Equivariant Cosheaves and Finite Group Representations in Graphic Statics
arxiv(2024)
摘要
This work extends the theory of reciprocal diagrams in graphic statics to
frameworks that are invariant under finite group actions by utilizing the
homology and representation theory of cellular cosheaves, recent tools from
applied algebraic topology. By introducing the structure of an equivariant
cellular cosheaf, we prove that pairs of self-stresses and reciprocal diagrams
of symmetric frameworks are classified by the irreducible representations of
the underlying group. We further derive the symmetry-aligned Euler
characteristics of a finite dimensional equivariant chain complex, which for
the force cosheaf yields a new formulation of the symmetry-adapted Maxwell
counting rule for detecting symmetric self-stresses and kinematic degrees of
freedom in frameworks. A freely available program is used to implement the
relevant cosheaf homologies and illustrate the theory with examples.
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