T in crystal

2ℓ♣wt♣TqqxwtT ✏ Qλ Graph structure implies Qλ is symmetric.

➓ Connected components for

skew shapes give LR rule for Qλ④µ.

2 1 1 1 2 1 1 2 2 1 1 2′ 3 1 1 1 2 1 2′ 2 2 1 1 3 3 1 1 2 2 1 2′ 3 3 1 2 2 3 1 1 3 3 1 1 3′ 3 2 2 2 3 1 2 3 3 1 2 3′ 3 2 2 3 3 2 2 3′ 3 1 3′ 3 3 2 3′ 3

Lattice walks of words

➓ Walk of w ✏ w1w2 ☎ ☎ ☎ wn P t1✶, 1, 2✶, 2✉n is a lattice walk in

first quadrant from ♣x0, y0q ✏ ♣0, 0q to ♣xn, ynq, with wi labeling the step ♣xi, yiq Ñ ♣xi1, yi1q. Directions:

1✶

Ý Ý Ñ

1

Ý Ý Ñ ➑ ➓

2✶

➑ ➓

2

if xiyi ✏ 0

1✶

Ý Ý Ñ ➓ ➒

1 2✶

Ð Ý Ý ➑ ➓

2

if xiyi ✘ 0

➓ Example: The walk of 1222✶11✶122 looks like: 1 2 2 2✶ 1 1✶ 1 2 2

Properties of lattice walks (G., Levinson, Purbhoo)

➓ Rectification: Endpoint ♣xn, ynq tells much about rect♣wq:

➓

♣♣

• q④

♣ ✁ ✁ q④ q

➓

♣♣ ✁ q④ ♣ ✁

• q④ q

1 2 2 2✶ 1 1✶ 1 2 2

1 1 1 1 2 2 2 2 2

➓

t ✶

✶

✉ ♣ q ✏

✶♣ q ✏ ➓

Properties of lattice walks (G., Levinson, Purbhoo)

➓ Rectification: Endpoint ♣xn, ynq tells much about rect♣wq:

➓ Shape is ♣♣n xn ynq④2, ♣n ✁ xn ✁ ynq④2q. ➓

♣♣ ✁ q④ ♣ ✁

• q④ q

1 2 2 2✶ 1 1✶ 1 2 2

1 1 1 1 2 2 2 2 2

➓

t ✶

✶

✉ ♣ q ✏

✶♣ q ✏ ➓

Properties of lattice walks (G., Levinson, Purbhoo)

➓ Rectification: Endpoint ♣xn, ynq tells much about rect♣wq:

➓ Shape is ♣♣n xn ynq④2, ♣n ✁ xn ✁ ynq④2q. ➓ Weight is ♣♣n xn ✁ ynq④2, ♣n ✁ xn ynq④2q.

1 2 2 2✶ 1 1✶ 1 2 2

1 1 1 1 2 2 2 2 2

➓

t ✶

✶

✉ ♣ q ✏

✶♣ q ✏ ➓

Properties of lattice walks (G., Levinson, Purbhoo)

➓ Rectification: Endpoint ♣xn, ynq tells much about rect♣wq:

➓ Shape is ♣♣n xn ynq④2, ♣n ✁ xn ✁ ynq④2q. ➓ Weight is ♣♣n xn ✁ ynq④2, ♣n ✁ xn ynq④2q.

1 2 2 2✶ 1 1✶ 1 2 2

1 1 1 1 2 2 2 2 2

➓ Highest weight: A word w with letters in t1✶, 1, 2✶, 2✉ has

E1♣wq ✏ E ✶

1♣wq ✏ ∅ iff its walk ends on the x-axis. ➓

Properties of lattice walks (G., Levinson, Purbhoo)

➓ Rectification: Endpoint ♣xn, ynq tells much about rect♣wq:

➓ Shape is ♣♣n xn ynq④2, ♣n ✁ xn ✁ ynq④2q. ➓ Weight is ♣♣n xn ✁ ynq④2, ♣n ✁ xn ynq④2q.

1 2 2 2✶ 1 1✶ 1 2 2

1 1 1 1 2 2 2 2 2

➓ Highest weight: A word w with letters in t1✶, 1, 2✶, 2✉ has

E1♣wq ✏ E ✶

1♣wq ✏ ∅ 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.

