Group representation
In mathematics, a group representation is a homomorphism from an abstract group G to the general linear group \mathrm{GL}(V) of invertible linear transformations on a vector space V over a field k, allowing the group's structure to be analyzed through concrete linear actions.[1] This framework, often over the complex numbers for finite groups, equates to a module over the group algebra k[G], where elements of G act linearly on V.[1] Equivalence of representations occurs via conjugation by an invertible linear map, preserving the group's action up to similarity.[2] Representation theory, the broader study encompassing these objects, originated in 1896 with Ferdinand Georg Frobenius's work on group characters, motivated by Richard Dedekind's inquiries into group determinants from multiplication tables.[1] Key developments include Maschke's theorem, which asserts that representations of finite groups over fields whose characteristic does not divide the group order are semisimple (decomposable into direct sums of irreducibles), and the Artin-Wedderburn theorem describing the structure of semisimple algebras.[1] Central concepts include irreducible representations (those with no nontrivial invariant subspaces), characters (traces of representation matrices, forming class functions constant on conjugacy classes), and operations like induction (extending representations from subgroups) and tensor products.[1][2] The theory applies across mathematics and physics, illuminating symmetries in quantum mechanics (e.g., via unitary representations), classifying finite simple groups through character tables, and connecting to algebraic geometry, number theory, and topology via tools like the Jordan-Hölder theorem on composition series uniqueness.[1] For infinite groups, such as Lie groups, representations extend to continuous homomorphisms, underpinning harmonic analysis and particle physics models.[1] Modern advances leverage computational methods and category theory, introduced by Samuel Eilenberg and Saunders Mac Lane in the 1940s, to unify representations of groups, algebras, and quivers.[1]Introduction
Historical overview
The origins of group representation theory can be traced to the early 19th century, when Augustin-Louis Cauchy began studying permutation groups in the context of solving polynomial equations, introducing early notions of group actions on sets in his 1812 memoir on substitutions.[1] Évariste Galois further advanced this in the 1830s by linking permutation groups to field extensions in his work on the solvability of equations by radicals, laying foundational ideas for understanding symmetries through group structures. By the late 19th century, Ferdinand Georg Frobenius made pivotal contributions, motivated by Richard Dedekind's 1896 query on the group determinant; in his 1896 papers, Frobenius developed the theory of group characters, computed the character table for the symmetric group S_3, and established the orthogonality relations for characters, marking the birth of modern representation theory for finite groups.[1][3] Key early 20th-century advancements built on Frobenius's framework. William Burnside's 1897 book Theory of Groups of Finite Order (revised in 1911) synthesized permutation group theory and applied early representation ideas to classify groups, while his 1904 theorem on the solvability of groups of order p^a q^b demonstrated the power of character theory.[4] Issai Schur, in his 1901 dissertation and subsequent papers from 1905 to 1911, introduced irreducibility criteria for representations, proved Schur's lemma on endomorphisms of irreducible representations, and extended character theory to integral representations, solidifying the algebraic foundations.[5] Heinrich Maschke complemented this in 1898 by proving that representations of finite groups over the complex numbers are completely reducible (Maschke's theorem), enabling the decomposition into irreducibles.[1] In the mid-20th century, the theory expanded to infinite and continuous groups. Hermann Weyl's 1925 book The Theory of Groups and Quantum Mechanics and papers from 1925–1926 developed unitary representations of compact Lie groups, introducing the highest weight construction for semisimple Lie algebras, while his 1931 work The Classical Groups unified finite and continuous cases through invariant theory.[6] Emil Artin's 1927 contributions in class field theory utilized group characters to define Artin L-functions, bridging representation theory with number theory by generalizing Dirichlet characters to non-abelian Galois groups.