Séminaire Lotharingien de Combinatoire, 93B.8 (2025), 12 pp.
Allen Knutson, Mario Sanchez and Melissa Sherman-Bennett
Permutahedral Subdivisions and Class Formulas from Coxeter Elements
Abstract.
The Bruhat interval polytope Pu,v is the convex hull of
the Bruhat interval [u,v] in Sn, where each permutation z is
interpreted as a vector (z(1), ..., z(n)) ∈ Rn. One
example is the permutahedron, which is
Pe,wo. We
explore the combinatorics of regular subdivisions of the permutahedron
into Bruhat interval polytopes. In particular, we identify 2n-2
finest such subdivisions, one for each Coxeter element of
Sn. For each subdivision, we provide an explicit height vector and
determine exactly the constituent Bruhat interval polytopes. We also
obtain an algebro-geometric counterpart of these subdivisions: for
each Coxeter element c ∈ Sn, we obtain a formula for the class of
the permutahedral variety as a sum of Richardson classes, where the
terms in the sum exactly correspond to maximal polytopes in the
subdivision. We further obtain formulas for the cohomology class of
more general subvarieties of G/B which include Hessenberg
varieties.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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