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Representation theory

Representation theory is a branch of that studies , such as groups, rings, and Lie algebras, by examining their actions on vector spaces through linear transformations, often representing these actions via homomorphisms to the general linear group GL(V). This approach translates otherwise intractable problems in into more manageable questions in linear algebra, particularly when the vector spaces are finite-dimensional, allowing elements of the algebraic structure to be expressed as matrices. The theory originated in the late 19th and early 20th centuries, with foundational work by mathematicians like Georg Frobenius and on representations of finite groups, motivated by the need to understand symmetries in and physics. A central goal is to classify representations up to , focusing on irreducible representations—those that cannot be decomposed into simpler non-trivial subrepresentations—as these form the building blocks for all others via direct sums. Key tools include , which uses traces of representation matrices to distinguish irreducibles and compute decomposition multiplicities, and concepts like induced and restricted representations to relate actions from subgroups to the full group. Representation theory extends beyond finite groups to infinite structures like Lie groups and algebras, where continuous representations play a crucial role in and . For instance, the representations of the sl(2,ℂ) underpin much of , including in . It also encompasses the study of modules over associative algebras and quiver representations, with classification results like Gabriel's theorem for finite-type using Dynkin diagrams. The subject has profound applications across mathematics and physics, serving as a cornerstone in areas such as (via modular representations), (through sheaf theory), and (in , , and condensed matter systems). By providing a linear lens on , representation theory bridges pure abstraction with concrete computation, influencing fields from to .

Foundations

Linear representations

A linear representation of a finite group G on a finite-dimensional vector space V over a field k is a group homomorphism \rho: G \to \mathrm{GL}(V), where \mathrm{GL}(V) denotes the general linear group of invertible linear endomorphisms of V. Equivalently, it equips V with a linear action of G such that \rho(gh) = \rho(g) \rho(h) for all g, h \in G and \rho(e) = \mathrm{id}_V for the identity e \in G. This framework allows groups to be studied through their actions on vector spaces, bridging abstract group theory with linear algebra. Explicit examples illustrate this concept. The regular representation of G arises from its left multiplication action on the group algebra k[G], the with basis \{ e_g \mid g \in G \}, where \rho(h) e_g = e_{hg} for h, g \in G; this yields a of |G|. Another fundamental example is the associated to a of G on a S: it acts linearly on the k^S with basis \{ e_s \mid s \in S \} by permuting the basis elements, \rho(g) e_s = e_{g \cdot s}. The notion extends naturally to associative algebras. A representation of an associative algebra A over k is a finite-dimensional vector space V equipped with a unital algebra homomorphism \rho: A \to \mathrm{End}_k(V), making V a left A-module where the action is k-linear and satisfies distributivity: a \cdot (v + w) = a \cdot v + a \cdot w and (a + b) \cdot v = a \cdot v + b \cdot v for a, b \in A, v, w \in V. A key result concerning such representations is , which characterizes the endomorphisms commuting with the . For an V of a G over an k, the \mathrm{End}_G(V) = \{ T \in \mathrm{End}_k(V) \mid T \rho(g) = \rho(g) T \ \forall g \in G \} is a over k; in particular, if k = \mathbb{C}, it is isomorphic to \mathbb{C}. Basic examples highlight diverse structures. The trivial representation assigns \rho(g) = \mathrm{id}_V for all g \in G, yielding the one-dimensional space where G acts invariantly. For the S_n, the sign representation is the one-dimensional \mathrm{sgn}: S_n \to \mathrm{GL}_1(k) \cong k^\times given by \mathrm{sgn}(\sigma) = (-1)^m, where m is the number of even-length cycles in \sigma (or equivalently, the parity of inversions). The general linear group \mathrm{GL}_n(k) admits the standard representation on the natural V = k^n, where A \in \mathrm{GL}_n(k) acts by left : A \cdot v = A v for v \in k^n.

Group actions and homomorphisms

A provides a fundamental way to study how a group G interacts with a set X, generalizing the notion of in algebraic structures. Formally, a left of G on X is a map \cdot: G \times X \to X satisfying two axioms: the acts trivially, so e \cdot x = x for all x \in X, and the action is compatible with the group operation, so g_1 \cdot (g_2 \cdot x) = (g_1 g_2) \cdot x for all g_1, g_2 \in G and x \in X. These axioms ensure that each group element induces a of X, preserving the set's structure under . For any x \in X, the orbit of x is the set \{g \cdot x \mid g \in G\}, which collects all points reachable from x via the action, and the stabilizer of x is the \{g \in G \mid g \cdot x = x\}. The orbit-stabilizer theorem relates these: if G is finite, then |G| = |\operatorname{orbit}(x)| \cdot |\operatorname{stabilizer}(x)| for any x \in X. This theorem highlights how the size of the group decomposes into the "reach" of the orbit and the "symmetries" fixing a point, providing a key tool for counting and classifying actions. Every group action corresponds to a homomorphism \rho: G \to \operatorname{Sym}(X), the on X, defined by \rho(g)(x) = g \cdot x. The homomorphism property follows directly from the action axioms: \rho(e) is the permutation, and \rho(g_1) \circ \rho(g_2) = \rho(g_1 g_2). Conversely, any such homomorphism defines an action via g \cdot x = \rho(g)(x). The action is faithful if the of \rho is trivial, meaning distinct group elements induce distinct permutations. Given actions of G on sets X and Y, an (or G-map) f: X \to Y is a function satisfying g \cdot f(x) = f(g \cdot x) for all g \in G and x \in X. Such maps preserve the structure, allowing comparison between different actions. Two actions are isomorphic if there exists a bijective equivariant map with an equivariant inverse, meaning the actions are essentially the same up to relabeling the set. Examples illustrate these concepts. A transitive action has a single orbit, so G can map any point in X to any other; for instance, G acts transitively on the cosets G/H by left multiplication. A free action has trivial stabilizers for all points, so no non-identity element fixes any point; the left regular action of G on itself is free. A core-free subgroup H of G is one whose core—the intersection of all conjugates gHg^{-1}—is trivial; the induced action of G on the cosets G/H is then faithful. Linear representations arise as a special case of group actions where X is a and the permutations are linear transformations.

Modules over algebras

In representation theory, a representation of an A over a k can be described as a left A- M, which is a over k equipped with a A \times M \to M, denoted (a, m) \mapsto a \cdot m, satisfying the associativity condition (ab) \cdot m = a \cdot (b \cdot m) for all a, b \in A and m \in M. This action induces a \rho: A \to \operatorname{End}_k(M), where \operatorname{End}_k(M) is the algebra of k-linear endomorphisms of M, making M into a module in the category of representations of A. Often, one restricts to unital modules, where the unit $1_A \in A acts as the identity: $1_A \cdot m = m for all m \in M. This ensures the representation respects the algebraic structure of A fully. A more general construction involves bimodules, which are vector spaces M with both left and right actions of A, satisfying (a \cdot m) \cdot b = a \cdot (m \cdot b) for a, b \in A and m \in M. In this case, Z(A) = \{ z \in A \mid za = az \ \forall a \in A \} acts centrally on M, meaning z \cdot m = m \cdot z for all z \in Z(A) and m \in M. Key notions in this framework include simple modules, which are non-zero left A-modules with no proper non-zero submodules, serving as the irreducible building blocks analogous to irreducible representations. A is defined as a of simple modules. Prominent examples illustrate these concepts. Modules over the group algebra k[G], where G is a , recover linear representations of G, as the left of group elements on M defines the representation homomorphism G \to \operatorname{GL}_k(M). For the full matrix algebra M_n(k), there is a unique simple module up to , namely k^n with the standard action by left on column vectors.