The operation F1 on words

➓ Let w P t1✶, 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 1F 1♣1✶q✝2✶ y ✏ 0 or y ✏ 1, x ➙ 1 2✶♣1✶q✝2 2F 1♣2q✝1✶ x ✏ 0 or x ✏ 1, y ➙ 1 2✶♣2q✝1 3F 1 y ✏ 0 2 4F 1✶ x ✏ 0 2✶ 5F 1 or 2✶ x ✏ 1, y ➙ 1 ∅

➓

☎ ☎ ☎

➓

♣ q ☎ ☎ ☎

➓

♣ q ✏

The operation F1 on words

➓ Let w P t1✶, 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 1F 1♣1✶q✝2✶ y ✏ 0 or y ✏ 1, x ➙ 1 2✶♣1✶q✝2 2F 1♣2q✝1✶ x ✏ 0 or x ✏ 1, y ➙ 1 2✶♣2q✝1 3F 1 y ✏ 0 2 4F 1✶ x ✏ 0 2✶ 5F 1 or 2✶ x ✏ 1, y ➙ 1 ∅

➓ Final substring is the F-critical substring wi ☎ ☎ ☎ wj with

largest j.

➓

♣ q ☎ ☎ ☎

➓

♣ q ✏

The operation F1 on words

➓ Let w P t1✶, 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 1F 1♣1✶q✝2✶ y ✏ 0 or y ✏ 1, x ➙ 1 2✶♣1✶q✝2 2F 1♣2q✝1✶ x ✏ 0 or x ✏ 1, y ➙ 1 2✶♣2q✝1 3F 1 y ✏ 0 2 4F 1✶ x ✏ 0 2✶ 5F 1 or 2✶ x ✏ 1, y ➙ 1 ∅

➓ Final substring is the F-critical substring wi ☎ ☎ ☎ wj with

largest j.

➓ F1♣wq: Replace wi ☎ ☎ ☎ wj with its transformation. ➓

♣ q ✏

The operation F1 on words

➓ Let w P t1✶, 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 1F 1♣1✶q✝2✶ y ✏ 0 or y ✏ 1, x ➙ 1 2✶♣1✶q✝2 2F 1♣2q✝1✶ x ✏ 0 or x ✏ 1, y ➙ 1 2✶♣2q✝1 3F 1 y ✏ 0 2 4F 1✶ x ✏ 0 2✶ 5F 1 or 2✶ x ✏ 1, y ➙ 1 ∅

➓ Final substring is the F-critical substring wi ☎ ☎ ☎ wj with

largest j.

➓ F1♣wq: Replace wi ☎ ☎ ☎ wj with its transformation. ➓ If no F-critical substrings, F1♣wq ✏ ∅.

Example

Type Substring Starting Location Transformation 1F 1♣1✶q✝2✶ y ✏ 0 or y ✏ 1, x ➙ 1 2✶♣1✶q✝2 2F 1♣2q✝1✶ x ✏ 0 or x ✏ 1, y ➙ 1 2✶♣2q✝1 3F 1 y ✏ 0 2 4F 1✶ x ✏ 0 2✶ 5F 1 or 2✶ x ✏ 1, y ➙ 1 ∅

The word w ✏ 1222✶11✶122 has a type 2F substring at 11✶, and this is its final F-critical substring. Thus F1♣wq ✏ 1222✶2✶1122.

1 2 2 2✶ 1 1✶ 1 2 2 F1

Ý Ñ

1 2 2 2✶ 2✶ 1 1 2 2

The operation E1 on words

➓ Let w P t1✶, 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 1E 2✶♣2q✝1 x ✏ 0 or x ✏ 1, y ➙ 1 1♣2q✝1✶ 2E 2✶♣1✶q✝2 y ✏ 0 or y ✏ 1, x ➙ 1 1♣1✶q✝2✶ 3E 2✶ y ✏ 0 1✶ 4E 2 x ✏ 0 1 5E 1 or 2✶ y ✏ 1, x ➙ 1 ∅

➓ Final substring is the E-critical substring wi ☎ ☎ ☎ wj with

largest i, breaking ties by largest j.

➓ E1♣wq defined by applying the appropriate transformation to

the final E-critical substring of w.

➓ If there are no E-critical substrings we define E1♣wq ✏ ∅.

Properties of E1 and F1 (G., Levinson, Purbhoo.)

• Theorem. The operators E1 and F1 are:

➓ Defined on tableaux: Applying E1 or F1 to the reading word

• f a skew shifted semistandard tableau preserves

semistandardness of the entries.