[7] In the 1940s, Claude Chevalley's 1941 lectures and 1946 book Theory of Lie Groups pioneered aspects of the study of algebraic groups over arbitrary fields, proving that semisimple algebraic groups have the same representation theory as their Lie algebras. Modern computational aspects emerged in the 1960s with the advent of computer algebra systems, enabling algorithmic computation of character tables and irreducible representations. Joachim Neubüser's 1960 paper initiated systematic computational group theory, leading to programs for permutation and matrix group computations by the late 1960s, such as those used in classifying finite simple groups.[8] A brief application in physics appeared with Eugene Wigner's 1931 book Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, where representations classified atomic states under symmetry groups.[9]| Year | Milestone | Contributor | Key Publication/THEOREM |
|---|---|---|---|
| 1812 | Early permutation group studies | Cauchy | Memoir on substitutions |
| 1830s | Permutation groups in Galois theory | Galois | Works on solvability by radicals |
| 1896 | Invention of group characters and orthogonality | Frobenius | "Über Gruppencharaktere" |
| 1898 | Complete reducibility over ℂ | Maschke | Maschke's theorem |
| 1904 | Solvability via characters | Burnside | Burnside's p^a q^b theorem |
| 1905–1911 | Irreducibility and Schur's lemma | Schur | Papers on linear substitutions |
| 1925 | Unitary representations of Lie groups | Weyl | Theory of Groups and Quantum Mechanics |
| 1927 | Characters in class field theory | Artin | Artin L-functions |
| 1941 | Algebraic groups and representations | Chevalley | Lectures on Lie groups |
| 1960 | Computational initiation | Neubüser | First computational paper on groups |
Motivations
Group representations provide a powerful framework for abstracting and analyzing group actions by linearizing symmetries into concrete matrix operations. Rather than studying abstract groups in isolation, representations embed them as subgroups of general linear groups acting on vector spaces, enabling the use of linear algebra to compute and understand group behaviors. This approach transforms potentially intractable problems about group elements and relations into manageable matrix manipulations, such as finding eigenvalues or solving systems of equations.[10][11] At their core, groups encode symmetries arising in geometry, physics, and algebra, and representations illuminate these by revealing invariant subspaces—subspaces preserved under the group action. These subspaces correspond to irreducible components of the representation, allowing the decomposition of complex actions into simpler building blocks that respect the underlying symmetry. Such decompositions highlight how group elements act consistently on geometric objects or algebraic structures, providing insight into the invariants that remain unchanged.[10] Representation theory unifies diverse mathematical themes, notably through its connections to invariant theory, as pioneered by David Hilbert in the 1890s. Hilbert's finiteness theorem demonstrated that the ring of polynomial invariants under a linearly reductive group action on a vector space is finitely generated, laying groundwork for studying symmetries via representations. This links to the decomposition of tensor products of representations into direct sums of irreducibles, which captures how combined symmetries behave and facilitates the classification of group elements through traces of representation matrices—quantities that are invariant under similarity and thus serve as signatures of conjugacy classes.[12][10] More broadly, representations enable the reduction of intricate problems to linear algebra, such as solving systems of equations that are invariant under group actions or computing cohomology groups via induced representations. By focusing on linear actions, this theory simplifies the analysis of group symmetries across disciplines, turning abstract algebraic questions into concrete computational tasks.[11][10]Core Definitions
Group homomorphisms to general linear groups
