The symmetry and Schur expansion of dual stable Grothendieck - PowerPoint PPT Presentation

The symmetry and Schur expansion of dual stable Grothendieck polynomials Pavel Galashin MIT October 7, 2015 Joint work with Gaku Liu and Darij Grinberg Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 1 / 25

Part 1: Symmetry

Young diagrams, skew-shapes, SSYT Skew shapes λ = ( 4 , 4 , 3 ) λ / µ = ( 4 , 4 , 3 ) / ( 2 , 1 ) Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 3 / 25

Young diagrams, skew-shapes, SSYT Skew shapes λ = ( 4 , 4 , 3 ) λ / µ = ( 4 , 4 , 3 ) / ( 2 , 1 ) Semi-standard Young tableau (SSYT) 1 3 1 4 2 2 4 2 2 4 2 6 6 7 6 6 Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 3 / 25

Reverse plane partitions (RPP) 1 3 1 3 1 2 3 1 2 2 2 2 2 3 2 2 Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 4 / 25

Reverse plane partitions (RPP) 1 3 1 3 1 2 3 1 2 2 2 2 2 3 2 2 SSYT is a special case of RPP! Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 4 / 25

Skew-Schur polynomials Definition If T is an SSYT then w ( T ) : = ( # T − 1 ( 1 ) , # T − 1 ( 2 ) , . . . , # T − 1 ( m )) , where # T − 1 ( i ) = [ the number of entries in T equal to i ] . Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 5 / 25

Skew-Schur polynomials Definition If T is an SSYT then w ( T ) : = ( # T − 1 ( 1 ) , # T − 1 ( 2 ) , . . . , # T − 1 ( m )) , where # T − 1 ( i ) = [ the number of entries in T equal to i ] . Example 1 3 x w ( T ) = x 1 1 x 3 2 x 1 3 x 1 4 x 0 5 x 2 T = , w ( T ) = ( 1 , 3 , 1 , 1 , 0 , 2 ) , 6 . 2 2 4 2 6 6 Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 5 / 25

Skew-Schur polynomials Definition If T is an SSYT then w ( T ) : = ( # T − 1 ( 1 ) , # T − 1 ( 2 ) , . . . , # T − 1 ( m )) , where # T − 1 ( i ) = [ the number of entries in T equal to i ] . Example 1 3 x w ( T ) = x 1 1 x 3 2 x 1 3 x 1 4 x 0 5 x 2 T = , w ( T ) = ( 1 , 3 , 1 , 1 , 0 , 2 ) , 6 . 2 2 4 2 6 6 Definition x w ( T ) . ∑ s λ / µ ( x 1 , . . . , x m ) = T is a SSYT of shape λ / µ with entries ≤ m Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 5 / 25

Example Example Let m = 2, λ = ( 3 , 2 ) , µ = ( 1 ) . Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 6 / 25

Example Example Let m = 2, λ = ( 3 , 2 ) , µ = ( 1 ) . 1 1 1 1 2 2 2 2 Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 6 / 25

Example Example Let m = 2, λ = ( 3 , 2 ) , µ = ( 1 ) . 1 1 1 2 1 1 1 2 1 2 1 2 2 2 2 2 Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 6 / 25

Example Example Let m = 2, λ = ( 3 , 2 ) , µ = ( 1 ) . 1 1 1 2 1 1 1 2 1 2 1 2 2 2 2 2 w ( T ) = ( 3 , 1 ) ( 2 , 2 ) ( 2 , 2 ) ( 1 , 3 ) Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 6 / 25

Example Example Let m = 2, λ = ( 3 , 2 ) , µ = ( 1 ) . 1 1 1 2 1 1 1 2 1 2 1 2 2 2 2 2 w ( T ) = ( 3 , 1 ) ( 2 , 2 ) ( 2 , 2 ) ( 1 , 3 ) x 3 + x 2 1 x 2 + x 2 1 x 2 + x 1 x 3 s λ / µ ( x 1 , x 2 ) = 2 . 1 x 2 2 2 Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 6 / 25

Problem Example Let m = 2, λ = ( 3 , 2 ) , µ = ( 1 ) . 1 1 1 2 1 1 1 2 2 2 1 1 2 2 2 “ w ( R ) = ” ( 3 , 0 ) ( 2 , 1 ) ( 2 , 1 ) ( 1 , 2 ) ( 0 , 3 ) Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 7 / 25

Dual stable Grothendieck polynomials Definition If R is an RPP then w ( R ) : = ( w 1 ( R ) , w 2 ( R ) , . . . , w m ( R )) , where w i ( R ) = [ the number of columns in R containing i ] . Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 8 / 25