➓ Agree with diagram on straight shapes:

2 1 1 1 1 2 1 1 1 2 2 1 1 2 2 2 1 1 1 2′ 2 1 1 2′ 2 2 1 2′ 2 2

F1 F1 F1 F1 F ′

1

F ′

1

F ′

1

➓ Coplactic: E1 and F1 commute with all sequences of inner or

• uter JDT slides. (Difficult!)

➓ Partial inverses: E1♣Tq ✏ T ✶ if and only if F1♣T ✶q ✏ T.

Primed operators E ✶

1 and F ✶ 1

➓ E ✶ 1♣wq is defined by changing the last 2✶ in w to a 1 if this

does not change the standardization. Otherwise E ✶

1♣wq ✏ ∅. ➓ F ✶ 1♣wq is defined by changing the last 1 in w to a 2✶ if this

does not change the standardization. Otherwise F ✶

1♣wq ✏ ∅. ➓ Two maximal F ✶ 1 chains:

12211✶

F ✶

1

Ý Ñ 1222✶1✶

F ✶

1

Ý Ñ ∅ 1111✶1✶

F ✶

1

Ý Ñ 1121✶1✶

F ✶

1

Ý Ñ 1221✶1✶

F ✶

1

Ý Ñ 22211✶

F ✶

1

Ý Ñ 2222✶1

F ✶

1

Ý Ñ 2222✶2✶

F ✶

1

Ý Ñ ∅

➓ ✶ ✶

➓ ➓

✶♣

q ✏

✶ ✶♣ ✶q ✏

➓

Primed operators E ✶

1 and F ✶ 1

➓ E ✶ 1♣wq is defined by changing the last 2✶ in w to a 1 if this

does not change the standardization. Otherwise E ✶

1♣wq ✏ ∅. ➓ F ✶ 1♣wq is defined by changing the last 1 in w to a 2✶ if this

does not change the standardization. Otherwise F ✶

1♣wq ✏ ∅. ➓ Two maximal F ✶ 1 chains:

12211✶

F ✶

1

Ý Ñ 1222✶1✶

F ✶

1

Ý Ñ ∅ 1111✶1✶

F ✶

1

Ý Ñ 1121✶1✶

F ✶

1

Ý Ñ 1221✶1✶

F ✶

1

Ý Ñ 22211✶

F ✶

1

Ý Ñ 2222✶1

F ✶

1

Ý Ñ 2222✶2✶

F ✶

1

Ý Ñ ∅

➓ Theorem. The operations E ✶ 1 and F ✶ 1 are:

➓ Coplactic and well-defined on skew shifted tableaux. ➓ Partial inverses: if E ✶

1♣Tq ✏ T ✶ then F ✶ 1♣T ✶q ✏ T.

➓ Have chains of length 2 unless the rectification shape has one

row; in the latter case they coincide with E1 and F1.

Properties

➓ E ✶ 1, E1, F ✶ 1, F1 all commute with each other. ➓ ✶

♣ ✁ q

✶

♣✁ q

✶

Properties

➓ E ✶ 1, E1, F ✶ 1, F1 all commute with each other. ➓ E1 and E ✶ 1 move the endpoint of the walk by ♣1, ✁1q, F1 and

F ✶

1 by ♣✁1, 1q. Example of repeated F1 followed by one F ✶ 1:

Properties

➓ E ✶ 1, E1, F ✶ 1, F1 all commute with each other. ➓ E1 and E ✶ 1 move the endpoint of the walk by ♣1, ✁1q, F1 and

F ✶

1 by ♣✁1, 1q. Example of repeated F1 followed by one F ✶ 1: 1 2 2 1 1✶ 1✶ 1 1 1

Properties

➓ E ✶ 1, E1, F ✶ 1, F1 all commute with each other. ➓ E1 and E ✶ 1 move the endpoint of the walk by ♣1, ✁1q, F1 and

F ✶

1 by ♣✁1, 1q. Example of repeated F1 followed by one F ✶ 1: 1 2 2 1 1✶ 1✶ 1 1 2

Properties

➓ E ✶ 1, E1, F ✶ 1, F1 all commute with each other. ➓ E1 and E ✶ 1 move the endpoint of the walk by ♣1, ✁1q, F1 and

F ✶

1 by ♣✁1, 1q. Example of repeated F1 followed by one F ✶ 1: 1 2 2 1 1✶ 1✶ 1 2 2

Properties

➓ E ✶ 1, E1, F ✶ 1, F1 all commute with each other. ➓ E1 and E ✶ 1 move the endpoint of the walk by ♣1, ✁1q, F1 and