In the abstract framework of group theory, a representation of a group G on a vector space V over a field F is defined as a group homomorphism \rho: G \to \mathrm{GL}(V), where \mathrm{GL}(V) denotes the general linear group of all invertible linear endomorphisms of V.[13] This homomorphism encodes how elements of G act linearly on V, preserving the group operation through composition of endomorphisms.[14] The associated group action is then expressed as g \cdot v = \rho(g)(v) for all g \in G and v \in V, ensuring that the action respects both the vector space structure and the group multiplication: (gh) \cdot v = g \cdot (h \cdot v) and e \cdot v = v, where e is the identity in G.[15] The dimension of the representation, denoted \dim V, is commonly referred to as the degree of the representation, which quantifies its complexity and plays a central role in decomposition theorems.[16] A particularly simple case is the trivial representation, where \rho(g) = \mathrm{Id}_V (the identity endomorphism) for every g \in G, meaning the action fixes every vector in V invariantly. Representations can also be classified by faithfulness: a representation is faithful if the homomorphism \rho is injective, embedding G as a subgroup of \mathrm{GL}(V) without kernel, whereas non-faithful representations have a non-trivial kernel, effectively representing a quotient group.[17] As an illustrative abstract example, consider the cyclic group C_n = \langle \sigma \mid \sigma^n = e \rangle. A representation \rho: C_n \to \mathrm{GL}(V) is fully determined by the image \rho(\sigma), which must be an invertible endomorphism of order dividing n, i.e., \rho(\sigma)^n = \mathrm{Id}_V, and extended by \rho(\sigma^k) = \rho(\sigma)^k for k = 0, \dots, n-1.[14] This setup highlights how the homomorphism property constrains the possible actions without specifying a basis or matrix form. Representations of this type are often analyzed up to equivalence, where two are equivalent if there exists an invertible linear map intertwining their actions.[18]Vector space representations
A vector space representation of a group G assigns to each element g \in G an invertible linear transformation \rho(g): [V](/page/V.) \to [V](/page/V.) on a finite-dimensional vector space [V](/page/V.) over a field F, typically \mathbb{C} or \mathbb{R}, such that the map \rho: G \to \mathrm{GL}(V) is a group homomorphism. This means \rho(g) preserves vector addition and scalar multiplication, and the assignment respects the group operation via \rho(gh) = \rho(g) \rho(h) for all g, h \in G.[1][19] To obtain a concrete matrix form, select a basis \{e_1, \dots, e_n\} for V, where n = \dim_F V. The action of \rho(g) is then encoded by an n \times n matrix (a_{ij}(g)) with entries in F, satisfying \rho(g) e_j = \sum_{i=1}^n a_{ij}(g) e_i for each j = 1, \dots, n. This matrix representation facilitates computations, as the homomorphism property translates to matrix multiplication: A(gh) = A(g) A(h), where A(g) = (a_{ij}(g)).[1][19] The choice of basis affects the specific matrices but not the underlying representation. If P is an invertible n \times n matrix representing a change of basis, the transformed matrices are given by similarity A'(g) = P^{-1} A(g) P for all g \in G, preserving the linear action on V. Equivalent representations in this sense yield isomorphic modules over the group algebra F[G].[1] Over F = \mathbb{C}, a representation is unitary if each \rho(g) is a unitary linear operator, meaning \rho(g)^* = \rho(g)^{-1} with respect to a Hermitian inner product on V, thereby preserving the inner product: \langle \rho(g) v, \rho(g) w \rangle = \langle v, w \rangle for all v, w \in V. For compact groups, every finite-dimensional complex representation is equivalent to a unitary one, obtained by averaging an inner product over the group.[1] Although finite-dimensional vector spaces form the core setting, the concept extends to infinite-dimensional Hilbert spaces, where unitary representations ensure continuity and completeness in the analysis of group actions, as in harmonic analysis on non-compact groups.[1]Fundamental Properties
Equivalence of representations
In representation theory, two representations \rho: [G](/page/G) \to \mathrm{[GL](/page/GL)}(V) and \sigma: [G](/page/G) \to \mathrm{[GL](/page/GL)}(W) of a group [G](/page/G) on vector spaces [V](/page/V.) and [W](/page/V.) over the same field are said to be equivalent if there exists an invertible linear map T: [V](/page/V.) \to [W](/page/V.) such that \sigma(g) = T \rho(g) T^{-1} for all g \in [G](/page/G).[20][21] This condition implies that the representations are related by a change of basis in the vector spaces, preserving the group action up to similarity transformation. Equivalence is an equivalence relation on the set of representations, partitioning them into classes where representations within the same class are structurally indistinguishable.[20] The invertible map T is known as an intertwining operator (or G-equivariant map) between the representations, satisfying the commutation relation T \rho(g) = \sigma(g) T for all g \in G.[22][23] The space of all such intertwining operators forms the Hom space \mathrm{Hom}_G(V, W), which is a vector space under pointwise addition and scalar multiplication. When T is invertible, the representations are isomorphic, meaning V and W are isomorphic as modules over the group algebra \mathbb{F}[G], where the group action is extended linearly.[24][23] This module-theoretic perspective underscores that equivalent representations capture the same abstract G-module structure. Representations are classified up to equivalence, with uniqueness holding in the sense that any two equivalent representations yield the same equivalence class, often used to study invariants like dimension or decomposition types.[21] For direct sum decompositions, two representations are equivalent if and only if their direct summands match up to equivalence and multiplicity; for instance, \rho \oplus \sigma is equivalent to \rho' \oplus \sigma' precisely when the pairs \{\rho, \sigma\} and \{\rho', \sigma'\} consist of equivalent components with the same multiplicities.[24][20] This ensures that non-matching decompositions, such as differing irreducible summands, yield non-equivalent representations.Subrepresentations and quotients
In the context of a representation \rho: [G](/page/G) \to \mathrm{[GL](/page/GL)}(V) of a group [G](/page/G) on a vector space V, a subrepresentation is defined as a subspace W \subseteq V such that \rho(g)W \subseteq W for all g \in [G](/page/G).[25] This condition ensures that the restriction of \rho to W, denoted \rho|_W: [G](/page/G) \to \mathrm{[GL](/page/GL)}(W), defines a valid representation on W.[26] Such a subspace W is also called a [G](/page/G)-invariant subspace or simply an invariant subspace, emphasizing the preservation of the subspace under the group action.[25] Given a subrepresentation W \subseteq V, the quotient space V/W inherits a natural representation structure from \rho, known as the quotient representation. This is defined by \overline{\rho}(g)(v + W) = \rho(g)v + W for all g \in G and v \in V, where the bar denotes the induced map on the quotient.[27] The quotient representation captures the action of G on the "cosets" of W within V, providing a way to factor out the subrepresentation while preserving the linear group action.[26] These constructions fit into the framework of exact sequences of representations. Specifically, for a subrepresentation W \subseteq V, there is a short exact sequence $0 \to W \xrightarrow{i} V \xrightarrow{q} V/W \to 0, where i is the inclusion map and q is the canonical quotient map, both G-equivariant (i.e., commuting with the representation actions).[26] This sequence is exact in the category of representations, meaning \ker i = 0, \mathrm{im} i = \ker q = W, and \mathrm{im} q = V/W, thus encoding the relationship between the subrepresentation, the original representation, and the quotient.[26] Representations can be classified based on their subrepresentation structure, particularly through the notions of simple and semisimple modules (or representations, viewed as modules over the group algebra). A representation is simple if it admits no proper nonzero subrepresentations, meaning the only invariant subspaces are \{0\} and V itself.[26] In contrast, a representation is semisimple if it decomposes as a direct sum of simple representations; here, subrepresentations and quotients behave particularly well, as every subrepresentation has a complementary invariant subspace, ensuring that quotient maps have precisely the expected kernels without additional complications from non-split extensions.[26] This distinction highlights how semisimple representations allow for clean decomposition into building blocks via subrepresentations and quotients.[19] The kernel of a representation \rho: G \to \mathrm{GL}(V) is the normal subgroup \ker \rho = \{ g \in G \mid \rho(g) = \mathrm{Id}_V \}, consisting of group elements that act trivially on V.[19] Since \rho is a group homomorphism, this kernel is normal in G, and the representation factors through the quotient group G / \ker \rho.[19] This kernel provides insight into the "ineffective" part of the group action and relates subrepresentations to the broader structure of group homomorphisms.[25]Examples
