Grothendieck expansions of symmetric polynomials Jason Bandlow - PowerPoint PPT Presentation

Grothendieck expansions of symmetric polynomials Jason Bandlow (joint work with Jennifer Morse) University of Pennsylvania August 3rd, 2010 – FPSAC San Francisco State University

Outline Symmetric functions Tableaux–Schur expansions Grothendieck functions and their dual basis Main theorem Examples Sketch of proof

The monomial basis The monomial symmetric functions are indexed by partitions λ = ( λ 1 , λ 2 , . . . , λ k ) λ i ≥ λ i +1 � x α m λ = α α a rearrangement of the parts of λ and infinitely many 0’s Example m 2 , 1 =( x 2 1 x 2 + x 1 x 2 2 ) + ( x 1 x 2 3 + x 2 1 x 3 ) + . . . + ( x 2 2 x 3 + x 2 x 2 3 ) + . . .

The complete homogeneous basis The (complete) homogeneous symmetric functions are defined by � h i = m λ λ ⊢ i h λ = h λ 1 h λ 2 . . . h λ k Example h 3 = m 3 + m 2 , 1 + m 1 , 1 , 1 h 4 , 2 , 1 = h 4 h 2 h 1

The Hall inner product Defined by � 1 if λ = µ � h λ , m µ � = 0 otherwise

The Hall inner product Defined by � 1 if λ = µ � h λ , m µ � = 0 otherwise Proposition If { f λ } , { f ∗ λ } and { g λ } , { g ∗ λ } are two pairs of dual bases with � f λ = M λ,µ g µ µ then g ∗ � M λ,µ f ∗ µ = λ λ

Semistandard Young tableaux A left-and-bottom justified, partition-shaped array of numbers, weakly increasing across rows and strictly increasing up columns. Example 7 7 5 6 6 8 3 3 4 7 1 1 2 2 2

Semistandard Young tableaux A left-and-bottom justified, partition-shaped array of numbers, weakly increasing across rows and strictly increasing up columns. Example 7 7 5 6 6 8 3 3 4 7 1 1 2 2 2 Shape: (5 , 4 , 4 , 2) Evaluation: (2 , 3 , 2 , 1 , 1 , 2 , 3 , 1) Reading word: 775668334711222

Knuth equivalence An equivalence relation on words generated by yxz ≡ yzx if x < y ≤ z xzy ≡ zxy if x ≤ y < z Key Fact Every word is Knuth equivalent to the reading word of exactly one tableau. Example � � 3 4 rw = 34123 ≡ 31423 ≡ 31243 ≡ 13243 ≡ 13423 1 2 3

The Schur basis Definition The Schur functions are given by � x ev ( T ) s λ = T ∈ SSYT ( λ )

The Schur basis Definition The Schur functions are given by � x ev ( T ) s λ = T ∈ SSYT ( λ ) Example x 2 x 1 x 2 s 2 , 1 = 1 x 2 + 2 + 2 x 1 x 2 x 3 + · · · 2 2 3 2 1 1 1 2 1 2 1 3 · · ·

The Schur basis Definition The Schur functions are given by � x ev ( T ) s λ = T ∈ SSYT ( λ ) Example x 2 x 1 x 2 s 2 , 1 = 1 x 2 + 2 + 2 x 1 x 2 x 3 + · · · 2 2 3 2 1 1 1 2 1 2 1 3 · · · Fact: Schur functions are a self-dual basis of the symmetric functions.

The Schur basis Using the fact that Schur functions are symmetric, we can rewrite the definition as � s λ = K λ,µ m µ µ where K λ,µ is the number of semistandard tableaux of shape λ and evaluation µ .