F ✶

1 by ♣✁1, 1q. Example of repeated F1 followed by one F ✶ 1: 1 2 2 2✶ 1 1✶ 1 2 2

Properties

➓ E ✶ 1, E1, F ✶ 1, F1 all commute with each other. ➓ E1 and E ✶ 1 move the endpoint of the walk by ♣1, ✁1q, F1 and

F ✶

1 by ♣✁1, 1q. Example of repeated F1 followed by one F ✶ 1: 1 2 2 2✶ 2✶ 1 1 2 2

Properties

➓ E ✶ 1, E1, F ✶ 1, F1 all commute with each other. ➓ E1 and E ✶ 1 move the endpoint of the walk by ♣1, ✁1q, F1 and

F ✶

1 by ♣✁1, 1q. Example of repeated F1 followed by one F ✶ 1: 1 2 2 2✶ 2✶ 1 2✶ 2 2

Application: Type B Schubert curves

➓ Orthogonal Grassmannian OG♣2n 1, nq: the type B

analog of Gr♣n, kq

➓

• ①♣ q ♣

q② ✏ ➦

✁ ➓

♣ q ✂

➓

Application: Type B Schubert curves

➓ Orthogonal Grassmannian OG♣2n 1, nq: the type B

analog of Gr♣n, kq

➓ Can be defined as the variety of n-dimensional isotropic

(self-orthogonal) subspaces V of C2n1 with respect to the symmetric inner form ①♣aiq, ♣bjq② ✏ ➦ aib2n1✁i.

➓

♣ q ✂

➓

Application: Type B Schubert curves

➓ Orthogonal Grassmannian OG♣2n 1, nq: the type B

analog of Gr♣n, kq

➓ Can be defined as the variety of n-dimensional isotropic

(self-orthogonal) subspaces V of C2n1 with respect to the symmetric inner form ①♣aiq, ♣bjq② ✏ ➦ aib2n1✁i.

➓ Schubert varieties Ωλ♣Fq defined for shifted partitions λ in

the n ✂ n staircase.

➓

Application: Type B Schubert curves

➓ Orthogonal Grassmannian OG♣2n 1, nq: the type B

analog of Gr♣n, kq

➓ Can be defined as the variety of n-dimensional isotropic

(self-orthogonal) subspaces V of C2n1 with respect to the symmetric inner form ①♣aiq, ♣bjq② ✏ ➦ aib2n1✁i.

➓ Schubert varieties Ωλ♣Fq defined for shifted partitions λ in

the n ✂ n staircase.

➓ Schubert curves: certain 1-dimensional intersections of

Schubert varieties

Schubert curves in the Orthogonal Grassmannian

➓ Real Schubert curves have a natural smooth covering of RP1,

monodromy operator given by a certain operation on highest weight skew tableau with a marked inner corner:

3 1′ 1 1 × 1′ 2′ 1 2

➓

❜ ✏ ❜ ❜ ✏ ✽ ❜

➓ ✶

Schubert curves in the Orthogonal Grassmannian

➓ Real Schubert curves have a natural smooth covering of RP1,

monodromy operator given by a certain operation on highest weight skew tableau with a marked inner corner:

3 1′ 1 1 × 1′ 2′ 1 2

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 ✶

Schubert curves in the Orthogonal Grassmannian

➓ Real Schubert curves have a natural smooth covering of RP1,

monodromy operator given by a certain operation on highest weight skew tableau with a marked inner corner:

3 1′ 1 1 × 1′ 2′ 1 2

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 Fi, F ✶ i give us a new easier rule that avoids

rectification!

Thank You!

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

• rder, 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.

➓

❜

✶

• ❜

3 1′ 1 1 × 1′ 2′ 1 2

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

• rder, 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.

➓

❜

✶

• ❜

3 1′ 1 1 1′ × 2′ 1 2

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

• rder, 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.

➓

❜

✶

• ❜

3 1′ 1 1 1′ 1 2′ × 2

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

• rder, 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.

➓

❜

✶

• ❜

3 1′ 1 1 1′ 1 × 2′ 2

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

• rder, 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

❜

✶

• ❜

3 1′ 1 1 1′ 1 2 2′ ×

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

• rder, 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 Fi, Fi1, . . . in that

• rder until only one entry is changing. Then replace that entry

with ❜.

3 1′ 1 1 1′ 1 2 2′ ×

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

• rder, 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 Fi, Fi1, . . . in that

• rder until only one entry is changing. Then replace that entry

with ❜.

3 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

• rder, 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 Fi, Fi1, . . . in that

• rder until only one entry is changing. Then replace that entry

with ❜.

