Quasisymmetric refinements of Schur functions Steph van Willigenburg - PowerPoint PPT Presentation

Quasisymmetric refinements of Schur functions Steph van Willigenburg University of British Columbia Triangle Lectures in Combinatorics, April 2011 with Christine Bessenrodt, Jim Haglund, Kurt Luoto, Sarah Mason 1

Compositions and partitions A composition α 1 . . . α k of n is a list of positive integers whose sum is n : 2213 � 8. A composition is a partition if α 1 ≥ α 2 ≥ . . . ≥ α k > 0: 3221 ⊢ 8. Any composition determines a partition: λ (2213) = 3221. α = α 1 . . . α k is a coarsening of β = β 1 . . . β l ( β is a refinement of α ) if β 1 + . . . + β i β i +1 + . . . + β j . . . β m + . . . + β l � �� � � �� � � �� � α 1 α 2 α k is true: 53 � 2213. 2

Quasisymmetric functions Let QSym be the algebra of quasisymmetric functions QSym := QSym 0 ⊕ QSym 1 ⊕ · · · ⊂ Q [ x 1 , x 2 , . . . ] QSym n := span Q { M α | α = α 1 . . . α k � n } = span Q { F α | α � n } � i 2 · · · x α k � x α 1 i 1 x α 2 M α := F α = M β i k i 1 <i 2 < ··· <i k α � β i 1 <i 2 <i 3 x 1 i 1 x 2 i 2 x 1 Example M 121 = � i 3 , F 121 = M 121 + M 1111 → QSym via m λ = � Remark Sym ֒ λ ( α )= λ M α . 3

Why quasisymmetric functions? • Generating functions for P-partitions, posets, matroids (Ges- sel 83, Ehrenborg 96, Stembridge 97, Petersen 07, Luoto 09, Billera-Jia-Reiner 09). • Combinatorial Hopf algebras (Ehrenborg 96, Aguiar-Bergeron- Sottile 06). • Dual to cd-index (Billera-Hsiao-vW 03). • Random walks (Stanley 01, Hsiao-Hersh 09). • Simplify Macdonald polys (Haglund-Luoto-Mason-vW 09). • Other types, coloured, shifted (Billey-Haiman 95, Hsiao-Petersen 10). 4

Diagrams and tableaux The diagram λ = λ 1 ≥ . . . ≥ λ k > 0 is the array of boxes with λ i boxes in row i from the top. 431 A (standard) reverse tableau T of shape λ is a filling of λ with (each first n ) 1 , 2 , 3 , . . . so rows weakly decrease and columns strictly decrease. 5

Diagrams and tableaux The diagram λ = λ 1 ≥ . . . ≥ λ k > 0 is the array of boxes with λ i boxes in row i from the top. 8 7 3 1 6 4 2 5 431 A (standard) reverse tableau T of shape λ is a filling of λ with (each first n ) 1 , 2 , 3 , . . . so rows weakly decrease and columns strictly decrease. 6

Composition diagrams and tableaux The composition diagram α = α 1 . . . α k > 0 is the array of boxes with α i boxes in row i from the top. 413 A (standard) composition tableau of shape α is a filling of α with (each first n ) 1 , 2 , 3 , . . . such that 7

Rules for composition tableaux • First column entries strictly increase top to bottom. • Rows weakly decrease left to right. • If b ≤ c then b < a . Example c a b 8

Rules for composition tableaux • First column entries strictly increase top to bottom. • Rows weakly decrease left to right. • If b ≤ c then b < a . Example c a b 9

Rules for composition tableaux • First column entries strictly increase top to bottom. • Rows weakly decrease left to right. • If b ≤ c then b < a . Example 5 4 3 1 6 8 7 2 10

Quasisymmetric Schur functions If x T := x #1 s x #2 s x #3 s . . . then QSym n = span Q {S α | α � n } 1 2 3 where � x T S α = T ∈ CT ( α ) Example 11

Quasisymmetric Schur functions If x T := x #1 s x #2 s x #3 s . . . then QSym n = span Q {S α | α � n } 1 2 3 where � x T S α = T ∈ CT ( α ) Example S 12 = x 1 x 2 2 + x 1 x 2 x 3 + x 1 x 2 3 + x 2 x 2 3 from 1 1 1 2 2 2 3 2 3 3 3 3 s λ = � λ ( α )= λ S α as m λ = � λ ( α )= λ M α . 12

Quasisymmetric Kostka numbers For λ ⊢ n � s λ = K λµ m µ µ where K λµ = number of reverse tableaux T of shape λ and µ 1 1s, µ 2 2s, . . . For α � n � S α = K αβ M β β where K αβ = number of composition tableaux T of shape α and β 1 1s, β 2 2s, . . . 13

Young’s lattice: L Y Partial order on partitions with covers • add 1 at end: 211 < 2111 • add 1 to leftmost part of size: 211 < 221, 211 < 311. saturated chains in L Y ↔ standard skew RT from µ to λ shape λ/µ Example 32 < 321 < 331 < 431 ↔ • • • 1 • • 2 3 14

Composition poset: L C Partial order on compositions with covers • add 1 at start: 121 < 1121 • add 1 to leftmost part of size: 121 < 221, 121 < 131. saturated chains in L C ↔ standard skew CT from α to β shape β//α Example 23 < 123 < 133 < 143 ↔ 3 • • 2 1 • • • 15

Descents and sets T standard (skew) tableau, Des ( T ) = { i | i + 1 weakly east } : 8 7 3 1 6 4 2 5 composition α 1 . . . α k � n ↔ subset { i 1 , . . . , i k − 1 } ⊆ [ n − 1] β 2312 � 8 ↔ { 2 , 5 , 6 } ⊆ [7] Set ( β ) 16

Quasisymmetric skew Schur functions Skew Schur functions � � c λ c λ s λ/µ = F δ = µν : +ve integers µν s ν ν where Set ( δ ) = Des ( T ), T ∈ SRT ( λ/µ ). Quasisymmetric skew Schur functions � � C γ C γ S γ//β = F δ = αβ S α αβ : +ve integers α where Set ( δ ) = Des ( T ), T ∈ SCT ( γ//β ). λ ( γ )= λ C γ If λ ( α ) = µ, λ ( β ) = ν then c λ µν = � αβ 17

What other Schur properties do S α have? • Z -basis for QSym . Expression in F β . ✔ • Quasisymmetric Pieri, LR rules. ✔ • Involution gives row strict versions (Mason-Remmel 10, Fer- reira 11) ✔ • Confirmed QSym over Sym has a stable basis (Lauve-Mason 10). “Just switch partition to composition” 18

Further properties? Other properties: • Jacobi-Trudi, Giambelli (quasi-) determinantal formulae? • Representation theoretic interpretation from F β ? • L C properties? • Normal or Kronecker (inner) product? Other applications: • Quasisymmetric Macdonald polynomials? • Skew Macdonald polynomials? • Product of Schubert polynomials? • Impact on QSym of different types? 19

Link to NC Schurs of Fomin and Greene P graded edge labelled poset, labels ( B, < ). For x ∈ P � x. h k = end ( ω ) ω ω : x b 1 b 2 b k → x 1 → · · · → x k = end ( ω ) for saturated ω , b 1 ≤ b 2 ≤ · · · ≤ b k ∈ B . For [ x, y ] of P � K [ x,y ] = � x. h α , y � M α � , � = δ ij α Example Skew Schur functions, Stanley symmetric functions, NC Schurs Fomin+Greene (Bergeron-Mykytiuk-Sottile-vW 00). 20

A new example Let L ′ C be the dual poset of L C edges labelled x ( − col, − row ) ˜ − → x and ( i, j ) < ( k, ℓ ) iff i < k or ( i = k = − 1 and j > ℓ ) or ( i = k < − 1 and j < ℓ ). Then K [ β,α ] = S β//α . 21

