Subword complexes via triangulations of root polytopes

Subword complexes are simplicial complexes introduced by Knutson and Miller to illustrate the combinatorics of Schubert polynomials and determinantal ideals. They proved that any subword complex is homeomorphic to a ball or a sphere and asked about their geometric realizations. We show that a family of subword complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of $\beta$-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We can also write the volume and Ehrhart series of root polytopes in terms of $\beta$-Grothendieck polynomials.Comment: 17 pages, 15 figure

Toric matrix Schubert varieties and root polytopes (extended abstract)

International audienceStart with a permutation matrix π and consider all matrices that can be obtained from π by taking downward row operations and rightward column operations; the closure of this set gives the matrix Schubert variety Xπ. We characterize when the ideal defining Xπ is toric (with respect to a 2n − 1-dimensional torus) and study the associated polytope of its projectivization. We construct regular triangulations of these polytopes which we show are geometric realizations of a family of subword complexes. We also show that these complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of β-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We also write the volume and Ehrhart series of root polytopes in terms of β-Grothendieck polynomials. Subword complexes were introduced by Knutson and Miller in 2004, who showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations

Brick polytopes, lattice quotients, and Hopf algebras

This paper is motivated by the interplay between the Tamari lattice, J.-L. Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf algebra on binary trees. We show that these constructions extend in the world of acyclic $k$-triangulations, which were already considered as the vertices of V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural surjection from the permutations to the acyclic $k$-triangulations. We show that the fibers of this surjection are the classes of the congruence $\equiv^k$ on $\mathfrak{S}_n$ defined as the transitive closure of the rewriting rule $U ac V_1 b_1 \cdots V_k b_k W \equiv^k U ca V_1 b_1 \cdots V_k b_k W$ for letters $a < b_1, \dots, b_k < c$ and words $U, V_1, \dots, V_k, W$ on $[n]$. We then show that the increasing flip order on $k$-triangulations is the lattice quotient of the weak order by this congruence. Moreover, we use this surjection to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on permutations, indexed by acyclic $k$-triangulations, and to describe the product and coproduct in this algebra and its dual in term of combinatorial operations on acyclic $k$-triangulations. Finally, we extend our results in three directions, describing a Cambrian, a tuple, and a Schr\"oder version of these constructions.Comment: 59 pages, 32 figure

Lattices of acyclic pipe dreams

We show that for any permutation $\omega$, the increasing flip graph on acyclic pipe dreams with exiting permutation $\omega$ is a lattice quotient of the interval $[e,\omega]$ of the weak order. We then discuss conjectural generalizations of this result to acyclic facets of subword complexes on arbitrary finite Coxeter groups.Comment: 34 pages, 19 figures. Version 2: New section 4.5 on nu-Tamari lattice

On Flow Polytopes, nu-Associahedra, and the Subdivision Algebra

This dissertation studies the geometry and combinatorics related to a flow polytope Fcar(ν) constructed from a lattice path ν, whose volume is given by the ν-Catalan numbers. It begins with a study of the ν-associahedron introduced by Ceballos, Padrol, and Sarmiento in 2019, but from the perspective of Schröder combinatorics. Some classical results for Schröder paths are extended to the ν-setting, and insights into the geometry of the ν-associahedron are obtained by describing its face poset with two ν-Schröder objects. The ν-associahedron is then shown to be dual to a framed triangulation of Fcar(ν), which is a geometric realization of the ν-Tamari complex. The dual graph of this triangulation is the Hasse diagram of the ν-Tamari lattice due to Préville-Ratelle and Viennot. The dual graph of a second framed triangulation of Fcar(ν) is shown to be the Hasse diagram of a principal order ideal of Young’s lattice generated by ν, and is used to show that the h∗-vector of Fcar(ν) is given by ν-Narayana numbers. This perspective serves to unify these two important lattices associated with ν-Dyck paths through framed triangulations of a flow polytope. Via an integral equivalence between Fcar(ν) and a subpolytope UI,J of a product of two simplices subdivisions of UI,J are shown to be obtainable with Mészáros’ subdivision algebra, which answers a question of Ceballos, Padrol, and Sarmiento. Building on this result, the subdivision algebra is extended to encode subdivisions of a product of two simplices, giving a new tool for their future study

Flow polytopes with Catalan volumes

The Chan-Robbins-Yuen polytope can be thought of as the flow polytope of the complete graph with netflow vector $(1, 0, \ldots, 0, -1)$. The normalized volume of the Chan-Robbins-Yuen polytope equals the product of consecutive Catalan numbers, yet there is no combinatorial proof of this fact. We consider a natural generalization of this polytope, namely, the flow polytope of the complete graph with netflow vector $(1,1, 0, \ldots, 0, -2)$. We show that the volume of this polytope is a certain power of $2$ times the product of consecutive Catalan numbers. Our proof uses constant term identities and further deepens the combinatorial mystery of why these numbers appear. In addition we introduce two more families of flow polytopes whose volumes are given by product formulas.Comment: 16 page