# Chapter 106. Representation Theory Basics # Chapter 106. Representation Theory Basics Representation theory studies algebraic objects by letting them act on vector spaces. Instead of studying a group, algebra, or Lie algebra only through its abstract definition, one represents its elements as linear transformations. This brings matrices, eigenvalues, invariant subspaces, decompositions, and linear algebra into the study of symmetry. Standard introductions describe representation theory as the study of groups or algebraic structures through their actions on vector spaces. ## 106.1 The Basic Idea Let \(G\) be a group. A representation of \(G\) assigns to each element \(g\in G\) an invertible linear map on a vector space \(V\). The assignment must respect the group multiplication. Thus a representation converts abstract multiplication in \(G\) into composition of linear maps. If $$ \rho : G \to GL(V) $$ is a representation, then $$ \rho(gh)=\rho(g)\rho(h) $$ for all \(g,h\in G\), and $$ \rho(e)=I_V. $$ Here \(e\) is the identity element of \(G\), and \(I_V\) is the identity map on \(V\). The vector space \(V\) is called the representation space. ## 106.2 Linear Actions A representation may also be described as a linear action. A group \(G\) acts linearly on \(V\) if each \(g\in G\) sends vectors in \(V\) to vectors in \(V\), and $$ g(v+w)=gv+gw, $$ $$ g(cv)=c(gv), $$ $$ e v=v, $$ and $$ g(hv)=(gh)v. $$ The notation \(gv\) means the action of \(g\) on the vector \(v\). The representation map and the action notation are equivalent. If $$ \rho : G\to GL(V), $$ then $$ gv=\rho(g)v. $$ Linear actions are often called \(G\)-modules. ## 106.3 Matrix Representations If \(V\) has finite dimension \(n\), then choosing a basis identifies $$ GL(V) $$ with $$ GL_n(F), $$ the group of invertible \(n\times n\) matrices. A representation can then be written as $$ \rho : G\to GL_n(F). $$ Each group element becomes a matrix. The condition $$ \rho(gh)=\rho(g)\rho(h) $$ says that group multiplication is represented by matrix multiplication. Thus representation theory studies abstract symmetry through concrete matrices. ## 106.4 First Example Let $$ G=\mathbb{Z}/2\mathbb{Z}=\{0,1\} $$ with addition modulo \(2\). Define a representation on \(\mathbb{R}^2\) by $$ \rho(0)= \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix}, $$ and $$ \rho(1)= \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix}. $$ The element \(1\) acts as reflection across the \(x\)-axis. Since $$ 1+1=0 $$ in \(\mathbb{Z}/2\mathbb{Z}\), we need $$ \rho(1)^2=\rho(0). $$ Indeed, $$ \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix}^2 = \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix}. $$ Thus this is a representation. ## 106.5 Trivial Representation Every group has a trivial representation on any vector space \(V\). It is defined by $$ \rho(g)=I_V $$ for every \(g\in G\). In this representation, every group element acts as the identity. The trivial representation contains no visible motion. It is still important because it often appears as a component inside larger representations. For example, invariant vectors form copies of the trivial representation. ## 106.6 Faithful Representations A representation $$ \rho:G\to GL(V) $$ is faithful if \(\rho\) is injective. This means different group elements act by different linear transformations. Equivalently, $$ \rho(g)=I_V $$ only when $$ g=e. $$ A faithful representation loses no group information. It realizes the group as a concrete group of linear transformations. A non-faithful representation collapses some group elements together. ## 106.7 Kernel of a Representation The kernel of a representation is $$ \ker \rho = \{g\in G:\rho(g)=I_V\}. $$ It consists of all group elements that act trivially on \(V\). The kernel is a normal subgroup of \(G\). A representation is faithful exactly when $$ \ker\rho=\{e\}. $$ Thus the kernel measures how much of the group the representation fails to see. ## 106.8 Subrepresentations Let \(V\) be a representation of \(G\). A subspace $$ W\subseteq V $$ is a subrepresentation if it is invariant under the action of \(G\): $$ gW\subseteq W $$ for every \(g\in G\). Equivalently, for every \(w\in W\), $$ gw\in W. $$ If \(W\) is invariant, then the action of \(G\) restricts to \(W\), so \(W\) becomes a representation in its own right. Invariant subspaces are the representation-theoretic analogue of subspaces preserved by a linear operator. ## 106.9 Irreducible Representations A nonzero representation \(V\) is irreducible if its only subrepresentations are $$ 0 $$ and $$ V. $$ An irreducible representation cannot be decomposed into smaller invariant pieces. Irreducible representations are the basic building blocks of representation theory. This is analogous to prime numbers in arithmetic or simple modules in algebra. A main problem in representation theory is to classify irreducible representations of a given algebraic object. ## 106.10 Direct Sums If \(V\) and \(W\) are representations of \(G\), their direct sum $$ V\oplus W $$ is also a representation. The action is defined by $$ g(v,w)=(gv,gw). $$ In matrix form, if \(g\) acts on \(V\) by \(A_g\) and on \(W\) by \(B_g\), then \(g\) acts on \(V\oplus W\) by the block diagonal matrix $$ \begin{bmatrix} A_g&0\\ 0&B_g \end{bmatrix}. $$ Direct sums