cc_math_roberta_ep01_definiendum

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• Let P be a G-poset. The strong compatibility graph of P, denoted by C_P, is the graph C_P with vertex set P, and two elements x, y∈ P are adjacent if there is an element g∈ G∖{e} such that x and g· y are comparable in P and y∉ [x], where [x]={g· x : g∈ G}.Copy
• A simplicial map f : X → Y between simplicial complexes X and Y is a map which sends vertices to vertices, and whenever vertices v_0, ..., v_k∈ X span a simplex σ of X then their images span a simplex τ of Y and we have f(σ) = τ. Therefore a simplicial map is determined by its values on the vertex set of X. A simplicial map is nondegenerate if it is injective on each simplex.Copy
• A vertical strip is a skew shape (either partition or composition) whose diagram contains at most one cell per row. A horizontal strip is a skew shape whose diagram contains at most one cell per column.Copy
• Permutations ω and π are C-equivalent , denoted ωπ, if ωQ∼π and (P(ω))=(P(π)). We denote the C-equivalence class of the permutation π by [π]_C. The rectified shape of [π]_C is (P(π)). Two SCT T and T' are C-equivalent T T' if they have the same skew shape and w_col(T) w_col(T'). We denote the C-equivalence class of T by [T]_C. The rectified shape of [T]_C is (T). We say that [T]_C is complete if { w_col(T') : T'∈ [T]_C } = [w_col(T)]_C.Copy