Divided symmetrization, quasisymmetric functions and Schubert - PowerPoint PPT Presentation

Divided symmetrization, quasisymmetric functions and Schubert polynomials Vasu Tewari University of Pennsylvania (joint with Philippe Nadeau, Universit´ e Lyon 1) AMS Sectional Meeting Gainesville, 3rd November 2019

Postnikov’s divided symmetrization Given f ∈ C [ x 1 , . . . , x n ], the divided symmetrization of f , denoted by � f � n , is: � � f � � f � n = σ · . � 1 ≤ i ≤ n − 1 ( x i − x i +1 ) σ ∈ S n Example 1 1 � 1 � 2 = + = 0 . x 1 − x 2 x 2 − x 1 x 1 x 2 � x 1 � 2 = + = 1 . x 1 − x 2 x 2 − x 1 x 2 � � 1 x 2 � x 2 � 1 x 2 � 3 = σ · = x 1 + x 2 + x 3 . ( x 1 − x 2 )( x 2 − x 3 ) σ ∈ S 3

Some basic observations � � f � � f � n = σ · . � 1 ≤ i ≤ n − 1 ( x i − x i +1 ) σ ∈ S n 1 � f � n is a symmetric polynomial in x 1 , . . . , x n . 2 If f = gh where g is symmetric, then � f � n = g � h � n . 3 deg f < n − 1 = ⇒ � f � n = 0. 4 deg f = n − 1 implies � f � n is a scalar!

Some basic observations � � f � � f � n = σ · . � 1 ≤ i ≤ n − 1 ( x i − x i +1 ) σ ∈ S n 1 � f � n is a symmetric polynomial in x 1 , . . . , x n . 2 If f = gh where g is symmetric, then � f � n = g � h � n . 3 deg f < n − 1 = ⇒ � f � n = 0. 4 deg f = n − 1 implies � f � n is a scalar! If f ∈ Z [ x 1 , . . . , x n ] and deg f = n − 1, does � f � n have a deeper meaning?

Usual permutahedra Definition (Usual permutahedra) For λ = ( λ 1 ≥ · · · ≥ λ n ) ∈ R n , the permutahedron P λ is the convex hull of the S n -orbit of λ .

Volumes of permutahedra P λ lies on the hyperplane defined by the sum of the λ i . Thus, the dimension of P λ is at most n − 1. Definition For a polytope P lying in a hyperplane in R n , define its volume vol ( P ) as the usual ( n − 1)-dimensional volume of the projection of P onto x n = 0. Given λ = ( λ 1 , . . . , λ n ), set V ( λ ) := vol ( P λ ) .

Permutahedron P 210

Permutahedron P 210

Divided symmetrization and volumes Theorem (Postnikov’05) n � λ i x i ) n − 1 � n ( n − 1)! V ( λ ) = � ( i =1 Example In the case n = 3 and λ 3 = 0, we get 2 V ( λ 1 , λ 2 , 0) = � λ 2 1 x 2 1 + 2 λ 1 λ 2 x 1 x 2 + λ 2 2 x 2 2 � 3 = λ 2 1 � x 2 1 � 3 + 2 λ 1 λ 2 � x 1 x 2 � 3 + λ 2 2 � x 2 2 � 3 = λ 2 1 + 2 λ 1 λ 2 − 2 λ 2 2 .

The class of the Peterson variety Alex Woo: Compute the class of the Peterson variety in terms of Schubert classes. We translate this question into an equivalent form that involves studying polynomials modulo a specific ideal.

The coinvariant algebra Let e k ( x 1 , . . . , x n ) denote the k th elementary symmetric polynomial. � e k ( x 1 , . . . , x n ) := x i 1 · · · x i k 1 ≤ i 1 < ··· < i k ≤ n I n = ideal in Z [ x 1 , . . . , x n ] generated by the e k . The (type A) coinvariant algebra is the quotient Z [ x 1 , . . . , x n ] / I n . By work of Borel, this quotient is isomorphic to the integer cohomology ring of the complete flag variety.

The class of the Peterson variety Theorem (Anderson-Tymoczko’07) The class of the Peterson variety is represented by � ( x i − x j ) . j − i > 1

The class of the Peterson variety Theorem (Anderson-Tymoczko’07) The class of the Peterson variety is represented by � ( x i − x j ) . j − i > 1 Here is A. Woo’s question reformulated: Reduce the product above mod I n and expand in terms of repre- sentatives of Schubert classes

The class of the Peterson variety Theorem (Anderson-Tymoczko’07) The class of the Peterson variety is represented by � ( x i − x j ) . j − i > 1 Here is A. Woo’s question reformulated: Reduce the product above mod I n and expand in terms of repre- sentatives of Schubert classes aka Schubert polynomials.

(Reduced) Pipe dreams aka rc-graphs To build a pipe dream for w ∈ S n , draw the staircase ( n , . . . , 1), enumerate its rows from 1 through n and columns from w (1) through w ( n ). Fill in the internal squares with crossing tiles or elbow tiles so that i connects to w ( i ), two strands intersect at most once. 1 4 3 2 1 2 3 4 Figure: A reduced pipe dream for w = 1432.

Reduced pipe dreams and associated monomials Given a reduced pipe dream D , set � x D := x row ( c ) crossings c ∈ D 1 4 3 2 1 2 3 4 Figure: A pipe dream D with x D = x 1 x 2 x 3 .

Bottom pipe dreams and codes The code of w ∈ S n is the weak composition ( c 1 , . . . , c n ) where c i = { j > i | w i > w j } . For instance, if w = 1432, then code ( w ) = (0 , 2 , 1 , 0). The bottom pipe dream for w is attached naturally to code ( w ). 1 4 3 2 1 2 3 4 Figure: The bottom pipe dream for w = 1432.

