brick polytopes, lattice quotients & hopf algebras - LIX-polytechnique

BRICK POLYTOPES, LATTICE QUOTIENTS & HOPF ALGEBRAS 1 2 3 4

4321 4312

3421

4132

2431 4213

1432

2413

1342

1423

++ −++

1 2 3 4 1 2 3 4

2341

1 2 3 4 1 2 3 4

1 2 3 4

3214 3124 2314

1243

1324 2134 1234

+− ++−

++ −+−

++ +−+

1 2 3 4

1 2 3 4

1 2 3 4

+− +−+

−+ −−+

1 2 3 4

1 2 3 4 1 2 3 4 1 2 3 4

+− +−−

−− −+−

−− +−+

−− −−+

+− −+−

−+ −+−

1 2 3 4

1 2 3 4

1 2 3 4

−+ −++ −+ +−+

1 2 3 4

2143

++ ++−

1 2 3 4

1 2 3 4

3142

4123

++ +++

1 2 3 4

3241

´ cole Polytechnique CNRS & E

1 2 3 4

1 2 3 4

3412

4231

Vincent PILAUD

−− +−− −− −−−

MOTIVATION Combinatorics

permutations

binary trees

binary sequences +++

4321 3421 3241 3214

4231

2431

2341 2314

4312

3412 3142

3124

4213 2413

2143

2134

4132 4123

1342

1324

−++

+−+

++−

−−+

−+−

+−−

1432

1423

1243

−−−

Algebra

1234

Malvenuto-Reutenauer algebra Loday-Ronco algebra FQSym = vect h Fτ | τ ∈ S i PBT = vect h PT | T ∈ BT i P P 0 0 Fτ · Fτ = Fσ PT · P T = PT 00

σ∈τ ¯ τ 0

P

4 Fσ =

σ∈τ ?τ 0 4321

4312

Fτ ⊗ F

τ0

4 Fγ =

P

Xη · Xη0 = Xη+η0 + Xη−η0

B(T, γ) ⊗ A(T, γ) 4Xη =

γ cut

+++

3421

2431 4213

3142

4123 1432

2413

1342

1423 2143

1243

B(η, γ) ⊗ A(η, γ)

γ cut ++−

3241 4132

P

3412

4231

Geometry

T 0 ≤ T 00 ≤ T - 0 % T T

Solomon algebra Rec = vect h Xη | η ∈ ±∗ i

−++

2341

−+−

3214 3124 2314

1324 2134 1234

+−+

+−− −−+ −−−

COMBINATORICS

k -TWISTS

(k, n)-twist = pipe dream in the trapezoidal shape of height n and width k

k -TWISTS 1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

(k, n)-twist = pipe dream in the trapezoidal shape of height n and width k

k -TWISTS 1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

(k, n)-twist = pipe dream in the trapezoidal shape of height n and width k contact graph of a twist T = vertices are pipes of T and arcs are elbows of T

1-TWISTS AND TRIANGULATIONS Correspondence elbow in row i and column j ←→ diagonal [i, j] of the (n + 2)-gon (1, n)-twist T ←→ triangulation T? of the (n + 2)-gon pth relevant pipe of T ←→ pth triangle of T? contact graph of T ←→ dual binary tree of T? elbow flips in T ←→ diagonal flips in T? 1 2 3 4 5

0

4 2 1

5 3

6 1

5 2

3

4

Woo. Catalan numbers and Schubert Polynomials for w = 1(n + 1) . . . 2. Unpub 2004 P. – Pocchiola. Multitriangulations, pseudotriangulations and primitive sorting networks. 2012

1-TWISTS AND TRIANGULATIONS Correspondence elbow in row i and column j ←→ diagonal [i, j] of the (n + 2)-gon (1, n)-twist T ←→ triangulation T? of the (n + 2)-gon pth relevant pipe of T ←→ pth triangle of T? contact graph of T ←→ dual binary tree of T? elbow flips in T ←→ diagonal flips in T? 1 2 3 4 5

0

2 4

1 3

6 1

5

5 2

3

4

Woo. Catalan numbers and Schubert Polynomials for w = 1(n + 1) . . . 2. Unpub 2004 P. – Pocchiola. Multitriangulations, pseudotriangulations and primitive sorting networks. 2012

k -TWISTS AND k -TRIANGULATIONS Correspondence elbow in row i and column j ←→ diagonal [i, j] of the (n + 2k)-gon (k, n)-twist T ←→ k -triangulation T? of the (n + 2k)-gon pth relevant pipe of T ←→ pth k -star of T? contact graph of T ←→ dual graph of T? elbow flips in T ←→ diagonal flips in T?

