Representation Theory¶
Job: linear actions of groups, algebras, and Lie objects — characters, complete reducibility, and the structure of finite-dimensional modules for semisimple Lie algebras and compact groups.
Core¶
Representation theory converts symmetry into linear algebra: \(G\to\operatorname{GL}(V)\) or \(\mathfrak{g}\to\mathfrak{gl}(V)\), then decomposes \(V\) and computes invariants. Finite groups over \(\mathbb{C}\), complex semisimple Lie algebras, and compact Lie groups form the classical triad.
| Layer | Content |
|---|---|
| Finite groups \(/\mathbb{C}\) | characters, orthogonality, Maschke, Artin–Wedderburn |
| Lie algebras | \(\mathfrak{sl}_2\), roots, weights, Cartan–Weyl classification sketch |
| Lie groups | \(\operatorname{Lie}\) functor, exp, Peter–Weyl for compact groups |
Reading map¶
| Page | Focus |
|---|---|
| Foundations | finite groups over \(\mathbb{C}\): characters, orthogonality, complete reducibility |
| Lie algebras | \(\mathfrak{sl}_2\), root systems, finite-dim semisimple modules |
| Lie groups | Lie algebra of a group, exponential, Peter–Weyl one-liner |
Dependency order¶
- Linear algebra + groups — group theory, modules.
- Finite group characters (no analysis).
- Lie algebras as algebraic objects; then Lie groups / compact analysis.
- Optional: algebraic groups, geometric representation theory — algebraic geometry.
Interfaces¶
| Area | Contact |
|---|---|
| Abstract algebra | \(k[G]\)-modules; semisimple algebras |
| Homological algebra | group cohomology \(H^*(G,M)\); Ext of modules |
| Category theory | monoidal categories of representations; Tannaka |
| Algebraic geometry | flag varieties, Borel–Weil–Bott |
| Higher Teichmüller theory | character varieties; Anosov / Hitchin / maximal representations of surface groups into semisimple \(G\) |
| Geometric Satake | the dual group \(\widehat{G}\) constructed (Tannakian) from \(G(\mathcal{O})\)-equivariant perverse sheaves on the affine Grassmannian |
| Algebraic topology | classifying spaces, characteristic classes of bundles |
Notation (paper-standard)¶
- \(\rho:G\to\operatorname{GL}(V)\), \(\chi_\rho(g)=\operatorname{tr}\rho(g)\), \(\langle\chi,\psi\rangle\)
- \(\mathfrak{g}\), \(\mathfrak{h}\), \(\Phi\), \(\Lambda\), \(V(\lambda)\), \(\mathfrak{sl}_n\), \(\mathfrak{g}_\alpha\)
- \(\operatorname{Lie}(G)\), \(\exp:\mathfrak{g}\to G\), \(L^2(G)\)
See it¶
Langlands / automorphic forms (pointer). Finite-dimensional representation theory of reductive groups and their duals organizes \(L\)-functions; the same root data that classify \(\mathfrak{g}\)-modules reappear in dual groups \({}^LG\). Compact-group harmonic analysis (Peter–Weyl) is the model for spectral decomposition of \(L^2(\Gamma\backslash G)\).
Quantum mechanics / particle physics. Angular momentum is \(\mathfrak{su}(2)\cong\mathfrak{so}(3)\) representation theory: spin \(j\) \(\leftrightarrow\) irreducible of dimension \(2j+1\), Clebsch–Gordan = tensor product decomposition — identical to \(\mathfrak{sl}_2(\mathbb{C})\) weight theory after complexification.
Check¶
Why does complete reducibility fail for infinite groups or modular characteristic?
\(\mathbb{Z}\) acting on \(\mathbb{C}^2\) by unipotent Jordan block: invariant line, no invariant complement. Over field char \(p\) dividing \(|G|\), Maschke fails: \(k[G]\) not semisimple (augmentation ideal).
What is the dimension of the irrep of \(S_3\) that is not the sign or trivial?
Degree 2 (standard representation on \(x+y+z=0\) in \(\mathbb{C}^3\)); character table has three irreps of dims \(1,1,2\) and \(1^2+1^2+2^2=6=|S_3|\).
Pitfalls¶
- Character determines irrep over \(\mathbb{C}\) for finite groups; false for general algebras without care.
- Real vs complex forms of Lie algebras: \(\mathfrak{su}(2)\) vs \(\mathfrak{sl}_2(\mathbb{R})\) vs \(\mathfrak{sl}_2(\mathbb{C})\) — different real rep theories.
- Projective representations (quantum mechanics) are linear reps of central extensions — not the same as \(\operatorname{GL}(V)\) reps.
Refs¶
- Fulton–Harris, Representation Theory: A First Course
- Hall, Lie Groups, Lie Algebras, and Representations
- Serre, Linear Representations of Finite Groups
- Humphreys, Introduction to Lie Algebras and Representation Theory
Next¶
Foundations · Lie algebras · Lie groups · Group theory · Modules · Homological algebra