Link to NC Schurs of Rosas and Sagan A set composition of [ n ] = { 1 , . . . , n } is an ordered partitioning of [ n ]: Φ = 36 / 489 / 2 / 157 � [9] with underlying composition α (Φ) = 2313. A set partition of [ n ] reorders by least element: ˜ Φ = 157 / 2 / 36 / 489 ⊢ [9] with underlying partition λ (Φ) = 3321. 22

Symmetric functions in noncommuting variables (Wolfe 36, Rosas-Sagan 06) NCSym := NCSym 0 ⊕ NCSym 1 ⊕ · · · ⊂ Q � x 1 , x 2 , . . . � where NCSym n := span Q { m π | π ⊢ [ n ] } � m π := x i 1 x i 2 · · · x i n and i j = i k iff j, k ∈ π m Example m 13 / 2 = x 1 x 2 x 1 + x 2 x 1 x 2 + x 1 x 3 x 1 + x 3 x 1 x 3 . . . 23

NC Schurs of Rosas and Sagan For T ∈ RT ( λ ) let ˙ T have 1 entry with k dots k = 1 , 2 , 3 . . . then x ˙ T = � � � S RS = µ ! K λµ m π λ µ T ∈ RT ( λ ) λ ( π )= µ where x ˙ T = monomial x i in position j if T has i with j dots. Example 2 ... ˙ 1 x 2 x 3 x 1 ❀ ¨ 3 24

Quasisymmetric functions in noncommuting variables (Aguiar-Majahan 06, Bergeron-Zabrocki 09) NCSym ⊂ NCQSym := NCQSym 0 ⊕ NCQSym 1 ⊕· · · ⊂ Q � x 1 , x 2 , . . . � where NCQSym n := span Q { M Π | Π � [ n ] } � M Π := x i 1 x i 2 · · · x i n • i j = i k iff j, k ∈ Π m • i j < i k iff j ∈ Π m 1 k ∈ Π m 2 and m 1 < m 2 . Example M 2 / 13 = x 2 x 1 x 2 + x 3 x 1 x 3 . . . 25

� � NC quasisymmetric Schurs Let x ˙ T = � � � S RS = β ! K αβ M Π α β T ∈ CT ( α ) α (Π)= β Furthermore � λ ( α )= λ S RS S RS α λ χ Rosas − Sagan ❀ χ n ! � λ ( α )= λ S α n ! s λ 26

Link to free Schurs of Poirier and Reutenauer (95) PR := span Q { T | T ∈ SRT } If T 1 ∈ SRT ( µ ) and T 2 ∈ SRT ( ν ) then T 1 ∗ T 2 = � SRT T where • T | µ = T 1 + | ν | • rect ( T \ µ ) = T 2 . Example 3 1 ∗ 1 = 4 2 1 + 4 2 + 4 2 2 3 3 1 3 1 27

Bijection (Mason 06): ρ : SCT → SRT The map takes 5 4 3 1 8 7 3 1 �→ 6 6 4 2 8 7 2 5 Connection: φ : PR → QSym ∗ If φ ( T ) = S ∗ α where ρ − 1 ( T ) ∈ SCT ( α ) then φ ( T 1 ∗ T 2 ) = φ ( T 2 ) ⋆ φ ( T 1 ) . 28

Further reading • Skew quasisymmetric Schur functions and noncommutative Schur functions (with Bessenrodt and Luoto), Adv. Math., 226:4492–4532 (2011) . • Refinements of the Littlewood-Richardson rule (with Haglund, Luoto and Mason), Trans. Amer. Math. Soc. 363:1665–1686 (2011). • Quasisymmetric Schur functions (with Haglund, Luoto and Mason), J. Combin. Theory Ser. A 118: 463–490 (2011). Thank you! 29