Permutation representations
A permutation representation of a finite group G arises from a left action of G on a finite set X. This action induces a linear representation \rho: G \to \mathrm{GL}(V) on the complex vector space V = \mathbb{C}^X of functions f: X \to \mathbb{C}, defined by (g \cdot f)(x) = f(g^{-1} x) for g \in G, f \in V, and x \in X.[1] The space V admits a natural permutation basis consisting of the Dirac delta functions \{\delta_x \mid x \in X\}, where \delta_x(y) = 1 if y = x and 0 otherwise; the group action permutes these basis vectors according to the action on X.[1] In this basis, the representing matrix \rho(g) is a permutation matrix with a 1 in position (x, g^{-1} x) for each x \in X, and 0s elsewhere.[1] Consequently, the trace of \rho(g), which counts the number of 1s on the diagonal, equals the number of fixed points of g on X, i.e., \operatorname{tr} \rho(g) = |\{ x \in X \mid g x = x \}|.[1] The permutation representation decomposes according to the G-orbits on X: since X is a disjoint union of orbits, V is the direct sum of the invariant subspaces spanned by the basis elements in each orbit, yielding a direct sum of transitive permutation representations (one for each orbit).[28] For the symmetric group S_n acting naturally on \{1, 2, \dots, n\}, the associated permutation representation admits a 1-dimensional subrepresentation known as the sign representation, given by \rho(\sigma) = \det(\rho_{\mathrm{perm}}(\sigma)) = \operatorname{sgn}(\sigma) = (-1)^{n - c(\sigma)}, where c(\sigma) is the number of cycles in \sigma (yielding +1 for even permutations and -1 for odd ones).[1] This representation is the unique nontrivial 1-dimensional representation of S_n.[1] A concrete example is the natural permutation representation of S_3 on the set \{1, 2, 3\}, which has dimension 3.[1] The character of this representation takes value 3 on the identity, 1 on each of the three transpositions, and 0 on each of the two 3-cycles.[1] By character orthogonality, it decomposes as the direct sum of the 1-dimensional trivial representation and the 2-dimensional irreducible representation of S_3.[1]Regular and induced representations
The regular representation of a finite group G over the complex numbers is the representation \rho: G \to \mathrm{GL}(\mathbb{C}[G]), where \mathbb{C}[G] is the group algebra with basis \{e_g \mid g \in G\} and G acts by left multiplication: \rho(h) \left( \sum_{g \in G} a_g e_g \right) = \sum_{g \in G} a_g e_{h g}.[29] This construction endows \mathbb{C}[G] with a natural G-module structure, capturing the group's action on itself.[4] The dimension of the regular representation is |G|, as the basis has one element per group element.[29] In the basis \{e_g\}, the matrix of \rho(h) is a permutation matrix corresponding to the left multiplication by h, which permutes the basis elements by shifting indices: the entry in position (e_k, e_g) is 1 if k = h g and 0 otherwise.[4] This permutation action highlights the regular representation as a faithful representation of G, embedding it into the symmetric group on |G| letters.[19] For finite G, the regular representation decomposes as a direct sum of all irreducible representations of G, where each irreducible representation V_i appears with multiplicity equal to \dim V_i.[29] This multiplicity follows from the orthogonality of characters and implies the sum-of-squares formula \sum_i (\dim V_i)^2 = |G|, providing a key tool for classifying irreducibles.[4] Given a subgroup H \leq G and a representation \sigma: H \to \mathrm{GL}(V) of H on a finite-dimensional complex vector space V, the induced representation \mathrm{Ind}_H^G \sigma is the representation of G on the vector space \mathbb{C}[G] \otimes_{\mathbb{C}[H]} V, with dimension \dim V \cdot |G:H|.[29] Equivalently, it can be described as the space of \mathbb{C}-valued functions f: G \to V satisfying f(h x) = \sigma(h) f(x) for all h \in H, x \in G, with G-action (g \cdot f)(x) = f(g^{-1} x).[4] The action of G on the induced space is given explicitly by (\mathrm{Ind}_H^G \sigma)(g) \left( \sum_{t \in T} e_t \otimes v_t \right) = \sum_{t \in T} e_{g t} \otimes v_t, where T is a set of coset representatives for H in G, adjusted for elements where g t \in H u for some u \in T via the H-action.[4] More generally, for an arbitrary element \sum a_h h \otimes v \in \mathbb{C}[G] \otimes_{\mathbb{C}[H]} V, the action is g \cdot (\sum a_h h \otimes v) = \sum a_h (g h) \otimes v, with identification under the right \mathbb{C}[H]-action.[29] The Frobenius reciprocity theorem relates induction and restriction: for representations V of G and W of H, there is a natural isomorphism \mathrm{Hom}_G(V, \mathrm{Ind}_H^G W) \cong \mathrm{Hom}_H(\mathrm{Res}_H^G V, W).[29] In terms of characters, this yields \langle \chi_V, \chi_{\mathrm{Ind}_H^G W} \rangle_G = \langle \chi_{\mathrm{Res}_H^G V}, \chi_W \rangle_H, equating multiplicities under induction and restriction.[4]Reducibility
Reducible representations
A representation \rho: G \to \mathrm{GL}(V) of a group G on a finite-dimensional vector space V over a field k (such as \mathbb{C}) is said to be reducible if there exists a proper nontrivial subspace W \subset V that is invariant under the action of all \rho(g) for g \in G, meaning \rho(g)W \subseteq W for every g.[30] This invariance implies that the representation restricts to a subrepresentation on W and induces a quotient representation on V/W.[30] A representation is completely reducible if it decomposes as a direct sum of irreducible representations, i.e., V = W_1 \oplus W_2 \oplus \cdots \oplus W_m where each W_i is an irreducible subrepresentation.[31] Over the complex numbers \mathbb{C} for finite groups G, every finite-dimensional representation is completely reducible, as guaranteed by Maschke's theorem, which states that any invariant subspace has a complementary invariant subspace, allowing full decomposition into irreducibles.[31] This property holds because the characteristic of \mathbb{C} does not divide the order of G, ensuring the group algebra \mathbb{C}[G] is semisimple.[32] Schur's lemma provides a key tool for understanding irreducible representations within reducible ones: if \rho is irreducible over an algebraically closed field like \mathbb{C}, then the endomorphism algebra \mathrm{End}_G(V) = \{ T \in \mathrm{End}(V) \mid T \rho(g) = \rho(g) T \ \forall g \in G \} consists precisely of scalar multiples of the identity, i.e., \mathrm{End}_G(V) \cong k \cdot \mathrm{Id}.[33] This implies that irreducible subrepresentations are rigid, with no nontrivial intertwiners, which aids in decomposing reducible representations by identifying distinct irreducible factors.[33] To explicitly decompose reducible representations, projection operators are constructed using group averages. The orthogonal projection onto the subspace of invariants V^G = \{ v \in V \mid \rho(g)v = v \ \forall g \in G \} is given by P = \frac{1}{|G|} \sum_{g \in G} \rho(g), which is idempotent (P^2 = P) and G-equivariant when |G| is invertible in k.[34] For finite G over \mathbb{C}, more generally, the projection onto the isotypic component V_\sigma corresponding to an irreducible representation \sigma (the sum of all subrepresentations isomorphic to \sigma) is P_\sigma = \frac{\dim \sigma}{|G|} \sum_{g \in G} \chi_\sigma(g^{-1}) \rho(g), where \chi_\sigma is the character of \sigma; this operator satisfies P_\sigma^2 = P_\sigma and projects V onto V_\sigma while annihilating other components.[34] These projections enable the explicit construction of the complete decomposition into isotypic components, each of which is a direct sum of copies of \sigma.[35]Irreducible representations
An irreducible representation of a group G on a vector space V over a field k is a representation \rho: G \to \mathrm{GL}(V) that admits no proper nontrivial subrepresentations, meaning the only G-invariant subspaces of V are \{0\} and V itself.[1] In the context of module theory, such a representation corresponds to a simple module over the group algebra k[G], where the module has no proper nontrivial submodules. A key criterion for irreducibility is given by Schur's lemma, which characterizes the endomorphism ring of an irreducible representation. Specifically, if \rho: G \to \mathrm{GL}(V) is an irreducible representation over an algebraically closed field k and \mathrm{End}_G(V) = \{\phi \in \mathrm{End}(V) \mid \phi \circ \rho(g) = \rho(g) \circ \phi \ \forall g \in G\} denotes the space of G-equivariant endomorphisms of V, then \mathrm{End}_G(V) = k \cdot \mathrm{Id}_V, the scalar multiples of the identity operator. To prove this, first note that for any \phi \in \mathrm{End}_G(V), the image \mathrm{im}(\phi) is a G-invariant subspace of V. Since \rho is irreducible and \phi \neq 0, it follows that \mathrm{im}(\phi) = V, so \phi is surjective. Similarly, the kernel \ker(\phi) is G-invariant, and since \dim V < \infty, surjectivity implies injectivity, hence \ker(\phi) = \{0\}. Thus, \phi is invertible, and the set of all such \phi forms a division algebra over k. By the assumption that k is algebraically closed, this division algebra must be k itself, so every \phi \in \mathrm{End}_G(V) is scalar. Conversely, if \mathrm{End}_G(V) = k \cdot \mathrm{Id}_V, then any nonzero \phi \in \mathrm{End}_G(V) is invertible, implying that no proper nontrivial G-invariant subspace exists, as the image of any such subspace under \phi would contradict irreducibility.