Core Properties

Subrepresentations and irreducibles

A subrepresentation of a representation \rho: G \to \mathrm{[GL](/page/GL)}(V) on a V is a W \subseteq V that is under the , meaning \rho(g)W \subseteq W for all g \in G. This invariance ensures that W itself carries a representation structure induced by the restriction of \rho to W. Given a subrepresentation W \subseteq V, the quotient space V/W inherits a natural representation structure defined by \rho(g)(v + W) = \rho(g)v + W for v \in V and g \in G. This quotient representation satisfies a : for any equivariant \phi: V \to U from the original to another U such that \ker(\phi) \supseteq W, there exists a unique equivariant \overline{\phi}: V/W \to U making the diagram commute. An is a nonzero representation with no proper nontrivial subrepresentations, meaning the only subspaces are \{0\} and V itself. In the language of module theory, correspond precisely to simple modules over the group \mathbb{C}[G]. In contrast, an indecomposable is one that cannot be expressed as a of two nontrivial subrepresentations. While every is indecomposable, the converse does not hold in general, though it does over algebraically closed fields of characteristic zero for finite-dimensional representations of finite groups. For finite-length modules, such as finite-dimensional representations of finite groups, the Jordan-Hölder theorem guarantees the existence of : chains of subrepresentations $0 = W_0 \subset W_1 \subset \cdots \subset W_n = V where each successive quotient W_{i+1}/W_i is irreducible. Any two such have the same length n and the same irreducible factors up to and permutation. This uniqueness underscores the role of irreducibles as the fundamental building blocks of representations.

Direct sums and complete reducibility

The direct sum of two representations \rho: G \to \mathrm{GL}(V) and \sigma: G \to \mathrm{GL}(W) of a group G over a field k is the representation \rho \oplus \sigma: G \to \mathrm{GL}(V \oplus W) defined by (\rho \oplus \sigma)(g)(v, w) = (\rho(g)v, \sigma(g)w) for all g \in G, v \in V, and w \in W. This construction equips the vector space direct sum V \oplus W with a G-action that acts componentwise, making it isomorphic to the external direct sum as kG-modules. A representation \rho: G \to \mathrm{GL}(V) is called completely reducible if V decomposes as a of irreducible subrepresentations, i.e., V = U_1 \oplus \cdots \oplus U_r where each U_i is an irreducible G-submodule of V. Equivalently, for every subrepresentation U \subseteq V, there exists a complementary subrepresentation W \subseteq V such that V = U \oplus W. In the language of modules, a kG-module V is completely reducible (or semisimple) if it is a of simple (irreducible) submodules. Maschke's theorem provides conditions under which all finite-dimensional representations of a are completely reducible. Specifically, if G is a and k is a whose does not divide |G|, then every finite-dimensional kG- is semisimple, i.e., a of simple modules. This holds in particular when \mathrm{char}(k) = 0, such as over \mathbb{C} or \mathbb{R}. The theorem implies that the group algebra kG is semisimple as a under these conditions. Semisimple modules over a semisimple A (such as kG under the hypotheses of Maschke's ) admit unique decompositions up to into direct sums of modules. The Artin-Wedderburn classifies such algebras: a finite-dimensional semisimple A over a k is isomorphic to a of algebras over s, A \cong \bigoplus_{i=1}^r M_{n_i}(D_i), where each D_i is a finite-dimensional over k and the n_i are positive integers. If k is algebraically closed (e.g., \mathbb{C}), then each D_i \cong k, so A \cong \bigoplus_{i=1}^r M_{n_i}(k). For the group \mathbb{C}G with G finite, this yields \mathbb{C}G \cong \bigoplus_{i=1}^r M_{n_i}(\mathbb{C}), where r is the number of irreducible representations of G (equal to the number of conjugacy classes) and \sum_{i=1}^r n_i^2 = |G|. Maschke's theorem fails when the characteristic p > 0 of the divides the group order, leading to non-semisimple modules. A standard counterexample is the H_p of order p^3 over a k of characteristic p, which can be realized as the group of $3 \times 3 upper-triangular matrices over \mathbb{F}_p with ones on the diagonal. This group admits p^2 one-dimensional irreducible representations and p-1 irreducible representations of dimension p (obtained by from the abelian of order p^2), but not all representations are completely reducible; for instance, the has a composition series of length greater than the number of simple summands would suggest in the semisimple case.

Tensor products and characters

In representation theory, the tensor product provides a fundamental construction for combining two representations of a group G over a k. Given representations \rho: G \to \mathrm{GL}(V) and \sigma: G \to \mathrm{GL}(W) on spaces V and W, the representation \rho \otimes \sigma acts on the space V \otimes_k W via the formula (\rho \otimes \sigma)(g)(v \otimes w) = \rho(g)v \otimes \sigma(g)w for all g \in G and v \in V, w \in W. This defines a representation over any k, preserving the bilinear structure of the . A key tool for analyzing representations, particularly for finite groups, is the character, defined as the trace of the representing matrices. For a representation \rho: G \to \mathrm{GL}(V), the character \chi_\rho: G \to k is given by \chi_\rho(g) = \mathrm{tr}(\rho(g)). Characters are class functions, meaning \chi_\rho(g) = \chi_\rho(hgh^{-1}) for all h \in G, so they remain constant on conjugacy classes of G. Characters exhibit linearity with respect to direct sums and multiplicativity with respect to tensor products. Specifically, for representations \rho on V and \sigma on W, the character of the direct sum satisfies \chi_{V \oplus W} = \chi_V + \chi_W, while the character of the tensor product is \chi_{V \otimes W}(g) = \chi_V(g) \chi_W(g). Over the complex numbers, the character of the dual representation \bar{V} (with action \rho(g^{-1})^t) satisfies \chi_{\bar{V}}(g) = \overline{\chi_V(g)}. For finite groups G, characters form a basis for the space of class functions, and an inner product can be defined to measure their . The inner product of two characters \chi and \psi is \langle \chi, \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \chi(g) \overline{\psi(g)}, which takes nonnegative integer values. This inner product equals the dimension of the space of G-equivariant homomorphisms: \dim \mathrm{Hom}_G(V, W) = \langle \chi_V, \chi_W \rangle. Thus, characters facilitate the decomposition of representations into irreducibles by computing multiplicities.

