Symmetry and Heilbronner modes

Scalar counting relations for sets of structural components can seen as expressions for characters under the identity operation of more general relations between representations of those sets. For example, the Euler relation in topology can be generalised to connect not only the numbers of edges, vertices and faces of a polyhedron, but also various symmetries associated with the structural features. The well-known Euler theorem 

Similarly, in mechanics, the extended Maxwell condition for rigidity of bar and joint assemblies can be used to generate a relation between the permutation representations of the bars and joints and those of the states of self-stress and mechanisms of the assembled framework [17a,b]. 

In the present case, the extension of the scalar counting rules for n(S) to symmetry theorems is straightforwardly achieved by replacing n(S), n(v), n(e) n( ) by the permutation representations F,T(S), r,T(v), r,T(e) and ra(vj) [13]. A permutation representation /j/v), of a set of objects x has character y(R) under operation (R) of the symmetry group of the undistorted framework, where x(R) is equal to the number of objects wnshifted under operation R. The subscript a is often dropped if there is no danger of confusion. With these replacements, equation (4) becomes 

The translation of equation (5) into symmetry-extended form requires more discussion. A bipartite graph is one in which the vertices can be partitioned into two sets, starred and unstarred, say, such that every starred vertex has only unstarred neighbours and vice versa. The origin of the +1 in the scalar equation (5) is that it is possible to define a vector of coefficients on the vertices of a bipartite graph such that 

The symmetry corresponding to the null constraint on the Heilbronner modes is the representation of the vector of 1 coefficients. This is a one-dimensional (ID) irreducible representation, 7. which has character +1 under those operations that permute vertices only within their starred and unstarred sets, and character - 1 under all the other operations, those that permute starred with unstarred vertices. The symmetry T is that of the inactive vertex constraint. With it, the scalar relation n(S) = n(e) — n(v) + 1 becomes 

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