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1. The arithmetic of Coxeter permutahedra

   https://doi.org/10.18257/raccefyn.1189  

   Ardila, Federico; Beck, Matthias; McWhirter, Jodi (December 2020, Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales)

   null (Ed.)

   Ehrhart theory mesures a polytope P discretely by counting the lattice points inside its dilates P, 2P, 3P, ..... We compute the Ehrhart theory of four families of polytopes of great importance in several areas of mathematics: the standard Coxeter permutahedra for the classical Coxeter groups An, Bn, Cn, Dn. A central tool, of independent interest, is a description of the Ehrhart theory of a rational translate of an integer projection of a cube. 

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2. Proper elements of Coxeter groups

   https://doi.org/10.1007/s40879-024-00746-0  

   Balogh, József; Brewster, David; Hodges, Reuven (June 2024, European Journal of Mathematics)
3. Rational noncrossing Coxeter–Catalan combinatorics

   https://doi.org/10.1112/plms.12643  

   Galashin, Pavel; Lam, Thomas; Trinh, Minh‐Tâm; Williams, Nathan (October 2024, Proceedings of the London Mathematical Society)

   Abstract We solve two open problems in Coxeter–Catalan combinatorics. First, we introduce a family of rational noncrossing objects for any finite Coxeter group, using the combinatorics of distinguished subwords. Second, we give a type‐uniform proof that these noncrossing Catalan objects are counted by the rational Coxeter–Catalan number, using the character theory of the associated Hecke algebra and the properties of Lusztig's exotic Fourier transform. We solve the same problems for rational noncrossing parking objects. 

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4. Convex cocompactness for Coxeter groups

   https://doi.org/10.4171/jems/1539  

   Danciger, Jeffrey; Guéritaud, François; Kassel, Fanny; Lee, Gye-Seon; Marquis, Ludovic (February 2025, Journal of the European Mathematical Society)

   We investigate representations of Coxeter groups into\mathrm{GL}(n,\mathbb{R})as geometric reflection groups which are convex cocompact in the projective space\mathbb{P}(\mathbb{R}^{n}). We characterize which Coxeter groups admit such representations, and we fully describe the corresponding spaces of convex cocompact representations as reflection groups, in terms of the associated Cartan matrices. The Coxeter groups that appear include all infinite word hyperbolic Coxeter groups; for such groups, the representations as reflection groups that we describe are exactly the projective Anosov ones. We also obtain a large class of nonhyperbolic Coxeter groups, thus providing many examples for the theory of nonhyperbolic convex cocompact subgroups in\mathbb{P}(\mathbb{R}^{n})developed by Danciger–Guéritaud–Kassel (2024). 

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5. Bargain Hunting in a Coxeter Group

   https://doi.org/10.1007/s00026-023-00670-2  

   Lewis, Joel Brewster; Tenner, Bridget Eileen (October 2023, Annals of Combinatorics)