The Prism Tableau Model for Schubert Polynomials Anna Weigandt - PowerPoint PPT Presentation

The Prism Tableau Model for Schubert Polynomials Anna Weigandt University of Illinois at Urbana-Champaign weigndt2@illinois.edu April 16th, 2016 Based on joint work with Alexander Yong arXiv:1509.02545 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

Overview The Prism Tableau Model for Schubert Polynomials Describe a tableau based combinatorial model for Schubert polynomials Give a description of the underlying geometric ideas of the proof Apply prism tableaux to study alternating sign matrix varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

The Ring of Symmetric Polynomials Λ n = { f ∈ Z [ x 1 , . . . , x n ] : w · f = f for all w ∈ S n } Schur polynomials { s λ } form a Z -linear basis for Λ n and have applications in geometry and representation theory Model for Schur polynomials as a sum over semistandard Young tableaux 1 + 1 2 2 2 1 s (2 , 1) ( x 1 , x 2 ) = x 1 x 2 2 + x 2 1 x 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

There is an inclusion: Λ n ֒ → Pol = Z [ x 1 , x 2 , ... ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

There is an inclusion: Λ n ֒ → Pol = Z [ x 1 , x 2 , ... ] Question: How do we lift the Schur polynomials to a basis of Pol ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

There is an inclusion: Λ n ֒ → Pol = Z [ x 1 , x 2 , ... ] Question: How do we lift the Schur polynomials to a basis of Pol ? An answer: Schubert polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

Schubert Polynomials Introduced by Lascoux and Sch¨ utzenberger in 1982 to study the cohomology of the complete flag variety Indexed by permutations, { S w : w ∈ S n } To find S w : S w 0 := x n − 1 x n − 2 . . . x n − 1 1 2 The rest are defined recursively by divided difference operators: ∂ i f := f − s i · f x i − x i +1 S ws i := ∂ i S w if w ( i ) > w ( i + 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

Schubert Polynomials for S 3 x 2 321 1 x 2 � ❅ � ❅ s 1 s 2 ∂ 1 ∂ 2 � ❅ � ❅ � ❅ � ❅ x 2 231 312 x 1 x 2 1 s 2 s 1 ∂ 2 ∂ 1 213 132 x 1 + x 2 x 1 ❅ � ❅ � ❅ � ❅ � ∂ 1 ∂ 2 s 1 s 2 ❅ � ❅ � 123 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

The Schubert Basis Schubert polynomials as a basis: ι There is a natural inclusion of symmetric groups S n → S n +1 − Schubert polynomials are stable under this inclusion: S w = S ι ( w ) { S w : w ∈ S ∞ } forms a Z -linear basis of Pol = Z [ x 1 , x 2 , . . . ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

The Schubert Basis Schubert polynomials as a basis: ι There is a natural inclusion of symmetric groups S n → S n +1 − Schubert polynomials are stable under this inclusion: S w = S ι ( w ) { S w : w ∈ S ∞ } forms a Z -linear basis of Pol = Z [ x 1 , x 2 , . . . ] Schubert polynomials as a lift of Schur polynomials: Every Schur polynomial is a Schubert polynomial for some w ∈ S ∞ S w is a Schur polynomial if and only if w is Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

Problem : Is there a combinatorial model for S w that is analogous to semistandard tableau? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

Problem : Is there a combinatorial model for S w that is analogous to semistandard tableau? Many earlier combinatorial models : A. Kohnert, S. Billey-C. Jockusch-R. Stanley, S. Fomin-A. Kirillov, S. Billey-N.Bergeron, ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

Problem : Is there a combinatorial model for S w that is analogous to semistandard tableau? Many earlier combinatorial models : A. Kohnert, S. Billey-C. Jockusch-R. Stanley, S. Fomin-A. Kirillov, S. Billey-N.Bergeron, ... A new solution: Prism Tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

What is a Prism Tableau? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

Some Definitions w = 35124 Each permutation has an associated: diagram : D ( w ) = { ( i , j ) : 1 ≤ i , j ≤ n , w ( i ) > j and w − 1 ( j ) > i } ⊂ n × n essential set : E ss ( w ) = { southeast-most boxes of each component of D ( w ) } rank function : r w ( i , j ) = the rank of the ( i , j ) NW submatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

The Shape Fix w ∈ S n . Each e = ( a , b ) ∈ E ss ( w ) indexes a color Let R e be an ( a − r w ( e )) × ( b − r w ( e )) rectangle in the n × n grid (left justified, bottom row in same row as e ) Define the shape : ∪ λ ( w ) = R e e ∈E ss ( w ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

The Shape Fix w ∈ S n . Each e = ( a , b ) ∈ E ss ( w ) indexes a color Let R e be an ( a − r w ( e )) × ( b − r w ( e )) rectangle in the n × n grid (left justified, bottom row in same row as e ) Define the shape : ∪ λ ( w ) = R e e ∈E ss ( w ) Example: w = 35142 e 1 e 2 ⇒ ⇒ λ ( w ) = R e 1 R e 2 e 3 R e 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

Prism Tableaux A prism tableau for w is a filling of λ ( w ) with colored labels, indexed by E ss ( w ) so that labels of color e : sit in boxes of R e weakly decrease along rows from left to right strictly increase along columns from top to bottom are flagged by row: no label is bigger than the row it sits in 1 1 122 21 1 2 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

The Weight of a Tableau Define the weight: x # of antidiagonals containing a label of number i ∏ wt ( T ) = i i Example: 1 1 wt ( T ) = x 2 T = 1 x 2 1 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

Minimal Prism Tableaux Fix a prism tableau T . T is minimal if the degree of wt ( T ) = ℓ ( w ). Example: w = 1432 11 1 21 1 3 3 wt ( T ) = x 2 wt ( T ) = x 2 1 x 3 1 x 2 x 3 minimal not minimal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

Unstable Triples We say labels ( ℓ c , ℓ d , ℓ ′ e ) in the same antidiagonal T form an unstable triple if ℓ < ℓ ′ and the tableau T ′ obtained by replacing ℓ c with ℓ ′ c is itself a prism tableau. 1 1 1 3 3 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

The Prism model for Schubert Polynomials Let Prism (w) be the set of minimal prism tableaux for w which have no unstable triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

The Prism model for Schubert Polynomials Let Prism (w) be the set of minimal prism tableaux for w which have no unstable triples . Theorem (W.- A. Yong 2015) ∑ S w = wt ( T ) . T ∈ Prism ( w ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

Example for w = 42513 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model

Example for w = 42513 (continued) 11 1 1 11 1 1 11 1 1 22 1 22 1 22 2 33 3 33 2 33 3 In Prism ( w ) In Prism ( w ) Not minimal 11 1 1 11 1 1 11 1 1 21 1 21 1 21 1 33 3 33 2 32 2 Unstable triple Unstable triple Not minimal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Weigandt Prism Tableau Model