| Title:
|
Identical groups of automorphisms of the permutohedron (English) |
| Author:
|
Studený, Milan |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
62 |
| Issue:
|
4 |
| Year:
|
2026 |
| Pages:
|
755-781 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper concerns a remarkable polytope, known as the permutohedron, studied already in 1911 by Schoute. In general, the symmetry group of a polytope, which is the group of its automorphisms, does depend on the considered class of isomorphisms. One can distinguish isometries, affine isomorphisms, face-lattice order isomorphisms, and graph isomorphisms of a polytope. We show that, in case of the permutohedron, these four groups of automorphisms are (group) isomorphic; we also provide their group-theoretical description. Specifically, if $N$ is the basic set, $|N|\geq 3$, then (each of) the automorphism group(s) is the direct product of the group of permutations of $N$ (= the symmetric group $S_{N}$) and of a two-element group generated by central symmetry. The result confirms an unproved statement formulated by Crisman in 2011. (English) |
| Keyword:
|
polytope |
| Keyword:
|
permutohedron |
| Keyword:
|
automorphism |
| Keyword:
|
symmetry group |
| MSC:
|
20B25 |
| MSC:
|
52B15 |
| DOI:
|
10.14736/kyb-2026-4-0755 |
| . |
| Date available:
|
2026-10-07T10:25:02Z |
| Last updated:
|
2026-10-07 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153782 |
| . |
| Reference:
|
[1] Billera, L. J., Sarangarajan, A.: The combinatorics of permutation polytopes..In: Formal Power Series and Algebraic Combinatorics, DISMACS Series in Discrete Mathematics and Theoretical Computer Science 24, AMS, Providence 1996, pp. 1-23. |
| Reference:
|
[2] Birkhoff, G.: Lattice Theory. Third edition..AMS Colloquium Publications 25, AMS, Providence 1995. |
| Reference:
|
[3] Brøndsted, A.: An Introduction to Convex Polytopes..Graduate Texts in Mathematics 90, Springer, New York 1983. |
| Reference:
|
[4] Crisman, K.-D.: The symmetry group of the permutahedron..Coll. Math. J. 42 (2011), 2, 136-139. |
| Reference:
|
[5] Kalai, G.: A simple way to tell a simple polytope from its graph..J. Combinator. Theory A 49 (1988), 381-383. |
| Reference:
|
[6] Mohammadi, F., Uhler, C., Wang, C., Yu, J.: Generalized permutohedra from probabilistic graphical models..SIAM Journal of Discrete Mathematics 32 (2018), 1, 640-93. |
| Reference:
|
[7] Morton, J., Pachter, L., Shiu, A., Sturmfels, B., Wienand, O.: Convex rank tests and semigraphoids..SIAM J. Discrete Math. 23 (2009), 3, 1117-1134. |
| Reference:
|
[8] Postnikov, A., Reiner, V., Williams, L.: Faces of generalized permutohedra..Documenta Mathematica 13 (2008), 207-273. |
| Reference:
|
[9] Schoute, P. H.: Analytic treatment of the polytopes regularly derived from the regular polytopes..Verhandelingen der Koninklijke Akademie van Wetenschappen 11 (3), Johannes Müller, Amsterdam 1911, 87 pages. |
| Reference:
|
[10] Studený, M., Kratochvíl, V.: Graphical view on linear extensions of finite posets..Submitted to Algebraic Statistics (2026), available on |
| Reference:
|
[11] Studený, M.: Semi-graphoids viewed as collections of posets..In: Proceedings of IPMU 2026, part III., 366-397, Springer. |
| Reference:
|
[12] Ziegler, G. M.: Lectures on Polytopes..Graduate Texts in Mathematics 152, Springer, New York 1995. Zbl 0823.52002 |
| . |
