×

zbMATH — the first resource for mathematics

A continuous family of marked poset polytopes. (English) Zbl 1445.52009
To a finite poset \((P, \preceq)\) Stanley [R. P. Stanley, Discrete Comput. Geom. 1, 9–23 (1986; Zbl 0595.52008)] famously associated two lattice polytopes: the order polytope \[ O_{P, \preceq} := \left\{ x \in [0,1]^P :\, p \preceq q \ \Rightarrow \ x_p \leq x_q \right\} \] and the chain polytope \[ C_{P, \preceq} := \left\{ x \in \mathbb{R}_{\ge 0}^P :\, p_1 \prec \cdots \prec p_m \text { maximal chain in } P \ \Rightarrow \ x_{p_1} + \cdots + x_{p_m} \le 1 \right\} . \] These polytopes have seen much research activity and were generalized to marked poset polytopes [F. Ardila et al., J. Comb. Theory, Ser. A 118, No. 8, 2454–2462 (2011; Zbl 1234.52009)], where now some of the poset elements are labelled with fixed integers. The paper under review defines and studies a continuous family of polytopes with marked order and chain polytope at extreme ends. The authors study the Ehrhart theory (lattice-point counting formulas) and the combinatorial type of these polytopes, using tools from discrete and tropical geometry.

MSC:
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
06A07 Combinatorics of partially ordered sets
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] F. Ardila, T. Bliem, and D. Salazar, Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann-Vinberg polytopes as marked poset polytopes, J. Combin. Theory Ser. A, 118 (2011), pp. 2454-2462, https://doi.org/10.1016/j.jcta.2011.06.004. · Zbl 1234.52009
[2] T. Backhaus and C. Desczyk, PBW filtration: Feigin-Fourier-Littelmann modules via Hasse diagrams, J. Lie Theory, 25 (2015), pp. 815-856. · Zbl 1359.17018
[3] G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier, and M. Reineke, Linear degenerations of flag varieties, Math. Z., 287 (2017), pp. 615-654, https://doi.org/10.1007/s00209-016-1839-y. · Zbl 1388.14145
[4] X. Fang and G. Fourier, Marked chain-order polytopes, European J. Combin., 58 (2016), pp. 267-282, https://doi.org/10.1016/j.ejc.2016.06.007. · Zbl 1343.05163
[5] E. Feigin, G. Fourier, and P. Littelmann, PBW filtration and bases for irreducible modules in type \(A_n\), Transform. Groups, 16 (2011), pp. 71-89, https://doi.org/10.1007/s00031-010-9115-4. · Zbl 1237.17011
[6] E. Feigin, G. Fourier, and P. Littelmann, Favourable modules: Filtrations, polytopes, Newton-Okounkov bodies and flat degenerations, Transform. Groups, 22 (2017), pp. 321-352, https://doi.org/10.1007/s00031-016-9389-2. · Zbl 06793800
[7] E. Feigin and I. Makhlin, Vertices of FFLV polytopes, J. Algebraic Combin., 45 (2017), pp. 1083-1110, https://doi.org/10.1007/s10801-016-0735-1. · Zbl 1370.05218
[8] A. Fink and F. Rincón, Stiefel tropical linear spaces, J. Combin. Theory Ser. A, 135 (2015), pp. 291-331, https://doi.org/10.1016/j.jcta.2015.06.001. · Zbl 1321.15044
[9] G. Fourier, Marked poset polytopes: Minkowski sums, indecomposables, and unimodular equivalence, J. Pure Appl. Algebra, 220 (2016), pp. 606-620, https://doi.org/10.1016/j.jpaa.2015.07.007. · Zbl 1328.52007
[10] I. M. Gelfand and M. L. Tsetlin, Finite-dimensional representations of the group of unimodular matrices, Dokl. Akad. Nauk SSSR, 71 (1950), pp. 825-828. · Zbl 0037.15301
[11] N. Gonciulea and V. Lakshmibai, Degenerations of flag and Schubert varieties to toric varieties, Transform. Groups, 1 (1996), pp. 215-248, https://doi.org/10.1007/BF02549207. · Zbl 0909.14028
[12] T. Hibi and N. Li, Unimodular equivalence of order and chain polytopes, Math. Scand., 118 (2016), pp. 5-12, https://doi.org/10.7146/math.scand.a-23291. · Zbl 1335.52026
[13] T. Hibi, N. Li, Y. Sahara, and A. Shikama, The numbers of edges of the order polytope and the chain polytope of a finite partially ordered set, Discrete Math., 340 (2017), pp. 991-994, https://doi.org/10.1016/j.disc.2017.01.005. · Zbl 1367.52012
[14] K. Jochemko and R. Sanyal, Arithmetic of marked order polytopes, monotone triangle reciprocity, and partial colorings, SIAM J. Discrete Math., 28 (2014), pp. 1540-1558, https://doi.org/10.1137/130944849. · Zbl 1321.52019
[15] C. Pegel, The face structure and geometry of marked order polyhedra, Order, 35 (2017), pp. 467-488, https://doi.org/10.1007/s11083-017-9443-2. · Zbl 1408.52014
[16] R. P. Stanley, Two poset polytopes, Discrete Comput. Geom., 1 (1986), pp. 9-23, https://doi.org/10.1007/BF02187680. · Zbl 0595.52008
[17] G. M. Ziegler, Lectures on Polytopes, Grad. Texts in Math. 152, Springer, New York, 1995, https://doi.org/10.1007/978-1-4613-8431-1.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.