Isocanted alcoved polytopes

María Jesús de la Puente; Pedro Luis Clavería

Applications of Mathematics (2020)

  • Volume: 65, Issue: 6, page 703-726
  • ISSN: 0862-7940

Abstract

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Through tropical normal idempotent matrices, we introduce isocanted alcoved polytopes, computing their f -vectors and checking the validity of the following five conjectures: Bárány, unimodality, 3 d , flag and cubical lower bound (CLBC). Isocanted alcoved polytopes are centrally symmetric, almost simple cubical polytopes. They are zonotopes. We show that, for each dimension, there is a unique combinatorial type. In dimension d , an isocanted alcoved polytope has 2 d + 1 - 2 vertices, its face lattice is the lattice of proper subsets of [ d + 1 ] and its diameter is d + 1 . They are realizations of d -elementary cubical polytopes. The f -vector of a d -dimensional isocanted alcoved polytope attains its maximum at the integer ⌊ d / 3 ⌋ .

How to cite

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de la Puente, María Jesús, and Clavería, Pedro Luis. "Isocanted alcoved polytopes." Applications of Mathematics 65.6 (2020): 703-726. <http://eudml.org/doc/296940>.

@article{delaPuente2020,
abstract = {Through tropical normal idempotent matrices, we introduce isocanted alcoved polytopes, computing their $f$-vectors and checking the validity of the following five conjectures: Bárány, unimodality, $3^d$, flag and cubical lower bound (CLBC). Isocanted alcoved polytopes are centrally symmetric, almost simple cubical polytopes. They are zonotopes. We show that, for each dimension, there is a unique combinatorial type. In dimension $d$, an isocanted alcoved polytope has $2^\{d+1\}-2$ vertices, its face lattice is the lattice of proper subsets of $[d+1]$ and its diameter is $d+1$. They are realizations of $d$-elementary cubical polytopes. The $f$-vector of a $d$-dimensional isocanted alcoved polytope attains its maximum at the integer $\lfloor d/3\rfloor $.},
author = {de la Puente, María Jesús, Clavería, Pedro Luis},
journal = {Applications of Mathematics},
keywords = {cubical polytope; isocanted; alcoved; centrally symmetric; almost simple; zonotope; $f$-vector; cubical $g$-vector; unimodal; flag; face lattice; log-concave sequence; tropical normal idempotent matrix; symmetric matrix},
language = {eng},
number = {6},
pages = {703-726},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Isocanted alcoved polytopes},
url = {http://eudml.org/doc/296940},
volume = {65},
year = {2020},
}

TY - JOUR
AU - de la Puente, María Jesús
AU - Clavería, Pedro Luis
TI - Isocanted alcoved polytopes
JO - Applications of Mathematics
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 6
SP - 703
EP - 726
AB - Through tropical normal idempotent matrices, we introduce isocanted alcoved polytopes, computing their $f$-vectors and checking the validity of the following five conjectures: Bárány, unimodality, $3^d$, flag and cubical lower bound (CLBC). Isocanted alcoved polytopes are centrally symmetric, almost simple cubical polytopes. They are zonotopes. We show that, for each dimension, there is a unique combinatorial type. In dimension $d$, an isocanted alcoved polytope has $2^{d+1}-2$ vertices, its face lattice is the lattice of proper subsets of $[d+1]$ and its diameter is $d+1$. They are realizations of $d$-elementary cubical polytopes. The $f$-vector of a $d$-dimensional isocanted alcoved polytope attains its maximum at the integer $\lfloor d/3\rfloor $.
LA - eng
KW - cubical polytope; isocanted; alcoved; centrally symmetric; almost simple; zonotope; $f$-vector; cubical $g$-vector; unimodal; flag; face lattice; log-concave sequence; tropical normal idempotent matrix; symmetric matrix
UR - http://eudml.org/doc/296940
ER -