4 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

• rder, 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 Fi, Fi1, . . . in that

• rder until only one entry is changing. Then replace that entry

with ❜.

5 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

• rder, 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 Fi, Fi1, . . . in that

• rder until only one entry is changing. Then replace that entry

with ❜.

6 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

• rder, 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 Fi, Fi1, . . . in that

• rder 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 ×

Application: Schubert curves

➓ Grassmannian: Gr♣n, kq is the variety of k-dimensional

subspaces of Cn.

➓

♣ q ✂ ♣ ✁ q

➓ ➓

⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ✏ ♣ ✁ q ✁ ✏ ♣ q ✏ ♣ q ❳ ♣ q ❳ ♣

✽q ✁

♣ q ÞÑ ♣ ☎ ☎ ☎

✁ q

Application: Schubert curves

➓ Grassmannian: Gr♣n, kq is the variety of k-dimensional

subspaces of Cn.

➓ Schubert varieties: Certain subvarieties Ωλ♣Fq where λ fits

in a k ✂ ♣n ✁ kq rectangle and F is a complete flag.

➓ ➓

⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ✏ ♣ ✁ q ✁ ✏ ♣ q ✏ ♣ q ❳ ♣ q ❳ ♣

✽q ✁

♣ q ÞÑ ♣ ☎ ☎ ☎

✁ q

Application: Schubert curves

➓ Grassmannian: Gr♣n, kq is the variety of k-dimensional

subspaces of Cn.

➓ Schubert varieties: Certain subvarieties Ωλ♣Fq where λ fits

in a k ✂ ♣n ✁ kq rectangle and F is a complete flag.

➓ Schubert curve: A one-dimensional intersection of Schubert

varieties.

➓

⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ✏ ♣ ✁ q ✁ ✏ ♣ q ✏ ♣ q ❳ ♣ q ❳ ♣

✽q ✁

♣ q ÞÑ ♣ ☎ ☎ ☎

✁ q

Application: Schubert curves

➓ Grassmannian: Gr♣n, kq is the variety of k-dimensional

subspaces of Cn.

➓ Schubert varieties: Certain subvarieties Ωλ♣Fq where λ fits

in a k ✂ ♣n ✁ kq 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

⑤α⑤ ⑤β⑤ ⑤γ⑤ ✏ k♣n ✁ kq ✁ 1. Define S ✏ S♣α, β, γq ✏ Ωα♣F0q ❳ Ωβ♣F1q ❳ Ωγ♣F✽q where the flag Ft is the maximally tangent flag at t of the rational normal curve in Pn✁1: ♣1 : tq ÞÑ ♣1 : t : t2 : ☎ ☎ ☎ : tn✁1q

Real geometry of S

➓

♣ q

✁

• ♣

q Ý Ý Ñ Ð Ý Ý ♣ q

➓

✏ ✆ ♣ q

Real geometry of S

Theorem(s). (Levinson, Speyer.) There is a degree-N map f : S Ñ P1 that makes S♣Rq a smooth covering of the circle RP1, with finite fibers of size N ✏ cα, ,β,γ.

➓

♣ q

✁

• ♣

q Ý Ý Ñ Ð Ý Ý ♣ q

➓

✏ ✆ ♣ q

Real geometry of S

➓ (Fiber over 0) Ø Tableaux of shape γc④α with one inner

corner ✂ and the rest a highest weight tableau of weight β.

➓ (Fiber over ✽) Ø Tableaux of shape γc④α with one outer

corner ✂ and the rest a highest weight tableau of weight β.

➓

♣ q

✁

• ♣

q Ý Ý Ñ Ð Ý Ý ♣ q

➓

✏ ✆ ♣ q

Real geometry of S

➓ The arcs of S♣Rq covering R✁ and R respectively induce the

shuffling and evacuation-shuffling bijections sh and esh: LR♣α, , β, γq

esh

Ý Ý Ñ Ð Ý Ý

sh

LR♣α, β, , γq

➓ Monodromy operator: ω ✏ sh ✆ esh. Cycles of ω correspond

to connected components of S♣Rq.

Real geometry of S

➓ The arcs of S♣Rq covering R✁ and R respectively induce the

shuffling and evacuation-shuffling bijections sh and esh: LR♣α, , β, γq

esh

Ý Ý Ñ Ð Ý Ý

sh

LR♣α, β, , γq

➓ Monodromy operator: ω ✏ sh ✆ esh. Cycles of ω correspond

to connected components of S♣Rq.

