The honest first answer is that you should not start with the famous book. Serre's Linear Representations of Finite Groups is under two hundred pages and it assumes on page one that you know what a character is, that modules are familiar, and that you will supply your own patience for the compressed proofs. Start instead with James and Liebeck's Representations and Characters of Groups, which teaches the same material with worked character tables and exercises and assumes no module theory.
Before even that, spend a day on the prerequisites, because this is a subject where the cliff is real and unannounced. Every book below silently assumes a full undergraduate course in linear algebra and one in abstract algebra through groups, modules and a little Galois theory. The two books in the first stage exist so you can find that out before spending money.
Checking what the rest of the path assumes
Groups and symmetry, by M. A. Armstrong, is the gentlest honest entry at 186 pages: it builds group theory out of the symmetries of physical objects, so a group arrives as something that acts before it arrives as an axiom list. Skip it if you can state the orbit-stabiliser theorem without looking it up.
Michael Artin's Algebra is the prerequisite check proper. It covers linear algebra, group theory and a first pass at representations and characters in one 552-page volume, and its chapter on representations of finite groups is the shortest complete statement of what the next stage needs. Our record is the second edition, which is the one to buy.
Finite groups and characters
Representations and Characters of Groups is the first course: Maschke's theorem, Schur's lemma, orthogonality relations, and enough constructed character tables that you can build one yourself. The record credits Gordon James alone; Martin Liebeck is the co-author, and the current second edition reorganises the later chapters.
Benjamin Steinberg's Representation Theory of Finite Groups is a short modern alternative, and it reaches applications — Fourier analysis on finite groups, random walks, card shuffling — unusually fast. Read it alongside James and Liebeck if the theory feels unmotivated rather than instead of it.
Then Serre. Linear Representations of Finite Groups is 188 pages in our record, credited there to Leonhard L. Scott, its translator, as well as to Serre. Its treatment of induced representations, Brauer's theorem and integrality is what most later books are quietly working from, which is why it sits third rather than first.
Character theory as a research subject
I. Martin Isaacs's Character theory of finite groups is the reference for using characters to prove structural theorems about groups rather than to tabulate them: Burnside's p-q theorem, Frobenius groups, Clifford theory. It is dense and exercise-heavy, and the widely available printing is the inexpensive Dover reissue of the 1976 text, 303 pages.
Barry Simon's Representations of Finite and Compact Groups is the bridge out. It handles finite groups and compact Lie groups with the same machinery — Haar measure, Peter-Weyl — so the next stage stops looking like a change of subject.
Lie algebras and Lie groups
John Stillwell's Naive lie theory is a deliberate on-ramp: the classical groups and their Lie algebras built from linear algebra and calculus, with no manifolds, in 217 pages. Skip it only if you have had a differential geometry course.
Brian Hall's Lie Groups, Lie Algebras, and Representations is the best-paced graduate introduction, doing matrix Lie groups first so the analysis stays elementary. Our record is the 2003 first edition; the second is substantially expanded and is the one now assigned. Humphreys's Introduction to Lie algebras and representation theory is the purely algebraic core — root systems, the Weyl group, Verma modules, the Weyl character formula, no analysis at all — unchanged since 1972 and still standard at 172 pages. Knapp's Lie groups beyond an introduction is where the path stops being a course and becomes a 708-page reference on real forms and noncompact structure theory.
The geometric and combinatorial picture
Fulton and Harris sits last, not first, despite its subtitle: it works the classical groups out explicitly before stating general theorems. Our catalogue holds it under the bare display title Representation theory, but it is the full 1991 Springer graduate text, 551 pages. Goodman and Wallach's Symmetry, representations, and invariants is the modern account of classical invariant theory and Schur-Weyl duality; the record lists only Roe Goodman, and Nolan Wallach is the co-author. Fulton's Young Tableaux closes it, making the character formulas computable by hand.
The full sequence, with every record and its length, is on the representation theory reading path.