[1] This criterion provides a practical test for irreducibility: a representation is irreducible if and only if its endomorphism ring consists solely of scalars. For finite groups, Maschke's theorem guarantees that representations decompose into irreducibles under suitable conditions. Precisely, if G is finite and k is a field whose characteristic does not divide |G|, then every finite-dimensional representation of G over k is semisimple, meaning it is isomorphic to a direct sum of irreducible representations.[36] The proof proceeds by showing that any subrepresentation has a complementary invariant subspace. Suppose W \subset V is a G-invariant subspace of a finite-dimensional representation (V, \rho). Equip V with an inner product \langle \cdot, \cdot \rangle, and define the projection P: V \to W by averaging over the group: more invariantly, the orthogonal projection onto W followed by group averaging P(v) = \frac{1}{|G|} \sum_{g \in G} \rho(g) P_0(\rho(g^{-1}) v), where P_0 is the orthogonal projection to W. Since \mathrm{char}(k) \nmid |G|, the averaging operator is well-defined and G-equivariant, and its image is W with kernel complementary to W. This yields V = W \oplus U with U G-invariant. Iterating this decomposition shows full semisimplicity.[1] Moreover, for finite G, the number of distinct irreducible representations (up to isomorphism) over an algebraically closed field of characteristic not dividing |G| equals the number of conjugacy classes in G.[1] This semisimplicity extends to compact groups via Weyl's unitary trick. For a compact Lie group G, every continuous finite-dimensional representation on a complex vector space is equivalent to a unitary representation with respect to some G-invariant Hermitian inner product, obtained by averaging an arbitrary inner product over G using the Haar measure. Since unitary representations preserve the inner product, any invariant subspace has an orthogonal complement that is also invariant, mirroring the finite-group case and implying complete reducibility into irreducibles.[1] This trick reduces the study of representations of compact groups to the unitary case, where orthogonality relations simplify analysis.Character Theory
Definition and basic properties of characters
In representation theory, the character of a representation \rho: G \to \mathrm{GL}(V) of a finite group G on a finite-dimensional complex vector space V is defined as the function \chi_\rho: G \to \mathbb{C} given by \chi_\rho(g) = \operatorname{tr}(\rho(g)) for each g \in G, where \operatorname{tr} denotes the trace of the matrix representing the linear operator \rho(g).[1] This definition extends to a linear functional on the group algebra \mathbb{C}[G] by linearity.[1] Characters possess several fundamental algebraic properties. First, \chi_\rho is a class function, meaning \chi_\rho(gh) = \chi_\rho(hg) for all g, h \in G, since the trace satisfies \operatorname{tr}(\rho(gh)) = \operatorname{tr}(\rho(g)\rho(h)) = \operatorname{tr}(\rho(h)\rho(g)) = \operatorname{tr}(\rho(hg)) and is invariant under simultaneous conjugation of the matrices.[1] Additionally, \chi_\rho(e) = \dim V, as \rho(e) is the identity operator whose trace equals the dimension of the space.[1] For representations that can be chosen unitary (which is always possible over \mathbb{C}), \chi_\rho(g^{-1}) = \overline{\chi_\rho(g)}, the complex conjugate, because the eigenvalues of \rho(g) are roots of unity and those of \rho(g^{-1}) are their conjugates.[1] Characters exhibit multiplicativity with respect to direct sums and tensor products of representations. For representations \rho: G \to \mathrm{GL}(V) and \sigma: G \to \mathrm{GL}(W), the character of the direct sum satisfies \chi_{\rho \oplus \sigma}(g) = \chi_\rho(g) + \chi_\sigma(g) for all g \in G, following from the additivity of the trace on block-diagonal matrices.