Polytopes from Posets

Poset から作られる多面体としては, まず Stanley [Sta86] により定義された chain polytope と order polytope がある。その変種も色々定義されている。

• chain polytope と order polytope
• marked chain polytope と marked order polytope [ABS11]
• chain-order polytope [Hib+19]
• marked chain-order polytope [FF16]
• enriched order polytope tol enriched chain polytope [OT21]
• signed chain polytope と signed order polytope [BH]

他に目についたものを挙げると, 次のようになる。

• order-chain polytope [HMT17]
• marked order-chain polytope [FF16]
• maximal chain polytope [Oda]
• poset associahedron [Gal24]
• linear order polytope [BKG99; CSS13; EY23]
• partial order polytope [Fio03]
• relative poset polytope [FM24]

References

[ABS11]

Federico Ardila, Thomas Bliem, and Dido Salazar. “Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann-Vinberg polytopes as marked poset polytopes”. In: J. Combin. Theory Ser. A 118.8 (2011), pp. 2454–2462. arXiv: 1008 . 2365. url: http://dx.doi.org/10.1016/j.jcta.2011.06.004.

[BH]

Matthias Beck and Max Hlavacek. Signed Poset Polytopes. arXiv: 2311.04409.

[BKG99]

G. Bolotashvili, M. Kovalev, and E. Girlich. “New facets of the linear ordering polytope”. In: SIAM J. Discrete Math. 12.3 (1999), pp. 326–336. url: https://doi.org/10.1137/S0895480196300145.

[CSS13]

Ilya Chevyrev, Dominic Searles, and Arkadii Slinko. “On the number of facets of polytopes representing comparative probability orders”. In: Order 30.3 (2013), pp. 749–761. arXiv: 1103 . 3938. url: https://doi.org/10.1007/s11083-012-9274-0.

[EY23]

Adolfo R. Escobedo and Romena Yasmin. “Derivations of large classes of facet defining inequalities of the weak order polytope using ranking structures”. In: J. Comb. Optim. 46.3 (2023), Paper No. 19, 45. arXiv: 2008.03799. url: https://doi.org/10.1007/s10878-023-01075-w.

[FF16]

Xin Fang and Ghislain Fourier. “Marked chain-order polytopes”. In: European J. Combin. 58 (2016), pp. 267–282. arXiv: 1508.02232. url: https://doi.org/10.1016/j.ejc.2016.06.007.

[Fio03]

Samuel Fiorini. “A combinatorial study of partial order polytopes”. In: European J. Combin. 24.2 (2003), pp. 149–159. url: https://doi.org/10.1016/S0195-6698(03)00009-X.

[FM24]

Evgeny Feigin and Igor Makhlin. “Relative poset polytopes and semitoric degenerations”. In: Selecta Math. (N.S.) 30.3 (2024), Paper No. 48. arXiv: 2112.05894. url: https://doi.org/10.1007/s00029-024-00935-5.

[Gal24]

Pavel Galashin. “\(P\)-associahedra”. In: Selecta Math. (N.S.) 30.1 (2024), Paper No. 6. arXiv: 2110.07257. url: https://doi.org/10.1007/s00029-023-00896-1.

[Hib+19]

Takayuki Hibi, Nan Li, Teresa Xueshan Li, Li Li Mu, and Akiyoshi Tsuchiya. “Order-chain polytopes”. In: Ars Math. Contemp. 16.2 (2019), pp. 299–317. arXiv: 1504 . 01706. url: https://doi.org/10.26493/1855-3974.1164.2f7.

[HMT17]

Takayuki Hibi, Kazunori Matsuda, and Akiyoshi Tsuchiya. “Quadratic Gröbner bases arising from partially ordered sets”. In: Math. Scand. 121.1 (2017), pp. 19–25. arXiv: 1506.00802. url: https://doi.org/10.7146/math.scand.a-26246.

[Oda]

Shinsuke Odagiri. Faces of maximal chain polytopes. arXiv: 2108. 11721.

[OT21]

Hidefumi Ohsugi and Akiyoshi Tsuchiya. “Enriched order polytopes and enriched Hibi rings”. In: Eur. J. Math. 7.1 (2021), pp. 48–68. arXiv: 1903.00909. url: https://doi.org/10.1007/s40879-020-00403-2.

[Sta86]

Richard P. Stanley. “Two poset polytopes”. In: Discrete Comput. Geom. 1.1 (1986), pp. 9–23. url: http://dx.doi.org/10.1007/BF02187680.