Representations of Finite Groups

Irreducible representations over complex numbers

For finite groups over the complex numbers, every representation is completely reducible into a of irreducible representations, which serve as the fundamental building blocks. The irreducible representations of a G are finite in number and correspond bijectively to the conjugacy classes of G, a result established through the theory of characters. These irreducible representations can be constructed explicitly using projection operators derived from characters. For an irreducible character \chi of dimension d = \dim \chi, the projection operator onto the \chi-isotypic component of a representation \rho: G \to \mathrm{GL}(V) is given by e_\chi = \frac{d}{|G|} \sum_{g \in G} \overline{\chi}(g) \, \rho(g), where |G| is the order of G. This operator is a central idempotent in the group algebra \mathbb{C}[G] and projects V onto the subspace where \chi appears, allowing the isolation of irreducible summands. The regular representation of G, which acts on \mathbb{C}[G] by left , provides a canonical example of complete reducibility. It decomposes as the \mathrm{Reg}(G) = \bigoplus_{\chi \in \mathrm{Irrep}(G)} \chi^{\oplus d}, where the sum is over all irreducible characters \chi and each appears with multiplicity equal to its d. This decomposition underscores the role of irreducibles in spanning the full representation theory of G. For abelian groups, all irreducible representations over \mathbb{C} are one-dimensional, corresponding to homomorphisms from G to the \mathbb{C}^\times, with one such representation per group element (since conjugacy classes are singletons). A non-abelian example is the S_3, which has three conjugacy classes and thus three irreducibles: the trivial representation (dimension 1), the sign representation (dimension 1), and the standard representation on the plane orthogonal to the trivial (dimension 2). Irreducible representations over \mathbb{C} further classify into real, quaternionic, or complex types via the Frobenius-Schur indicator \nu_2(\chi) = \frac{1}{|G|} \sum_{g \in G} \chi(g^2), which equals 1 for real type (realizable over \mathbb{R}), -1 for quaternionic type (requiring a quaternionic structure), and 0 for type (not realizable over \mathbb{R} or \mathbb{H}). This indicator determines the Schur index and the minimal over which the representation is defined.

Character theory and orthogonality relations

Character theory provides a powerful for analyzing representations of finite groups over the numbers by associating to each a known as its . The \chi_\rho of a \rho: G \to \mathrm{GL}(V) of a G on a finite-dimensional V is defined by \chi_\rho(g) = \mathrm{Tr}(\rho(g)) for all g \in G. Since \rho can be chosen unitary, characters satisfy \chi_\rho(g^{-1}) = \overline{\chi_\rho(g)} and are constant on conjugacy classes, making them s. Introduced by Georg Frobenius in his paper on group characters, this concept allows representations to be studied via their traces without explicit matrix forms. The orthogonality relations for irreducible characters form the cornerstone of character theory, enabling the classification and decomposition of representations. For distinct irreducible characters \chi and \psi of G, the row orthogonality relation states that \sum_{g \in G} \chi(g) \overline{\psi(g)} = |G| \delta_{\chi\psi}, where \delta_{\chi\psi} = 1 if \chi = \psi and 0 otherwise; this holds with equality to |G| when \chi = \psi. These relations, fully established by Issai Schur in 1905 building on Frobenius's work, arise from the unitarity of representations and the inner product on class functions defined by \langle f, h \rangle = \frac{1}{|G|} \sum_{g \in G} f(g) \overline{h(g)}, under which irreducible characters are orthonormal. A complementary column orthogonality relation governs sums over characters at elements from distinct conjugacy classes. Let g_i and g_j be representatives of conjugacy classes of G; then \sum_{\chi} \chi(g_i) \overline{\chi(g_j)} = |C_G(g_i)| \delta_{ij}, where the sum runs over all irreducible \chi, C_G(g_i) is the centralizer of g_i in G, and \delta_{ij} = 1 if g_i and g_j are conjugate (i.e., i = j) and 0 otherwise. Equivalently, if g and h are conjugate, the sum \sum_{\chi} \chi(g) \overline{\chi(h)} = |C_G(g)|; otherwise, it vanishes. This relation reflects the structure of the character table, whose columns (indexed by conjugacy classes) satisfy a scaled tied to centralizer sizes. The completeness of the irreducible characters follows directly from these orthogonality relations: they form an for the of all class functions on G with respect to the inner product \langle \cdot, \cdot \rangle. The dimension of this space equals the number of conjugacy classes of G, implying that the number of irreducible complex representations (and thus irreducible characters) also equals the number of conjugacy classes; this is , a key application linking representation theory to group structure. These tools facilitate the decomposition of any finite-dimensional complex representation V of G into irreducibles. The character \chi_V of V decomposes uniquely as \chi_V = \sum_{\chi} m_{\chi} \chi, where the sum is over irreducible characters \chi and the multiplicity m_{\chi} of the corresponding in V is given by the inner product m_{\chi} = \langle \chi_V, \chi \rangle. This formula, derived from the of characters, allows explicit computation of decomposition numbers using the character table without constructing the representation matrices.

Induced representations and Frobenius reciprocity

In , the restriction functor provides a way to descend representations from a group to a . For a finite group G and H \leq G, the restriction of a \mathbb{C}G-module W to H, denoted \operatorname{Res}_H^G W, is simply W viewed as a \mathbb{C}H-module by restricting the action to elements of H. To construct representations of G from those of H, one uses . Given a \mathbb{C}H- V, the \operatorname{Ind}_H^G V is the \mathbb{C}G- \mathbb{C}G \otimes_{\mathbb{C}H} V, where \mathbb{C}G and \mathbb{C}H are the complex group algebras. The of \operatorname{Ind}_H^G V is [G:H] \cdot \dim V. Equivalently, \operatorname{Ind}_H^G V can be realized as the space of functions f: G \to V satisfying f(hg) = h \cdot f(g) for all h \in H, g \in G, with G- given by (g' \cdot f)(g) = f(g g'). The of an admits an explicit formula. If \chi_V is the of V, then the \chi_{\operatorname{Ind}_H^G V} of \operatorname{Ind}_H^G V is \chi_{\operatorname{Ind}_H^G V}(g) = \frac{1}{|H|} \sum_{t \in G} \hat{\chi}_V(t^{-1} g t), where \hat{\chi}_V(x) = \chi_V(x) if x \in H and $0 otherwise. A fundamental relation between and restriction is given by Frobenius reciprocity. As functors between the categories of , is left to restriction: for \mathbb{C}H-modules U and \mathbb{C}G-modules V, \operatorname{Hom}_{\mathbb{C}G}(\operatorname{Ind}_H^G U, V) \cong \operatorname{Hom}_{\mathbb{C}H}(U, \operatorname{Res}_H^G V). In terms of , if \chi is the of a of G and \lambda is the of a of H, then \langle \chi_{\operatorname{Ind}_H^G \lambda}, \chi \rangle_G = \langle \lambda, \operatorname{Res}_H^G \chi \rangle_H, where \langle \cdot, \cdot \rangle_K denotes the standard inner product on class functions of K. For compact groups, the theory of induced representations extends analogously when H is an open subgroup of finite index, and such inductions from finite subgroups yield representations whose spans are dense in the space of square-integrable functions on the group in the Peter-Weyl sense.

Representations of Lie Structures

Representations of Lie algebras

A representation of a Lie algebra \mathfrak{g} over an algebraically closed field of characteristic zero, such as \mathbb{C}, on a vector space V is defined as a Lie algebra homomorphism \rho: \mathfrak{g} \to \mathfrak{gl}(V) that preserves the Lie bracket, meaning [\rho(x), \rho(y)] = \rho([x, y]) for all x, y \in \mathfrak{g}. This linear action allows elements of \mathfrak{g} to act as endomorphisms on V, capturing the infinitesimal symmetries encoded by the bracket structure of \mathfrak{g}. Representations are often studied in the context of semisimple Lie algebras, where \mathfrak{g} admits a Cartan–Killing form that is nondegenerate. Such a representation is equivalent to a left structure on V over the universal enveloping algebra U(\mathfrak{g}), the generated by \mathfrak{g} with relations reflecting the Lie bracket via the . The Poincaré–Birkhoff–Witt (PBW) asserts that if \{x_i\} is a basis for \mathfrak{g}, then U(\mathfrak{g}) possesses a basis consisting of all ordered monomials x_1^{k_1} \cdots x_n^{k_n} with k_i \in \mathbb{N}, providing a PBW basis that identifies U(\mathfrak{g}) with the S(\mathfrak{g}) modulo the relations from the bracket. This equivalence facilitates the study of representations through techniques while preserving the Lie . The adjoint representation offers a canonical example, where \mathfrak{g} acts on itself via \mathrm{ad}_x(y) = [x, y] for x, y \in \mathfrak{g}, yielding a homomorphism \mathrm{ad}: \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g}). For a semisimple Lie algebra \mathfrak{g}, a Cartan subalgebra \mathfrak{h} is a maximal toral subalgebra (ad-diagonalizable and abelian), and the roots \Delta \subset \mathfrak{h}^* are the nonzero eigenvalues of \mathrm{ad}_h for h \in \mathfrak{h}, forming the root system. This leads to the root space decomposition \mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Delta} \mathfrak{g}_\alpha, where each \mathfrak{g}_\alpha = \{ x \in \mathfrak{g} \mid \mathrm{ad}_h(x) = \alpha(h) x \ \forall h \in \mathfrak{h} \} is one-dimensional. In representations of semisimple \mathfrak{g}, the action of \mathfrak{h} diagonalizes V into a direct sum of weight spaces V_\lambda = \{ v \in V \mid h \cdot v = \lambda(h) v \ \forall h \in \mathfrak{h} \} for weights \lambda \in \mathfrak{h}^*, assuming V is a weight module. The root system \Delta governs the action of root vectors, shifting weights by roots: if v \in V_\lambda, then e_\alpha v \in V_{\lambda + \alpha} for positive root generators e_\alpha. Highest weight modules provide a fundamental class of representations, generated as a cyclic U(\mathfrak{g})-module by a vector v of weight \lambda \in \mathfrak{h}^* that is annihilated by the nilpotent subalgebra \mathfrak{n}_+ spanned by positive root spaces (\mathfrak{n}_+ v = 0). Verma modules exemplify these, serving as universal highest weight modules: for a Borel subalgebra \mathfrak{b} = \mathfrak{h} \oplus \mathfrak{n}_+, the Verma module M(\lambda) is U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} \mathbb{C}_\lambda, where \mathbb{C}_\lambda is the one-dimensional \mathfrak{b}-module with \mathfrak{n}_+ acting trivially and \mathfrak{h} by the character \lambda. Each M(\lambda) decomposes into weight spaces M(\lambda) = \bigoplus_{\mu \in \lambda - \mathbb{N} \Delta^+} M(\lambda)_\mu with \dim M(\lambda)_\lambda = 1, generated by the image of $1 \otimes 1. These modules are indecomposable and possess a unique irreducible quotient, central to the structure theory of representations.

Representations of Lie groups

A differentiable representation of a Lie group G on a finite-dimensional complex vector space V is a smooth Lie group homomorphism \rho: G \to \mathrm{GL}(V). Such representations are automatically analytic, meaning \rho is holomorphic when G is a complex Lie group or real analytic otherwise. The differential d\rho: \mathfrak{g} \to \mathrm{gl}(V) at the identity provides the associated , linking the global group action to the infinitesimal algebra action. The exponential map \exp: \mathfrak{g} \to G plays a key role in constructing representations of G from those of \mathfrak{g}, as every element near the identity in G arises from \exp(X) for X \in \mathfrak{g}. For finite-dimensional representations, the image under \rho \circ \exp yields the group action on analytic vectors, which in this context coincide with the entire space V due to the smoothness of \rho. This integration ensures that finite-dimensional representations of the Lie algebra extend uniquely to smooth representations of the simply connected Lie group covering G. For compact Lie groups, Weyl's unitary trick asserts that every finite-dimensional representation is equivalent to a unitary one with respect to some invariant positive definite Hermitian form on V. This follows from averaging the form over the group using the , which produces an invariant inner product. Consequently, representations of compact groups are completely reducible into irreducibles, facilitating their classification. For semisimple Lie groups, the finite-dimensional irreducible representations are parametrized by dominant integral weights \lambda in the weight lattice, extending the highest weight classification from the Lie algebra level. The character \chi_\lambda of such a representation is given by the Weyl character formula: \chi_\lambda = \frac{\sum_{w \in W} \epsilon(w) e^{w(\lambda + \rho)}}{\sum_{w \in W} \epsilon(w) e^{w \rho}}, where \rho is half the sum of positive roots, W is the Weyl group, and \epsilon(w) = \sgn(w). This formula computes the trace of \rho(g) explicitly without constructing the representation. Representative examples include the irreducible representations of \mathrm{SL}(2, \mathbb{C}), labeled by spin j = 0, 1/2, 1, \dots, where the dimension is $2j + 1 and the highest weight is $2j times the fundamental weight; these arise as symmetric powers of the standard 2-dimensional representation. For classical groups, fundamental representations include the defining n-dimensional representation of \mathrm{SL}(n, \mathbb{C}), the standard representation of \mathrm{SO}(n, \mathbb{C}), and the $2m-dimensional representation of \mathrm{Sp}(2m, \mathbb{C}), each corresponding to minuscule weights.

Infinite-dimensional unitary representations

A unitary representation of a G on a H is a continuous \rho: G \to U(H) into the group of unitary operators on H, satisfying \langle \rho(g)v, \rho(g)w \rangle = \langle v, w \rangle for all g \in G and v, w \in H, where \langle \cdot, \cdot \rangle denotes the inner product. This preserves the Hilbert space structure and ensures the representation is strongly continuous. For Lie groups, such representations often arise in and , where infinite-dimensional s like L^2(\mathbb{R}^n) are common. The Stone-von Neumann theorem exemplifies the uniqueness of certain irreducible unitary representations for nilpotent groups. For the H_n, defined as upper triangular $3 \times 3 matrices with ones on the diagonal over \mathbb{R}^n, the theorem states that every irreducible unitary satisfying the canonical commutation relations [Q_j, P_k] = -i \delta_{jk} \hbar (with position Q_j and momentum P_k operators) is unitarily equivalent to the Schrödinger representation on L^2(\mathbb{R}^n). This uniqueness holds up to unitary equivalence, resolving foundational questions in from the 1920s–1930s. For compact Lie groups G, the decomposes the on L^2(G) into a (or ) over irreducible unitary representations. Specifically, for f \in L^2(G), \|f\|^2 = \sum_{\pi} d(\pi) \|\pi(f \, dh)\|^2_{HS}, where the sum is over equivalence classes of irreducible representations \pi, d(\pi) is the of \pi, \|\cdot\|_{HS} is the Hilbert-Schmidt , and dh is the normalized . This formula integrates the contributions from each irreducible with multiplicity given by the Haar measure on the . In contrast, non-compact semisimple groups lack complete reducibility but admit specific classes of infinite-dimensional unitary representations. series representations are square-integrable irreducible unitary representations whose matrix coefficients lie in L^2(G), embedding discretely into the Plancherel decomposition of L^2(G). They exist precisely when the real rank of G equals the rank of its maximal compact subgroup K, and are classified by parameters \lambda \in i \mathfrak{t}^* that are analytically integral with respect to the half-sum \rho of positive roots. Principal series representations, on the other hand, are constructed by unitary from characters of parabolic subgroups P = [MAN](/page/The_Man) (with M compact, A abelian, N ), yielding \pi_{\tau, \sigma} = \mathrm{Ind}_P^G (\eta_{\tau, \sigma}) where \eta_{\tau, \sigma}(man) = \tau(m) e^{i \sigma(\log a)} for a representation \tau of M and \sigma \in \mathfrak{a}^*. These form a continuous family parameterizing much of the unitary dual for groups like SL(2, \mathbb{R}). A notable example is Bargmann's realization of the oscillator representation for SL(2, \mathbb{R}), which provides an explicit unitary model on the H_F of entire functions on \mathbb{C} square-integrable with respect to the e^{-|w|^2} dw / \pi. Here, the creation operator a^\dagger = w and annihilation a = d/dw satisfy [a, a^\dagger] = 1, generating actions of \mathfrak{sp}(2, \mathbb{R}) \cong \mathfrak{sl}(2, \mathbb{R}) such as \pi'(q^2 - p^2) = -i/2 ((a^\dagger)^2 + a^2), which integrate to a projective of SL(2, \mathbb{R}) or a true representation on its metaplectic cover. This realization connects to the discrete series via the Bargmann transform, unitarily equivalent to the Schrödinger model. Such representations differentiate to actions, linking infinite-dimensional unitary to finite-dimensional algebraic ones.

Modular and Specialized Representations

Modular representations and characteristic p

Modular representations of a finite group G are studied over an k of p > 0, where the group algebra k[G] generally fails to be semisimple, as Maschke's theorem does not hold when p divides |G|. In this setting, representations—equivalently, finite-dimensional k[G]-s—need not be completely reducible, leading to indecomposable modules that are not simple and extensions between simple modules. Reducing an from a field of zero (e.g., \mathbb{C}) p involves selecting a G-invariant lattice and applying a decomposition map, but irreducibility is typically not preserved; the reduced module may have a nontrivial with multiple simple factors. To analyze these reductions, Brauer characters are defined as the traces of of p-regular elements (those whose order is coprime to p) on a , taken over the \overline{k}. Unlike ordinary characters, Brauer characters are not class functions on all of G, but only on the p- conjugacy classes, and they provide a complete set of linearly independent invariants for the k[G]-, numbering equal to the number of p- classes. The matrix D relates the ordinary irreducible characters (rows) to the Brauer characters of modular representations (columns), with entries d_{\chi,\phi} giving the multiplicity of the modular corresponding to \phi in the of the reduction of the ordinary for \chi. This matrix is upper triangular with 1s on the diagonal after suitable and has nonnegative integer entries, capturing how ordinary representations modularly. The group algebra k[G] decomposes into a of indecomposable two-sided ideals called blocks, each corresponding to a central idempotent and on a of the simple modules; these blocks are the "indecomposable components" governing the modular structure. Each block B has a defect group P (a p- maximal with respect to trivially on the head of the block projective), and the block theory links representations within the block to p-local structure. Green's correspondence establishes a bijection between the indecomposable k[G]-modules in a block B with vertex (stabilizer) a p- Q and the indecomposable modules over the block of the normalizer N_G(P) (where P is a Sylow p-) with the same vertex, preserving composition factors and projectivity relative to subgroups. This correspondence, originally proved using relative projectivity, facilitates computing modular representations by relating global and local (Sylow-normalizer) data. A concrete example occurs for the S_3 over a of 2. The ordinary irreducible representations of S_3 are the trivial (dimension 1), the (dimension 1), and the (dimension 2). In characteristic 2, the representation coincides with the trivial representation. Reducing modulo 2, both the trivial and representations yield the 1-dimensional trivial simple module, while the 2-dimensional representation remains irreducible. Thus, there are two simple modules: the trivial (dimension 1) and the (dimension 2). The decomposition matrix is thus \begin{pmatrix} 1 & 0 \\ 1 & 0 \\ 0 & 1 \end{pmatrix}, with rows for trivial, , and , and columns for the modular trivial and simples.

Representations of algebraic groups

A representation of an algebraic group G over an k is a rational \rho: G \to \mathrm{GL}_n(k), where the entries of the matrices \rho(g) are rational functions in the coordinates of g that are defined everywhere on G. Such representations are also called rational representations, and they form the \mathrm{Rep}_k(G) of finite-dimensional representations of G. Over an , every algebraic group can be embedded as a closed of \mathrm{GL}_n(k) for some n, making rational representations equivalent to matrix representations defined by polynomial equations. For a reductive algebraic group G over k with \mathrm{[char](/page/Char)}(k) = 0, every finite-dimensional rational is completely reducible. This follows from the linear reductivity of connected reductive groups in characteristic zero, which ensures that every rational is semisimple. The Cartier criterion characterizes linearly reductive groups as those for which every rational on a finite-dimensional admits a rational complement to any , a property satisfied by reductive groups over algebraically closed fields of characteristic zero. In positive characteristic, complete reducibility holds only for linearly reductive groups, such as tori, but not generally for semisimple groups. The irreducible rational representations of a semisimple algebraic group G are classified using highest weight theory, analogous to the case but adapted to the algebraic setting. For a T \subset G, the irreducible representations are parametrized by dominant weights in the lattice X(T) \otimes \mathbb{Z}_{\geq 0}, where X(T) is the of T. The Weyl module \Delta(\lambda) for a dominant weight \lambda is the universal with highest weight \lambda, generated by a highest weight vector annihilated by positive Borel subgroups, and the simple head L(\lambda) of \Delta(\lambda) is the irreducible representation of highest weight \lambda. In characteristic zero, the Weyl modules coincide with the irreducibles, recovering the classical highest weight classification. As the characteristic approaches zero, these representations specialize to those of the of G. In characteristic p > 0, the Frobenius map F: G \to G^{(p)}, which raises coordinates to the p-th power, induces the Frobenius twist functor on representations: for a rational G-module V, the twisted module V^{(p)} = F^* V has the same underlying vector space but with G-action twisted by the p-th power map on scalars. This twist links rational representations of G to those in characteristic p, facilitating the study of modular representations by reducing to finite groups of Lie type via restriction to Frobenius kernels. Iterated twists V^{(p^r)} connect to the theory of restricted representations and Steinberg's tensor product theorem for decomposing irreducibles. A fundamental example is the general linear group \mathrm{GL}_n(k), whose rational representations are precisely the polynomial representations on finite-dimensional modules, realized as subspaces of the k[X_{ij}] on n \times n matrices with \mathrm{GL}_n- by . The irreducible polynomial representations of \mathrm{GL}_n(k) in characteristic zero are the Schur modules S^\lambda(V), indexed by partitions \lambda, corresponding to highest weights with at most n parts. For Chevalley groups, such as \mathrm{SL}_n(k), the representation \mathrm{St}_G is the unique irreducible of dimension q^l (where q = |k| if finite, or formal in general), appearing in the decomposition of the permutation representation on the flag variety and serving as a building block for all irreducibles via tensor products. In positive characteristic, the module for Chevalley groups is simple and projective, playing a key role in the .

Quantum groups and Hopf algebras

A Hopf algebra over a k is a H equipped with a linear antipode map S: H \to H such that the convolution product \mu \circ (S \otimes \mathrm{id}) \circ [\Delta](/page/Delta) = \eta \circ \varepsilon = \mu \circ (\mathrm{id} \otimes S) \circ [\Delta](/page/Delta), where \mu is the multiplication, \Delta the comultiplication, \eta the unit, and \varepsilon the counit; this structure generalizes both algebras and coalgebras, enabling a duality between representations and corepresentations. Classic examples include the group algebra k[G] for a G, where the comultiplication is [\Delta](/page/Delta)(g) = g \otimes g and the antipode is S(g) = g^{-1}, and the universal enveloping algebra U(\mathfrak{g}) of a \mathfrak{g}, with [\Delta](/page/Delta)(x) = x \otimes 1 + 1 \otimes x for x \in \mathfrak{g} and S(x) = -x. Representations of Hopf algebras extend beyond ordinary modules: a left H-module M becomes a Hopf module if it is also a right H-comodule with compatibility \rho(m \cdot h) = \sum m_{(0)} \cdot h_{(1)} \otimes m_{(1)} h_{(2)}, where \rho is the comodule map and Sweedler notation is used for \Delta(h) = \sum h_{(1)} \otimes h_{(2)}; corepresentations arise dually as right comodules over H, often denoted as V with \Delta_V: V \to V \otimes H. This framework unifies and comodule actions, allowing tensor products of representations to incorporate the Hopf structure via the comultiplication, which acts as a on representations. Quantum groups, specifically the Drinfeld-Jimbo quantized enveloping algebras U_q(\mathfrak{g}) for a semisimple Lie algebra \mathfrak{g}, deform the classical U(\mathfrak{g}) by introducing a parameter q \in k^\times, typically generic or a root of unity; they are Hopf algebras generated by elements E_i, F_i, K_i, K_i^{-1} (for simple roots i) satisfying relations like K_i E_j - q^{\langle \alpha_i, \alpha_j \rangle} E_j K_i = 0 and a q-Serre relation, with coproduct \Delta(E_i) = E_i \otimes 1 + K_i \otimes E_i. Quantum groups provide a deformation of Lie algebras in the sense that setting q=1 recovers U(\mathfrak{g}). For generic q, the representation category \mathrm{Rep}(U_q(\mathfrak{g})) is semisimple with highest weight modules analogous to classical ones, while at roots of unity, it features tilting modules and a braided tensor structure. These Hopf algebras are often quasi-triangular, meaning there exists an invertible universal -matrix R \in U_q(\mathfrak{g}) \otimes U_q(\mathfrak{g}) satisfying \Delta^{\mathrm{op}}(a) R = R \Delta(a) for all a \in U_q(\mathfrak{g}) and the quantum Yang-Baxter R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}; this R-matrix defines a braiding on tensor products of representations via \hat{R}(v \otimes w) = \sum \mathrm{ev}_1(v \otimes w_{(1)}) \otimes \mathrm{ev}_2(w_{(2)} \otimes v_{(1)}) \otimes v_{(2)}, endowing \mathrm{Rep}(U_q(\mathfrak{g})) with a braided structure. The quasi-triangular property ensures consistency in braided tensor products, crucial for studying invariants in representation theory. A prominent example is U_q(\mathfrak{sl}_2), generated by E, F, K, K^{-1} with relations KE = q^2 EK, KF = q^{-2} FK, EF - FE = (K - K^{-1})/(q - q^{-1}), and coproduct \Delta(E) = E \otimes 1 + K \otimes E; its finite-dimensional irreducible representations V(n) (for n \in \mathbb{N}) have q-dimension [n+1]_q = (q^{n+1} - q^{-(n+1)})/(q - q^{-1}), generalizing classical dimensions and vanishing at certain roots of unity to yield nontrivial modular categories. For irreducible representations of U_q(\mathfrak{g}), crystal bases provide a combinatorial model: introduced independently by Lusztig as bases and Kashiwara as bases, they consist of a basis B(\lambda) over \mathbb{Z} for the irreducible module L(\lambda) together with a graph encoding the action of E_i, F_i at q=0, satisfying strict relations like e_i b = 0 if no edge from b in the i-. Crystal bases for U_q(\mathfrak{sl}_2) recover the Weyl modules combinatorially, with vertices labeled by weights and edges by root vectors, facilitating decomposition of tensor products via Kashiwara .

Applications

Harmonic analysis on groups

Harmonic analysis on groups employs the machinery of representation theory to decompose functions on a group G into components transforming under irreducible representations, generalizing classical . This approach allows the study of operators and transforms on L^2(G) or related spaces, revealing the spectral structure of the group. For compact groups, the theory provides a complete orthogonal decomposition, while for non-compact cases, it involves more subtle notions of temperedness and distributions. For a compact topological group G, the Peter-Weyl theorem establishes that the L^2(G) decomposes as a over irreducible unitary \pi of G: L^2(G) \cong \bigoplus_{\pi \in \hat{G}} V_\pi \otimes V_\pi^*, where V_\pi is the representation of \pi, and the summands are spanned by the matrix coefficients of \pi, which are the functions g \mapsto \langle \pi(g) v, w \rangle for v \in V_\pi, w \in V_\pi^*. This decomposition implies that the matrix coefficients form an for L^2(G), enabling a expansion of square-integrable functions. The theorem, proved using the completeness of irreducible representations in the , underpins non-abelian on compact groups. The Fourier transform on a compact group G extends this decomposition to general functions. For f \in L^1(G), the Fourier transform at an irreducible representation \pi is the operator-valued integral \hat{f}(\pi) = \int_G f(g) \pi(g)^* \, dg, where dg is the normalized Haar measure and \pi(g)^* is the adjoint. This transform is inverted via the Plancherel formula, which provides an L^2-isometry between L^2(G) and a direct sum of Hilbert-Schmidt operator spaces over the irreducibles, weighted by the formal dimension of \pi. Convolution on L^1(G) corresponds to multiplication in the Fourier domain, facilitating the analysis of multipliers and pseudodifferential operators on G. When G is abelian, the irreducible unitary representations are one-dimensional characters, and Pontryagin duality identifies the dual group \hat{G} with the set of continuous homomorphisms from G to the circle group \mathbb{T}. This duality theorem states that for any locally compact abelian group G, the double dual map G \to \hat{\hat{G}} is a topological , allowing the \hat{f}(\chi) = \int_G f(g) \overline{\chi(g)} \, dg for \chi \in \hat{G} to invert via integration over \hat{G} with its dual . Classical examples include the on \mathbb{R} (with dual \mathbb{R}) and on \mathbb{Z} (with dual \mathbb{T}), where the theory recovers the standard for periodic or aperiodic functions. For non-compact groups, such as semisimple groups, requires handling infinite-dimensional representations and conditions. The Harish-Chandra \mathcal{C}(G) consists of functions on G that decay rapidly along geodesics in the symmetric G/K (where K is a maximal compact ), together with all derivatives under a family of differential operators including the . This is stable under and Fourier inversion, and the maps it isometrically onto a of hyperfunctions or distributions on the dual, enabling the for tempered representations. Tempered distributions on G are then continuous linear functionals on \mathcal{C}(G), extending the analysis to singular functions. Convolution algebras like L^1(G) play a central role, where the group law induces a Banach algebra structure via (f * h)(g) = \int_G f(gh^{-1}) h(h) \, dh. Representation theory classifies the closed ideals of L^1(G) through the kernels of irreducible representations or primitive ideals in the associated C^*-algebra, with the Gelfand spectrum corresponding to the unitary dual of G. For unimodular G, the ideals relate to coideals in the group von Neumann algebra, providing tools to study approximate identities and spectral synthesis.

