Subword complexes via triangulations of root polytopes
Algebraic Combinatorics, Volume 1 (2018) no. 3, pp. 395-414.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/alco.17
Classification: 52B20, 05E45
Keywords: subword complex, pipedream, triangulation, root polytope

Escobar, Laura 1; Mészáros, Karola 2

1 University of Illinois at Urbana-Champaign, Department of Mathematics, 1409 W. Green Street, Urbana, IL 61801, USA
2 Cornell University, Department of Mathematics, 310 Malott Hall, Ithaca, NY 14850, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{ALCO_2018__1_3_395_0,
     author = {Escobar, Laura and M\'esz\'aros, Karola},
     title = {Subword complexes via triangulations of root polytopes},
     journal = {Algebraic Combinatorics},
     pages = {395--414},
     publisher = {MathOA foundation},
     volume = {1},
     number = {3},
     year = {2018},
     doi = {10.5802/alco.17},
     mrnumber = {3856530},
     zbl = {1393.52010},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.17/}
}
TY  - JOUR
AU  - Escobar, Laura
AU  - Mészáros, Karola
TI  - Subword complexes via triangulations of root polytopes
JO  - Algebraic Combinatorics
PY  - 2018
SP  - 395
EP  - 414
VL  - 1
IS  - 3
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.17/
DO  - 10.5802/alco.17
LA  - en
ID  - ALCO_2018__1_3_395_0
ER  - 
%0 Journal Article
%A Escobar, Laura
%A Mészáros, Karola
%T Subword complexes via triangulations of root polytopes
%J Algebraic Combinatorics
%D 2018
%P 395-414
%V 1
%N 3
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.17/
%R 10.5802/alco.17
%G en
%F ALCO_2018__1_3_395_0
Escobar, Laura; Mészáros, Karola. Subword complexes via triangulations of root polytopes. Algebraic Combinatorics, Volume 1 (2018) no. 3, pp. 395-414. doi : 10.5802/alco.17. https://alco.centre-mersenne.org/articles/10.5802/alco.17/

[1] Bergeron, Nantel; Billey, Sara RC-graphs and Schubert polynomials, Exp. Math., Volume 2 (1993) no. 4, pp. 257-269 | DOI | MR | Zbl

[2] Bergeron, Nantel; Ceballos, Cesar; Labbé, Jean-Philippe Fan realizations of type A subword complexes and multi-associahedra of rank 3, Discrete Comput. Geom., Volume 54 (2015) no. 1, pp. 195-231 | DOI | MR | Zbl

[3] Ceballos, Cesar On associahedra and related topics, Ph. D. Thesis, Freie Universität Berlin (Germany) (2012)

[4] Ceballos, Cesar; Labbé, Jean-Philippe; Stump, Christian Subword complexes, cluster complexes, and generalized multi-associahedra, J. Algebr. Comb., Volume 39 (2014) no. 1, pp. 17-51 | DOI | MR | Zbl

[5] Escobar, Laura Brick manifolds and toric varieties of brick polytopes, Electron. J. Comb., Volume 23 (2016) no. 2, Paper no. P2.25, 18 pages | MR | Zbl

[6] Fomin, Sergey; Kirillov, Anatol N. Grothendieck polynomials and the Yang-Baxter equation, Formal power series and algebraic combinatorics, DIMACS, 1994, pp. 183-189 | MR

[7] Gelfand, Israel M.; Graev, Mark I.; Postnikov, Alexander Combinatorics of hypergeometric functions associated with positive roots, The Arnold-Gelfand mathematical seminars, Birkhäuser, 1997, pp. 205-221 | DOI | MR | Zbl

[8] Grinberg, Darij t-unique reductions for Mészáros’s subdivision algebra (2017) (https://arxiv.org/abs/1704.00839) | Zbl

[9] Kirillov, Anatol N. On some quadratic algebras, Dunkl elements, Schubert, Grothendieck, Tutte and reduced polynomials (2014) (RIMS preprint)

[10] Knutson, Allen; Miller, Ezra Subword complexes in Coxeter groups, Adv. Math., Volume 184 (2004) no. 1, pp. 161-176 | DOI | MR | Zbl

[11] Knutson, Allen; Miller, Ezra Gröbner geometry of Schubert polynomials, Ann. Math., Volume 161 (2005) no. 3, pp. 1245-1318 | DOI | MR | Zbl

[12] Mészáros, Karola Root polytopes, triangulations, and the subdivision algebra. I, Trans. Am. Math. Soc., Volume 363 (2011) no. 8, pp. 4359-4382 | DOI | MR | Zbl

[13] Mészáros, Karola Root polytopes, triangulations, and the subdivision algebra, II, Trans. Am. Math. Soc., Volume 363 (2011) no. 11, pp. 6111-6141 | DOI | MR | Zbl

[14] Mészáros, Karola h-polynomials via reduced forms, Electron. J. Comb., Volume 22 (2015) no. 4, Paper no. P4.18, 17 pages | MR | Zbl

[15] Mészáros, Karola Product formulas for volumes of flow polytopes, Proc. Am. Math. Soc., Volume 143 (2015) no. 3, pp. 937-954 | DOI | MR | Zbl

[16] Mészáros, Karola h-polynomials of reduction trees, SIAM J. Discrete Math., Volume 30 (2016) no. 2, pp. 736-762 | DOI | MR | Zbl

[17] Mészáros, Karola Pipe dream complexes and triangulations of root polytopes belong together, SIAM J. Discrete Math., Volume 30 (2016) no. 1, pp. 100-111 | DOI | MR | Zbl

[18] Mészáros, Karola; Morales, Alejandro H. Flow polytopes of signed graphs and the Kostant partition function, Int. Math. Res. Not. (2015) no. 3, pp. 830-871 | DOI | MR | Zbl

[19] Mészáros, Karola; St. Dizier, Avery From generalized permutahedra to Grothendieck polynomials via flow polytopes (2017) (https://arxiv.org/abs/1705.02418) | Zbl

[20] Pilaud, Vincent; Pocchiola, Michel Multitriangulations, pseudotriangulations and primitive sorting networks, Discrete Comput. Geom., Volume 48 (2012) no. 1, pp. 142-191 | DOI | MR | Zbl

[21] Pilaud, Vincent; Santos, Francisco The brick polytope of a sorting network, Eur. J. Comb., Volume 33 (2012) no. 4, pp. 632-662 | DOI | MR | Zbl

[22] Serrano, Luis; Stump, Christian Maximal fillings of moon polyominoes, simplicial complexes, and Schubert polynomials, Electron. J. Comb., Volume 19 (2012) no. 1, Paper no. P16, 18 pages | MR | Zbl

[23] Stanley, Richard P. Combinatorics and Commutative Algebra, Progress in Mathematics, 41, Birkhäuser, 1996, vi+164 pages | MR | Zbl

[24] Stump, Christian A new perspective on k-triangulations, J. Comb. Theory, Ser. A, Volume 118 (2011) no. 6, pp. 1794-1800 | DOI | MR | Zbl

Cited by Sources: