Let be an irreducible crystallographic root system and its root polytope, i.e., the convex hull of . We provide a uniform construction, for all root types, of a triangulation of the facets of . We also prove that, on each orbit of facets under the action of the Weyl group, the triangulation is unimodular with respect to a root sublattice that depends on the orbit.
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.7
Keywords: Root system, root polytope, triangulation, Borel subalgebra, abelian ideal, abelian nilradical
@article{ALCO_2018__1_1_115_0, author = {Cellini, Paola}, title = {Triangulations of root polytopes}, journal = {Algebraic Combinatorics}, pages = {115--145}, publisher = {MathOA foundation}, volume = {1}, number = {1}, year = {2018}, doi = {10.5802/alco.7}, mrnumber = {3857162}, zbl = {06882337}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/alco.7/} }
Cellini, Paola. Triangulations of root polytopes. Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 115-145. doi : 10.5802/alco.7. http://archive.numdam.org/articles/10.5802/alco.7/
[1] Root polytopes and growth series of root lattices, SIAM J. Discrete Math., Volume 25 (2011), pp. 360-378 | DOI | MR | Zbl
[2] Groupes et Algèbre de Lie, Chapitres 4–6, Hermann, Paris, 1968 | Zbl
[3] Groupes et Algèbre de Lie, Chapitres 7–8, Hermann, Paris, 1975 | Zbl
[4] Root polytopes and abelian ideals, J. of Algebraic Combinatorics, Volume 39 (2014) no. 3, pp. 607-645 | DOI | MR | Zbl
[5] Polar root polytopes that are zonotopes, SLC, Volume 73 (2015), pp. 1-10 | MR | Zbl
[6] Root polytopes and Borel subalgebras, International Mathematics Research Notices, Volume 2015 (2015) no. 12, pp. 4392-4420 | DOI | MR | Zbl
[7] Compatible discrete series, Pacific J. Math., Volume 212 (2003) no. 2, pp. 201-230 | DOI | MR | Zbl
[8] ad-Nilpotent Ideals of a Borel Subalgebra, Journal of Algebra, Volume 225 (2000), pp. 130-141 | DOI | MR | Zbl
[9] Abelian ideals of Borel subalgebras and affine Weyl groups, Advances in Math., Volume 187 (2004), pp. 320-361 | DOI | MR | Zbl
[10] Root polytopes and partitions, J. of Algebraic Combinatorics, Volume 41 (2015) no. 1, pp. 49-71 | DOI | MR | Zbl
[11] Semisimple subalgebras of semisimple Lie algebras, Trans. Am. Math. Soc. Ser.2, Volume 6 (1957), pp. 111-244 Mat. Sb. (N.S), 30(72):2 (1952), 349–462 | MR | Zbl
[12] Combinatorics of hypergeometric functions associated with positive roots, Arnold-Gelfand Mathematical Seminars: Geometry and Singularity Theory, Birkhäuser, Boston, 1996, pp. 205-221 | MR | Zbl
[13] Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York-Heidelberg-Berlin, 1972 | DOI | MR | Zbl
[14] Reflection Groups and Coxeter Groups, Cambridge Univ. Press, Cambridge, 1990 | DOI | MR | Zbl
[15] On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math., Volume 25 (1965), pp. 5-48 | DOI | Numdam | MR | Zbl
[16] Eigenvalues of a Laplacian and commutative Lie subalgebras, Topology, Volume 3 (1965) no. Supplement 2, pp. 147-159 | DOI | MR | Zbl
[17] The set of Abelian ideals of a Borel subalgebra, Cartan decomposition, and discrete series representations, International Mathematics Research Notices, Volume 1998 (1998) no. 5, pp. 225-252 | DOI | MR | Zbl
[18] Root polytopes, triangulations, and the subdivision algebra, I, Trans. Amer. Math. Soc., Volume 363 (2011), pp. 4359-4382 | DOI | MR | Zbl
[19] Root polytopes, triangulations, and the subdivision algebra, II, Trans. Amer. Math. Soc., Volume 363 (2011), pp. 6111-6141 | DOI | MR | Zbl
[20] Abelian ideals of a Borel subalgebra and long positive roots, International Mathematics Research Notices, Volume 2003 (2003) no. 35, pp. 1889-1913 | DOI | MR | Zbl
[21] Abelian ideals in a Borel subalgebras of a complex simple Lie algebra, Invent. Math., Volume 156 (2004), pp. 175-221 | DOI | MR | Zbl
[22] On certain commutative subalgebras of a universal enveloping algebra, Math. USSR Izv., Volume 36 (1991) no. 1, pp. 1-22 | DOI | MR | Zbl
Cited by Sources: