Mathematics / advanced
Representations and characters of finite groups
Translate finite symmetry into linear actions, characters, induction, and decompositions with concrete calculations.
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Module 01 advanced
Representations and characters of finite groups
Translate finite symmetry into linear actions, characters, induction, and decompositions with concrete calculations.
- 01
Linear representations and intertwiners
Turn group multiplication into compatible linear maps and identify the maps that preserve an action.
29 min - 02
Invariant subspaces and Maschke’s theorem
Average an arbitrary projection to build an invariant complement and identify the precise characteristic obstruction.
31 min - 03
Schur’s lemma and irreducibility
Use invariant kernels and eigenvalues to constrain intertwiners, with a real-field counterexample to the scalar conclusion.
30 min - 04
Character inner products and orthogonality
Derive multiplicities from an averaged Hom-space action and compute a complete decomposition from three trace values.
32 min - 05
Regular representations and group algebras
Use the regular character to count every irreducible and identify the matrix blocks inside the complex group algebra.
32 min - 06
Character tables and class functions
Assemble orthogonal character rows, check completeness, and distinguish arbitrary class functions from genuine characters.
31 min - 07
Tensor products and dual representations
Combine symmetry actions, track inverse-transpose duals, and separate symmetric from alternating tensor sectors.
33 min - 08
Induced representations and Frobenius reciprocity
Extend a subgroup action with explicit left-coset conventions and compute multiplicities by restriction.
35 min - 09
Symmetric-group representations and Young diagrams
Label complex symmetric-group irreducibles by partitions and compute dimensions using actual hook products.
35 min - 10
Fourier analysis on finite groups
Encode functions as irreducible matrix blocks with explicit transform, convolution, inversion, and energy conventions.
36 min