References

top
  1. Adin, R. M., 10.1016/S0012-365X(96)83003-2, Discrete Math. 157 (1996), 3-14. (1996) Zbl0861.52007MR1417283DOI10.1016/S0012-365X(96)83003-2
  2. Adin, R. M., Kalmanovich, D., Nevo, E., 10.1090/proc/14380, Proc. Am. Math. Soc. 147 (2019), 1851-1866. (2019) Zbl07046511MR3937665DOI10.1090/proc/14380
  3. Barvinok, A., 10.4171/052, Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zürich (2008). (2008) Zbl1154.52009MR2455889DOI10.4171/052
  4. Bisztriczky, T., McMullen, P., Schneider, R., (eds.), A. I. Weiss, 10.1007/978-94-011-0924-6, NATO ASI Series. Series C. Mathematical and Physical Sciences 440. Kluwer Academic Publishers, Dordrecht (1994). (1994) Zbl0797.00016MR1322054DOI10.1007/978-94-011-0924-6
  5. Blind, G., Blind, R., 10.1007/BF02574048, Discrete Comput. Geom. 13 (1995), 321-345. (1995) Zbl0824.52013MR1318781DOI10.1007/BF02574048
  6. Blind, G., Blind, R., 10.1016/s0012-365x(97)00159-3, Discrete Math. 184 (1998), 25-48. (1998) Zbl0956.52008MR1609343DOI10.1016/s0012-365x(97)00159-3
  7. Brenti, F., 10.1090/conm/178, Jerusalem combinatorics '93 Contemporary Mathematics 178. American Mathematical Society, Providence (1994), 71-89. (1994) Zbl0813.05007MR1310575DOI10.1090/conm/178
  8. Brugallé, E., 10.1051/quadrature/2009015, Quadrature 74 (2009), 10-22 French. (2009) Zbl1202.14055DOI10.1051/quadrature/2009015
  9. Brugallé, E., Some aspects of tropical geometry, Eur. Math. Soc. Newsl. 83 (2012), 23-28. (2012) Zbl1285.14069MR2934649
  10. Butkovič, P., 10.1007/978-1-84996-299-5, Springer Monographs in Mathematics. Springer, London (2010). (2010) Zbl1202.15032MR2681232DOI10.1007/978-1-84996-299-5
  11. Puente, M. J. de la, On tropical Kleene star matrices and alcoved polytopes, Kybernetika 49 (2013), 897-910. (2013) Zbl1297.15029MR3182647
  12. Puente, M. J. de la, 10.14736/kyb-2014-3-0408, Kybernetika 50 (2014), 408-335. (2014) Zbl1321.14050MR3245538DOI10.14736/kyb-2014-3-0408
  13. Puente, M. J. de la, 10.1080/03081087.2019.1572065, Linear Multilinear Algebra 68 (2020), 2110-2142. (2020) MR4160431DOI10.1080/03081087.2019.1572065
  14. Develin, M., Santos, F., Sturmfels, B., On the rank of a tropical matrix, Combinatorial and computational geometry Mathematical Sciences Research Institute Publications 52. Cambridge University Press, Cambridge (2005), 213-242. (2005) Zbl1095.15001MR2178322
  15. Develin, M., Sturmfels, B., Tropical convexity, Doc. Math. 9 (2004), 1-27 corrigendum ibid. 9 2004 205-206. (2004) Zbl1054.52004MR2054977
  16. Grünbaum, B., 10.1007/978-1-4613-0019-9, John Wiley & Sons, London (1967). (1967) Zbl0163.16603MR0226496DOI10.1007/978-1-4613-0019-9
  17. Guillon, P., Izhakian, Z., Mairesse, J., Merlet, G., 10.1016/j.jalgebra.2015.02.026, J. Algebra 437 (2015), 222-248. (2015) Zbl1316.15030MR3351964DOI10.1016/j.jalgebra.2015.02.026
  18. Henk, M., Richter-Gebert, J., Ziegler, G. M., Basic properties of convex polytopes, Handbook of Discrete and Computational Geometry J. E. Goodman et al. CRC Press Series on Discrete Mathematics and Its Applications. CRC Press, Boca Raton (1997), 243-270. (1997) Zbl0911.52007MR1730169
  19. Jiménez, A., Puente, M. J. de la, Six combinatorial clases of maximal convex tropical polyhedra, (2012), 40 pages Available at https://arxiv.org/abs/1205.4162. (2012) 
  20. Jockusch, W., 10.1007/BF02189315, Discrete Comput. Geom. 9 (1993), 159-163. (1993) Zbl0771.52005MR1194033DOI10.1007/BF02189315
