Linear Representations of Finite Groups

Serre, J. (1977). Linear Representations of Finite Groups. Springer.

Type

Book

Intellectual base

Review or monograph

Year

1977

DOI

10.1007/978-1-4684-9458-7

ISBN

9781468494600

Link

https://doi.org/10.1007/978-1-4684-9458-7

Cited from

mathematics

Cited by

10 citations across 10 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Equivariance
  • Formally, f(g·x) = g·f(x), a relation whose modern abstract form descends from the representation-theoretic notion of an intertwining map between group actions, as Serre (1977) develops in his canonical treatment of linear representations.

    This sourceCanonical treatment of group representations, intertwining (equivariant) linear maps, and Schur's lemma.

Domain-specific

• Character Theory
  • This basis-invariant compression often retains enough information to decide equivalence and decompose finite-dimensional complex representations.
• Frobenius–Schur indicator
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Induced representation
  • The construction does not literally extend the same action on the same vector space; it builds a generally larger \(G\)-module with a universal relation to restriction.
• Minimal Polynomial (Linear Algebra)
  • Elementary and abstract linear algebra. It unifies annihilating equations, Cayley–Hamilton consequences, diagonalizability, projections, involutions, nilpotence, and cyclic operators. Matrix canonical forms. It is the largest invariant factor in rational canonical form and encodes the maximum Jordan-chain length for each eigenvalue when Jordan form exists over the field. Primary decomposition. Factorization into coprime primary powers separates invariant subspaces and supports projection operators obtained through Bézout identities. Representation theory. A group element acting linearly has a minimal polynomial constrained by its order or defining relations; repeated factors reveal possible nonsemisimple behavior in the field's characteristic

    This sourceSerre supplies both halves in their standard form: an element of finite order n is annihilated by x^n - 1, and complete reducibility holds away from the modular case, failing in characteristic p dividing the group order where Brauer's theory takes over; the repeated-factor reading of that failure is the article's own.

• Quaternionic representation
  • A quaternionic representation is self-conjugate but admits an invariant skew-symmetric rather than symmetric structure, characterized for irreducibles by Frobenius–Schur indicator −1.
• Real Representation
  • Analogous definitions apply to algebras and Lie algebras.
• Representation ring
  • The ring depends on the group, base field and representation category; in modular characteristic semisimplicity may fail and Green-ring conventions must distinguish split from exact Grothendieck relations.
• Symplectic Representation
  • For a finite group with irreducible character $\chi$, the Frobenius–Schur indicator distinguishes absence of a self-dual form from symmetric and alternating types.

    This sourceStandard source for invariant forms, characters, and Frobenius–Schur theory.

• Trivial Representation
  • The one-dimensional version over the base field is the canonical irreducible trivial representation; higher-dimensional trivial representations are direct sums of it.

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