A representation translates a group operation into linear transformations, allowing algebraic questions to be studied with invariant subspaces and characters. Etingof and collaborators provide an entry through algebras and finite-group representations. Etingof's Lie-group notes bring smooth structure into the picture. Choose the finite-group route while you are learning decomposition and character calculations; add the Lie route when tangent spaces and smooth maps are already familiar.
What to work on
Find invariant subspaces and intertwiners of a finite-dimensional representation.
Apply Maschke's theorem over a field of suitable characteristic.
Compute a finite group's character table and tensor-product multiplicities.
Verify the sl(2) relations in an explicit matrix representation.
A starting text
Introduction to Representation Theory — Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob and Elena Yudovina. Read section 2.3 for representations; 4.1-4.5 and 4.8-4.9 for finite groups and characters. For a first Lie-algebra example use sections 2.9 and 2.15, then the problems in 2.16.
Choosing what to read
Write matrices for a small group action and verify every defining relation. Find invariant subspaces directly before using a decomposition theorem. Compute characters of a cyclic group and a small nonabelian group, checking dimensions and orthogonality where the hypotheses permit. Record the ground field and its characteristic: averaging arguments depend on them. When moving to Lie algebras, differentiate a concrete matrix action and compare the resulting bracket with matrix commutators.
Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob and Elena Yudovina
Build explicit representations and practice irreducible decomposition.
How it reads
Algebraic introduction with finite-group characters and early Lie-algebra examples.
What it expects
Linear algebra and abstract algebra; careful attention to field characteristic.
What to keep in mind
Finite-dimensional examples do not supply the analytic foundations of infinite-dimensional representations.
Practice
Exercises provided; public solutions not verified. Section 4.12 Problems, printed page 83; section 2.16 Problems on Lie algebras, page 39.
Complete public PDF.
Edition, reading range and source
Read section 2.3 for representations; 4.1-4.5 and 4.8-4.9 for finite groups and characters. For a first Lie-algebra example use sections 2.9 and 2.15, then the problems in 2.16.
Understand how a smooth group action gives an infinitesimal representation.
How it reads
Graduate course connecting Lie groups, Lie algebras and representations.
What it expects
Abstract algebra, topology and smooth manifolds.
What to keep in mind
Initial manifold material helps orientation but does not remove the geometry prerequisite.
Practice
Exercise and public solution coverage were not recorded for this catalogue item; verify a matrix-group example as an editorial task.
Complete 223-page MIT author PDF.
Edition, reading range and source
Lecture 1, sections 1.1-1.7, for topological groups, manifolds and tangent spaces; lecture 2, sections 2.1-2.6, for Lie groups and coverings. Contents checked; later representation reading requires a separate selection.
Finite-group character and sl(2) problems. Section 4.12 Problems, printed page 83; section 2.16 Problems on Lie algebras, page 39.. Exercises provided; public solutions not verified.
Try these before moving on
Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.
For E=[[0,1],[0,0]], F=[[0,0],[1,0]] and H=diag(1,-1), verify [H,E]=2E, [H,F]=-2F and [E,F]=H. Prove the resulting action on C squared is irreducible.
A hint
Use [A,B]=AB-BA. Any nonzero proper subspace of C squared is a line; an H-invariant line must be an eigenspace of H.
Work through the solution
Direct multiplication gives HE=E, EH=−E, HF=−F, FH=F, EF=diag(1,0) and FE=diag(0,1). Subtracting proves the three stated commutator identities, with the convention [A,B]=AB−BA. These are the defining brackets for the standard basis of sl2(C).
If a nonzero proper subspace W⊂C2 were invariant under the action, it would be a line invariant under H. For a nonzero generator w, Hw=λw, so W would be one of the two eigenspaces Ce1 or Ce2, since H has distinct eigenvalues 1,−1. But Fe1=e2 excludes the first line, and Ee2=e1 excludes the second. No such W exists, proving irreducibility.
If this was the sticking point:
Linear Algebra — Review invariant subspaces, eigenspaces and matrix multiplication.
Already comfortable with this? Skip finite-group review if you can build the S3 table and explain the characteristic restriction in Maschke's theorem. The sl(2) example can be skipped if weights and raising/lowering operators are already usable in computations.
Introduces representations using algebra and many exercises. Appropriate for strong undergraduates with linear algebra; the pace assumes comfort with abstract structures.
Preparation: Linear algebra; groups, rings and fields.
A graduate course connecting Lie groups, Lie algebras and representations. Initial manifold material helps orientation but does not remove the need for mathematical maturity.
Preparation: Abstract algebra; topology and smooth manifolds.
Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik · Monograph · Advanced
Develops the categorical setting for representation theory. A specialist continuation rather than an introductory algebra text; link to the authorized author-hosted edition.
Gordon James and Martin Liebeck · Textbook · Intermediate · English · Purchase / institutional access
Purchase or library subscription required; previews are not the complete book.
A concrete introduction to finite-group representations and character tables. A useful starting branch before Lie groups, modular representation theory or geometric representation theory.
戚天成 · Course notes · Intermediate · Chinese · Online text
Online lecture text; chapters can be read separately.在线讲义，可按章节阅读。
A short course approaching finite-group representations through the structure of algebras, rather than starting only with character computations.从代数结构的观点进入有限群表示论的短篇讲义，不仅围绕特征标计算展开。
Preparation: Linear algebra; finite groups; rings and modules.线性代数、有限群、环与模。
Use: Use it to connect a first representation-theory course with semisimple algebra.用来连接初等表示论与半单代数。
Roe Goodman and Nolan R. Wallach · Textbook · Graduate · English · Purchase / institutional access
Purchase or library subscription required; previews are not the complete book.
A substantial reference connecting classical groups, highest weights and invariant theory. Select chapters according to the intended representation-theoretic problem.
François Bruhat · Long lecture notes · Graduate · English · PDF
Complete institutional PDF, authorized for individual use only. Copyright remains with TIFR; retypesetting may retain errors. This is not a redistribution licence.
Links Lie-group structure with representation theory on locally compact groups. Develop integration and topological-group foundations before the specialized arguments.
Thomas J. Enright · Long lecture notes · Research · English · PDF
Complete institutional PDF, authorized for individual use only. Copyright remains with TIFR; retypesetting may retain errors. This is not a redistribution licence.
A specialist continuation from semisimple Lie algebras to representation-theoretic structure. Follow roots, weights and basic representations first.