1 2 3 4 5

1

5 2

3

4

P. – Pocchiola. Multitriangulations, pseudotriangulations and primitive sorting networks. 2012

NUMEROLOGY THM.

The (k, n)-twists are counted by det(Cn+2k−i−j )i,j∈[k], where Cm =

2m 1 m+1 m

.

Jonsson. Generalized triangulations and diagonal-free subsets of stack polyominoes. 2005 P. – Pocchiola. Multitriangulations, pseudotriangulations and primitive sorting networks. 2012 Stump. A new perspective on k -triangulations. 2011

QU.

What is the number of acyclic (k, n)-twists? k\ n 0 1 2 3 4 5 6 7 8 9

1 2 3

4

5

6

7

8

9

10

1 1 1 1 1 1 1 1 1 1 . 2 5 14 42 132 429 1430 4862 16796 . . 6 22 92 420 2042 10404 54954 298648 . . . 24 114 612 3600 22680 150732 1045440 . . . . 120 696 4512 31920 242160 1942800 . . . . . 720 4920 37200 305280 2680800 . . . . . . 5040 39600 341280 3175200 . . . . . . . 40320 357840 3457440 . . . . . . . . 362880 3588480 . . . . . . . . . 3628800

k -TWIST INSERTION Input: a permutation τ = τ1 · · · τn Algo: Insert pipes one by one (from right to left) as northwest as possible Output: an acyclic (k, n)-twist insk (τ ) Exm: Insertion of τ = 31542

5 4 3 2 1

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5 5 4 3 2 1

1 2 3 4 5

5 4 3 2 1

5 4 3 2 1


5 4 3 2 1

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5 5 4 3 2 1

1 2 3 4 5

5 4 3 2 1

5 4 3 2 1


5 4 3 2 1

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5 5 4 3 2 1

1 2 3 4 5

5 4 3 2 1

5 4 3 2 1


5 4 3 2 1

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5 5 4 3 2 1

1 2 3 4 5

5 4 3 2 1

5 4 3 2 1


5 4 3 2 1

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5 5 4 3 2 1

1 2 3 4 5

5 4 3 2 1

5 4 3 2 1


5 4 3 2 1

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5 5 4 3 2 1

1 2 3 4 5

5 4 3 2 1

5 4 3 2 1


5 4 3 2 1

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5 5 4 3 2 1

THM. insk

1 2 3 4 5

5 4 3 2 1

5 4 3 2 1

is a surjection from permutations of [n] to acyclic (k, n)-twists. fiber of a (k, n)-twist T = linear extensions of its contact graph T#.

Exm: insertion in binary search trees

k -TWIST CONGRUENCE congruence = equivalence relation ≡k on Sn defined as the transitive closure of the rewriting rule

DEF. k -twist

U acV1b1V2b2 · · · Vk bk W ≡k U caV1b1V2b2 · · · Vk bk W

if a < bi < c for all i ∈ [k].

4321 3421 3241 3214

2314

4231

2431

2341

2134

4213 2413

2143 1324 1234

PROP.

4312

3412 3142

3124

4321 3421 4132 4123 1342 1243

1423

3241 1432

3214

2431

2341 2314

4231 3412 3142

3124 2134

4312 4213 2413

2143 1324 1234

For any τ, τ 0 ∈ Sn, we have τ ≡k τ 0 ⇐⇒ insk (τ ) = insk (τ 0).

4132 4123

1342 1243

1423

1432

LATTICE CONGRUENCES DEF.

