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Alcoved polytopes. II. (English) Zbl 1414.52007
Kac, Victor G. (ed.) et al., Lie groups, geometry, and representation theory. A tribute to the life and work of Bertram Kostant. Cham: Birkhäuser. Prog. Math. 326, 253-272 (2018).
Summary: This is the second of two papers where we study polytopes arising from affine Coxeter arrangements. Our results include a formula for their volumes, and also compatible definitions of hypersimplices, descent numbers and major index for all Weyl groups. We give a $$q$$-analogue of Weyl’s formula for the order of the Weyl group. For $$A_n$$, $$C_n$$ and $$D_4$$, we give a Gröbner basis which induces the triangulation of alcoved polytopes.
For Part I, see [the authors, Discrete Comput. Geom. 38, No. 3, 453–478 (2007; Zbl 1134.52019)].
For the entire collection see [Zbl 1412.22001].

##### MSC:
 52B12 Special polytopes (linear programming, centrally symmetric, etc.) 52B11 $$n$$-dimensional polytopes 05E10 Combinatorial aspects of representation theory 20F55 Reflection and Coxeter groups (group-theoretic aspects) 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
##### Keywords:
convex polytopes; affine Weyl group; hypersimplex
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