The Schur basis Using the fact that Schur functions are symmetric, we can rewrite the definition as � s λ = K λ,µ m µ µ where K λ,µ is the number of semistandard tableaux of shape λ and evaluation µ . Using the proposition about dual bases, we get � h µ = K λ,µ s λ λ

The Schur basis Using the fact that Schur functions are symmetric, we can rewrite the definition as � s λ = K λ,µ m µ µ where K λ,µ is the number of semistandard tableaux of shape λ and evaluation µ . Using the proposition about dual bases, we get � h µ = K λ,µ s λ λ which can be rewritten as � h µ = s sh ( T ) T ∈ T µ where T µ is the set of all semistandard tableaux of evaluation µ .

Tableaux–Schur expansions Elements of a family { f α } of symmetric functions have tableaux–Schur expansions if there exist sets T α of semistandard tableaux and weight functions wt α such that � f α = wt α ( T ) s sh ( T ) T ∈ T α

Tableaux–Schur expansions Elements of a family { f α } of symmetric functions have tableaux–Schur expansions if there exist sets T α of semistandard tableaux and weight functions wt α such that � f α = wt α ( T ) s sh ( T ) T ∈ T α Goal: find appropriate sets and modifications of wt so that � f α = wt α ( S ) G sh ( S ) S ∈ S α � f α = wt α ( R ) g sh ( R ) R ∈ R α where G and g are, respectively, the Grothendieck and dual-Grothendieck functions.

Tableaux–Schur expansions Examples

Tableaux–Schur expansions Examples ◮ h µ = � T ∈ T µ s sh ( T )

Tableaux–Schur expansions Examples ◮ h µ = � T ∈ T µ s sh ( T ) T ∈ T µ t ch ( T ) s sh ( T ) ◮ H µ [ X ; t ] = �

Tableaux–Schur expansions Examples ◮ h µ = � T ∈ T µ s sh ( T ) T ∈ T µ t ch ( T ) s sh ( T ) ◮ H µ [ X ; t ] = � ◮ s λ s µ = � T ∈ T λ,µ s sh ( T )

Tableaux–Schur expansions Examples ◮ h µ = � T ∈ T µ s sh ( T ) T ∈ T µ t ch ( T ) s sh ( T ) ◮ H µ [ X ; t ] = � ◮ s λ s µ = � T ∈ T λ,µ s sh ( T ) ◮ F σ = � T ∈ T σ s sh ( T )

Tableaux–Schur expansions Examples ◮ h µ = � T ∈ T µ s sh ( T ) T ∈ T µ t ch ( T ) s sh ( T ) ◮ H µ [ X ; t ] = � ◮ s λ s µ = � T ∈ T λ,µ s sh ( T ) ◮ F σ = � T ∈ T σ s sh ( T ) ◮ A ( k ) T ∈ T k ,µ t ch ( T ) s sh ( T ) µ [ X ; t ] = �

Tableaux–Schur expansions Examples ◮ h µ = � T ∈ T µ s sh ( T ) T ∈ T µ t ch ( T ) s sh ( T ) ◮ H µ [ X ; t ] = � ◮ s λ s µ = � T ∈ T λ,µ s sh ( T ) ◮ F σ = � T ∈ T σ s sh ( T ) ◮ A ( k ) T ∈ T k ,µ t ch ( T ) s sh ( T ) µ [ X ; t ] = � T ∈ T (1 n ) t ch µ ( T ) s sh ( T ) ◮ H µ [ X ; 1 , t ] = �

Tableaux–Schur expansions Examples ◮ h µ = � T ∈ T µ s sh ( T ) T ∈ T µ t ch ( T ) s sh ( T ) ◮ H µ [ X ; t ] = � ◮ s λ s µ = � T ∈ T λ,µ s sh ( T ) ◮ F σ = � T ∈ T σ s sh ( T ) ◮ A ( k ) T ∈ T k ,µ t ch ( T ) s sh ( T ) µ [ X ; t ] = � T ∈ T (1 n ) t ch µ ( T ) s sh ( T ) ◮ H µ [ X ; 1 , t ] = � ◮ H µ [ X ; q , t ] ? T ∈ T (1 n ) q a µ ( T ) t b µ ( T ) s sh ( T ) = �

Grothendieck polynomials ◮ Introduced by Lascoux-Sch¨ utzenberger (1982) ◮ Represent K -theory classes of structure sheaves of Schubert varieties in GL n ◮ Given by elements of Z [[ x 1 , . . . , x n ]] ◮ Analogous to Schubert polynomials ◮ Sign-alternating by degree, equal to Schubert polynomial in bottom degree

Stable Grassmannian Grothendieck functions Stable: ◮ Due to Fomin-Kirillov ◮ Limit as number of variables → ∞

Stable Grassmannian Grothendieck functions Stable: ◮ Due to Fomin-Kirillov ◮ Limit as number of variables → ∞ Grassmannian: ◮ Indexed by partitions ◮ Power series of symmetric functions ◮ Sign-alternating by degree in the Schur basis ◮ Analog of Schur functions

Grassmannian Grothendieck functions Theorem (Buch) � ε ( S ) x ev ( S ) G λ = S ∈ SVT ( λ ) where SVT ( λ ) is the set of set-valued tableaux of shape λ .

Grassmannian Grothendieck functions Theorem (Buch) � ε ( S ) x ev ( S ) G λ = S ∈ SVT ( λ ) where SVT ( λ ) is the set of set-valued tableaux of shape λ . Example G 1 = s 1 − s 1 , 1 + s 1 , 1 , 1 . . . 123 1 12 · · · 234 23 · · · . . . . . .

Set-valued tableaux Example 234 45 1 12 3

Set-valued tableaux Example 234 45 1 12 3 shape ( S ) = (3 , 2) evaluation ( S ) = (2 , 2 , 2 , 2 , 1) ε ( S ) = ( − 1) | ev ( S ) | + | sh ( S ) | = 1

Set-valued tableaux Example 234 45 1 12 3 shape ( S ) = (3 , 2) evaluation ( S ) = (2 , 2 , 2 , 2 , 1) ε ( S ) = ( − 1) | ev ( S ) | + | sh ( S ) | = 1 From � ε ( S ) x ev ( S ) G λ = S ∈ SVT ( λ ) we see G λ = s λ ± higher degree terms

Set-valued tableaux We introduce the reading word of a set-valued tableau, defined by: ◮ Proceed row by row, from top to bottom ◮ In each row, first ignore the smallest element of each cell. Then read the remaining elements from right to left , and from largest to smallest within each cell. ◮ Read the smallest elements of each cell from left to right. Example 234 45 1 12 3 has reading word: 543242113.

Dual Grothendieck functions We denote the dual basis to the Grothendieck by g Theorem (Lam, Pylyavskyy) � x ev ( R ) g λ = R ∈ RPP ( λ ) where RPP ( λ ) is the set of reverse plane partitions of shape λ .

Dual Grothendieck functions We denote the dual basis to the Grothendieck by g Theorem (Lam, Pylyavskyy) � x ev ( R ) g λ = R ∈ RPP ( λ ) where RPP ( λ ) is the set of reverse plane partitions of shape λ . Example g 2 , 1 = s 2 , 1 + s 2 2 1 1 1 1 1 2 1 1 3 1 2 . . . . . .

Reverse plane partitions Example 2 3 2 2 4 1 2 2 1 1 1 1 shape ( R ) = (4 , 3 , 3 , 2) evaluation ( R ) = (4 , 3 , 1 , 1)

Reverse plane partitions Example 2 3 2 2 4 1 2 2 1 1 1 1 shape ( R ) = (4 , 3 , 3 , 2) evaluation ( R ) = (4 , 3 , 1 , 1) From � x ev ( R ) g λ = R ∈ RPP ( λ ) we see g λ = s λ ± lower degree terms

Dual Grothendieck functions We define the reading word of a reverse plane partition to be a subsequence of the usual reading word, where we only take the bottommost occurence of every letter in each column. Example 2 3 2 2 4 1 2 2 1 1 1 1 has reading word 324221111.