Order congruence = equivalence relation ≡ on a poset P such that:

(i) Every equivalence class under ≡ is an interval of P . (ii) The projection π↓ : P → P (resp. π ↑ : P → P ), which maps an element of P to the minimal (resp. maximal) element of its equivalence class, is order preserving. poset quotient = X ≤ Y in P/ ≡ ⇐⇒ ∃ x ∈ X, y ∈ Y such that x ≤ y in P . If moreover P is a lattice, ≡ is automatically a lattice congruence, compatible with meets and joins: x ≡ x0 and y ≡ y 0 ⇒ x ∧ y ≡ x0 ∧ y 0 and x ∨ y ≡ x0 ∨ y 0. lattice quotient = X ∧ Y and X ∨ Y are the congruence classes of x ∧ y and x ∨ y for arbitrary representatives x ∈ X and y ∈ Y . THM.

The k -twist congruence is a lattice quotient of the weak order.

INCREASING FLIP LATTICE flip in a k -twist = exchange an elbow with the unique crossing between its two pipes increasing flip = the elbow is southwest of the crossing increasing flip order = transitive closure of the increasing flip graph 1 2 3 4 5

5 4 3 2 1

−→

1 2 3 4 5

5 4 3 2 1

−→

1 2 3 4 5

5 4 3 2 1

INCREASING FLIP LATTICE 1 2 3 4

1 2 3 4

1 2 3 4 1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4 1 2 3 4

1 2 3 4

1 2 3 4 1 2 3 4

1 2 3 4

1 2 3 4

INCREASING FLIP LATTICE flip in a k -twist = exchange an elbow with the unique crossing between its two pipes increasing flip = the elbow is southwest of the crossing increasing flip order = transitive closure of the increasing flip graph 1 2 3 4 5

5 4 3 2 1 PROP.

−→

1 2 3 4 5

5 4 3 2 1

−→

1 2 3 4 5

5 4 3 2 1

The increasing flip order on acyclic k -twists is isomorphic to:

• the quotient lattice of the weak order by the k -twist congruence ≡k , • the sublattice of the weak order induced by the permutations of Sn avoiding the pattern 1(k + 2) – (σ1 + 1) – · · · – (σk + 1) for all σ ∈ Sk .

k -RECOILS G (n) = graph with vertex set [n] and edge set {i, j} ∈ [n] i < j ≤ i + k

k

( k

number of acyclic orientations of G (n) =

2

n!

if n ≤ k

k! (k + 1)n−k

if n ≥ k

k -recoils scheme of τ ∈ Sn = acyclic orientation reck (τ ) of Gk (n) with edge i → j for all i, j ∈ [n] such that |i − j| ≤ k and τ −1(i) < τ −1(j)

k -RECOILS k -recoil congruence = equivalence relation ≈k on Sn defined as the transitive closure of the rewriting rule U ijV ≈k U jiV if i + k < j . 4321 3421 3241 3214

2314

4231

2431

2341

2134

4213 2413

2143 1324 1234

PROP.

4312

3412 3142

3124

4321 3421 4132 4123 1342 1243

1423

3241 1432

3214

2431

2341 2314

4231 3412 3142

3124 2134

4312 4132

4213 2413

2143 1324

4123 1342

1432

1423

1243

1234

For any τ, τ 0 ∈ Sn, we have τ ≈k τ 0 ⇐⇒ reck (τ ) = reck (τ 0). Novelli, Reutenauer, Thibon. Generalized descent patterns in permutations and associated Hopf Algebras. 2011

THM.

The k -recoil congruence is a lattice quotient of the weak order.

k -CANOPY The maps insk , cank , and reck define a commutative diagram of lattice homomorphisms: reck

Sn insk

AOk (n) cank

AT k (n)

4321 3421

3214

2341 2314

4231

2431

3241

2134

4312

3412 3142

3124

4321

4213 2413

2143 1324 1234

3421 4132

4123 1342 1243

1423

3241 1432

3214

2431

3124 2134

4312

3412 3142

2341 2314

4231

4213 2413

2143 1324 1234

4132 4123

1342 1243

1423

1432

ALGEBRA

SHUFFLE AND CONVOLUTION For n, n0 ∈ N, consider the set of perms of Sn+n0 with at most one descent, at position n: (n,n0 ) :