  21. Kalai, G., 10.1007/BF01788696, Graphs Comb. 5 (1989), 389-391. (1989) Zbl1168.52303MR1554357DOI10.1007/BF01788696
  22. Kalai, G., (eds.), M. G. Ziegler, 10.1007/978-3-0348-8438-9, DMV Seminar 29. Birkhäuser, Basel (2000). (2000) Zbl0944.00089MR1785290DOI10.1007/978-3-0348-8438-9
  23. Lam, T., Postnikov, A., 10.1007/s00454-006-1294-3, Discrete Comput. Geom. 38 (2007), 453-478. (2007) Zbl1134.52019MR2352704DOI10.1007/s00454-006-1294-3
  24. Litvinov, G. L., (eds.), V. P. Maslov, 10.1090/conm/377, Contemporary Mathematics 377. American Mathematical Society, Providence (2005). (2005) Zbl1069.00011MR2145152DOI10.1090/conm/377
  25. Litvinov, G. L., (eds.), S. N. Sergeev, 10.1090/conm/495, Contemporary Mathematics 495. American Mathematical Society, Providence (2009). (2009) Zbl1172.00019MR2581510DOI10.1090/conm/495
  26. Mikhalkin, G., What is ... a tropical curve?, Notices Am. Math. Soc. 54 (2007), 511-513. (2007) Zbl1142.14300MR2305295
  27. Richter-Gebert, J., Sturmfels, B., Theobald, T., 10.1090/conm/377, Idempotent Mathematics and Mathematical Physics Contemporary Mathematics 377. American Mathematical Society, Providence (2005), 289-317. (2005) Zbl1093.14080MR2149011DOI10.1090/conm/377
  28. Sanyal, R., Werner, A., Ziegler, G. M., 10.1007/s00454-008-9104-8, Discrete Comput. Geom. 41 (2009), 183-198. (2009) Zbl1168.52013MR2471868DOI10.1007/s00454-008-9104-8
  29. Schmitt, M. W., Ziegler, G. M., 10.1007/978-0-387-92714-5_22, Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination Springer, New York (2013), 279-289. (2013) Zbl1267.52002MR3087288DOI10.1007/978-0-387-92714-5_22
  30. (ed.), M. Senechal, 10.1007/978-0-387-92714-5, Springer, New York (2013). (2013) Zbl1267.52002MR3087288DOI10.1007/978-0-387-92714-5
  31. Sergeev, S., 10.1090/conm/495, Tropical and Idempotent Mathematics Contemporary Mathematics 495. American Mathematical Society, Providence (2009), 317-342. (2009) Zbl1179.15033MR2581526DOI10.1090/conm/495
  32. Sloane, N. J. A., The On-Line Encyclopedia of Integer Sequences, Available at http://oeis.org/ (2020). (2020) MR3822822
  33. Speyer, D., 10.1137/080716219, SIAM J. Discrete Math. 22 (2008), 1527-1558. (2008) Zbl1191.14076MR2448909DOI10.1137/080716219
  34. Speyer, D., Sturmfels, B., 10.1515/advg.2004.023, Adv. Geom. 4 (2004), 389-411. (2004) Zbl1065.14071MR2071813DOI10.1515/advg.2004.023
  35. Stanley, R. P., 10.1111/j.1749-6632.1989.tb16434.x, Graph Theory and Its Applications: East and West Annals of the New York Academy of Sciences 576. New York Academy of Sciences, New York (1989), 500-535. (1989) Zbl0792.05008MR1110850DOI10.1111/j.1749-6632.1989.tb16434.x
  36. Tran, N. M., 10.1016/j.jcta.2017.03.011, J. Comb. Theory, Ser. A 151 (2017), 1-22. (2017) Zbl06744864MR3663485DOI10.1016/j.jcta.2017.03.011
  37. Werner, A., Yu, J., 10.37236/3646, Electron. J. Comb. 21 (2014), Article ID 1.20, 14 pages. (2014) Zbl1302.52014MR3177515DOI10.37236/3646
  38. Yu, B., Zhao, X., Zeng, L., 10.1016/j.laa.2018.06.027, Linear Algebra Appl. 555 (2018), 321-335. (2018) Zbl1396.15022MR3834207DOI10.1016/j.laa.2018.06.027
  39. Ziegler, G. M., 10.1007/978-1-4613-8431-1, Graduate Texts in Mathematics 152. Springer, Berlin (1995). (1995) Zbl0823.52002MR1311028DOI10.1007/978-1-4613-8431-1
  40. Ziegler, G. M., 10.1090/pcms/013/10, Geometric Combinatorics IAS/Park City Mathematics Series 13. American Mathematical Society, Providence (2007). (2007) Zbl1134.52018MR2383133DOI10.1090/pcms/013/10

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