Linear Representations of Finite Groups

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

Type

Book

Intellectual base

Review or monograph

Year

1977

Volume

42

Pages cited

12-13

DOI

10.1007/978-1-4684-9458-7

ISBN

9781468494600

DOI registrant

Springer Nature

Link

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

Cited from

mathematics

Cited by

13 citations across 12 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.

    Supported in partVerified against the publisher's abstract

    Serre's own book abstract confirms GTM 42 is a canonical treatment of linear representations and characters, but it never mentions intertwining or equivariant maps, so it backs the book's standing, not the provenance clause.

    “This book consists of three parts, rather different in level and purpose: The first part was originally written for quantum chemists. It describes the correspondence, due to Frobenius, between linear representations and charac ters.”

    Read by an automated reader; not a human review. How support was checked

Domain-specific

• Character Theory
  • This basis-invariant compression often retains enough information to decide equivalence and decompose finite-dimensional complex representations.
• Conjugacy class sum
  • Class sums are invariant under conjugation, lie in the group-algebra center and form its basis over a splitting field under standard hypotheses, connecting character theory with central multiplication constants.
• 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

    Supported in partVerified against the source

    Read by an automated reader; not a human review. How support was checked

• 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.
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• 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.
• Subrepresentation
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• 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.

Verification

Does it exist? Confirmed. This work's DOI resolves to a registered record, which fixes its identity. That is all it fixes.

Does it back the claim? Read against the text for 2 of 13 citations: 2 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.

Support is checked per citation rather than per work — the same source can be cited soundly in one article and wrongly in another. Per-citation recording began recently, so a citation with no recorded check is a gap in the record rather than evidence it went unchecked.

See how references were verified.

Links previously used in the corpus

Before the registry existed this work was also linked 1 other way.

• https://link.springer.com/book/10.1007/978-1-4684-9458-7 ×1

Registry ID ref:30df32d94c54 · see in the full table