M Theory Lesson 304

The special $M$ matrices map the h-vector to the face vector as follows. First turn an h-vector into a g-vector by taking differences of entries. For example, $(1,3,1)$ will become $(1,2)$, because we imagine a dummy zero to the left of the h-vector. The vectors are shortened, because it is unnecessary to include the information that there is one cell for the polytope in the highest dimension. In dimension $d$, the $M$ matrix is defined by which leads to the odd convention of writing, for $d = 3$, The pentagon formula is given by which shows that a pentagon has $5$ vertices and $5$ edges. Our favourite associahedron in dimension $3$ is described by These $M$ matrices work for any polytopes. Another example is the cyclohedron face vector, given by Note that the usual description of the $M_d$ matrices does not define them as square matrices, but square matrices work fine.