Transition matrices for symmetric and quasisymmetric Hall-Littlewood - PowerPoint PPT Presentation

Transition matrices for symmetric and quasisymmetric Hall-Littlewood polynomials Nick Loehr – Virginia Tech and U.S. Naval Academy Luis Serrano – LaCIM, UQ` AM Greg Warrington – University of Vermont FPSAC – SFCA Paris, France June 27, 2013 1 / 41

Original motivation Schur − → fundamental quasisymmetric s 31 = F 31 + F 22 + F 13 2 / 41

Original motivation Schur − → fundamental quasisymmetric s 31 = F 31 + F 22 + F 13 monomial symmetric − → monomial quasisymmetric m 31 = M 31 + M 13 2 / 41

Original motivation Schur − → fundamental quasisymmetric s 31 = F 31 + F 22 + F 13 monomial symmetric − → monomial quasisymmetric m 31 = M 31 + M 13 Hall-Littlewood polynomials P λ ( x ; t ) P λ ( x ; 0) = s λ P λ ( x ; 1) = m λ 2 / 41

Original motivation Schur − → fundamental quasisymmetric s 31 = F 31 + F 22 + F 13 monomial symmetric − → monomial quasisymmetric m 31 = M 31 + M 13 Hall-Littlewood polynomials P λ ( x ; t ) P λ ( x ; 0) = s λ P λ ( x ; 1) = m λ Question: Is there some quasisymmetric expansion of P λ which: ◮ at t = 0 gives us the fundamental expansion of Schur functions, and ◮ at t = 1 gives us the monomial quasisymmetric expansion of monomial symmetric functions? 2 / 41

Prism of bases Hall-Littlewood t=0 t=1 Symmetric Functions Schur monomial Hivert t=0 t=1 Quasisymmetric Functions Fundamental Monomial 3 / 41

Prism of bases Hall-Littlewood Symmetric Functions Schur monomial Hivert Quasisymmetric Functions Fundamental Monomial 4 / 41

Symmetric functions monomial: � x 4 i x 2 j x 1 m 421 = k i , j , k 5 / 41

Symmetric functions monomial: � x 4 i x 2 j x 1 m 421 = k i , j , k Schur: x 2 x 1 x 2 s 21 = 1 x 2 + + x 1 x 2 x 3 + x 1 x 2 x 3 · · · 2 2 2 3 2 1 1 1 2 1 2 1 3 5 / 41

Prism of bases Hall-Littlewood Symmetric Functions SSYT Schur monomial Hivert Quasisymmetric Functions Fundamental Monomial 6 / 41

Prism of bases Hall-Littlewood Symmetric Functions SRHT Schur monomial (E˘ gecio˘ glu – Remmel) Hivert Quasisymmetric Functions Fundamental Monomial 7 / 41

Quasisymmetric functions Monomial: � x 1 i x 4 j x 2 M 142 = k i < j < k 8 / 41

Quasisymmetric functions Monomial: � x 1 i x 4 j x 2 M 142 = k i < j < k Fundamental (refinement order – Gessel): F 23 = M 23 + M 221 + M 212 + M 2111 + M 113 + M 1121 + M 1112 + M 11111 8 / 41

Prism of bases Hall-Littlewood Symmetric Functions Schur monomial Hivert Quasisymmetric Functions refinement Fundamental Monomial (Gessel) 9 / 41

Prism of bases Hall-Littlewood Symmetric Functions Schur monomial Hivert Quasisymmetric Functions M¨ obius Fundamental inversion Monomial 10 / 41

Prism of bases Hall-Littlewood Symmetric Functions Schur monomial descents unsort Hivert (Gessel) Quasisymmetric Functions Fundamental Monomial 11 / 41

Symmetric to quasisymmetric Schur to fundamental, sum over descent compositions (Gessel) s 31 = F 31 + F 22 + F 13 4 3 2 1 2 3 1 2 4 1 3 4 12 / 41

Symmetric to quasisymmetric Schur to fundamental, sum over descent compositions (Gessel) s 31 = F 31 + F 22 + F 13 4 3 2 1 2 3 1 2 4 1 3 4 Monomial symmetric to monomial quasisymmetric (unsort) m 321 = M 321 + M 312 + M 213 + M 231 + M 132 + M 123 12 / 41

Prism of bases Hall-Littlewood Symmetric Functions Schur monomial (Egge-Loehr- sort Hivert Warrington, Garsia) Quasisymmetric Functions Fundamental Monomial 13 / 41

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia) Schur expansion of F 31 + F 22 + F 13 ? 14 / 41

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia) Schur expansion of F 31 + F 22 + F 13 ? F 31 + F 22 + F 13 = s 31 + s 22 + s 13 14 / 41

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia) Schur expansion of F 31 + F 22 + F 13 ? F 31 + F 22 + F 13 = s 31 + s 22 + s 13 (First year graduate student’s dream) 14 / 41

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia) Schur expansion of F 31 + F 22 + F 13 ? F 31 + F 22 + F 13 = s 31 + s 22 + s 13 (First year graduate student’s dream) What is s 13 ???? 14 / 41

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia) Schur expansion of F 31 + F 22 + F 13 ? F 31 + F 22 + F 13 = s 31 + s 22 + s 13 (First year graduate student’s dream) What is s 13 ???? s 13 = − s 22 . 14 / 41

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia) Schur expansion of F 31 + F 22 + F 13 ? F 31 + F 22 + F 13 = s 31 + s 22 + s 13 (First year graduate student’s dream) What is s 13 ???? s 13 = − s 22 . What is s α if α is not a partition? 14 / 41