combine representations without mixing their components. ## 106.11 Decomposable and Indecomposable Representations A representation \(V\) is decomposable if it can be written as a direct sum $$ V=W_1\oplus W_2 $$ where \(W_1\) and \(W_2\) are nonzero subrepresentations. If no such decomposition exists, \(V\) is indecomposable. Every irreducible representation is indecomposable. The converse may fail. A representation may have nontrivial subrepresentations but still resist splitting as a direct sum. This distinction becomes important over fields or algebras where complete reducibility fails. ## 106.12 Complete Reducibility A representation is completely reducible if it is a direct sum of irreducible representations. Thus $$ V\cong V_1\oplus\cdots\oplus V_r, $$ where each \(V_i\) is irreducible. For finite groups over fields of characteristic \(0\), every finite-dimensional representation is completely reducible. This is Maschke's theorem. Complete reducibility allows representation theory to reduce many questions to irreducible representations. Without complete reducibility, extensions between representations become part of the theory. ## 106.13 Homomorphisms of Representations Let \(V\) and \(W\) be representations of \(G\). A homomorphism of representations is a linear map $$ T:V\to W $$ such that $$ T(gv)=gT(v) $$ for every \(g\in G\) and every \(v\in V\). Such maps are also called equivariant maps or intertwining maps. They preserve both the vector-space structure and the group action. In terms of representation maps, $$ T\rho_V(g)=\rho_W(g)T. $$ This equation says that \(T\) commutes with the action of \(G\). ## 106.14 Isomorphic Representations Two representations \(V\) and \(W\) are isomorphic if there exists an invertible representation homomorphism $$ T:V\to W. $$ That means $$ T(gv)=gT(v) $$ and \(T\) is a vector-space isomorphism. In matrix terms, isomorphic representations differ only by a change of basis. If $$ \rho_W(g)=P^{-1}\rho_V(g)P $$ for every \(g\in G\), then the two representations are equivalent. Thus representation theory classifies actions up to change of coordinates. ## 106.15 Schur's Lemma Schur's lemma is one of the fundamental results of representation theory. Let \(V\) and \(W\) be irreducible representations over an algebraically closed field. If $$ T:V\to W $$ is a representation homomorphism, then either $$ T=0 $$ or \(T\) is an isomorphism. In particular, if \(V\) is irreducible, then every endomorphism $$ T:V\to V $$ that commutes with the group action is scalar multiplication: $$ T=\lambda I. $$ Schur's lemma explains why irreducible representations behave like atomic objects. ## 106.16 Characters For a finite-dimensional representation $$ \rho:G\to GL(V), $$ the character is the function $$ \chi_\rho:G\to F $$ defined by $$ \chi_\rho(g)=\operatorname{tr}(\rho(g)). $$ The character records the trace of each representing matrix. Characters are useful because trace is unchanged under change of basis. Thus isomorphic representations have the same character. For many classes of finite groups, characters provide a powerful way to decompose and classify representations. ## 106.17 Example of a Character Consider the representation of $$ \mathbb{Z}/2\mathbb{Z} $$ on \(\mathbb{R}^2\) from Section 106.4. We have $$ \rho(0)= \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix}, \qquad \rho(1)= \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix}. $$ The character is $$ \chi(0)=\operatorname{tr}(\rho(0))=2, $$ and $$ \chi(1)=\operatorname{tr}(\rho(1))=0. $$ The value at the identity equals the dimension of the representation. ## 106.18 Representations of Lie Algebras Representation theory also applies to Lie algebras. If \(\mathfrak g\) is a Lie algebra, a representation of \(\mathfrak g\) on \(V\) is a Lie algebra homomorphism $$ \rho:\mathfrak g\to\mathfrak{gl}(V). $$ This means $$ \rho([x,y]) = [\rho(x),\rho(y)] $$ for all \(x,y\in\mathfrak g\). The bracket on the right is the commutator bracket: $$ [A,B]=AB-BA. $$ Thus Lie algebra elements act as linear operators whose commutators reproduce the Lie bracket. This is the standard definition for Lie algebra representations. ## 106.19 The Adjoint Representation Every Lie algebra \(\mathfrak g\) has a natural representation on itself. Define $$ \operatorname{ad}_x:\mathfrak g\to\mathfrak g $$ by $$ \operatorname{ad}_x(y)=[x,y]. $$ The map $$ \operatorname{ad}:\mathfrak g\to\mathfrak{gl}(\mathfrak g) $$ is called the adjoint representation. The Jacobi identity ensures that $$ \operatorname{ad}_{[x,y]} = [\operatorname{ad}_x,\operatorname{ad}_y]. $$ Thus the adjoint representation is a genuine Lie algebra representation. ## 106.20 Representations of Associative Algebras Let \(A\) be an associative algebra over \(F\). A representation of \(A\) on a vector space \(V\) is an algebra homomorphism $$ \rho:A\to \operatorname{End}_F(V). $$ Equivalently, \(V\) is an \(A\)-module. This means elements of \(A\) act as linear maps on \(V\), and multiplication in \(A\) corresponds to composition of linear maps. Group representations can often be studied through the group algebra \(F[G]\). Lie algebra representations can often be studied through the universal enveloping algebra. This connects group, Lie, and associative algebra representation theories. ## 106.21 