Cylindric Schur Functions R e t r o s p e c t i v e I - PowerPoint PPT Presentation

Cylindric Schur Functions R e t r o s p e c t i v e I n C o m b i n a t o r i c s : H o n o r i n g t h S T A N L E Y ’ S 6 0 b i R t h - D a y 24 June 2004 Peter McNamara Slides and forthcoming paper available from www.lacim.uqam.ca/~mcnamara Peter McNamara – p.1/11

Cylindric skew Schur functions • Infinite skew shape C • Invariant under translation • Identify ( x, y ) and k n−k ( x + k, y − n + k ) . Peter McNamara – p.2/11

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 2 4 n−k ( x + k, y − 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 = · · · . 1 2 T T Straightforward: s C is a symmetric function Peter McNamara – p.2/11

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 Peter McNamara – p.3/11

Motivation 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”: Peter McNamara – p.4/11

Motivation 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”: Find an algebraic or combinatorial proof of this fact. Peter McNamara – p.4/11

What’s cylindric got to do with it? T HEOREM (Postnikov) C ν,d � s λ/d/µ ( x 1 , . . . , x k ) = λµ s ν ( x 1 , . . . , x k ) . ν ⊆ k × ( n − k ) Conclusion: Want to understand expansions of cylindric skew Schur functions into Schur functions. C OROLLARY s λ/d/µ ( x 1 , x 2 , . . . , x k ) is Schur-positive. Known: s λ/d/µ ( x 1 , x 2 , . . . ) need not be Schur-positive. T HEOREM (McN.) For any cylindric shape C , s C ( x 1 , x 2 , . . . ) is Schur-positive ⇔ C is a skew shape . Peter McNamara – p.5/11

Example: Cylindric ribbons E XAMPLE C: k n−k � s C ( x 1 , x 2 , . . . ) = c ν s ν + s n − k, 1 k − s n − k − 1 , 1 k +1 ν ⊆ k × ( n − k ) + s n − k − 2 , 1 k +2 − · · · + ( − 1) n − k s 1 n . Schur-positive with k + 1 variables Not Schur-positive with ≥ k + 2 variables General cylindric skew shape: ≥ k + 2 + l variables Toric shapes: ≥ 2 k + 1 variables Peter McNamara – p.6/11

Example: Cylindric ribbons C: k n−k � s C ( x 1 , x 2 , . . . ) = c ν s ν + s n − k, 1 k − s n − k − 1 , 1 k +1 ν ⊆ k × ( n − k ) + s n − k − 2 , 1 k +2 − · · · + ( − 1) n − k s 1 n . Peter McNamara – p.7/11

Example: Cylindric ribbons C: H: k k n−k n−k � s C ( x 1 , x 2 , . . . ) = c ν s ν + s n − k, 1 k − s n − k − 1 , 1 k +1 ν ⊆ k × ( n − k ) + s n − k − 2 , 1 k +2 − · · · + ( − 1) n − k s 1 n . However, s C ( x 1 , x 2 , . . . ) = � c ν s ν + s H . ν ⊆ k × ( n − k ) s C : cylindric skew Schur function s H : cylindric Schur function We say that s C is cylindric Schur positive. Peter McNamara – p.7/11

A Conjecture C ONJECTURE For any cylindric shape C , s C is cylindric Schur positive. Peter McNamara – p.8/11

Tool: Cylindric skew Schur functions as alternating sums of skew Schurs Bertram, Ciocan-Fontanine, Fulton: Nice description in terms of ribbons � � Only for toric shapes, certain terms ✁ ✁ Gessel, Krattenthaler: Works for all cylindric shapes ✂ ✂ Not as nice a description ✄ ✄ We can get the best of both worlds: A technique for expanding a cylindric skew Schur function in terms of skew Schur functions that Works for all cylindric shapes like G-K and has a nice description like B-CF-F Peter McNamara – p.9/11

Tool: Cylindric skew Schur functions as alternating sums of skew Schurs E XAMPLE − + = + k n−k Peter McNamara – p.10/11

Tool: Cylindric skew Schur functions as alternating sums of skew Schurs E XAMPLE − + = + k n−k Peter McNamara – p.10/11

Tool: Cylindric skew Schur functions as alternating sums of skew Schurs E XAMPLE − + = + k n−k Peter McNamara – p.10/11

Tool: Cylindric skew Schur functions as alternating sums of skew Schurs E XAMPLE − + = + k n−k Peter McNamara – p.10/11

Tool: Cylindric skew Schur functions as alternating sums of skew Schurs E XAMPLE − + = + k n−k Peter McNamara – p.10/11

Tool: Cylindric skew Schur functions as alternating sums of skew Schurs E XAMPLE − + = + k n−k Peter McNamara – p.10/11

Tool: Cylindric skew Schur functions as alternating sums of skew Schurs E XAMPLE − + = + k n−k = s 33321 / 21 − s 3222111 / 21 + s 321111111 / 21 s C = s 333 + 2 s 3321 + s 33111 + s 3222 − s 321111 + s 3111111 − s 22221 − 2 s 222111 + 2 s 21111111 + s 111111111 . Peter McNamara – p.10/11

St.-Jean-Baptiste Day Peter McNamara – p.11/11

St.-Jean-Baptiste Day Special Session in Algebraic Combinatorics Canadian Mathematical Society Winter Meeting Saturday, December 11 - Monday, December 13 McGill University, Montréal http://www.lacim.uqam.ca/ ∼ biagioli/CMS/cms.html bergeron.francois@uqam.ca François Bergeron biagioli@lacim.uqam.ca Riccardo Biagioli mcnamara@lacim.uqam.ca Peter McNamara christo@lacim.uqam.ca Christophe Reutenauer Peter McNamara – p.11/11