Schubert polynomials, pipe dreams, and associahedra Evgeny Smirnov - PowerPoint PPT Presentation

Schubert polynomials, pipe dreams, and associahedra Evgeny Smirnov Higher School of Economics Department of Mathematics Laboratoire J.-V. Poncelet Moscow, Russia Askoldfest, Moscow, June 4, 2012 Evgeny Smirnov (HSE & Labo Poncelet) Schubert polynomials and combinatorics Moscow, June 4, 2012 1 / 18

Outline General definitions 1 Flag varieties Schubert varieties and Schubert polynomials Pipe dreams and Fomin–Kirillov theorem Numerology of Schubert polynomials 2 Permutations with many pipe dreams Catalan numbers and Catalan–Hankel determinants Combinatorics of Schubert polynomials 3 Pipe dream complexes Generalizations for other Weyl groups Open questions 4 Evgeny Smirnov (HSE & Labo Poncelet) Schubert polynomials and combinatorics Moscow, June 4, 2012 2 / 18

Flag varieties G = GL n ( C ) B ⊂ G upper-triangular matrices Fl ( n ) = { V 0 ⊂ V 1 ⊂ · · · ⊂ V n | dim V i = i } ∼ = G / B Theorem (Borel, 1953) Z [ x 1 , . . . , x n ] / ( x 1 + · · · + x n , . . . , x 1 . . . x n ) ∼ = H ∗ ( G / B , Z ) . This isomorphism is constructed as follows: V 1 , . . . , V n tautological vector bundles over G / B ; L i = V i / V i − 1 ( 1 ≤ i ≤ n ); x i �→ − c 1 ( L i ) ; The kernel is generated by the symmetric polynomials without constant term. Evgeny Smirnov (HSE & Labo Poncelet) Schubert polynomials and combinatorics Moscow, June 4, 2012 3 / 18

Flag varieties G = GL n ( C ) B ⊂ G upper-triangular matrices Fl ( n ) = { V 0 ⊂ V 1 ⊂ · · · ⊂ V n | dim V i = i } ∼ = G / B Theorem (Borel, 1953) Z [ x 1 , . . . , x n ] / ( x 1 + · · · + x n , . . . , x 1 . . . x n ) ∼ = H ∗ ( G / B , Z ) . This isomorphism is constructed as follows: V 1 , . . . , V n tautological vector bundles over G / B ; L i = V i / V i − 1 ( 1 ≤ i ≤ n ); x i �→ − c 1 ( L i ) ; The kernel is generated by the symmetric polynomials without constant term. Evgeny Smirnov (HSE & Labo Poncelet) Schubert polynomials and combinatorics Moscow, June 4, 2012 3 / 18

Flag varieties G = GL n ( C ) B ⊂ G upper-triangular matrices Fl ( n ) = { V 0 ⊂ V 1 ⊂ · · · ⊂ V n | dim V i = i } ∼ = G / B Theorem (Borel, 1953) Z [ x 1 , . . . , x n ] / ( x 1 + · · · + x n , . . . , x 1 . . . x n ) ∼ = H ∗ ( G / B , Z ) . This isomorphism is constructed as follows: V 1 , . . . , V n tautological vector bundles over G / B ; L i = V i / V i − 1 ( 1 ≤ i ≤ n ); x i �→ − c 1 ( L i ) ; The kernel is generated by the symmetric polynomials without constant term. Evgeny Smirnov (HSE & Labo Poncelet) Schubert polynomials and combinatorics Moscow, June 4, 2012 3 / 18

Schubert varieties G / B = � w ∈ S n B − wB / B — Schubert decomposition ; X w = B − wB / B , where B − the opposite Borel subgroup; H ∗ ( G / B , Z ) ∼ w ∈ S n Z · [ X w ] as abelian groups. � = Question Are there any “nice” representatives of [ X w ] in Z [ x 1 , . . . , x n ] ? Answer: Schubert polynomials S w ( x 1 , . . . , x n − 1 ) ∈ Z [ x 1 , . . . , x n ] ; w ∈ S n � S w �→ [ X w ] ∈ H ∗ ( G / B , Z ) under the Borel isomorphism; Introduced by J. N. Bernstein, I. M. Gelfand, S. I. Gelfand (1978), A. Lascoux and M.-P. Sch¨ utzenberger, 1982; Combinatorial description: S. Billey and N. Bergeron, S. Fomin and An. Kirillov, 1993–1994. Evgeny Smirnov (HSE & Labo Poncelet) Schubert polynomials and combinatorics Moscow, June 4, 2012 4 / 18

Schubert varieties G / B = � w ∈ S n B − wB / B — Schubert decomposition ; X w = B − wB / B , where B − the opposite Borel subgroup; H ∗ ( G / B , Z ) ∼ w ∈ S n Z · [ X w ] as abelian groups. � = Question Are there any “nice” representatives of [ X w ] in Z [ x 1 , . . . , x n ] ? Answer: Schubert polynomials S w ( x 1 , . . . , x n − 1 ) ∈ Z [ x 1 , . . . , x n ] ; w ∈ S n � S w �→ [ X w ] ∈ H ∗ ( G / B , Z ) under the Borel isomorphism; Introduced by J. N. Bernstein, I. M. Gelfand, S. I. Gelfand (1978), A. Lascoux and M.-P. Sch¨ utzenberger, 1982; Combinatorial description: S. Billey and N. Bergeron, S. Fomin and An. Kirillov, 1993–1994. Evgeny Smirnov (HSE & Labo Poncelet) Schubert polynomials and combinatorics Moscow, June 4, 2012 4 / 18

