Algebra 2. The symmetric groups Sn by René Schoof

By René Schoof

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A polytope has an interior if and only if it is full dimensional. For instance the interior of C2 is the set int(C2 ) = {(x1 , x2 ) ∈ R2 : 0 < x1 < 1, 0 < x2 < 1}. The line segment conv({(1, 0), (0, 1)}) ⊂ R2 does not have an interior since there is no point on this segment such that a 2-dimensional ball centered at this point will be contained in the line segment. However, this line segment does have an interior if we think of it as a polytope in its affine hull, where it is a full dimensional polytope.

Schlegel Diagrams 31 We are interested in polyhedral complexes since a polytope P gives rise to some natural polyhedral complexes which play an important role in the study of polytopes. 17. Let P be a polytope. (1) The complex C(P ) of the polytope P is the polyhedral complex of all faces of P . The face poset of C(P ) is the face lattice of P . (2) The boundary complex ∂(C(P )) is the polyhedral complex of all the proper faces of P along with the empty face. (3) A polytopal subdivision of P is a polyhedral complex C with support P in which all the polyhedra are polytopes.

In this chapter, we will see that we can actually visualize even higher dimensional polytopes as long as they do not have too many vertices. We do this via a tool called the Gale diagram of the polytope. Consider n points v1 , . . , vn in Rd−1 whose affine hull has dimension d − 1, and the matrix A := 1 v1 1 v2 ··· ··· 1 vn ∈ Rd×n . A basic fact of affine linear algebra is that the vectors v1 , . . , vn are affinely independent (see below) if and only if the vectors (1, v1 ), . . , (1, vn ) are linearly independent.

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