Shuffling

➓ Shuffling, or JDT: Do an outer jeu de taquin slide with the

❜ as the empty square to get an element of LR♣α, , β, γq.

1 1 1 1 2 2 2 2 3 3 1 3 4 4 3 4 5 × α γ

Shuffling

➓ Shuffling, or JDT: Do an outer jeu de taquin slide with the

❜ as the empty square to get an element of LR♣α, , β, γq.

1 1 1 1 2 2 2 2 3 3 1 3 4 4 3 4 5 ×

Shuffling

➓ Shuffling, or JDT: Do an outer jeu de taquin slide with the

❜ as the empty square to get an element of LR♣α, , β, γq.

1 1 1 1 2 2 2 2 3 3 1 3 4 4 3 4 5 ×

Shuffling

➓ Shuffling, or JDT: Do an outer jeu de taquin slide with the

❜ as the empty square to get an element of LR♣α, , β, γq.

1 1 1 1 2 2 2 2 3 3 1 3 4 3 4 4 5 ×

Shuffling

➓ Shuffling, or JDT: Do an outer jeu de taquin slide with the

❜ as the empty square to get an element of LR♣α, , β, γq.

1 1 1 1 2 2 2 2 3 1 3 3 4 3 4 4 5 ×

Shuffling

➓ Shuffling, or JDT: Do an outer jeu de taquin slide with the

❜ as the empty square to get an element of LR♣α, , β, γq.

1 1 1 1 2 2 2 2 3 1 3 3 4 3 4 4 5 ×

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ ➓

✂

1 1 1 2 2 1 2 3 × 1 1 1 2 2 1 2 3 ×

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ ➓

✂

1 1 1 2 2 1 2 3 × 1 1 1 2 2 1 2 3 ×

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ ➓

✂

1 1 1 2 2 1 2 3 × 1 1 1 2 2 2 1 3 ×

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ ➓

✂

1 1 1 2 2 1 2 3 × 1 1 1 2 2 2 1 3 ×

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ ➓

✂

1 1 1 2 2 1 2 3 × 1 1 1 2 2 2 1 3 ×

1

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ ➓

✂

1 1 1 2 2 1 2 3 × 1 1 1 1 2 2 2 3 ×

1

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ ➓

✂

1 1 1 2 2 1 2 3 × 1 1 1 1 2 2 2 3 ×

1 2

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ ➓

✂

1 1 1 2 2 1 2 3 × 1 1 1 1 2 2 2 3 ×

1 2

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ ➓

✂

1 1 1 2 2 1 2 3 × 1 1 1 2 2 2 3 × 1

1 2

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ ➓

✂

1 1 1 2 2 1 2 3 × 1 1 2 2 2 3 × 1 1

1 2

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ Shuffling ➓

✂

1 1 1 2 2 1 2 3 × 1 2 2 2 3 × 1 1 1

1 2 3

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ Shuffling ➓

✂

1 1 1 2 2 1 2 3 × 1 1 1 2 2 2 3 × 1

1 2 3

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ Shuffling ➓

✂

1 1 1 2 2 1 2 3 × 1 1 1 2 2 2 3 × 1

1 2 3

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ Shuffling ➓

✂

1 1 1 2 2 1 2 3 × 1 1 1 2 2 2 3 × 1

1 2 3

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ Shuffling ➓ Un-rectification: Treat ✂ as largest entry.

1 1 1 2 2 1 2 3 × 1 1 1 2 2 2 3 × 1

1 2 3

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ Shuffling ➓ Un-rectification: Treat ✂ as largest entry.

1 1 1 2 2 1 2 3 × 1 1 1 2 2 2 3 × 1

1 2

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ Shuffling ➓ Un-rectification: Treat ✂ as largest entry.

1 1 1 2 2 1 2 3 × 1 1 1 2 2 2 3 × 1

1 2

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ Shuffling ➓ Un-rectification: Treat ✂ as largest entry.

1 1 1 2 2 1 2 3 × 1 1 1 2 2 2 3 × 1

1 2

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ Shuffling ➓ Un-rectification: Treat ✂ as largest entry.

1 1 1 2 2 1 2 3 × 1 1 1 1 2 2 2 3 ×

1 2

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ Shuffling ➓ Un-rectification: Treat ✂ as largest entry.

1 1 1 2 2 1 2 3 × 1 1 1 1 2 2 2 3 ×

1

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ Shuffling ➓ Un-rectification: Treat ✂ as largest entry.

1 1 1 2 2 1 2 3 × 1 1 1 1 2 2 2 3 ×

1

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ Shuffling ➓ Un-rectification: Treat ✂ as largest entry.

