Cylindric Skew Schur Functions University of Minnesota Combinatorics - PowerPoint PPT Presentation

Cylindric Skew Schur Functions University of Minnesota Combinatorics Seminar 5 November 2004 Peter McNamara LaCIM, UQÀM Slides and preprint available from www.lacim.uqam.ca/~mcnamara . – p.1/23

Schur functions • Partition λ = ( λ 1 , λ 2 , . . . , λ l ) . • Example: (4 , 4 , 3 , 1) . – p.2/23

Schur functions • Partition λ = ( λ 1 , λ 2 , . . . , λ l ) . 7 • Example: (4 , 4 , 3 , 1) 5 6 6 4 4 4 9 • Semistandard Young tableau 1 3 3 4 (SSYT) Schur function s λ in the variables x = ( x 1 , x 2 , . . . ) defined by x #1 ′ s in T x #2 ′ s in T x T = � � s λ ( x ) = · · · . 1 2 SSYT T SSYT T s 4431 ( x ) = x 1 x 2 3 x 4 4 x 5 x 2 6 x 7 x 9 + · · · . . – p.2/23

Skew Schur functions • Partition λ = ( λ 1 , λ 2 , . . . , λ l ) . • µ fits inside λ : form λ/µ . 7 • Example: (4 , 4 , 3 , 1) / (3 , 1) 5 6 6 4 4 9 • Semistandard Young tableau 4 (SSYT) Skew Schur function s λ/µ in the variables x = ( x 1 , x 2 , . . . ) defined by x #1 ′ s in T x #2 ′ s in T x T = � � s λ/µ ( x ) = · · · . 1 2 SSYT T SSYT T s 4431 ( x ) = x 1 x 2 3 x 4 4 x 5 x 2 6 x 7 x 9 + · · · . . – p.2/23

Do we care? For Schur! • Schur functions are symmetric functions • Schur functions s λ form a basis for the symmetric functions. • Arise in: representation theory of the symmetric group S n . • They are the characters of the irreducible representations of GL( n, C ) . • Correspond to Schubert classes in H ∗ (Gr kn ) . . – p.3/23

For skew Schur? • Skew Schur functions are symmetric functions c ν � s λ/µ ( x ) = λµ s ν ( x ) . ν λµ : Littlewood-Richardson coefficients c ν • Since c ν λµ ≥ 0 , they are Schur-positive . s 4431 / 31 = s 44 + 2 s 431 + s 422 + s 4211 + s 332 + s 3311 . • Schur-positive symmetric functions are significant in the representation theory of S n . . – p.4/23

Cylindric skew Schur functions • Infinite skew shape C • Invariant under translation • Identify ( x, y ) and k ( x − n + k, y + k ) . n−k . – p.5/23

Cylindric skew Schur functions • Infinite skew shape C 6 6 4 4 9 • Invariant under 1 3 3 5 6 6 2 4 4 4 9 translation 1 3 3 5 6 6 2 4 4 4 9 • Identify ( x, y ) and k 1 3 3 5 ( x − n + k, y + k ) . 2 4 n−k • Entries weakly increasing in each row Strictly increasing up each column • Alternatively: SSYT with relations between entries in first and last columns x #1 ′ s in T x #2 ′ s in T x T = � � s C ( x ) = · · · . 1 2 T T • s C is a symmetric function . – p.5/23

Cylindric skew Schur functions E XAMPLE k n−k • Gessel,Krattenthaler: “Cylindric Partitions” • Bertram, Ciocan-Fontanine, Fulton: “Quantum Multiplication of Schur Polynomials” • Postnikov: “Affine Approach to Quantum Schubert Calculus” math.CO/0205165 • Stanley: “Recent Developments in Algebraic Combinatorics” math.CO/0211114 . – p.6/23

Motivation 1: P -partitions and an old conjecture of Stanley . – p.7/23

Motivation 1: P -partitions and an old conjecture of Stanley 5 5 1 P : partially ordered set 2 3 2 3 (poset) 4 2 ω : P → { 1 , 2 , . . . , | P |} 2 2 bijective labelling 3 D EFINITION (R. Stanley) Given a labelled poset ( P, ω ) , a ( P, ω ) -partition is a map f : P → P with the following properties: • f is order-preserving : If x ≤ y in P then f ( x ) ≤ f ( y ) • If x ⋖ y in P and ω ( x ) > ω ( y ) then f ( x ) < f ( y ) . – p.7/23

Motivation 1: P -partitions and an old conjecture of Stanley 5 5 1 P : partially ordered set 2 3 2 3 (poset) 4 2 ω : P → { 1 , 2 , . . . , | P |} 2 2 bijective labelling 3 D EFINITION (R. Stanley) Given a labelled poset ( P, ω ) , a ( P, ω ) -partition is a map f : P → P with the following properties: • f is order-preserving : If x ≤ y in P then f ( x ) ≤ f ( y ) • If x ⋖ y in P and ω ( x ) > ω ( y ) then f ( x ) < f ( y ) . – p.7/23

Motivation 1: P -partitions and an old conjecture of Stanley 5 5 1 P : partially ordered set 2 3 2 3 (poset) 4 2 ω : P → { 1 , 2 , . . . , | P |} 2 2 bijective labelling 3 D EFINITION (R. Stanley) Given a labelled poset ( P, ω ) , a ( P, ω ) -partition is a map f : P → P with the following properties: • f is order-preserving : If x ≤ y in P then f ( x ) ≤ f ( y ) • If x ⋖ y in P and ω ( x ) > ω ( y ) then f ( x ) < f ( y ) x # f − 1 (1) x # f − 1 (2) x f = � � K P,ω ( x ) = · · · . 1 2 f f . – p.7/23