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia) Schur expansion of F 31 + F 22 + F 13 ? F 31 + F 22 + F 13 = s 31 + s 22 + s 13 (First year graduate student’s dream) What is s 13 ???? s 13 = − s 22 . What is s α if α is not a partition? s 154 = s 433 14 / 41

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia) Schur expansion of F 31 + F 22 + F 13 ? F 31 + F 22 + F 13 = s 31 + s 22 + s 13 (First year graduate student’s dream) What is s 13 ???? s 13 = − s 22 . What is s α if α is not a partition? s 154 = s 433 ( − 1) 2 s 433 s 154 = 14 / 41

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia) Schur expansion of F 31 + F 22 + F 13 ? F 31 + F 22 + F 13 = s 31 + s 22 + s 13 (First year graduate student’s dream) What is s 13 ???? s 13 = − s 22 . What is s α if α is not a partition? s 154 = s 433 ( − 1) 2 s 433 s 154 = o o o o o o o o o o 14 / 41

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia) Schur expansion of F 31 + F 22 + F 13 ? F 31 + F 22 + F 13 = s 31 + s 22 + s 13 (First year graduate student’s dream) What is s 13 ???? s 13 = − s 22 . What is s α if α is not a partition? s 154 = s 433 ( − 1) 2 s 433 s 154 = o o o o o o o o o o o o o o o o o o o o 14 / 41

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia) s 12 = 0 15 / 41

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia) s 12 = 0 o o o 15 / 41

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia) s 13 = − s 22 o o o o o o o o 16 / 41

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia) s 13 = − s 22 o o o o o o o o F 31 + F 22 + F 13 = s 31 + s 22 + s 13 = s 31 + s 22 − s 22 = s 31 16 / 41

Moral of this talk 17 / 41

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Moral Function hard Schur Symmetric Functions Quasisymmetric Functions 19 / 41

(Counterexample to the triangle inequality) Symmetric Functions Function hard Schur E-L-W,G easy (easy) Quasisymmetric Functions Fundamental 20 / 41

(Counterexample to the triangle inequality) Symmetric Functions Function hard Schur E-L-W,G easy (easy) Quasisymmetric Functions Fundamental Side effects may include hallucinations, such as an apparent loss of Schur positivity. 20 / 41

Prism of bases Hall-Littlewood algebraic Tournament matrices Symmetric Functions (Carbonara) (Macdonald) Schur monomial Hivert Quasisymmetric Functions Fundamental Monomial 21 / 41

Prism of bases Hall-Littlewood charge Symmetric Functions (Lascoux-Sch¨ utzenberger) Schur monomial Hivert Quasisymmetric Functions Fundamental Monomial 22 / 41

New Transition matrices (LSW) Hall-Littlewood Symmetric Functions Schur monomial Hivert Quasisymmetric Functions Fundamental Monomial 23 / 41

Hall-Littlewood to Fundamental Hall-Littlewood Symmetric Functions Schur monomial Hivert Quasisymmetric Functions Fundamental Monomial 24 / 41

Hall-Littlewood polynomials The P λ ( x ; t ) satisfy: P λ ( x ; 0) = s λ ( x ) P λ ( x ; 1) = m λ ( x ) 25 / 41

Hall-Littlewood polynomials The P λ ( x ; t ) satisfy: P λ ( x ; 0) = s λ ( x ) P λ ( x ; 1) = m λ ( x ) Theorem (Loehr-S.-Warrington) � sgn( S ∗ ) t tstat( S ∗ ) F Asc ′ ( S ∗ ) ( x ) . P λ/µ ( x ; t ) = S ∗ ∈ SYT ∗ ( λ/µ ) 25 / 41

Hall-Littlewood polynomials The P λ ( x ; t ) satisfy: P λ ( x ; 0) = s λ ( x ) P λ ( x ; 1) = m λ ( x ) Theorem (Loehr-S.-Warrington) � sgn( S ∗ ) t tstat( S ∗ ) F Asc ′ ( S ∗ ) ( x ) . P λ/µ ( x ; t ) = S ∗ ∈ SYT ∗ ( λ/µ ) t 2 F 111 . P 21 ( t ) = F 21 − tF 111 + F 12 − 3 3 2 2 1 2 1 2 ∗ 1 3 1 3 ∗ 25 / 41

Starred tableaux λ = 65211 11 10 S ∗ = 5 12 6 7 ∗ 13 ∗ 14 4 3 8 ∗ 9 15 1 2 ∗ 26 / 41

Starred tableaux λ = 65211 11 10 S ∗ = 5 12 6 7 ∗ 13 ∗ 14 4 3 8 ∗ 9 15 1 2 ∗ Ascents: ◮ When i + 1 is above i (or i ∗ ). 26 / 41

Starred tableaux λ = 65211 11 10 S ∗ = 5 12 6 7 ∗ 13 ∗ 14 4 3 8 ∗ 9 15 1 2 ∗ Ascents: ◮ When i + 1 is above i (or i ∗ ). ◮ When ( i + 1) ∗ is in the next column to the right of i (or i ∗ ). 26 / 41

Starred tableaux λ = 65211 11 10 S ∗ = 5 12 6 7 ∗ 13 ∗ 14 4 3 8 ∗ 9 15 1 2 ∗ Ascents: ◮ When i + 1 is above i (or i ∗ ). ◮ When ( i + 1) ∗ is in the next column to the right of i (or i ∗ ). Asc( S ∗ ) = { 3 , 4 , 6 , 7 , 9 , 10 , 14 } 26 / 41