Truncated Stanley symmetric functions and amplituhedron cells - PowerPoint PPT Presentation

Truncated Stanley symmetric functions and amplituhedron cells Thomas Lam June 2014

Reduced words The symmetric group S n is generated by s 1 , s 2 , . . . , s n − 1 with relations s 2 i = 1 s i s j = s j s i if | i − j | ≥ 2 s i s i +1 s i = s i +1 s i s i +1 A reduced word i for w ∈ S n is a sequence i = i 1 i 2 · · · i ℓ ∈ { 1 , 2 , . . . , n − 1 } ℓ such that w = s i 1 s i 2 · · · s i ℓ and ℓ = ℓ ( w ) is minimal.

Stanley symmetric functions Let R ( w ) denote the set of reduced words of w ∈ S n . Definition (Stanley symmetric function) � � F w ( x 1 , x 2 , . . . ) := x a 1 x a 2 · · · x a ℓ 1 ≤ a 1 ≤ a 2 ≤···≤ a ℓ i = i 1 i 2 ··· i ℓ ∈ R ( w ) i j < i j +1 = ⇒ a j +1 > a j The coefficient of x 1 x 2 · · · x ℓ in F w is | R ( w ) | .

Stanley symmetric functions Let R ( w ) denote the set of reduced words of w ∈ S n . Definition (Stanley symmetric function) � � F w ( x 1 , x 2 , . . . ) := x a 1 x a 2 · · · x a ℓ 1 ≤ a 1 ≤ a 2 ≤···≤ a ℓ i = i 1 i 2 ··· i ℓ ∈ R ( w ) i j < i j +1 = ⇒ a j +1 > a j The coefficient of x 1 x 2 · · · x ℓ in F w is | R ( w ) | . Example n = 3 and w = w 0 = 321. We have R ( w ) = { 121 , 212 } , so F w = ( x 1 x 2 2 + x 1 x 2 x 3 + · · · ) + ( x 2 1 x 2 + x 1 x 2 x 3 + · · · ) = m 21 + 2 m 111 = s 21

Symmetry and Schur-positivity Theorem (Stanley) F w is a symmetric function. Theorem (Stanley) Let w 0 = n ( n − 1) · · · 1 be the longest permutation in S n . Then � n � ! 2 | R ( w 0 ) | = 1 n − 1 3 n − 2 5 n − 3 · · · (2 n − 3) 1 Theorem (Edelman-Greene, Lascoux-Sch¨ utzenberger) F w is Schur-positive.

Affine Stanley symmetric functions The affine symmetric group ˜ S n is generated by s 0 , s 1 , s 2 , . . . , s m − 1 with relations s 2 i = 1 s i s j = s j s i if | i − j | ≥ 2 s i s i +1 s i = s i +1 s i s i +1 where indices are taken modulo n . The affine Stanley symmetric function ˜ F w is defined by introducing a notion of cyclically decreasing factorizations for ˜ S n . Theorem (L.) 1 ˜ F w is a symmetric function. 2 ˜ F w is “affine Schur”-positive.

Postnikov’s TNN Grassmannian Take integers 1 ≤ k ≤ n . The Grassmannian Gr ( k , n ) is the set of k -dimensional subspaces of C n .   · · · a 11 a 12 a 1 n a 21 a 22 · · · a 2 n   X =  . . .  ... . . .   . . .   · · · a k 1 a k 2 a kn

Postnikov’s TNN Grassmannian Take integers 1 ≤ k ≤ n . The Grassmannian Gr ( k , n ) is the set of k -dimensional subspaces of C n .   · · · a 11 a 12 a 1 n a 21 a 22 · · · a 2 n   X =  . . .  ... . . .   . . .   · · · a k 1 a k 2 a kn Definition (Totally nonnegative Grassmannian) The totally nonnegative Grassmannian Gr ( k , n ) ≥ 0 is the locus in the real Grassmannian representable by X such that all k × k minors are nonnegative. Also studied by Lusztig, with a different definition.

Gr ( k , n ) ≥ 0 is like a simplex Let k = 1. Then Gr (1 , n ) = P n − 1 and Gr (1 , n ) ≥ 0 = { ( a 1 , a 2 , . . . , a n ) � = 0 | a i ∈ R ≥ 0 } modulo scaling by R > 0 which can be identified with the simplex ∆ n − 1 := { ( a 1 , a 2 , . . . , a n ) | a i ∈ [0 , 1] and a 1 + a 2 + · · · + a n = 1 } .

Polytopes and amplituhedra A convex polytope in R d with vertices v 1 , v 2 , . . . , v n is the image of a simplex ∆ n = conv ( e 1 , e 2 , . . . , e n ) ⊂ R n +1 under a projection map Z : R n → R d where Z ( e i ) = v i . Definition (Arkani-Hamed and Trnka’s amplituhedron) An amplituhedron A ( k , n , d ) in Gr ( k , d ) is the image of Gr ( k , n ) ≥ 0 under a (positive) projection map Z : R n → R d inducing Z Gr : Gr ( k , n ) → Gr ( k , d ). (Caution: Z Gr is not defined everywhere.)