1 1 1 2 2 1 2 3 × 1 1 1 1 2 2 2 3 ×

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ Shuffling ➓ Un-rectification: Treat ✂ as largest entry.

1 1 1 2 2 1 2 3 × 1 1 1 1 2 2 2 3 ×

T

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ Shuffling ➓ Un-rectification: Treat ✂ as largest entry.

1 1 1 2 2 1 2 3 × 1 1 1 1 2 2 2 3 ×

T esh♣Tq

Evacuation-shuffling

➓ Conjugation of shuffling by JDT rectification. ➓ Rectification: Treat ✂ as 0. ➓ Shuffling ➓ Un-rectification: Treat ✂ as largest entry.

1 1 1 2 2 1 2 3 × 1 1 1 1 2 2 2 3 ×

T esh♣Tq

➓ Shuffle again to compute ω ✏ sh ✆ esh:

ωT:

1 1 1 1 2 2 2 3 ×

Local rule for evacuation-shuffling

➓ Recall: esh consists of rectifying, shuffling, and un-rectifying. ➓ (G., Levinson.) Local rule, without rectifying: Start at i ✏ 1.

➓

❜ ❜

➓

❜

• ❜

1 1 1 × 1 1 2 2 1 2 2 3 3 3 4 4 4 2 3 5

Local rule for evacuation-shuffling

➓ Recall: esh consists of rectifying, shuffling, and un-rectifying. ➓ (G., Levinson.) Local rule, without rectifying: Start at i ✏ 1.

➓ Phase 1. Switch ❜ with the nearest i prior to it in reading order, if

• ne exists. Increment i by 1 and repeat.

❜

➓

❜

• ❜

1 1 1 × 1 1 2 2 1 2 2 3 3 3 4 4 4 2 3 5

Local rule for evacuation-shuffling

➓ Recall: esh consists of rectifying, shuffling, and un-rectifying. ➓ (G., Levinson.) Local rule, without rectifying: Start at i ✏ 1.

➓ Phase 1. Switch ❜ with the nearest i prior to it in reading order, if

• ne exists. Increment i by 1 and repeat.

❜

➓

❜

• ❜

1 1 1 1 1 1 2 2 × 2 2 3 3 3 4 4 4 2 3 5

Local rule for evacuation-shuffling

➓ Recall: esh consists of rectifying, shuffling, and un-rectifying. ➓ (G., Levinson.) Local rule, without rectifying: Start at i ✏ 1.

➓ Phase 1. Switch ❜ with the nearest i prior to it in reading order, if

• ne exists. Increment i by 1 and repeat.

❜

➓

❜

• ❜

1 1 1 1 1 1 2 2 2 2 2 3 3 3 4 4 4 × 3 5

Local rule for evacuation-shuffling

➓ Recall: esh consists of rectifying, shuffling, and un-rectifying. ➓ (G., Levinson.) Local rule, without rectifying: Start at i ✏ 1.

➓ Phase 1. Switch ❜ with the nearest i prior to it in reading order, if

• ne exists. Increment i by 1 and repeat.

If the ❜ precedes all of the i’s in reading order, go to Phase 2.

➓

❜

• ❜

1 1 1 1 1 1 2 2 2 2 2 3 3 3 4 4 4 × 3 5

Local rule for evacuation-shuffling

➓ Recall: esh consists of rectifying, shuffling, and un-rectifying. ➓ (G., Levinson.) Local rule, without rectifying: Start at i ✏ 1.

➓ Phase 1. Switch ❜ with the nearest i prior to it in reading order, if

• ne exists. Increment i by 1 and repeat.

If the ❜ precedes all of the i’s in reading order, go to Phase 2.

➓ Phase 2. Replace the ❜ with i and apply Fi, Fi1, . . . in that order

until only one entry is changing. Then replace that entry with ❜.

1 1 1 1 1 1 2 2 2 2 2 3 3 3 4 4 4 × 3 5

Local rule for evacuation-shuffling

➓ Recall: esh consists of rectifying, shuffling, and un-rectifying. ➓ (G., Levinson.) Local rule, without rectifying: Start at i ✏ 1.

➓ Phase 1. Switch ❜ with the nearest i prior to it in reading order, if

• ne exists. Increment i by 1 and repeat.

If the ❜ precedes all of the i’s in reading order, go to Phase 2.

➓ Phase 2. Replace the ❜ with i and apply Fi, Fi1, . . . in that order

until only one entry is changing. Then replace that entry with ❜.