A non-symmetric example x # f − 1 (1) x # f − 1 (2) x T = � � K P,ω ( x ) = · · · . 1 2 f f E XAMPLE 3 d 2 1 c b 1 a Coefficient of x 2 1 x 2 x 3 = 1 Coefficient of x 1 x 2 x 2 3 = 0 ⇒ not symmetric . – p.8/23

Schur labelled skew shape posets and Stanley’s P -partitions Conjecture 1 2 3 7 3 5 8 2 4 5 7 1 4 6 6 8 Bijection: SSYT of shape λ/µ ↔ ( P, ω ) -partitions Furthermore, K P,ω ( x ) = s λ/µ ( x ) . B IG Q UESTION What other labelled posets ( P, ω ) have symmetric K P,ω ( x ) ? . – p.9/23

Schur labelled skew shape posets and Stanley’s P -partitions Conjecture 1 2 3 7 3 5 8 2 4 5 7 1 4 6 6 8 Bijection: SSYT of shape λ/µ ↔ ( P, ω ) -partitions Furthermore, K P,ω ( x ) = s λ/µ ( x ) . B IG Q UESTION What other labelled posets ( P, ω ) have symmetric K P,ω ( x ) ? C ONJECTURE (Stanley, c.1971) K P,ω ( x ) is symmetric if and only if ( P, ω ) is isomorphic to a (Schur labelled) skew shape poset. . – p.9/23

Connection to cylindric skew Schur functions E XAMPLE d c b a We can check that K P,ω ( x ) is symmetric. So does it obey Stanley’s conjecture? a d b c a d b c a d b c . – p.10/23

Connection to cylindric skew Schur functions E XAMPLE d c b a We can check that K P,ω ( x ) is symmetric. So does it obey Stanley’s conjecture? a d b c a d b c a d b c . – p.10/23

Connection to cylindric skew Schur functions E XAMPLE d c b a We can check that K P,ω ( x ) is symmetric. So does it obey Stanley’s conjecture? a d b c a d b c a d b c . – p.10/23

Connection to cylindric skew Schur functions E XAMPLE d c b a We can check that K P,ω ( x ) is symmetric. So does it obey Stanley’s conjecture? a d b c a d b c a d b c . – p.10/23

Connection to cylindric skew Schur functions E XAMPLE d c b a We can check that K P,ω ( x ) is symmetric. So does it obey Stanley’s conjecture? a d b c a d b c a d b c . – p.10/23

Connection to cylindric skew Schur functions E XAMPLE d c b a We can check that K P,ω ( x ) is symmetric. So does it obey Stanley’s conjecture? a d b c a d b c a d b c . – p.10/23

Connection to cylindric skew Schur functions E XAMPLE d c b a We can check that K P,ω ( x ) is symmetric. So does it obey Stanley’s conjecture? a d b c a d b c a d b c . – p.10/23

Connection to cylindric skew Schur functions E XAMPLE d c b a We can check that K P,ω ( x ) is symmetric. So does it obey Stanley’s conjecture? a d b c a d b c a d b c . – p.10/23

Connection to cylindric skew Schur functions E XAMPLE d c b a We can check that K P,ω ( x ) is symmetric. So does it obey Stanley’s conjecture? a d b c a d b c a d b c ω ( a ) > ω ( c ) > ω ( b ) > ω ( d ) > ω ( a ) Yikes! . – p.10/23

Connection to cylindric skew Schur functions E XAMPLE d c b a We can check that K P,ω ( x ) is symmetric. So does it obey Stanley’s conjecture? a d b c a d b c a d b c ω ( a ) > ω ( c ) > ω ( b ) > ω ( d ) > ω ( a ) Yikes! Oriented Poset . – p.10/23

( P, O ) -partitions Labelled poset ( P, ω ) Oriented poset ( P, O ) K P,ω ( x ) K P,O ( x ) skew shape posets cylindric skew shape posets skew Schur functions cylindric skew Schur functions 7 7 2 3 7 3 3 1 1 3 3 3 2 3 7 1 1 3 2 2 1 2 3 7 1 1 1 1 3 1 . – p.11/23

Malvenuto’s reformulation T HEOREM (C. Malvenuto, c. 1995) A labelled poset is a skew shape poset if and only if every connected component has no forbidden convex subposets T HEOREM (McN.) An oriented poset is a cylindric skew shape poset if and only if every connected component has no forbidden convex subposets C ONJECTURE (Stanley) K P,ω ( x ) is symmetric if and only if every connected component of ( P, ω ) is isomorphic to a skew shape poset. C ONJECTURE (Stanley’s conjecture extended to oriented posets) K P,O ( x ) is symmetric if and only if every connected component of ( P, O ) is isomorphic to a cylindric skew shape poset. . – p.12/23

Extended version is false! . – p.13/23

Motivation 2: Positivity of Gromov- Witten invariants In H ∗ ( Gr kn ) , c ν � σ λ σ µ = λµ σ ν . ν ⊆ k × ( n − k ) In QH ∗ ( Gr kn ) , q d C ν,d � � σ λ ∗ σ µ = λµ σ ν . d ≥ 0 ν ⊢| λ | + | µ |− dn ν ⊆ k × ( n − k ) C ν,d λµ = 3-point Gromov-Witten invariants = # { rational curves of degree d in Gr kn that meet fixed generic translates of the Schubert varieties Ω ν ∨ , Ω λ and Ω µ } . Key point: C ν,d λµ ≥ 0 . “Fundamental Open Problem”: . – p.14/23