[1] Similarly, the character of the tensor product is \chi_{\rho \otimes \sigma}(g) = \chi_\rho(g) \chi_\sigma(g), as the trace of the Kronecker product of matrices multiplies accordingly.[1] The space of class functions on G carries a Hermitian inner product defined by \langle \chi, \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \chi(g) \overline{\psi(g)} for characters \chi, \psi.[1] This inner product is positive definite and induces a Hilbert space structure, with \langle \chi_\rho, \chi_\sigma \rangle = \dim \mathrm{Hom}_G(V, W) measuring the intertwining dimension between representations.[1] The kernel of a character \chi, defined as \ker \chi = \{ g \in G \mid \chi(g) = \chi(e) \}, forms a normal subgroup of G.[37] This subgroup consists of elements acting as scalar multiples of the identity on the representation space, up to the character's trace value. For a representation \sigma of a subgroup H \leq G, the character of the induced representation \chi_{\mathrm{Ind}_H^G \sigma} is given by the formula \chi_{\mathrm{Ind}_H^G \sigma}(g) = \frac{1}{|H|} \sum_{\substack{k \in G \\ k^{-1} g k \in H}} \sigma(k^{-1} g k) for g \in G, where the sum is over those k such that the conjugate k^{-1} g k lies in H (and \sigma is extended by zero outside H).[1] This expression, known as the Frobenius formula, arises from averaging the action over cosets.[1]Orthogonality and decomposition
One of the key features of character theory for finite groups is the orthogonality relations satisfied by the characters of irreducible representations. These relations arise from the inner product on the space of class functions F_c(G, \mathbb{C}), defined by \langle \chi, \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \chi(g) \overline{\psi(g)}, where G is a finite group and \chi, \psi are class functions (constant on conjugacy classes). For characters \chi_V and \chi_W of irreducible representations V and W, this inner product equals \dim \Hom_G(V, W), which is 1 if V \cong W and 0 otherwise.[38] Thus, the column orthogonality relation states that \sum_{g \in G} \chi_V(g) \overline{\chi_W(g)} = |G| \delta_{V,W}, where \delta_{V,W} is the Kronecker delta. This orthogonality implies that distinct irreducible characters are linearly independent.[38] The row orthogonality relation provides another perspective, focusing on sums over irreducible characters for fixed group elements. For elements g, h \in G, \sum_V \chi_V(g) \overline{\chi_V(h)} = |C_G(g)| \quad \text{if } g \text{ and } h \text{ are conjugate, and } 0 \text{ otherwise}, where the sum is over a complete set of irreducible representations V and C_G(g) is the centralizer of g in G. Since the size of the conjugacy class of g is |G|/|C_G(g)|, this relation shows that the columns of the character table (indexed by conjugacy classes) are orthogonal when appropriately weighted by class sizes.[38] These orthogonality relations establish the completeness of the irreducible characters: they form an orthonormal basis for the vector space of class functions F_c(G, \mathbb{C}) with respect to the inner product above. The dimension of this space equals the number of conjugacy classes, which matches the number of irreducible representations by basic properties of characters. This basis property allows any class function, including the character of an arbitrary representation, to be uniquely expressed as a linear combination of irreducible characters.[38] A central application is the decomposition of any finite-dimensional representation \rho: G \to \GL(V) into a direct sum of irreducible representations. The multiplicity m_i of the irreducible representation with character \chi_i in the decomposition \chi = \sum_i m_i \chi_i is given by the inner product m_i = \langle \chi, \chi_i \rangle = \frac{1}{|G|} \sum_{g \in G} \chi(g) \overline{\chi_i(g)}. Since characters determine representations up to isomorphism over \mathbb{C}, this formula completely classifies the representation via its character table projection.[38] Character tables tabulate these values for all conjugacy classes and irreducible characters, facilitating computations. For example, the symmetric group S_3 has three conjugacy classes: the identity (size 1), transpositions like (1\,2) (size 3), and 3-cycles like (1\,2\,3) (size 2). Its three irreducible representations yield the following character table:[38]| Representation | Id (size 1) | (1 2) (size 3) | (1 2 3) (size 2) |
|---|---|---|---|
| Trivial (C_+) | 1 | 1 | 1 |
| Sign (C_-) | 1 | -1 | 1 |
| Standard (C_2) | 2 | 0 | -1 |