Toric orbifolds associated with partitioned weight polytopes in classical types

• Tatsuya Horiguchi ^[1] ; Mikiya Masuda ^[3] ; John Shareshian ^[4] ; Jongbaek Song ^[2]
  1. [1] National Institute Of Technology

     National Institute Of Technology

     Japón

     
  2. [2] Pusan National University

     Pusan National University

     Corea del Sur

     
  3. [3] Osaka Central Advanced Mathematical Institute, Osaka Metropolitan University, Japan
  4. [4] Osaka Central Advanced Mathematical Institute, Osaka Metropolitan University, USA

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 5, 2024
• Idioma: inglés
• DOI: 10.1007/s00029-024-00977-9
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• Resumen

  • Given a root system Φ of type An, Bn, Cn, or Dn in Euclidean space E, let W be the associated Weyl group. For a point p ∈ E not orthogonal to any of the roots in Φ, we consider the W-permutohedron PW , which is the convex hull of the Worbit of p. The representation of W on the rational cohomology ring H∗(XΦ) of the toric variety X Φ associated to (the normal fan to) PW has been studied by various authors. Let {s1,...,sn} be a complete set of simple reflections in W. For K ⊆ [n], let WK be the standard parabolic subgroup of W generated by {sk : k ∈ K}. We show that the fixed subring H∗(XΦ)WK is isomorphic to the cohomology ring of the toric variety XΦ (K) associated to a polytope obtained by intersecting PW with half-spaces bounded by reflecting hyperplanes for the given generators of WK . We also obtain explicit formulas for h-vectors of these polytopes. By a result of Balibanu–Crooks, the cohomology rings H∗(X Φ (K)) are isomorphic with cohomology rings of certain regular Hessenberg varieties.

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