1 1 1 1 1 1 2 2 2 2 2 3 3 3 4 4 4 3 3 5

Local rule for evacuation-shuffling

➓ Recall: esh consists of rectifying, shuffling, and un-rectifying. ➓ (G., Levinson.) Local rule, without rectifying: Start at i ✏ 1.

➓ Phase 1. Switch ❜ with the nearest i prior to it in reading order, if

• ne exists. Increment i by 1 and repeat.

If the ❜ precedes all of the i’s in reading order, go to Phase 2.

➓ Phase 2. Replace the ❜ with i and apply Fi, Fi1, . . . in that order

until only one entry is changing. Then replace that entry with ❜.

1 1 1 1 1 1 2 2 2 2 2 3 3 3 4 4 4 3 4 5

Local rule for evacuation-shuffling

➓ Recall: esh consists of rectifying, shuffling, and un-rectifying. ➓ (G., Levinson.) Local rule, without rectifying: Start at i ✏ 1.

➓ Phase 1. Switch ❜ with the nearest i prior to it in reading order, if

• ne exists. Increment i by 1 and repeat.

If the ❜ precedes all of the i’s in reading order, go to Phase 2.

➓ Phase 2. Replace the ❜ with i and apply Fi, Fi1, . . . in that order

until only one entry is changing. Then replace that entry with ❜.

1 1 1 1 1 1 2 2 2 2 2 3 3 3 5 4 4 3 4 5

Local rule for evacuation-shuffling

➓ Recall: esh consists of rectifying, shuffling, and un-rectifying. ➓ (G., Levinson.) Local rule, without rectifying: Start at i ✏ 1.

➓ Phase 1. Switch ❜ with the nearest i prior to it in reading order, if

• ne exists. Increment i by 1 and repeat.

If the ❜ precedes all of the i’s in reading order, go to Phase 2.

➓ Phase 2. Replace the ❜ with i and apply Fi, Fi1, . . . in that order

until only one entry is changing. Then replace that entry with ❜.

1 1 1 1 1 1 2 2 2 2 2 3 3 3 6 4 4 3 4 5

Local rule for evacuation-shuffling

➓ Recall: esh consists of rectifying, shuffling, and un-rectifying. ➓ (G., Levinson.) Local rule, without rectifying: Start at i ✏ 1.

➓ Phase 1. Switch ❜ with the nearest i prior to it in reading order, if

• ne exists. Increment i by 1 and repeat.

If the ❜ precedes all of the i’s in reading order, go to Phase 2.

➓ Phase 2. Replace the ❜ with i and apply Fi, Fi1, . . . in that order

until only one entry is changing. Then replace that entry with ❜.

1 1 1 1 1 1 2 2 2 2 2 3 3 3 7 4 4 3 4 5

Local rule for evacuation-shuffling

➓ Recall: esh consists of rectifying, shuffling, and un-rectifying. ➓ (G., Levinson.) Local rule, without rectifying: Start at i ✏ 1.

➓ Phase 1. Switch ❜ with the nearest i prior to it in reading order, if

• ne exists. Increment i by 1 and repeat.

If the ❜ precedes all of the i’s in reading order, go to Phase 2.

➓ Phase 2. Replace the ❜ with i and apply Fi, Fi1, . . . in that order

until only one entry is changing. Then replace that entry with ❜.

1 1 1 1 1 1 2 2 2 2 2 3 3 3 × 4 4 3 4 5

Geometric consequences (G., Levinson)

➓ Connections to K-theory: Pechenik and Yong’s “genomic

tableaux” that are used to compute in K♣Gr♣n, kqq appear naturally as steps in the algorithm.

➓ ➓

Geometric consequences (G., Levinson)

➓ Connections to K-theory: Pechenik and Yong’s “genomic

tableaux” that are used to compute in K♣Gr♣n, kqq appear naturally as steps in the algorithm.

➓ Schubert curves can have arbitrarily high arithmetic genus

(connected ω-orbits with many genomic tableaux appearing).

➓

Geometric consequences (G., Levinson)

➓ Connections to K-theory: Pechenik and Yong’s “genomic

tableaux” that are used to compute in K♣Gr♣n, kqq appear naturally as steps in the algorithm.

➓ Schubert curves can have arbitrarily high arithmetic genus

(connected ω-orbits with many genomic tableaux appearing).

➓ Schubert curves can have arbitrarily many connected

components, and in fact can be a disjoint union of arbitrarily many copies of P1 (when all tableaux of the given shape and content are fixed by ω).