S

= {τ ∈ Sn+n0 | τ (1) < · · · < τ (n) and τ (n + 1) < · · · < τ (n + n0)}

For τ ∈ Sn and τ 0 ∈ Sn0 , define shifted concatenation τ τ¯0 = [τ (1), . . . , τ (n), τ 0 (1) + n, . . . , τ 0(n0) + n] ∈ Sn+n0 0 0) 0 −1 (n,n shifted shuffle product τ ¯ τ = (τ τ¯ ) ◦ π π∈S ⊂ Sn+n0 0 convolution product τ ? τ 0 = π ◦ (τ τ¯0) π ∈ S(n,n ) ⊂ Sn+n0

Exm: 12 ¯ 231 = {12453, 14253, 14523, 14532, 41253, 41523, 41532, 45123, 45132, 45312} 12 ? 231 = {12453, 13452, 14352, 15342, 23451, 24351, 25341, 34251, 35241, 45231}

5 4 3 2 1

5 4 3 2 1 1 2 3 4 5 concatenation

5 4 3 2 1 1 2 3 4 5 shuffle

1 2 3 4 5 convolution

MALVENUTO & REUTENAUER’S HOPF ALGEBRA ON PERMUTATIONS Combinatorial Hopf Algebra = combinatorial vector space B endowed with product · : B ⊗ B → B coproduct 4 : B → B ⊗ B which are “compatible”, ie.

DEF.

·

B⊗B

B

4

B⊗B

4⊗4

·⊗·

B⊗B⊗B⊗B

I ⊗ swap ⊗ I

B⊗B⊗B⊗B

Malvenuto-Reutenauer algebra = Hopf algebra FQSym with basis (Fτ )τ ∈S and where X X Fσ and 4 Fσ = Fτ ⊗ Fτ 0 Fτ · Fτ 0 =

σ∈τ ¯ τ 0

σ∈τ ?τ 0

Malvenuto-Reutenauer. Duality between quasi-symmetric functions and the Solomon descent algebra. 1995

HOPF SUBALGEBRA k -Twist algebra = vector subspace Twistk of FQSym generated by

PT :=

X

Fτ =

τ ∈S insk (τ )=T

X

Fτ ,

τ ∈L(T# )

for all acyclic k -twists T. Exm: P

1 2 3 4 5

=

X τ ∈S5

THEO. Twistk

Fτ

P

1 2 3 4 5

= F13542 + F15342 + F31542 + F51342 + F35142 + F53142 + F35412 + F53412

P

1 2 3 4 5

= F31542 + F35142

P

1 2 3 4 5

= F31542.

is a subalgebra of FQSym Loday-Ronco. Hopf algebra of the planar binary trees. 1998 Hivert-Novelli-Thibon. The algebra of binary search trees. 2005

GAME: Explain the product and coproduct directly on the k -twists...

PRODUCT P

1 2 3 4

·P

1 2

= (F1423 + F4123 ) · F21           F F142635  142653    F F F 146523 164523 165423  + F146235  F164253  F164235       + F146253   F142365  ++ F614235 ++ F412653 ++ F614253 ++ F416523 ++ F614523 ++ F615423  = + + F 412635    + F461523  + F641523  + F651423  + F412365  + F416235  + F641235 + F416253  + F641253 + F654123 + F465123 + F645123 + F461253 + F461235 = P 1 2 3 4 5 6+ P 1 2 3 4 5 6 + P 1 2 3 4 5 6 + P 1 2 3 4 5 6 + P 1 2 3 4 5 6 + P 1 2 3 4 5 6 + P 1 2 3 4 5 6 + P 1 2 3 4 5 6

k

0

k

0

P

For T ∈ AT (n) and T ∈ AT (n ) acyclic k -twists, PT · PT0 = S PS, where S runs over the interval between T\T0 and T/T0 in the (k, n + n0)-twist lattice.

PROP.

1 2 3 4

1 2

1 2 3 4 5 6

1 2 3 4 5 6

GEOMETRY

PERMUTAHEDRON Permutahedron Permk (n) = conv {(τ (1), . . . , τ (n)) | τ ∈ Sn} \ X |J| + 1 n = H ∩ x ∈ R x ≥ j 4321 2 3421 j∈J ∅6=J ([n] X 4312

3412

4231

= 11 +

3241 4132

2431 4213

3142

4123 1432

2413

1342

1423 2143

1243

2341

3214 3124 2314

1324 2134 1234

[ei, ej ]

1≤i