Noncrossing sets and a Graßmann associahedron

Transcription

1 Noncrossing sets and a Graßmann associahedron Francisco Santos, Christian Stump, Volkmar Welker (in partial rediscovering work of T. K. Petersen, P. Pylyavskyy, and D. E. Speyer, 2008) (in partial rediscovering even earlier work of A. Mihailovs, 1998) July 1, 2014

2 What do we want to do? Study a binary (noncrossing) relation on k-subsets of [n], and the simplicial complex NC of pairwise noncrossing collections of k-subsets. This complex generalizes the simplicial complex of triangulations with similar algebraic properties: the Stanley-Reisner ideal of NC provides a particularly nice squarefree monomial initial ideal of the defining ideal of the Graßmannian geometric properties: NC is a regular, unimodular and Gorenstein triangulation of the order polytope O of a square grid (n k) k and thus generalizing the associahedron combinatorial properties: the oriented dual graph of NC generalizes the Tamari order 1 / 14

3 The simple definition We (and a few people before) considered the following very simple definition: I = (i 1 <... < i k ), J = (j 1 <... < j k ) are noncrossing if for all indices a < b with i l = j l for a < l < b, the arcs (i a < i b ) and (j a < j b ) do not cross. Example (1, 4, 6, 8, 9), (2, 3, 6, 7, 8) are crossing since (3, 7), (4, 8) cross. (1, 4, 5, 8, 9), (2, 3, 6, 7, 8) are noncrossing. The (multidimensional) noncrossing complex NC is the flag simplicial complex with vertices being k-subsets of [n] and with faces being collections of noncrossing k-subsets. For k = 2, this reduces to the usual complex of triangulations of a convex n-gon. Well-known: I, J are nonnesting if for all indices a < b, the arcs (i a < i b ) and (j a < j b ) do not nest. NN is analogously defined. 2 / 14

4 First properties NC has the following symmetries. (i) The map a n + 1 a induces an automorphism on NC. (ii) The map I [n] \ I induces an isomorphism NC NC n. (iii) I, J are noncrossing if and only if they are noncrossing when restricting to the symmetric difference I J = (I J) \ (I J). (iv) For b [n] The restriction of NC to vertices with b I yields NC The restriction of NC to vertices with b I yields NC 1. k 1,n 1. (v) The n k-subsets obtained by cyclic rotations of the vertex (1, 2,..., k) do not cross any other vertex in V and hence are contained in every maximal face of NC. BUT: NC is not symmetric (and cannot be symmetric) under rotation of the outer polygon. 3 / 14

5 Let P be a finite poset. The order polytope of P can be given as O(P) = conv { χ F : F order filter of P }, where χ F N P is the characteristic vector of the order filter F of P. Example P antichain: O(P) cube P chain: O(P) simplex Identifying k-subsets of [n] and order filters in P = [k] [n k]: { (1, 3), (1, 4), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4) } (2, 4, 5) / 14

6 Theorem (Petersen-Pylyavskyy-Speyer 2008, Santos-S-Welker 2014) The noncrossing complex NC is a flag, regular, unimodular and Gorenstein triangulation of the order polytope O. Corollary NC sphere decomposes as a join of an (n 1)-dimensional simplex and a flag polytopal NC of dimension k(n k) n. Corollary NC has all desired properties from the first page. 5 / 14

7 Corollary NC is pure and its maximal faces are counted by the (n k, k)th multidimensional Catalan number 0! 1! (k 1)! ( ) k(n k)! (n 1)! (n 2)! (n k)! also counting standard Young tableaux of shape [(n k) k ]. Moreover, its h-vector (h 0,..., h n(n k) ) is given by the multidimensional Narayana numbers h () i = # { standard Young tableaux of shape [(n k) k ] with i peaks where T has a peak in position a if a + 1 is placed in a row lower than a. 6 / 14

8 Combinatorics of the noncrossing complex We encode a multiset of k-subsets of [n] into columns of a table ordered lexicographically and record its summing tableau as a weakly increasing filling of a k (n k) matrix: Observe that the set of all weakly increasing tableaux is the set of lattice points in the cone over O(P). Theorem Let T be a tableau. There is a unique multiset ϕ NN (T ) and a unique multiset ϕ NC (T ) of { k-subsets } \{(12... k)} whose summing tableaux are T, 0such that the vectors in ϕ NN (T ) are mutually nonnesting, and the vectors in ϕ NC (T ) are mutually noncrossing. 7 / 14

9 Combinatorics of the noncrossing complex T = ϕ NN (T ) = ϕ NC (T ) = / 14

10 Combinatorics of the noncrossing complex Observation Observation ϕ NN (T ) and ϕ NC (T ) are facets of NN and NC, respectively, every column contains a unique mark. ϕ NN (T ) and ϕ NC (T ) are codimension one faces of NN and NC, respectively, every column contains unique mark except one that contains exactly two marks. Observation Starting with a table of mutually noncrossing (resp. nonnesting) k-subsets, pushing marked entries in a column containing more than one mark yields a new column noncrossing (resp. nonnesting) with all other columns. 9 / 14

11 Combinatorics of the noncrossing complex Corollary The simplicial complexes NN and NC are pure of dimension k(n k). Corollary Every face of the reduced noncrossing complex contained in exactly two maximal faces. NC of codimension one is 10 / 14

12 A Graßmann-Tamari order The dual graph G of NC is given by the maximal faces of NC, and where two maximal faces F 1 and F 2 share an edge if they intersect in a face of codimension one. Corollary The pushing procedure yields a natural orientation of the dual graph, a Graßmann-Tamari digraph. Theorem The Graßmann-Tamari digraph is acyclic, and hence yields a Graßmann-Tamari order on facets of NC. The Graßmann-Tamari order has the following symmetries: The map induced on T by a n + 1 a is order reversing. In particular, T is self-dual. The map from T to T n induced by I [n] \ I is order reversing. In particular, T = Tn. 11 / 14

13 A Graßmann-Tamari order / 14

14 Open combinatorial problems Conjecture The Graßmann-Tamari order T is a lattice. Observe that we have h () i = {T NC : T has i upper covers in T }, are the multidimensional Narayana numbers and thus equal to the number of standard Young tableaux with exactly i peaks. Question Is there an operation on peaks of standard Young tableaux that describes the Graßmann-Tamari order directly on standard Young tableaux, thus also providing a bijection between maximal faces of NN and of NC, which generalizes the Tamari order as defined on ordinary Dyck paths? 13 / 14

15 Thank you page 14 / 14