A Pieri rule for sk ew shapes Peter McNamara Bucknell University - PowerPoint PPT Presentation

A Pieri rule for sk ew shapes Peter McNamara Bucknell University Joint work with: Sami Assaf MIT 4 August 2010 Full paper available from www.facstaff.bucknell.edu/pm040/ A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 1

A Pieri rule for sk ew shapes Peter McNamara Bucknell University Joint work with: Sami Assaf MIT 4 August 2010 Full paper available from www.facstaff.bucknell.edu/pm040/ A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 1

Outline ◮ Background on skew Schur functions and Pieri rule ◮ Main result ◮ Some highlights of the combinatorial proof ◮ 3 further-development nuggets A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 2

Schur functions Cauchy, 1815 ◮ Partition λ = ( λ 1 , λ 2 , . . . , λ ℓ ) ◮ Young diagram Example: = ( 4 , 4 , 3 , 1 ) λ A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 3

Schur functions Cauchy, 1815 ◮ Partition λ = ( λ 1 , λ 2 , . . . , λ ℓ ) 7 5 6 6 ◮ Young diagram Example: 4 4 4 9 = ( 4 , 4 , 3 , 1 ) λ 1 3 3 4 ◮ Semistandard Young tableau (SSYT) A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 3

Schur functions Cauchy, 1815 ◮ Partition λ = ( λ 1 , λ 2 , . . . , λ ℓ ) 7 5 6 6 ◮ Young diagram Example: 4 4 4 9 = ( 4 , 4 , 3 , 1 ) λ 1 3 3 4 ◮ Semistandard Young tableau (SSYT) The Schur function s λ in the variables x = ( x 1 , x 2 , . . . ) is then defined by x # 1’s in T x # 2’s in T � s λ = · · · . 1 2 SSYT T Example: = x 1 x 2 3 x 4 4 x 5 x 2 s 4431 6 x 7 x 9 + · · · . A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 3

Skew Schur functions Cauchy, 1815 ◮ Partition λ = ( λ 1 , λ 2 , . . . , λ ℓ ) ◮ µ fits inside λ 7 ◮ Young diagram 5 6 6 Example: 4 4 9 λ/µ = ( 4 , 4 , 3 , 1 ) / ( 3 , 1 ) 4 ◮ Semistandard Young tableau (SSYT) The skew Schur function s λ/µ in the variables x = ( x 1 , x 2 , . . . ) is then defined by x # 1’s in T x # 2’s in T � s λ/µ = · · · . 1 2 SSYT T Example: x 3 4 x 5 x 2 s 4431 / 31 = 6 x 7 x 9 + · · · . A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 3

Skew Schur functions Example: s 3 ( x 1 , x 2 , . . . ) A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 4

Skew Schur functions Example: i j k � s 3 ( x 1 , x 2 , . . . ) = x i x j x k i ≤ j ≤ k A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 4

Skew Schur functions Example: i j k � s 3 ( x 1 , x 2 , . . . ) = x i x j x k i ≤ j ≤ k Question: Why do we care about skew Schur functions? A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 4

Skew Schur functions Example: i j k � s 3 ( x 1 , x 2 , . . . ) = x i x j x k i ≤ j ≤ k Question: Why do we care about skew Schur functions? ◮ Fact: Skew Schur functions are symmetric in x 1 , x 2 , . . . . A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 4

Skew Schur functions Example: i j k � s 3 ( x 1 , x 2 , . . . ) = x i x j x k i ≤ j ≤ k Question: Why do we care about skew Schur functions? ◮ Fact: Skew Schur functions are symmetric in x 1 , x 2 , . . . . ◮ Fact: The Schur functions form a basis for the algebra of symmetric functions. A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 4

Skew Schur functions Example: i j k � s 3 ( x 1 , x 2 , . . . ) = x i x j x k i ≤ j ≤ k Question: Why do we care about skew Schur functions? ◮ Fact: Skew Schur functions are symmetric in x 1 , x 2 , . . . . ◮ Fact: The Schur functions form a basis for the algebra of symmetric functions. ◮ Strong connections with representation theory of S n and GL ( n , C ) , Schubert Calculus, eigenvalues of Hermitian matrices, .... A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 4

The Pieri rule The (classical) Pieri rule expands s λ s n in terms of { s µ } . + + + + = = A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 5

The Pieri rule The (classical) Pieri rule expands s λ s n in terms of { s µ } . Theorem [Pieri, 1893]: For a partition λ and positive integer n , � s λ s n = s λ + , λ + /λ n − hor . strip where the sum is over all λ + such that λ + /λ is a horizontal strip with n boxes. + + + + = = A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 5

The Pieri rule The (classical) Pieri rule expands s λ s n in terms of { s µ } . Theorem [Pieri, 1893]: For a partition λ and positive integer n , � s λ s n = s λ + , λ + /λ n − hor . strip where the sum is over all λ + such that λ + /λ is a horizontal strip with n boxes. Example: s 322 s 2 = s 3222 + s 3321 + s 4221 + s 432 + s 522 . + + + + = = A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 5