Invariant theory and symmetries

is a branch of representation theory that studies the or functions that remain unchanged under the action of a group G on a V, providing insights into the symmetries preserved by group representations. In the context of representations on polynomial spaces, where G acts linearly on V over a k, the coordinate k[V] becomes a representation, and the invariants form a capturing the structure. The of invariants, denoted k[V]^G = \{f \in k[V] \mid \rho(g)^* f = f \ \forall g \in G\}, consists of all polynomials fixed by the induced action on functions, where \rho(g) is the representation on V and \rho(g)^* its dual pullback. For reductive groups, Hilbert's finiteness theorem guarantees that k[V]^G is finitely generated as a k-algebra when the action is rational. However, Hilbert's 14th problem asks whether the subring is always finitely generated for arbitrary algebraic group actions on finitely generated algebras; Nagata's 1959 showed this is false in general, though it holds for finite groups and many classical cases. For finite groups G, the structure of k[V]^G is particularly tractable. Molien's theorem provides a formula for the Hilbert series of the invariant ring: the dimension of the degree-d invariants is \dim k[V]^G_d = \frac{1}{|G|} \sum_{g \in G} \frac{1}{\det(1 - \rho(g) t)} \big|_{t^d}, enabling explicit computation of generators via averaging over group elements. The Reynolds operator, defined as the projection E: k[V] \to k[V]^G by E(f) = \frac{1}{|G|} \sum_{g \in G} \rho(g)^* f, is a key tool for extracting invariants; it is idempotent, k-linear, and projects onto the fixed subspace, facilitating decompositions like Reynolds' theorem for primary components in characteristic zero. Geometric invariant theory (GIT), developed by Mumford, constructs quotients for actions on projective varieties, associating to an affine action on V the Proj construction \mathbb{P}(k[V]^G) as the categorical quotient V // G, which parametrizes closed orbits. For actions of \mathrm{SL}_n, stability conditions define semistable points whose orbits have finite stabilizers, ensuring the quotient is a geometric invariant space; Hilbert-Mumford criterion characterizes instability via one-parameter subgroups bounding weights negatively. Noether normalization asserts that k[V]^G is a finitely generated module over a polynomial subring in the transcendence degree variables, providing a geometric integral extension for studying singularities in quotients. Classic examples illustrate these concepts. For the action of \mathrm{SL}_2(k) on forms of n, the of is generated by finitely many covariants, such as the for quadrics (n=2) or catalecticants for higher degrees, resolving the orbit space into weighted . In cases, Noether yields explicit parameters, like traces or power sums, generating the as a , as seen in actions on representations.

Connections to number theory and automorphic forms

Automorphic representations play a central role in connecting representation theory to number theory, particularly through their study on the adele ring \mathbb{Q}_\mathbb{A} of the rationals. For the general linear group GL_n, an automorphic representation \pi is a smooth irreducible representation of GL_n(\mathbb{Q}_\mathbb{A}) that is generated by automorphic forms, which are functions on GL_n(\mathbb{Q}_\mathbb{A}) satisfying certain invariance and growth conditions under the action of GL_n(\mathbb{Q}). Cuspidal automorphic representations, a key subclass, are those whose associated automorphic forms vanish at infinity in a suitable sense, ensuring they contribute to the discrete spectrum of the group. These representations often admit Whittaker models, which realize \pi as a space of functions W on GL_n(\mathbb{Q}_\mathbb{A}) transforming under a specific additive character \psi and the maximal unipotent subgroup, facilitating the study of Fourier coefficients and L-functions. The establishes a profound between these automorphic representations and Galois representations, linking number-theoretic objects to analytic ones. Specifically, it conjectures a bijection between irreducible n-dimensional Galois representations \rho: \Gal(\overline{\mathbb{Q}}/\mathbb{Q}) \to \GL_n(\mathbb{C}) and cuspidal automorphic representations \pi of GL_n(\mathbb{Q}_\mathbb{A}), matched via their local behaviors at finite primes through the Artin and Langlands local correspondences. This global implies that L-functions attached to \rho coincide with those of \pi, providing a unifying framework for reciprocity laws and functoriality principles in . Partial realizations of this program, such as for n=2 via modular forms, have led to breakthroughs like the proof of . In July 2024, a team of nine mathematicians, including Gaitsgory, announced a proof of the geometric Langlands , a key geometric analog and major milestone in the program. A cornerstone of this connection is the Ramanujan conjecture, which imposes bounds on the Satake parameters of unramified automorphic s. For a cuspidal automorphic \pi of GL_n(\mathbb{Q}_\mathbb{A}), at an unramified prime p, the Satake parameters \{\alpha_{i,p}\}_{i=1}^n are the eigenvalues of the semisimple Frobenius conjugacy class in the dual group, and the conjecture asserts that |\alpha_{i,p}| = 1 for all i and p. This unitarity condition generalizes Ramanujan's observation on the partition function and ensures the and of the associated L(s, \pi). While proven for n=2 by Deligne using , the general case remains open but has been established for many higher-rank cases via techniques like base change and . The Arthur-Selberg trace formula provides a powerful tool for analyzing these representations, equating spectral data from automorphic forms to geometric data via orbital integrals. In the context of GL_n(\mathbb{Q}_\mathbb{A}), Arthur's refinement of Selberg's original formula expresses the trace of a convolution operator on the space of automorphic forms as a weighted sum over conjugacy classes in GL_n(\mathbb{Q}), with terms involving orbital integrals that encode volumes of quotients and unipotent contributions. This identity is instrumental for computing global L-functions, as the spectral side decomposes into contributions from individual automorphic representations, allowing extraction of parameters and bounds through matching geometric terms. Applications include the computation of Tamagawa numbers and the study of functoriality transfers between groups. Concrete examples illustrate these connections, such as classical modular forms, which arise as automorphic representations of SL_2(\mathbb{Z}). A holomorphic cusp form of weight k for SL_2(\mathbb{Z}) generates a cuspidal automorphic representation \pi on GL_2(\mathbb{Q}_\mathbb{A}) via adelization, where the local component at the archimedean place corresponds to a discrete series representation of GL_2(\mathbb{R}). , on the other hand, exemplify induced representations: the non-holomorphic E(z, s) for SL_2(\mathbb{Z}) is induced from a character on the , yielding a representation \pi = \Ind_{B(\mathbb{A})}^{GL_2(\mathbb{A})} \chi that is not cuspidal but contributes to the continuous spectrum and generates the full space of modular forms of a given level. These structures underpin the modularity theorem, linking elliptic curves to such s.