Schubert varieties G / B = � w ∈ S n B − wB / B — Schubert decomposition ; X w = B − wB / B , where B − the opposite Borel subgroup; H ∗ ( G / B , Z ) ∼ w ∈ S n Z · [ X w ] as abelian groups. � = Question Are there any “nice” representatives of [ X w ] in Z [ x 1 , . . . , x n ] ? Answer: Schubert polynomials S w ( x 1 , . . . , x n − 1 ) ∈ Z [ x 1 , . . . , x n ] ; w ∈ S n � S w �→ [ X w ] ∈ H ∗ ( G / B , Z ) under the Borel isomorphism; Introduced by J. N. Bernstein, I. M. Gelfand, S. I. Gelfand (1978), A. Lascoux and M.-P. Sch¨ utzenberger, 1982; Combinatorial description: S. Billey and N. Bergeron, S. Fomin and An. Kirillov, 1993–1994. Evgeny Smirnov (HSE & Labo Poncelet) Schubert polynomials and combinatorics Moscow, June 4, 2012 4 / 18

Pipe dreams Let w ∈ S n . Consider a triangular table filled by and � , such that: the strands intertwine as prescribed by w ; no two strands cross more than once ( reduced pipe dream). Pipe dreams for w = (1432) 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 � � � � � � � � � � � � � � � � 1 1 1 1 1 � � � � � � � � � 2 2 2 2 2 � � � � � � � 3 3 3 3 3 � � � � � 4 4 4 4 4 monomial x d ( P ) = x d 1 2 . . . x d n − 1 1 x d 2 Pipe dream P n − 1 , � d i = # { ’s in the i -th row } x 2 x 2 x 1 x 2 x 2 2 x 3 x 1 x 2 x 3 1 x 3 1 x 2 2 Evgeny Smirnov (HSE & Labo Poncelet) Schubert polynomials and combinatorics Moscow, June 4, 2012 5 / 18

Pipe dreams Let w ∈ S n . Consider a triangular table filled by and � , such that: the strands intertwine as prescribed by w ; no two strands cross more than once ( reduced pipe dream). Pipe dreams for w = (1432) 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 � � � � � � � � � � � � � � � � 1 1 1 1 1 � � � � � � � � � 2 2 2 2 2 � � � � � � � 3 3 3 3 3 � � � � � 4 4 4 4 4 monomial x d ( P ) = x d 1 2 . . . x d n − 1 1 x d 2 Pipe dream P n − 1 , � d i = # { ’s in the i -th row } x 2 x 2 x 1 x 2 x 2 2 x 3 x 1 x 2 x 3 1 x 3 1 x 2 2 Evgeny Smirnov (HSE & Labo Poncelet) Schubert polynomials and combinatorics Moscow, June 4, 2012 5 / 18

Pipe dreams Let w ∈ S n . Consider a triangular table filled by and � , such that: the strands intertwine as prescribed by w ; no two strands cross more than once ( reduced pipe dream). Pipe dreams for w = (1432) 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 � � � � � � � � � � � � � � � � 1 1 1 1 1 � � � � � � � � � 2 2 2 2 2 � � � � � � � 3 3 3 3 3 � � � � � 4 4 4 4 4 monomial x d ( P ) = x d 1 2 . . . x d n − 1 1 x d 2 Pipe dream P n − 1 , � d i = # { ’s in the i -th row } x 2 x 2 x 1 x 2 x 2 2 x 3 x 1 x 2 x 3 1 x 3 1 x 2 2 Evgeny Smirnov (HSE & Labo Poncelet) Schubert polynomials and combinatorics Moscow, June 4, 2012 5 / 18

Pipe dreams Let w ∈ S n . Consider a triangular table filled by and � , such that: the strands intertwine as prescribed by w ; no two strands cross more than once ( reduced pipe dream). Pipe dreams for w = (1432) 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 � � � � � � � � � � � � � � � � 1 1 1 1 1 � � � � � � � � � 2 2 2 2 2 � � � � � � � 3 3 3 3 3 � � � � � 4 4 4 4 4 monomial x d ( P ) = x d 1 2 . . . x d n − 1 1 x d 2 Pipe dream P n − 1 , � d i = # { ’s in the i -th row } x 2 x 2 x 1 x 2 x 2 2 x 3 x 1 x 2 x 3 1 x 3 1 x 2 2 Evgeny Smirnov (HSE & Labo Poncelet) Schubert polynomials and combinatorics Moscow, June 4, 2012 5 / 18

Pipe dreams and Schubert polynomials Theorem (S. Fomin, An. Kirillov, 1994) Let w ∈ S n . Then � x d ( P ) , S w ( x 1 , . . . , x n − 1 ) = w ( P )= w where the sum is taken over all reduced pipe dreams P corresponding to w . Example S 1432 ( x 1 , x 2 , x 3 ) = x 2 2 x 3 + x 1 x 2 x 3 + x 2 1 x 3 + x 1 x 2 2 + x 2 1 x 2 . Evgeny Smirnov (HSE & Labo Poncelet) Schubert polynomials and combinatorics Moscow, June 4, 2012 6 / 18

Pipe dreams and Schubert polynomials Theorem (S. Fomin, An. Kirillov, 1994) Let w ∈ S n . Then � x d ( P ) , S w ( x 1 , . . . , x n − 1 ) = w ( P )= w where the sum is taken over all reduced pipe dreams P corresponding to w . Example S 1432 ( x 1 , x 2 , x 3 ) = x 2 2 x 3 + x 1 x 2 x 3 + x 2 1 x 3 + x 1 x 2 2 + x 2 1 x 2 . Evgeny Smirnov (HSE & Labo Poncelet) Schubert polynomials and combinatorics Moscow, June 4, 2012 6 / 18