Characterizing the universal rigidity of generic tensegrities

A tensegrity is a structure made from cables, struts, and stiff bars. A d-dimensional tensegrity is universally rigid if it is rigid in any dimension $$d'$$ d ′ with $$d'\ge d$$ d ′ ≥ d . The celebrated super stability condition due to Connelly gives a sufficient condition for a tensegrity to be universally rigid. Gortler and Thurston showed that super stability characterizes universal rigidity when the point configuration is generic and every member is a stiff bar. We extend this result in two directions. We first show that a generic universally rigid tensegrity is super stable. We then extend it to tensegrities with point group symmetry, and show that this characterization still holds as long as a tensegrity is generic modulo symmetry. Our strategy is based on the block-diagonalization technique for symmetric semidefinite programming problems, and our proof relies on the theory of real irreducible representations of finite groups.

Splitting fields of real irreducible representations of finite groups

We show that any irreducible representation ρ \rho of a finite group G G of exponent n n , realisable over R \mathbb {R} , is realisable over the field E ≔ Q ( ζ n ) ∩ R E≔\mathbb {Q}(\zeta _n)\cap \mathbb {R} of real cyclotomic numbers of order n n , and describe an algorithmic procedure transforming a realisation of ρ \rho over Q ( ζ n ) \mathbb {Q}(\zeta _n) to one over E E .

Study of conditions of existence of irreducible representations of finite groups by means of computer algebra system GAP

We consider the criterion of existence of exact irreducible representations of finite groups and construct the algorithm that allows to verify it by means of computer algebra system GAP.

CONNECTIVITY PROPERTIES OF MCKAY QUIVERS

We present several results on the connectivity of McKay quivers of finite-dimensional complex representations of finite groups, with no restriction on the faithfulness or self-duality of the representations. We give examples of McKay quivers, as well as quivers that cannot arise as McKay quivers, and discuss a necessary and sufficient condition for two finite groups to share a connected McKay quiver.

The generalized Gelfand–Graev characters of GLn(Fq)

International audience Introduced by Kawanaka in order to find the unipotent representations of finite groups of Lie type, gener- alized Gelfand–Graev characters have remained somewhat mysterious. Even in the case of the finite general linear groups, the combinatorics of their decompositions has not been worked out. This paper re-interprets Kawanaka's def- inition in type A in a way that gives far more flexibility in computations. We use these alternate constructions to show how to obtain generalized Gelfand–Graev representations directly from the maximal unipotent subgroups. We also explicitly decompose the corresponding generalized Gelfand–Graev characters in terms of unipotent representations, thereby recovering the Kostka–Foulkes polynomials as multiplicities.

Almost cyclic elements in cross-characteristic representations of finite groups of Lie type

This paper is a significant contribution to a general programme aimed to classify all projective irreducible representations of finite simple groups over an algebraically closed field, in which the image of at least one element is represented by an almost cyclic matrix (that is, a square matrix M of size n over a field {\mathbb{F}} with the property that there exists {\alpha\in\mathbb{F}} such that M is similar to {\operatorname{diag}(\alpha\cdot\mathrm{Id}_{k},M_{1})}, where {M_{1}} is cyclic and {0\leq k\leq n}). While a previous paper dealt with the Weil representations of finite classical groups, which play a key role in the general picture, the present paper provides a conclusive answer for all cross-characteristic projective irreducible representations of the finite quasi-simple groups of Lie type and their automorphism groups.