Scattering amplitudes Arkani-Hamed and Trnka assert that the scattering amplitude (at tree level) in N = 4 super Yang-Mills is the integral of a“volume form” ω SYM of an amplituhedron (for d = k + 4), and that this form can be calculated by studying “triangulations” of A ( k , n , d ): � ω SYM = ω Y f cells Y f in a triangulation of A ( k , n , d ) where ω Y f ’s can be considered known. p 3 p 2 p 1 p n � Scattering amplitude = A ( p 1 , p 2 , . . . , p n ) “=” ω SYM

Triangulating a quadrilateral Cells of a triangulations of a polytope Z (∆ n ) can be obtained by looking at the images Z ( F ) of lower-dimensional faces F of ∆ n . R 3 or P 3 ( R ) R 2 or P 2 ( R )

Positroid cells Postnikov described the facial structure of Gr ( k , n ) ≥ 0 : � Gr ( k , n ) ≥ 0 = (Π f ) > 0 f ∈ Bound ( k , n ) where (Π f ) > 0 ≃ R d > 0 are called positroid cells and Bound ( k , n ) ⊂ ˜ S ′ n is the set of bounded affine permutations, certain elements in the extended affine symmetric group ˜ S ′ n . Postnikov gave many objects to index these strata: Grassmann necklaces, decorated permutations, Le-diagrams,...

Partial order The closure partial order for positroid cells was described by Postnikov and Rietsch. Theorem (Knutson-L.-Speyer, after Postnikov and Rietsch) � (Π f ) > 0 = (Π g ) > 0 g ≥ f where ≥ is Bruhat order for the affine symmetric group restricted to Bound ( k , n ) . For k = 1, the set Bound (1 , n ) is in bijection with nonempty subsets of [ n ], which index faces of the simplex. The partial order is simply containment of subsets.

Triangulations of the amplituhedron Define the amplituhedron cell ( Y f ) > 0 := Z Gr ((Π f ) > 0 ) . The map Z Gr exhibits some features that are not present in the polytope case: 1 Even when Z : R n → R d is generic, the image Z Gr ((Π f ) > 0 ) may not have the expected dimension. 2 Even in the dimension-preserving case, the map Z Gr : (Π f ) > 0 �− → ( Y f ) > 0 can have degree greater than one.

Triangulations of the amplituhedron Define the amplituhedron cell ( Y f ) > 0 := Z Gr ((Π f ) > 0 ) . The map Z Gr exhibits some features that are not present in the polytope case: 1 Even when Z : R n → R d is generic, the image Z Gr ((Π f ) > 0 ) may not have the expected dimension. 2 Even in the dimension-preserving case, the map Z Gr : (Π f ) > 0 �− → ( Y f ) > 0 can have degree greater than one. These questions bring us into the realm of Schubert calculus!

Cohomology of the Grassmannian The cohomology ring H ∗ ( Gr ( k , n )) can be identified with a quotient of the ring of symmetric functions. � H ∗ ( Gr ( k , n )) = Z · s λ . λ ⊂ ( n − k ) k Each irreducible subvariety X ⊂ Gr ( k , n ) has a cohomology class [ X ]. The Schur function s λ is the cohomology classes of the Schubert variety X λ ⊂ Gr ( k , n ).

Cohomology of the Grassmannian The cohomology ring H ∗ ( Gr ( k , n )) can be identified with a quotient of the ring of symmetric functions. � H ∗ ( Gr ( k , n )) = Z · s λ . λ ⊂ ( n − k ) k Each irreducible subvariety X ⊂ Gr ( k , n ) has a cohomology class [ X ]. The Schur function s λ is the cohomology classes of the Schubert variety X λ ⊂ Gr ( k , n ). Cohomology classes know about: 1 dimension 2 degree (expected number of points of intersection with a generic hyperspace) When k = 1, the cohomology class [ L ] of a linear subspace L ⊂ Gr (1 , n ) = P n − 1 is simply its dimension.

Cohomology class of a positroid variety The positroid variety Π f is the Zariski-closure of (Π f ) > 0 in the (complex) Grassmannian Gr ( k , n ). Each Π f is an intersection of rotated Schubert varieties: Π f = X I 1 ∩ χ ( X I 2 ) ∩ · · · ∩ χ n − 1 ( X I n ) where χ denotes rotation. Theorem (Knutson-L.-Speyer) The cohomology class [Π f ] ∈ H ∗ ( Gr ( k , n )) can be identified with an affine Stanley symmetric function ˜ F f .

Cohomology class of a positroid variety The positroid variety Π f is the Zariski-closure of (Π f ) > 0 in the (complex) Grassmannian Gr ( k , n ). Each Π f is an intersection of rotated Schubert varieties: Π f = X I 1 ∩ χ ( X I 2 ) ∩ · · · ∩ χ n − 1 ( X I n ) where χ denotes rotation. Theorem (Knutson-L.-Speyer) The cohomology class [Π f ] ∈ H ∗ ( Gr ( k , n )) can be identified with an affine Stanley symmetric function ˜ F f . All faces of ∆ n of the same dimension “look” the same. The faces of Gr ( k , n ) ≥ 0 of the same dimension are abstractly homeomorphic, but don’t “look” the same when considered as embedded subsets of the Grassmannian.

Truncation Suppose � a λ s λ ∈ H ∗ ( Gr ( k , n )) . G = λ ⊂ ( n − k ) k Define the truncation � a µ + s µ ∈ H ∗ ( Gr ( k , d )) τ d ( G ) = µ ⊂ ( d − k ) k where µ + is obtained from µ by adding n − d columns of length k to the left of µ µ = µ + =

An example Example Let k = 2 , n = 8 , d = 6. For w = s 1 s 3 s 5 s 7 we have F w = ( x 1 + x 2 + · · · ) 4 = s + 3 s + 2 s + 3 s + s and τ d ( F w ) = 2 . This is the smallest “physical” example, where the amplituhedron cell is mapped onto with degree 2.