QSym over Sym has a stable basis Aaron Lauve and Sarah Mason 3 - PowerPoint PPT Presentation

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis QSym over Sym has a stable basis Aaron Lauve and Sarah Mason 3 August 2010 Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Sym n ֒ → Q [ x ] free module coinvariant space Q [ x ] / Sym n of dimension n ! Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Sym n ֒ → Q [ x ] free module coinvariant space Q [ x ] / Sym n of dimension n ! → QSym n ֒ → Q [ x ] Sym n ֒ Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric polynomials Symmetric and quasisymmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer basis The Bergeron-Reutenauer conjectures Symmetric polynomials in n variables f ( x 1 , x 2 , . . . , x n ) ∈ Sym n ⇐ ⇒ π f = f ∀ π ∈ S n Example ( Sym 3 ) x 1 x 2 + x 1 x 3 + x 2 x 3 x 2 1 x 2 + x 2 1 x 3 + x 2 2 x 3 x 2 1 x 2 + x 1 x 2 2 x 2 1 x 2 + x 2 1 x 3 + x 2 2 x 3 + x 1 x 2 2 + x 1 x 2 3 + x 2 x 2 3 x 3 1 x 2 + x 2 3 Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric polynomials Symmetric and quasisymmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer basis The Bergeron-Reutenauer conjectures Partitions A partition of a positive integer n is a weakly decreasing sequence of positive integers which sum to n . Example: 13 = 5 + 3 + 3 + 2 λ = (5 , 3 , 3 , 2) Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric polynomials Symmetric and quasisymmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer basis The Bergeron-Reutenauer conjectures Semi-standard Young tableaux (SSYT) A semi-standard Young tableau is a filling of the entries in a partition diagram with positive integers so that the rows are weakly increasing from left to right and the columns are strictly increasing from top to bottom. The weight of a SSYT T , denoted x T is the i x # of i ′ s product � . i Example: T = 1 2 2 4 7 3 4 7 4 6 9 7 8 x T = x 1 x 2 2 x 3 x 3 4 x 6 x 3 7 x 8 x 9 Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric polynomials Symmetric and quasisymmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer basis The Bergeron-Reutenauer conjectures Schur function basis The Schur function s λ is the polynomial obtained by summing the weights of all SSYT of shape λ : � x T s λ ( x 1 , x 2 , . . . , x n ) = T ∈ SSYT ( λ ) s 2 , 1 ( x 1 , x 2 , x 3 ) = 1 1 1 1 1 2 1 2 1 3 1 3 2 2 2 3 2 3 2 3 2 3 3 3 x 2 1 x 2 + x 2 1 x 3 + x 1 x 2 + x 1 x 2 3 + x 2 2 x 3 + x 2 x 2 2 + 2 x 1 x 2 x 3 3 Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric polynomials Symmetric and quasisymmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer basis The Bergeron-Reutenauer conjectures Quasiymmetric polynomials in n variables f ( x 1 , x 2 , . . . , x n ) ∈ QSym n ⇐ ⇒ ( ∀ i 1 < i 2 < . . . < i k ) coeff of x a 1 1 x a 2 2 . . . x a k k = coeff of x a 1 i 1 x a 2 i 2 · · · x a k i k Example ( QSym 3 ) x 1 x 2 + x 1 x 3 + x 2 x 3 x 2 1 x 2 + x 2 1 x 3 + x 2 2 x 3 x 2 1 x 2 + x 1 x 2 2 x 2 1 x 2 + x 2 1 x 3 + x 2 2 x 3 + x 1 x 2 2 + x 1 x 2 3 + x 2 x 2 3 x 3 1 x 2 + x 2 3 Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric polynomials Symmetric and quasisymmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer basis The Bergeron-Reutenauer conjectures Compositions A composition of a positive integer n is a sequence of positive integers which sum to n . Example: 14 = 2 + 4 + 1 + 4 + 3 α = (2 , 4 , 1 , 4 , 3) Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric polynomials Symmetric and quasisymmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer basis The Bergeron-Reutenauer conjectures Composition tableau (CT) A filling, F , of the cells of a composition such that: 1 The leftmost column entries strictly increase top to bottom. 2 The row entries weakly decrease from L to R. 3 The entries satisfy the triple rule: for i < j , ( F ( j , k ) � = 0 and F ( j , k ) ≥ F ( i , k )) = ⇒ F ( j , k ) > F ( i , k − 1) . F= 1 1 α = (2 , 4 , 1 , 4 , 3) 3 2 2 2 x F = x 3 1 x 3 2 x 3 x 2 4 x 3 6 x 2 4 7 6 6 6 1 7 7 4 Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric polynomials Symmetric and quasisymmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer basis The Bergeron-Reutenauer conjectures Quasisymmetric Schur function basis � x T , QS α = T ∈ CT ( α ) where CT ( α ) is the set of all composition tableau of shape α . Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric polynomials Symmetric and quasisymmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer basis The Bergeron-Reutenauer conjectures QS 3 , 1 , 2 ( x 1 , x 2 , x 3 , x 4 ) = 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3 4 3 4 4 4 4 x 3 1 x 2 x 2 x 3 + x 3 1 x 2 x 2 x 3 1 x 3 x 2 + 1 x 2 x 3 x 4 + 3 4 4 2 1 1 2 2 1 2 2 2 3 3 3 4 4 4 4 4 4 + x 2 1 x 2 x 3 x 2 + x 1 x 2 2 x 3 x 2 + x 3 2 x 3 x 2 4 4 4 Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric polynomials Symmetric and quasisymmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer basis The Bergeron-Reutenauer conjectures The quasisymmetric Schur functions: form a basis for all quasisymmetric functions � refine the Schur functions in a natural way ( s λ = QS α ) α = λ ˜ decompose into positive sums of Gessel’s fundamental quasisymemtric functions enjoy many nice combinatorial properties, including a Littlewood-Richardson style multiplication rule Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric polynomials Symmetric and quasisymmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer basis The Bergeron-Reutenauer conjectures � QS α × s λ = QS β β A few definitions: contre-lattice A word is contre-lattice if the number of i ′ s in any prefix of the resulting word is greater than or equal to the number of ( i − 1)’s in the prefix for all i . Example: 3 2 3 1 2 3 3 1 2, 3 2 1 2 1 Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric polynomials Symmetric and quasisymmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer basis The Bergeron-Reutenauer conjectures β/α (def by example) β = (3 , 1 , 4 , 2) α = (3 , 2)                     � � � � � � β/α = , ,         � � � � �           � � � � Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric polynomials Symmetric and quasisymmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer basis The Bergeron-Reutenauer conjectures reading word The column reading word ( w col ) of a skew diagram is the word obtained by reading the entries in the available cells down columns, working from right to left. Littlewood-Richardson composition tableau (LRCT) A Littlewood-Richardson composition tableau (LRCT) is a composition tableau filling of a skew composition diagram with positive integers so that the column reading word is contre-lattice. Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric polynomials Symmetric and quasisymmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer basis The Bergeron-Reutenauer conjectures Littlewood-Richardson composition tableaux (LRCT) 7 7 2 10 10 10 1 6 6 6 3 9 1 3 1 8 8 8 8 2 4 4 4 7 7 2 6 6 6 3 3 3 LRCT of shape LRCT of shape , (3,4,2,3)/(2,3,3) (4 , 2 , 5 , 3 , 6) / (3 , 1 , 4 , 2 , 3) w col = 3213 w col = 3231321 Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric polynomials Symmetric and quasisymmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer basis The Bergeron-Reutenauer conjectures Theorem (Haglund, Luoto, M, van Willigenburg 08) c β � QS α · s λ = α,λ QS β , β where c β α,λ is the number of (LRCT) of shape in β/α with content λ ∗ . Example (3 variables): QS 1 , 2 · s 1 , 1 = QS 2 , 3 + QS 1 , 1 , 3 + QS 1 , 3 , 1 5 1 6 5 (2 , 1) 4 4 2 1 4 4 2 4 4 2 1 Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric polynomials Symmetric and quasisymmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer basis The Bergeron-Reutenauer conjectures Sym n ֒ → Q [ x 1 , . . . , x n ] = Q [ x ] A classical result The following are equivalent: 1 Sym n is a polynomial ring, generated by the elementary symmetric polynomials E n = { e 1 ( x ) , . . . , e n ( x ) } ; 2 the ring Q [ x ] is a free Sym n -module; 3 the coinvariant space Q [ x ] S n = Q [ x ] / ( E n ) has dimension n ! and is isomorphic to the regular representation of S n . Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric polynomials Symmetric and quasisymmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer basis The Bergeron-Reutenauer conjectures Sym n ֒ → QSym n ֒ → Q [ x ] Bergeron-Reutenauer conjectures The ring QSym n is a free module over Sym n ; the dimension of the coinvariant space QSym n / ( E n ) is n !; the set of pure and inverting compositions form the indexing set for a basis for QSym n / ( E n ). Aaron Lauve and Sarah Mason QSym/Sym