Dual stable Grothendieck polynomials Definition If R is an RPP then w ( R ) : = ( w 1 ( R ) , w 2 ( R ) , . . . , w m ( R )) , where w i ( R ) = [ the number of columns in R containing i ] . Definition x w ( R ) . ∑ g λ / µ ( x 1 , . . . , x m ) = R is a RPP of shape λ / µ with entries ≤ m Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 8 / 25

Dual stable Grothendieck polynomials Definition If R is an RPP then w ( R ) : = ( w 1 ( R ) , w 2 ( R ) , . . . , w m ( R )) , where w i ( R ) = [ the number of columns in R containing i ] . Definition x w ( R ) . ∑ g λ / µ ( x 1 , . . . , x m ) = R is a RPP of shape λ / µ with entries ≤ m Example 1 1 1 2 1 1 1 2 2 2 1 1 2 2 2 w ( R ) = ( 2 , 0 ) ( 1 , 1 ) ( 2 , 1 ) ( 1 , 2 ) ( 0 , 2 ) Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 8 / 25

Properties of g λ / µ Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 9 / 25

Properties of g λ / µ “represent the classes in K-homology of the ideal sheaves of the boundaries of Schubert varieties” (see [Lam, Pylyavskyy (2007)]); Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 9 / 25

Properties of g λ / µ “represent the classes in K-homology of the ideal sheaves of the boundaries of Schubert varieties” (see [Lam, Pylyavskyy (2007)]); SSYT ( λ / µ , ≤ m ) ⊂ RPP ( λ / µ , ≤ m ) and the top-degree homogeneous component of g λ / µ is s λ / µ ; Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 9 / 25

Properties of g λ / µ “represent the classes in K-homology of the ideal sheaves of the boundaries of Schubert varieties” (see [Lam, Pylyavskyy (2007)]); SSYT ( λ / µ , ≤ m ) ⊂ RPP ( λ / µ , ≤ m ) and the top-degree homogeneous component of g λ / µ is s λ / µ ; g λ / µ are symmetric (see [Lam, Pylyavskyy (2007)]); Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 9 / 25

Properties of g λ / µ “represent the classes in K-homology of the ideal sheaves of the boundaries of Schubert varieties” (see [Lam, Pylyavskyy (2007)]); SSYT ( λ / µ , ≤ m ) ⊂ RPP ( λ / µ , ≤ m ) and the top-degree homogeneous component of g λ / µ is s λ / µ ; g λ / µ are symmetric (see [Lam, Pylyavskyy (2007)]); there exist involutions B i : RPP ( λ / µ , ≤ m ) → RPP ( λ / µ , ≤ m ) such that w ( B i ( R )) = s i w ( R ) ; Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 9 / 25

Properties of g λ / µ “represent the classes in K-homology of the ideal sheaves of the boundaries of Schubert varieties” (see [Lam, Pylyavskyy (2007)]); SSYT ( λ / µ , ≤ m ) ⊂ RPP ( λ / µ , ≤ m ) and the top-degree homogeneous component of g λ / µ is s λ / µ ; g λ / µ are symmetric (see [Lam, Pylyavskyy (2007)]); there exist involutions B i : RPP ( λ / µ , ≤ m ) → RPP ( λ / µ , ≤ m ) such that w ( B i ( R )) = s i w ( R ) ; B i restricted to SSYT ( λ / µ , ≤ m ) are classical Bender-Knuth involutions. Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 9 / 25

s λ / µ and g λ / µ are symmetric! Bender-Knuth involutions Want to construct B i : RPP ( λ / µ , ≤ m ) → RPP ( λ / µ , ≤ m ) . Note that it is enough to consider the case i = 1 , m = 2: Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 10 / 25

s λ / µ and g λ / µ are symmetric! Bender-Knuth involutions Want to construct B i : RPP ( λ / µ , ≤ m ) → RPP ( λ / µ , ≤ m ) . Note that it is enough to consider the case i = 1 , m = 2: Reduction to the case m = 2 Let i = 5. 1 5 5 2 6 7 1 3 3 7 8 1 1 5 6 6 8 9 5 5 6 6 9 6 7 7 8 Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 10 / 25

The symmetry and Schur expansion of dual stable Grothendieck - PowerPoint PPT Presentation

The symmetry and Schur expansion of dual stable Grothendieck polynomials Pavel Galashin MIT October 7, 2015 Joint work with Gaku Liu and Darij Grinberg Pavel Galashin (MIT) Dual stable Grothendieck polynomials October 7, 2015 1 / 25 Part

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