Colored matroids

#The assignment-vector embedding

A coloring can be represented by its zero-one vector in the coordinates \(E\times C.\) The convex hull of the vectors belonging to a family of permutations of a fixed word is its assignment polytope. Fixed row and color sums supply immediate affine equations; the exchange axiom imposes additional geometry.

#Edges and cycle moves

Cyclic exchanges give sparse directions with entries in \(\{-1,0,1\}.\) This chapter will separate the proved implication from symmetric cycle exchange to simple-cycle edge directions from the open converse for arbitrary nonempty families of permutations of a fixed word.

#Inequality descriptions

The principal question is whether every assignment polytope arising from symmetric cycle exchange admits a complete description, including its affine hull, with coefficients in \(\{-1,0,1\}.\) The site will record established classes, reductions, counterexamples to stronger formulations, and the exact remaining obligations.