Page 186 - Geometric Modeling and Algebraic Geometry

188 I. Ivrissimtzis and H.-P. Seidel
and the eigenvectors (which depend only on G) are given by

v χ =(χ(g)) g∈G (10.11)

where χ is an (irreducible) character of G.
The characters of a group are complex valued functions

χ : G → C (10.12)

If G is abelian, then there is an 1-1 correspondence between the characters and the
elements of G. From Proposition 1 we can easily compute the eigenvalues and eigen-
vectors of a G-circulant matrix, as long as we know the characters of G. Given that
every finite abelian group is the direct product of cyclic groups, the following two
propositions giving the characters of a cyclic group and the characters of a direct
product group suffice for our purposes.

Proposition 2. The characters of the cyclic group Z n are given by
jk
χ j : χ j (k)= ω , ∀k ∈ Z n (10.13)
2πi
where j is an element of Z n and ω = e n
Proposition 3. If a group G has a direct product structure, then its characters are
tensor products of the characters of its components.

10.3.1 G-circulant matrices and geometric transformations

G-circulant matrices can be used to study linear transformations of a mesh. We first
index the rows and the columns of the matrix by the vertices of the mesh. Similarly
to the case of circulant matrices and polygons in section 10.2, if the position of the

new vertex P is computed by the first row of the matrix, and if P is mapped on P i
0 0
by a symmetry g of the mesh, we expect that the row giving P is the permutation of

i
the first row by g.
In a general setting, we assume that a group of symmetries G acts on the vertices
of a mesh in a way that every point is mapped on any other point by a unique ele-
ment of G. In other words we have a free transitive action. This is the case with the
n
n-dimensional cube and the prism, and the groups Z and Z 2 × Z n , respectively.
2
Notice that the choice of the group G is not necessarily unique. For example, for the
quadrilateral we can either use the Z or the Z 4 , while for the hexahedron we can
2
2
use the Z or the Z 2 × Z 4 . With either choice, the transformations and the matrices
3
2
are the same up to a labeling, but Proposition 1 yields different sets of eigenvectors
and thus, we obtain different geometric interpretations.