Schubert polynomial (BJS or BB definition) PD ( w ) := { reduced pipe dreams for w } . Definition For w ∈ S n , the Schubert polynomial S w ( x 1 , . . . , x n ) is defined as � S w ( x 1 , . . . , x n ) := x D . D ∈ PD ( w )

Reduced pipe dreams for w = 1432 1 4 3 2 1 4 3 2 1 4 3 2 1 1 1 2 2 2 3 3 3 4 4 4 1 4 3 2 1 4 3 2 1 1 2 2 3 3 4 4 S 1432 = x 2 2 x 3 + x 1 x 2 x 3 + x 1 x 2 2 + x 2 1 x 3 + x 2 1 x 2

Some empirical observations Henceforth, for w ∈ S n with ℓ ( w ) = n − 1, set a w := � S w � n . Theorem (Nadeau-T.’19) The a w give coefficients for Schuberts in the class of the Peterson.

Some empirical observations Henceforth, for w ∈ S n with ℓ ( w ) = n − 1, set a w := � S w � n . Theorem (Nadeau-T.’19) The a w give coefficients for Schuberts in the class of the Peterson. Geometry says a w ≥ 0.

Some empirical observations Henceforth, for w ∈ S n with ℓ ( w ) = n − 1, set a w := � S w � n . Theorem (Nadeau-T.’19) The a w give coefficients for Schuberts in the class of the Peterson. Geometry says a w ≥ 0. In fact a w > 0 conjecturally. Also conjectured by Harada et al. a w = a w − 1 conjecturally. a w = a w 0 ww 0 . Straightforward to establish.

An example: n = 3 We need to reduce � ( x i − x j ) = x 1 − x 3 j − i > 1 modulo ideal generated by x 1 + x 2 + x 3 , x 1 x 2 + x 1 x 3 + x 2 x 3 , and x 1 x 2 x 3 .

An example: n = 3 We need to reduce � ( x i − x j ) = x 1 − x 3 j − i > 1 modulo ideal generated by x 1 + x 2 + x 3 , x 1 x 2 + x 1 x 3 + x 2 x 3 , and x 1 x 2 x 3 . x 1 − x 3 ≡ x 1 +( x 1 + x 2 ) = 1 S 213 + 1 S 132 .

An example: n = 3 We need to reduce � ( x i − x j ) = x 1 − x 3 j − i > 1 modulo ideal generated by x 1 + x 2 + x 3 , x 1 x 2 + x 1 x 3 + x 2 x 3 , and x 1 x 2 x 3 . x 1 − x 3 ≡ x 1 +( x 1 + x 2 ) = 1 S 213 + 1 S 132 . � S 231 � 3 = � x 1 x 2 � 3 = 1 � S 312 � 3 = � x 2 1 � 3 = 1 .

The real question Find a manifestly positive combinatorial rule for a w . Bonus points if it reflects invariance under inverses and/or conjugation by w 0 .

The real question Find a manifestly positive combinatorial rule for a w . Bonus points if it reflects invariance under inverses and/or conjugation by w 0 . Naive idea: Use divided symmetrization of monomials and BJS expansion of Schuberts. Results in a signed formula, and yet instructive.

Catalan permutations Call w ∈ S n with ℓ ( w ) = n − 1 a Catalan permutation if the bottom pipe dream of w has at least i crosses in the first i diagonals for 1 ≤ i ≤ n − 1. 1 2 3 4 5 2 4 1 5 3 Figure: The bottom pipe dream for w = 24153.

Catalan permutations Call w ∈ S n with ℓ ( w ) = n − 1 a Catalan permutation if the bottom pipe dream of w has at least i crosses in the first i diagonals for 1 ≤ i ≤ n − 1. 1 2 3 4 5 2 4 1 5 3 Figure: The bottom pipe dream for w = 24153.

Catalan permutations Call w ∈ S n with ℓ ( w ) = n − 1 a Catalan permutation if the bottom pipe dream of w has at least i crosses in the first i diagonals for 1 ≤ i ≤ n − 1. 1 2 3 4 5 2 4 1 5 3 Figure: The bottom pipe dream for w = 24153.

Catalan permutations Call w ∈ S n with ℓ ( w ) = n − 1 a Catalan permutation if the bottom pipe dream of w has at least i crosses in the first i diagonals for 1 ≤ i ≤ n − 1. 1 2 3 4 5 2 4 1 5 3 Figure: The bottom pipe dream for w = 24153.

Catalan permutations Call w ∈ S n with ℓ ( w ) = n − 1 a Catalan permutation if the bottom pipe dream of w has at least i crosses in the first i diagonals for 1 ≤ i ≤ n − 1. 1 2 3 4 5 2 4 1 5 3 Figure: The bottom pipe dream for w = 24153.

Catalan permutations Theorem (Nadeau-T.’19) If w ∈ S n is Catalan, then � S w � n = | PD ( w ) | = S w (1 n ) .

Catalan permutations Theorem (Nadeau-T.’19) If w ∈ S n is Catalan, then � S w � n = | PD ( w ) | = S w (1 n ) . If w is Catalan then so is w − 1 , and so � S w � n = � S w − 1 � n says that PD ( w ) = PD ( w − 1 ).

Schuberts of grassmannian permutations (Schurs) Given partition λ = ( λ 1 ≤ · · · ≤ λ k ), a semistandard Young tableau T of shape λ is a filling of the Young diagram of λ so that rows increase weakly and columns increase strictly. 4 2 2 1 1 3 Figure: A semistandard Young tableau of shape (1 , 2 , 3). � x cont ( T ) . s λ ( x 1 , . . . , x m ) := T ∈ SSYT ≤ m ( λ )