Subword Complexes and Nil-Hecke Moves
https://doi.org/10.18255/1818-1015-2013-6-121-128
Abstract
For a finite Coxeter group W, a subword complex is a simplicial complex associated with a pair (Q, ρ), where Q is a word in the alphabet of simple reflections, ρ is a group element. We describe the transformations of such a complex induced by nil-moves and inverse operations on Q in the nil-Hecke monoid corresponding to W. If the complex is polytopal, we also describe such transformations for the dual polytope. For W simply-laced, these descriptions and results of [5] provide an algorithm for the construction of the subword complex corresponding to (Q, ρ) from the one corresponding to (δ(Q), ρ), for any sequence of elementary moves reducing the word Q to its Demazure product δ(Q). The former complex is spherical or empty if and only if the latter one is empty. The article is published in the author’s wording.
Supporting agencies: DIM RDM-IdF of the Région Île-de-France
About the Author
M. A. Gorsky
Steklov Mathematical Institute;
Université Paris Diderot;
Institut de Mathématiques de Jussieu
Russian Federation
Russian Federation
Gubkina str., 8, Moscow, 119991, Russia;
Paris 7;
Paris Rive Gauche, Bât. Sophie Germain, 75205 Paris Cedex 13, France
References
1. Björner A., Brenti F. Combinatorics of Coxeter groups // Graduate Texts in Mathematics 231, Springer, 2005.
2. Buchstaber V. M., Volodin V. D. Combinatorial 2-truncated cubes and applications, in “Associahedra, Tamari Lattices, and Related Structures. Tamari Memorial Festschrift” // Progress in Mathematics, 2012, 299, P. 161–186.
3. Ceballos C., Labbé J. P., Stump C. Subword complexes, cluster complexes, and generalized multi-associahedra // Journal of Algebraic Combinatoric, to appear; arXiv:1108.1776, 2011.
4. Fomin S., Zelevinsky A., Y -systems and generalized associahedra. Annals of Math., 2003, 158, P. 977–1018.
5. Gorsky M. A. Subword complexes and edge subdivisions. arXiv preprint, arXiv:1305.5499.
6. Knutson A., Miller E. Groebner geometry of Schubert polynomials // Ann. of Math. (2), 2005, 161, No. 3, P. 1245–1318.
7. Knutson A., Miller E. Subword complexes in Coxeter groups // Advances in Mathematics, 2004, 184 (1), P. 161–176.
8. Pilaud V., Stump C. Brick polytopes of spherical subword complexes: A new approach to generalized associahedra. arXiv preprint, arXiv:1111.3349.
Review
For citations:
Gorsky M.A. Subword Complexes and Nil-Hecke Moves. Modeling and Analysis of Information Systems. 2013;20(6):121-128. https://doi.org/10.18255/1818-1015-2013-6-121-128
Views:
937
Email this article 