The Pieri rule The (classical) Pieri rule expands s λ s n in terms of { s µ } . Theorem [Pieri, 1893]: For a partition λ and positive integer n , � s λ s n = s λ + , λ + /λ n − hor . strip where the sum is over all λ + such that λ + /λ is a horizontal strip with n boxes. Example: s 322 s 2 = s 3222 + s 3321 + s 4221 + s 432 + s 522 . + + + + = = A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 5

The Pieri rule The (classical) Pieri rule expands s λ s n in terms of { s µ } . Theorem [Pieri, 1893]: For a partition λ and positive integer n , � s λ s n = s λ + , λ + /λ n − hor . strip where the sum is over all λ + such that λ + /λ is a horizontal strip with n boxes. Example: s ( 322 ) ∗ ( 2 ) = s 322 s 2 = s 3222 + s 3321 + s 4221 + s 432 + s 522 . + + + + = = A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 5

Pieri-type rules in other settings ◮ k -Schur functions [Lapointe–Morse] ◮ Schubert polynomials [Lascoux–Schützenberger, Lenart–Sottile, Manivel, Sottile, Winkel] ◮ LLT polynomials [Lam] ◮ Schubert classes in the affine Grassmannian [Lam–Lapointe–Morse–Shimozono] ◮ Hall-Littlewood polynomials [Morris] ◮ Jack polynomials [Lassalle, Stanley] ◮ Macdonald polynomials [Koornwinder, Macdonald] ◮ Quasisymmetric Schur functions [Haglund, Luoto, Mason, van Willigenburg] ◮ Grothendieck polynomials [Lenart–Sottile] ◮ Factorial Grothendieck polynomials [McNamara (no relation!)] ◮ .... Notably absent: skew Schur functions A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 6

The skew Pieri rule The skew Pieri rule expands s λ/µ s n in terms of { s τ/σ } . + + + + = + A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 7

The skew Pieri rule The skew Pieri rule expands s λ/µ s n in terms of { s τ/σ } . Theorem [Assaf–McN.]: For a skew shape λ/µ and positive integer n , n � ( − 1 ) k � s λ/µ s n = s λ + /µ − , λ + /λ ( n − k ) -hor . strip k = 0 µ/µ − k -vert . strip where λ + /λ is a horizontal strip with n − k boxes and µ/µ − is a vertical strip with k boxes. + + + + = + A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 7

The skew Pieri rule The skew Pieri rule expands s λ/µ s n in terms of { s τ/σ } . Theorem [Assaf–McN.]: For a skew shape λ/µ and positive integer n , n � ( − 1 ) k � s λ/µ s n = s λ + /µ − , λ + /λ ( n − k ) -hor . strip k = 0 µ/µ − k -vert . strip where λ + /λ is a horizontal strip with n − k boxes and µ/µ − is a vertical strip with k boxes. Example: + + + + = + A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 7

The skew Pieri rule The skew Pieri rule expands s λ/µ s n in terms of { s τ/σ } . Theorem [Assaf–McN.]: For a skew shape λ/µ and positive integer n , n � ( − 1 ) k � s λ/µ s n = s λ + /µ − , λ + /λ ( n − k ) -hor . strip k = 0 µ/µ − k -vert . strip where λ + /λ is a horizontal strip with n − k boxes and µ/µ − is a vertical strip with k boxes. Example: + + + + = + A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 7

The skew Pieri rule The skew Pieri rule expands s λ/µ s n in terms of { s τ/σ } . Theorem [Assaf–McN.]: For a skew shape λ/µ and positive integer n , n � ( − 1 ) k � s λ/µ s n = s λ + /µ − , λ + /λ ( n − k ) -hor . strip k = 0 µ/µ − k -vert . strip where λ + /λ is a horizontal strip with n − k boxes and µ/µ − is a vertical strip with k boxes. Example: + + + + = + A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 7

The combinatorial proof n � ( − 1 ) k � s λ/µ s n = s λ + /µ − , λ + /λ ( n − k ) -hor . strip k = 0 µ/µ − k -vert . strip + + + + = + A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 8

The combinatorial proof n � ( − 1 ) k � s λ/µ s n = s λ + /µ − , λ + /λ ( n − k ) -hor . strip k = 0 µ/µ − k -vert . strip + + + + = + Technique: a sign-reversing involution on SSYT that: ◮ Preserves entries appearing in each SSYT; ◮ Has fixed points with k = 0 in bijection with SSYT of shape ( λ/µ ) ∗ ( n ) ; ◮ (Remaining SSYT with k even) ← → (SSYT with k odd). A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 8

Robinson-Schensted-Knuth insertion A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 9

Robinson-Schensted-Knuth insertion 5 5 7 4 7 8 9 4 7 8 9 2 2 4 7 7 2 4 7 7 2 1 1 3 4 7 1 1 3 4 7 A Pieri Rule for Skew Shapes Sami Assaf & Peter McNamara 9