Rate My Professor Corey Carito

In June 2022, Corey Carito completed her Master of Arts thesis at San Francisco State University, titled 'Signed Permutations and the Signed Permutahedron.' Advised by Dr. Emily Clader and with committee members Serkan Hosten and Alexander Schuster, the thesis delves into geometric and combinatorial structures associated with permutation groups. It begins by recalling the permutahedron Δ_n, a convex polytope in ℝ^n whose vertices are in one-to-one correspondence with the elements of the symmetric group S_n, with edges connecting vertices that differ by adjacent transpositions. The work extends this construction to the broader family of groups G(r,n), which generalize S_n = G(1,n) as finite subgroups of complex reflection groups. These groups consist of n×n matrices with exactly one nonzero entry per row and column, where that entry is an r-th root of unity. Focusing specifically on the case r=2, which corresponds to signed permutations incorporating sign flips (±1), Carito explicitly constructs the signed permutahedron Δ_{2,n}. She proves Theorem 1: the vertices of Δ_{2,n} are in bijection with the elements of G(2,n), realized as signed permutations of the standard basis vectors. Theorem 2 establishes that two vertices are adjacent if and only if the corresponding group elements differ by right multiplication by an element of the generating set T, consisting of adjacent transpositions and a sign flip on the first coordinate. The polytope is defined by the inequalities ∑_{i∈I} a_i x_i ≤ δ_{|I| } for subsets I ⊆ {1,...,n} and signs a_i ∈ {±1}, where δ_k = inom{n}{k} k!/2 or equivalent summation. Centered at the origin, Δ_{2,n} has dimension n and features vertices that are signed permutations of (1,2,...,n). Detailed examples for n=2 (8 vertices, octagonal faces) and n=3 (48 vertices) illustrate the structure, confirming the combinatorial encoding of faces analogous to lower-dimensional permutahedra. This research establishes Δ_{2,n} as the natural polytope counterpart to the permutahedron for signed permutations, with potential extensions to higher r or face lattices.