arXiv subjects

Laura Escobar

Publications and source records attributed to Laura Escobar.

24 records · Page 2Linked to original sources

arXiv2017

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 $β$-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 $β$-Grothendieck polynomials.

math.CO↗

arXiv2017

Newton polytopes and symmetric Grothendieck polynomials

Symmetric Grothendieck polynomials are inhomogeneous versions of Schur polynomials that arise in combinatorial $K$-theory. A polynomial has saturated Newton polytope (SNP) if every lattice point in the polytope is an exponent vector. We show Newton polytopes of these Grothendieck polynomials and their homogeneous components have SNP. Moreover, the Newton polytope of each homogeneous component is a permutahedron. This addresses recent conjectures of C. Monical-N. Tokcan-A. Yong and of A. Fink-K. Mészáros-A. St. Dizier in this special case.

math.CO↗

arXiv2017

The multidegree of the multi-image variety

The multi-image variety is a subvariety of Gr(1,P^3)^n that models taking pictures with n rational cameras. We compute its cohomology class in the cohomology of Gr(1,P^3)^n, and from there its multidegree as a subvariety of (P^5)^n under the Plücker embedding.

math.AG↗

arXiv2016

Rhombic tilings and Bott-Samelson varieties

S.~Elnitsky (1997) gave an elegant bijection between rhombic tilings of $2n$-gons and commutation classes of reduced words in the symmetric group on $n$ letters. P.~Magyar (1998) found an important construction of the Bott-Samelson varieties introduced by H.C.~Hansen (1973) and M.~Demazure (1974). We explain a natural connection between S.~Elnitsky's and P.~Magyar's results. This suggests using tilings to encapsulate Bott-Samelson data (in type $A$). It also indicates a geometric perspective on S.~Elnitsky's combinatorics. We also extend this construction by assigning desingularizations to the zonotopal tilings considered by B.~Tenner (2006).

math.CO↗

arXiv2015

Toric matrix Schubert varieties and their polytopes

Given a matrix Schubert variety $\overline{X_π}$, it can be written as $\overline{X_π}=Y_π\times \mathbb{C}^q$ (where $q$ is maximal possible). We characterize when $Y_π$ is toric (with respect to a $(\mathbb{C}^*)^{2n-1}$-action) and study the associated polytope $Φ(\mathbb{P}(Y_π))$ of its projectivization. We construct regular triangulations of $Φ(\mathbb{P}(Y_π))$ which we show are geometric realizations of a family of subword complexes. Subword complexes were introduced by Knutson and Miller in 2004, who also showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations.

math.CO↗

arXiv2014

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 will see 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 and Lange. These stories connect in a nice way: 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.

math.AG↗