Representation Theory of Finite Groups

Representation theory of finite groups studies how group elements can be realised as invertible linear transformations of vector spaces over a field. This discipline translates abstract group operations into matrices and uses characters—trace functions of those matrices—to probe group structure. Central to the subject is the decomposition of the group algebra into simple components, organised by block theory, which reveals intricate local-to-global phenomena tied to primes dividing the group order. Classic results such as Maschke’s theorem guarantee complete reducibility in characteristic zero, while modular representation theory in positive characteristic introduces defect groups and p-blocks that capture the extent of non-semisimplicity. Induction and restriction of representations are governed by Frobenius reciprocity, linking subgroup data to ambient groups, and Deligne–Lusztig theory connects representations of finite groups of Lie type to geometry. Modern advances draw on cohomological methods and categorical frameworks like fusion systems, enriching our understanding of conjectures that relate global character counts to local subgroup structure. Beyond pure algebra, these ideas find applications in coding theory, cryptography and theoretical physics, where symmetry underpins system behaviour.

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Technical terms

Group representation: A homomorphism from a finite group to the group of invertible linear transformations of a vector space.

Character: A function assigning to each group element the trace of its action in a representation, serving as a key invariant.

Block: A primitive idempotent summand of the group algebra that organises representations sharing common local structure.

p-block: A block defined in characteristic p, associated with defect groups that measure its departure from semisimplicity.

Sylow p-subgroup: A maximal p-subgroup of a finite group, central to many local–global conjectures.

Fusion system: A category capturing conjugacy relations among the p-subgroups of a p-group, encoding block-theoretic local data.

McKay conjecture: A prediction that the number of irreducible characters of p′-degree for a finite group equals that for the normaliser of a Sylow p-subgroup.

References

1. Modular Conjectures for Direct Product of Finite Groups †. Symmetry (2023).
2. Weight conjectures for fusion systems. Advances in Mathematics (2019).
3. On Héthelyi–Külshammer’s conjecture forprincipal blocks. Algebra & Number Theory (2023).