Proceedings of the 28-th International Conference on Formal Power Series and Algebraic Combinatorics
4-8 Jul 2016 Vancouver, British Columbia (Canada)

Extended abstracts listed by author > Escobar Laura

Toric matrix Schubert varieties and root polytopes (extended abstract)
Laura Escobar  1  , Karola Mészáros  2  
1 : Department of Mathematics [Urbana]  -  Website
Department of Mathematics 1409 W. Green Street Urbana, IL 61801 -  États-Unis
2 : Department of Mathematics [Cornell]  -  Website
Cornell UniversityMalot Hall Ithaca, NY 14853-2401 -  États-Unis

Start 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. 

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