Permutohedra, Associahedra, and Beyond or Three Formulas for - PowerPoint PPT Presentation

Permutohedra, Associahedra, and Beyond or Three Formulas for Volumes of Permutohedra by Alex Postnikov Massachusetts Institute of Technology June 26, 2004 on the occasion of Richard P. Stanley’s Birthday 1

Permutohedron P n ( x 1 , . . . , x n +1 ) := ConvexHull(( x w (1) , . . . , x w ( n +1) ) | w ∈ S n +1 ) This is a convex n -dimensional polytope in H ⊂ R n +1 . Example: n = 2 (type A 2 ) ( x 1 , x 2 , x 3 ) P 2 ( x 1 , x 2 , x 3 ) = More generaly, for a Weyl group W , P W ( x ) := ConvexHull( w ( x ) | w ∈ W ). 2

Question: What is the volume V n := Vol P n ? Volume form is normalized so that the volume of a parallelepiped formed by generators of the lattice Z n +1 ∩ H is 1. Question: What is the number of lattice points N n := P n ∩ Z n +1 ? We will see that V n and N n are polynomials in x 1 , . . . , x n +1 of degree n . The polynomial V n is the top homogeneous part of N n . The Ehrhart polynomial of P n is E ( t ) = N n ( tx 1 , . . . , tx n ), and V n is its top coefficient. We will give 3 totally different formulas for these polynomials. 3

Special Case: P n ( n + 1 , n, . . . , 1) = ConvexHull(( w (1) , ..., w ( n + 1)) | w ∈ S n +1 ) is the most symmetric permutohedron. regular hexagon subdivided into 3 rhombi It is a zonotope = Minkowsky sum of line intervals. Well-known facts: ➠ V n ( n + 1 , . . . , 1) = ( n + 1) n − 1 is the number of trees on n + 1 labelled vertices. P n ( n + 1 , . . . , 1) can be subdivided into parallelepipeds of unit volume associated with trees. This works for any zonotope. ➠ N n ( n + 1 , . . . , 1) is the number of forests on n + 1 labelled vertices. 4

First Formula Fix any distinct numbers λ 1 , . . . , λ n +1 such that λ 1 + · · · + λ n +1 = 0. ( λ w (1) x 1 + · · · + λ w ( n +1) x n +1 ) n V n ( x 1 , . . . , x n +1 ) = 1 � n ! ( λ w (1) − λ w (2) )( λ w (2) − λ w (3) ) · · · ( λ w ( n ) − λ w ( n +1) ) w ∈ S n +1 Notice that the symmetrization in RHS does not depends on λ i ’s. Use Khovansky-Puchlikov’s method: Idea of Proof ➠ Instead of just counting the number of lattice points in P , define [ P ] = sum of formal exponents e a over lattice points a ∈ P ∩ Z n . ➠ Now we can work with unbounded polytopes. For example, for a simplicial cone C , the sum [ C ] is given by a simple rational expression. ➠ Any polytope P can be explicitly presented as an alternating sum of simplicial cones: [ P ] = [ C 1 ] ± [ C 2 ] ± · · · . Applying this procedure to the permutohedron, we obtain . . . 5

Let α 1 , . . . , α n be a system of simple roots for Weyl group W , and let L be the root lattice. Theorem: For a dominant weight µ , e w ( µ ) e a = � � [ P W ( µ )] := (1 − e − w ( α 1 ) ) · · · (1 − e − w ( α n ) ) w ∈ W a ∈ P W ( µ ) ∩ ( L + µ ) Compare this with Weyl’s character formula! Note: LHS is obtained from the character ch V µ of the irrep V µ by replacing all nonzero coefficients with 1. In type A , ch V µ = Schur polynomial s µ . From this expression, one can deduce the First Formula and also its gen- eralizations to other Weyl groups. 6

Second Formula Let us use the coordinates y 1 , . . . , y n +1 related x 1 , . . . , x n +1 by  y 1 = − x 1      y 2 = − x 2 + x 1     y 3 = − x 3 + 2 x 2 − x 1   · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·      � n � n � n � � �  y n +1 = − x n + x n − 1 − · · · ± x 1  0 1 n and write V n = Vol P n ( x 1 , . . . , x n +1 ) as a polynomial in y 1 , . . . , y n +1 . Examples: V 1 = Vol ([( x 1 , x 2 ) , ( x 2 , x 1 )]) = x 1 − x 2 = y 2 2 + 3 y 2 y 3 + 1 V 2 = · · · = 3 y 2 2 y 2 3 7

Theorem: V n ( x 1 , . . . , x n +1 ) = 1 � y | S 1 | · · · y | S n | , n ! ( S 1 ,...,S n ) where the sum is over ordered collections of subsets S 1 , . . . , S n ⊂ [ n + 1] such that either of the following equivalent conditions is satisfied: ➠ For any distinct i 1 , . . . , i k , we have | S i 1 ∪ · · · ∪ S i k | ≥ k + 1 (cf. Hall’s Marriage Theorem) ➠ For any j ∈ [ n + 1], there is a system of distinct representatives in S 1 , . . . , S n that avoids j . Thus n ! V n is a polynomial in y 2 , . . . , y n +1 with positive integer coefficients. 8

This formula can be extended to generalized permutohedra a generalized permutohedron Generalized permutohedra are obtained from usual permutohedra by mov- ing faces while preserving all angles. this is also a generalized permutohedron 9

Generalized Permutohedra Coordinate simplices in R n +1 : ∆ I = ConvexHull( e i | i ∈ I ), for I ⊆ [ n +1]. Let Y = { Y I } be the collection of variables Y I ≥ 0 associated with all subsets I ⊂ [ n + 1]. Define � P n ( Y ) := Y I · ∆ I (Minkowsky sum) I ⊂ [ n +1] Its combinatorial type depends only on the set of I ’s for which Y I � = 0. Examples: ➠ If Y I = y | I | , then P n ( Y ) is a usual permutohedron. ➠ If Y I � = 0 iff I is a consecutive interval, then P n ( Y ) is an associahedron. ➠ If Y I � = 0 iff I is a cyclic interval, then P n ( Y ) is a cyclohedron. ➠ If Y I � = 0 iff I is a connected set in Dynkin diagram, then P n ( Y ) is a generalized associahedron related to DeConcini-Procesi’s work. (Do not confuse with Fomin-Zelevinsky’s generalized associahedra!) ➠ If Y I � = 0 iff I is an initial interval { 1 , . . . , i } , then P n ( Y ) is the Stanley-Pitman polytope. 10

Theorem: The volume of the generalized permutohedron is given by Vol P n ( Y ) = 1 � Y S 1 · · · Y S n , n ! ( S 1 ,...,S n ) where S 1 , . . . , S n satisfy the same condition. Theorem: The # of lattice points in the generalized permutohedron is P n ( Y ) ∩ Z n +1 = 1 � { Y S 1 · · · Y S n } , n ! ( S 1 ,...,S n ) �� � , where Y { a } = Y ( Y +1) · · · ( Y + a − 1) . := ( Y [ n +1] +1) { a [ n +1] } � Y { a I } Y a I I I I I � =[ n +1] This extends a formula from [Stanley-Pitman] for the volume of their polytope. In this case, the above summation is over parking functions. 11

We also have a combinatorial description of face structure of generalized permutohedra in terms of nested collections of subsets in [ n + 1]. This is related to DeConcini-Procesi’s wonderful arrangements. Not enough time for this now. The most interesting part of the talk is . . . 12

Third Formula Let use the coordinates z 1 , . . . , z n related to x 1 , . . . , x n +1 by z 1 = x 1 − x 2 , z 2 = x 2 − x 3 , · · · , z n = x n − x n +1 These coordinates are canonically defined for an arbitrary Weyl group. Then the permutohedron P n is the Minkowsky sum P n = z 1 ∆ 1 n + z 2 ∆ 2 n + · · · + z n ∆ nn of hypersimplices ∆ kn = P n (1 , . . . , 1 , 0 , . . . , 0) (with k 1’s). + = 13

This implies z c 1 c 1 ! · · · z c n � 1 n Vol P n = A c 1 ,...,c n c n ! , c 1 ,...,c n where the sum is over c 1 , . . . , c n ≥ 0, c 1 + · · · + c n = n , and A c 1 ,...,c n = MixedVolume(∆ c 1 1 n , . . . , ∆ c n nn ) ∈ Z > 0 In particular, n ! V n is a positive integer polynomial in z 1 , . . . , z n . Let us call the integers A c 1 ,...,c n the Mixed Eulerian numbers. Examples: V 1 = 1 z 1 V 2 = 1 z 2 2 + 2 z 1 z 2 + 1 z 2 1 2 2 V 3 = 1 z 3 3! + 2 z 2 2 + 4 z 3 3! + 3 z 2 z 2 2 z 2 + 4 z 1 2 z 3 + 6 z 1 z 2 z 3 + 1 1 2 1 +4 z 2 z 2 z 2 2 + 1 z 3 2 z 3 + 3 z 1 2 + 2 z 2 2 3 3 3 3! (The mixed Eulerian numbers are marked in red.) 14

Properties of Mixed Eulerian numbers: ➠ A c 1 ,...,c n are positive integers defined for c 1 , . . . , c n ≥ 0, c 1 + · · · + c n = n . c 1 ! ··· c n ! A c 1 ,...,c n = ( n + 1) n − 1 . 1 ➠ � ➠ A 0 ,..., 0 ,n, 0 ,..., 0 ( n is in k -th position) is the usual Eulerian number A kn = # permutations in S n with k descents = n ! Vol ∆ kn . ➠ A 1 ,..., 1 = n ! � n � ➠ A k, 0 ,..., 0 ,n − k = k ➠ A c 1 ,...,c n = 1 c 1 2 c 2 · · · n c n if c 1 + · · · + c i ≥ i , for i = 1 , . . . , n . � 2 n 1 � There are exactly C n = such sequences ( c 1 , . . . , c n ). n +1 n When I showed these numbers to Richard Stanley, he conjectured that ➠ � A c 1 ,...,c n = n ! C n . Moreover, he conjectured that . . . 15

One can subdivide all sequences ( c 1 , . . . , c n ) into C n classes such that the sum of mixed Eulerian numbers for each class is n !. For example, A 1 ,..., 1 = n ! and A n, 0 ,..., 0 + A 0 ,n, 0 ,..., 0 + A 0 , 0 ,n,..., 0 + · · · + A 0 ,..., 0 ,n = n !, because the sum of Eulerian numbers � k A kn is n !. Let us write ( c 1 , . . . , c n ) ∼ ( c ′ 1 , . . . , c ′ n ) iff ( c 1 , . . . , c n , 0) is a cyclic shift of ( c ′ 1 , . . . , c ′ n , 0). Stanley conjectured that, for fixed ( c 1 , . . . , c n ), we have � n = n ! A c ′ 1 ,...,c ′ ( c ′ n ) ∼ ( c 1 ,...,c n ) 1 ,...,c ′ Exercise: Check that there are exactly C n equivalence classes of sequences. Every equivalence class contains exactly one sequence ( c 1 , . . . , c n ) such that c 1 + · · · + c i ≥ i , for i = 1 , . . . , n . (For this sequence, A c 1 ,...,c n = 1 c 1 · · · n c n .) These conjectures follow from . . . 16