The ν-Tamari Lattice via ν-Trees, ν-Bracket Vectors, and Subword Complexes

We give a new interpretation of the $\nu$-Tamari lattice of Préville-Ratelle and Viennot in terms of a rotation lattice of $\nu$-trees. This uncovers the relation with known combinatorial objects such as north-east fillings, \mbox{tree-like} tableaux and subword complexes. We provide a simple description of the lattice property using certain bracket vectors of $\nu$-trees, and show that the Hasse diagram of the $\nu$-Tamari lattice can be obtained as the facet adjacency graph of certain subword complex. Finally, this point of view generalizes to multi $\nu$-Tamari complexes, and gives (conjectural) insight on their geometric realizability via polytopal subdivisions of multiassociahedra.

Brick Manifolds and Toric Varieties of Brick Polytopes

Bott-Samelson varieties are a twisted product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational to the image; however in this paper we study a fiber of this map when it is not birational. We prove that in some cases the general fiber, which we christen a brick manifold, is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into one in terms of the "subword complexes" of Knutson and Miller. Pilaud and Stump realized certain subword complexes as the dual of the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg, Lange and Thomas. These stories connect in a nice way: we show that the moment polytope of the brick manifold is the brick polytope. In particular, we give a nice description of the toric variety of the associahedron. We give each brick manifold a stratification dual to the subword complex. In addition, we relate brick manifolds to Brion's resolutions of Richardon varieties.

Fan realizations of type $A$ subword complexes and multi-associahedra of rank 3

International audience We present complete simplicial fan realizations of any spherical subword complex of type $A_n$ for $n\leq 3$. This provides complete simplicial fan realizations of simplicial multi-associahedra $\Delta_{2k+4,k}$, whose facets are in correspondence with $k$-triangulations of a convex $(2k+4)$-gon. This solves the first open case of the problem of finding fan realizations where polytopality is not known. The techniques presented in this paper work for all finite Coxeter groups and we hope that they will be useful to construct fans realizing subword complexes in general. In particular, we present fan realizations of two previously unknown cases of subword complexes of type $A_4$, namely the multi-associahedra $\Delta_{9,2}$ and $\Delta_{11,3}$. Nous construisons des éventails simpliciaux complets ayant la combinatoire des complexes de sous-mots de type $A_n$ pour $n\leq 3$. Par conséquent, nous obtenons des constructions d’éventails des multi-associaèdres $\Delta_{2k+4,k}$, dont les facettes correspondent aux $k$-triangulations d’un $(2k+4)$-gone. Cette construction confirme l’existence d’éventails ayant la combinatoire du multi-associaèdres pour une famille dont la polytopalité n’est pas confirmée. Les techniques utilisées fonctionnent pour tous les groupes de Coxeter et nous espérons qu’elles seront utiles afin de construire des éventails réalisant les complexes de sous-mots en général. En particulier, nous présentons des éventails pour deux complexes de sous-mots de type $A_4$ dont l’existence était inconnue: les multi-associaèdres $\Delta_{9,2}$ et $\Delta_{11,3}$.

Bott-Samelson Varieties, Subword Complexes and Brick Polytopes

International audience Bott-Samelson varieties factor the flag variety $G/B$ into a product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational; however in this paper we study fibers of this map when it is not birational. We will see that in some cases this fiber is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into a purely combinatorial one in terms of a subword complex. These simplicial complexes, defined by Knutson and Miller, encode a lot of information about reduced words in a Coxeter system. Pilaud and Stump realized certain subword complexes as the dual to the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg and Lange. These stories connect in a nice way: the moment polytope of a fiber of the Bott-Samelson map is the Brick polytope. In particular, we give a nice description of the toric variety of the associahedron.

EL-labelings and canonical spanning trees for subword complexes

International audience We describe edge labelings of the increasing flip graph of a subword complex on a finite Coxeter group, and study applications thereof. On the one hand, we show that they provide canonical spanning trees of the facet-ridge graph of the subword complex, describe inductively these trees, and present their close relations to greedy facets. Searching these trees yields an efficient algorithm to generate all facets of the subword complex, which extends the greedy flip algorithm for pointed pseudotriangulations. On the other hand, when the increasing flip graph is a Hasse diagram, we show that the edge labeling is indeed an EL-labeling and derive further combinatorial properties of paths in the increasing flip graph. These results apply in particular to Cambrian lattices, in which case a similar EL-labeling was recently studied by M. Kallipoliti and H. Mühle.