--- title: "Decomposing a Representation from Its Character" description: "Build the character table of a small finite group over the complex numbers and use weighted character inner products to decompose finite-dimensional representations." canonical: https://lightmysky.com/learn/mathematics/decomposing-a-representation-from-its-character-mt_ccEcWW9xAa source: https://lightmysky.com/learn/mathematics/decomposing-a-representation-from-its-character-mt_ccEcWW9xAa.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Decomposing a Representation from Its Character Build the character table of a small finite group over the complex numbers and use weighted character inner products to decompose finite-dimensional representations. Subject: Mathematics · Area: Abstract Algebra · Ages 23 to 24 Page: https://lightmysky.com/learn/mathematics/decomposing-a-representation-from-its-character-mt_ccEcWW9xAa ## Ready when they can - Construct the character table of a small non-abelian group - Decompose finite-dimensional complex representations, including permutation representations, using inner products with irreducible characters, and check dimensions from the multiplicities - Read a normal subgroup off the table from the kernel of a character ## Lesson: Reading a group from its character table Work with finite-dimensional complex representations of a finite group G. As in the previous topic, characters are traces and are constant on conjugacy classes. A character table has one row per irreducible character and one column per conjugacy class. The identity entry chi(e) is the representation's dimension. For classes C_j with representatives c_j, define =(1/|G|) sum_j |C_j| chi(c_j) conjugate(psi(c_j)). Conjugation changes a+bi to a-bi and leaves real values unchanged. If psi is irreducible, is its multiplicity in the representation with character chi. Align classes, conjugate psi, multiply by class size, sum and divide by |G|. Multiplicities are nonnegative integers. S3 permutes three letters. Its classes are identity, transpositions and 3-cycles, of sizes 1,3,2. Use that order throughout. A transposition swaps two letters; a 3-cycle moves all three cyclically. The permutation character counts fixed letters, giving P=(3,1,0). The trivial row is (1,1,1). The sign row is (1,-1,1), since transpositions are odd and 3-cycles are even. The permutation space splits into constant vectors and the plane x1+x2+x3=0. Subtract trivial (1,1,1) from P to get S=(2,0,-1). For a known representation character, squared norm equals the sum of squared irreducible multiplicities, so norm 1 means irreducible. Here =(4+0+2)/6=1. Trivial and sign are distinct one-dimensional irreducibles. S supplies a third, and three conjugacy classes mean the table is complete. *(drawing: Each cell shows a row. Entries follow identity, transpositions, 3-cycles, with class sizes 1,3,2.)* **Example.** For S3, class sizes are 1,3,2 and the group order is 6. Take chi=(5,1,2). Pairing with trivial (1,1,1) gives (5+3+4)/6=2 copies. Pairing with sign (1,-1,1) gives (5-3+4)/6=1 copy. Pairing with standard (2,0,-1) gives (10+0-4)/6=1 copy. Thus chi contains two trivial copies, one sign copy and one standard copy. Multiplicity counts copies; chi(e) gives total dimension. If irreducible chi_i has dimension d_i and multiplicity m_i, the number of constituents, counting copies, is sum_i m_i. Dimension is sum_i m_i d_i. The trivial character has value 1 everywhere, so its multiplicity is (1/|G|) sum_j |C_j| chi(c_j), not generally chi(e). The regular representation permutes basis vectors indexed by G through left multiplication. Its character is |G| at the identity and 0 elsewhere. Pairing it with an irreducible of dimension d gives |G|*d/|G|=d copies. Since the regular representation has dimension |G|, the squares of all irreducible dimensions sum to |G|. This checks the dimension data of a completed table. For a character chi of dimension d, ker(chi)={g: chi(g)=d} equals the representation kernel and is a normal subgroup. To see the equality, each representing matrix has finite order, so it is diagonalizable with eigenvalues of modulus 1. Their sum can equal d only when every eigenvalue is 1. Diagonalizability then makes the matrix the identity. Conversely, the identity matrix has trace d. The character kernel equals the representation kernel. If rho(g)=I, then rho(hgh^-1)=rho(h)I rho(h)^-1=I, so conjugation preserves the kernel. As a homomorphism kernel, it is a subgroup and therefore normal. An element lies in a direct-sum kernel exactly when it acts as the identity on every summand, so intersect the summand kernels. For a kernel K, the quotient group has |G|/|K| elements. In S3, the trivial kernel is all of S3. The sign kernel is A3={e,(123),(132)}, because those are exactly its elements with value 1. The standard kernel is {e}, since only its identity value equals its dimension 2. A single irreducible row need not reveal every normal subgroup; intersections of row kernels may be needed. The norm-1 irreducibility test assumes a known representation character. A class function can have norm 1 without being a character. Its expansion in irreducible rows must have nonnegative integer coefficients to be a representation character. If valid character data give a fractional multiplicity, check class alignment, conjugation, class-size weights, division by |G| and arithmetic. **Recap.** Weighted character inner products count irreducible copies. Dimensions weight those copies, and character kernels are normal subgroups. *This lesson, its practice questions and flashcards were written for this dataset.* ## Practice 18 questions on this page, each with its working shown. ## Needs first - [Characters and the Orthogonality Relations](https://lightmysky.com/learn/mathematics/characters-and-the-orthogonality-relations-mt_RleR7YGXjX) - [Normal Subgroups and Quotient Groups](https://lightmysky.com/learn/mathematics/normal-subgroups-and-quotient-groups-mt_RU7A-t7wQ5)