Invariant Vectors Let \(V\) be a representation of \(G\). A vector \(v\in V\) is invariant if $$ gv=v $$ for every \(g\in G\). The set of invariant vectors is $$ V^G = \{v\in V:gv=v\text{ for all }g\in G\}. $$ This is a subspace of \(V\). Invariant vectors form the part of the representation on which the group acts trivially. In applications, invariant vectors often correspond to conserved quantities, symmetric tensors, or fixed features. ## 106.22 Dual Representations If \(V\) is a representation of \(G\), then the dual space $$ V^* $$ also carries a representation. For \(f\in V^*\), define $$ (gf)(v)=f(g^{-1}v). $$ The inverse is needed so that the group action law holds. The dual representation describes how linear functionals transform when vectors transform. Dual representations are essential in tensor algebra, invariant theory, and differential geometry. ## 106.23 Tensor Product Representations If \(V\) and \(W\) are representations of \(G\), then $$ V\otimes W $$ is also a representation. The action is defined on pure tensors by $$ g(v\otimes w)=(gv)\otimes(gw). $$ This extends linearly to all of \(V\otimes W\). Tensor products allow one to build new representations from old ones. They also lead to decomposition problems: given \(V\otimes W\), determine its irreducible components. ## 106.24 Symmetric and Exterior Powers If \(V\) is a representation of \(G\), then the symmetric powers $$ S^k(V) $$ and exterior powers $$ \Lambda^k(V) $$ are also representations. The group action is induced by $$ g(v_1\cdots v_k) = (gv_1)\cdots(gv_k) $$ on symmetric powers and $$ g(v_1\wedge\cdots\wedge v_k) = (gv_1)\wedge\cdots\wedge(gv_k) $$ on exterior powers. These constructions are central in geometry and algebra. They show how representation theory interacts with multilinear algebra. ## 106.25 Regular Representation Let \(G\) be a finite group. The regular representation is the representation of \(G\) on the vector space with basis vectors $$ \{e_g:g\in G\}. $$ The action is $$ h e_g = e_{hg}. $$ Thus each group element permutes the basis according to left multiplication. The regular representation is important because it contains all irreducible representations of \(G\) over suitable fields. It is one of the main tools for understanding finite group representations. ## 106.26 Permutation Representations Suppose \(G\) acts on a finite set \(X\). Let \(V\) be the vector space with basis $$ \{e_x:x\in X\}. $$ Define $$ g e_x=e_{gx}. $$ This gives a representation called a permutation representation. Permutation representations translate actions on finite sets into linear algebra. They appear in combinatorics, group actions, graph symmetry, and spectral methods. ## 106.27 Decomposing a Permutation Representation Let \(G=S_3\) act on $$ X=\{1,2,3\}. $$ The associated permutation representation acts on \(\mathbb{R}^3\) by permuting coordinates. The line $$ L=\operatorname{span} \left\{ \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} \right\} $$ is invariant. It is a copy of the trivial representation. The plane $$ W= \left\{ \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix} : x_1+x_2+x_3=0 \right\} $$ is also invariant. Thus $$ \mathbb{R}^3=L\oplus W. $$ The representation decomposes into a one-dimensional trivial part and a two-dimensional standard part. ## 106.28 Why Representation Theory Matters Representation theory is useful because linear algebra is highly developed. Once an algebraic object acts on a vector space, we can use: | Linear-algebra tool | Representation-theoretic use | |---|---| | Eigenvalues | Study action of elements | | Trace | Define characters | | Invariant subspaces | Find subrepresentations | | Direct sums | Decompose representations | | Tensor products | Build new representations | | Dual spaces | Study covariant structures | | Matrices | Compute explicitly | Representation theory therefore turns symmetry into linear structure. ## 106.29 Applications Representation theory appears in many fields. | Area | Role | |---|---| | Group theory | Study groups through matrices | | Lie theory | Study infinitesimal symmetries | | Quantum mechanics | Symmetry actions on state spaces | | Chemistry | Molecular symmetry | | Fourier analysis | Decomposition into frequency modes | | Number theory | Automorphic representations | | Geometry | Symmetry of spaces and tensors | | Machine learning | Equivariant models | The same formal language applies because all these areas involve structure-preserving linear actions. ## 106.30 Summary Representation theory studies algebraic structures by their actions on vector spaces. | Concept | Meaning | |---|---| | Representation | Homomorphism into linear transformations | | Representation space | Vector space being acted on | | Subrepresentation | Invariant subspace | | Irreducible representation | Representation with no nontrivial subrepresentations | | Intertwiner | Linear map preserving the action | | Character | Trace of the representing matrices | | Tensor product representation | Combined action on \(V\otimes W\) | | Faithful representation | Injective action | The basic principle is that abstract algebra becomes more tractable when expressed through linear transformations. Representation theory uses matrices and vector spaces to study symmetry, structure, decomposition, and invariance.