Historical Development

Origins in group theory

The foundations of representation theory emerged in the mid-19th century through the study of permutation groups and abstract group actions. In 1854, published his seminal paper "On the theory of groups, as depending on the symbolic equation θ^n = 1," which introduced the modern abstract notion of a group and emphasized their realization via permutations on sets, providing the initial framework for permutation representations that would later evolve into linear representations. A pivotal advancement occurred in 1896 when Richard Dedekind, in a letter to dated March 25, examined the group determinant for the S_3 over a field of characteristic 2, observing that it failed to factor into linear terms as it did in characteristic 0; this provided the first explicit example of a modular representation where complete reducibility does not hold, highlighting the role of the field's characteristic in representation structure. Prompted by this correspondence, Frobenius developed the theory of group characters in his 1896 paper "Über Gruppencharaktere," initially for the symmetric groups S_n, where he defined characters as traces of representation matrices and demonstrated their utility in factoring the group determinant into irreducible factors, laying the cornerstone for character theory of finite groups. In 1897, William Burnside published Theory of Groups of Finite Order, a comprehensive text that formalized the representation of finite groups as subgroups of linear groups via substitutions, including detailed treatments of representations and their into transitive components, thus establishing basic tools for analyzing group actions linearly. Burnside extended these ideas from 1898 to 1904, developing concepts akin to blocks in the of representations of symmetric groups and exploring properties of characters to classify irreducible constituents. Issai Schur's contributions from 1901 to 1911 marked a deepening of the field. In his 1901 dissertation "Über eine Klasse von Matrizen, die sich einer gegebenen Matrix von beliebiger endlicher Ordnung zuordnen lassen," Schur initiated the study of rational representations of the general linear group, focusing on integrality conditions. He subsequently proved Schur's lemma in 1904, asserting that endomorphisms of an irreducible complex representation of a finite group are scalar multiples of the identity, which underpins the rigidity of irreducible modules. Over the next decade, Schur established the Schur index theorem, quantifying the minimal extension degree for realizing irreducible characters over number fields, and provided bounds on normalizer orders in permutation groups to control representation dimensions.

Key 20th-century advances

In the mid-1920s, Hermann Weyl advanced the representation theory of compact Lie groups by employing the unitary trick, which exploits the density of compact subgroups to study general representations through their unitary counterparts, thereby simplifying the analysis of finite-dimensional irreducible representations. Weyl further classified these irreducible representations via the highest weight theorem, parametrizing them by dominant integral weights in the dual of the Cartan subalgebra, a framework that unified discrete and continuous aspects of the theory. Élie Cartan's early 20th-century classification of semisimple Lie algebras over the and real numbers laid essential groundwork for their structure theory, including the use of root decompositions relative to Cartan subalgebras. Building on this, Weyl in the 1920s integrated root systems explicitly into representation theory, elucidating actions and reflections that govern invariants under transformations. In the 1940s and 1950s, axiomatized root systems, enabling the systematic construction of semisimple Lie algebras and bridging representation theory with . Cartan's work in the 1920s and 1930s on symmetric spaces, where root systems help classify irreducible factors of invariant differential operators, further connected these ideas to . During the 1930s, Richard Brauer pioneered for finite groups over fields of characteristic p, introducing modular characters as traces of representations modulo p and developing the theory of blocks, which partition the irreducible modular representations into indecomposable components linked by Brauer relations. This work resolved key challenges in understanding representations when the group order is divisible by p, providing tools to decompose modules and compute decomposition numbers for symmetric and alternating groups. In the 1950s, established the Plancherel formula for semisimple Lie groups, decomposing the on L²(G) into a direct integral of irreducible unitary representations weighted by a explicit measure on the . He also introduced discrete series representations, square-integrable irreducibles parameterized by modules, which form the discrete spectrum in the Plancherel decomposition for groups like SL(2,ℝ). The 1960s saw Nagayoshi Iwahori develop the theory of BN-pairs for reductive algebraic groups over finite fields, axiomatizing the Bruhat-Tits decomposition and linking it to Hecke algebras generated by operators. This framework classified representations of finite groups of Lie type and connected them to spherical functions and intertwining operators in the p-adic setting. From the late 1960s through the 1970s, formulated the functoriality conjectures, positing homomorphisms between L-groups that transfer automorphic representations between groups, thereby linking Galois representations in to cuspidal automorphic forms on adelic groups. These conjectures, outlined in his 1967 correspondence and subsequent papers, established a non-abelian framework, influencing reciprocity laws and the trace formula. A related development in the 1960s was the Milnor-Moore theorem, characterizing connected graded Hopf algebras as enveloping algebras of their primitive elements' s.

Modern extensions and categorification

In the late 1990s and early 2000s, representation theory saw significant advancements through categorification, a process that lifts algebraic structures like or categories to higher-dimensional objects such as complexes or 2-categories, providing richer homological interpretations. A seminal example is Mikhail Khovanov's 2000 construction of a bigraded theory for links, which categorifies the Jones polynomial—a quantum associated with the \mathfrak{sl}(2)—using chain complexes of graded vector spaces. This approach not only recovers the original as the but also introduces torsion elements that distinguish links beyond polynomial invariants, influencing subsequent developments in and higher representation theory. Parallel to these combinatorial categorifications, geometric methods emerged to reinterpret representations via algebro-geometric objects. In the 1990s, , , and Robert MacPherson provided a geometric realization of the quantum deformation of \mathrm{GL}_n, embedding representations into the structure sheaf of varieties and paving the way for the geometric . This equates the of perverse sheaves on the affine with representations of the Langlands group, offering a geometric counterpart to classical Satake correspondence and facilitating proofs of deep properties in through sheaf-theoretic tools. Wolfgang Soergel's introduction of bimodules in the early further bridged algebraic and geometric representation theory by categorifying Hecke algebras associated to Coxeter groups. Soergel bimodules, constructed as certain Tor-independent bimodules over polynomial rings, form a whose recovers the , with indecomposables corresponding to Kazhdan-Lusztig basis elements. This framework, refined in the , has applications in proving conjectures on category \mathcal{O} and extends to diagrammatic categories, enabling combinatorial computations of representation-theoretic data. Derived categorical perspectives have also evolved, revealing equivalences that unify linear representations with geometric data. The bounded derived category D^b(\mathrm{Rep}\, G) of finite-dimensional representations of a reductive group G admits an with the derived category of coherent sheaves on the flag variety G/B, realized through localization functors that invert the action of the universal enveloping algebra. This , building on Beilinson-Bernstein localization, allows on representations to be studied via sheaf on flag varieties, illuminating decomposition theorems and support varieties in positive characteristic. More recent developments in the explore abelianization processes in representation categories, constructing d-abelian structures from derived categories to model higher homological dimensions. For instance, abelian Hall categories arise from quivers, categorifying preprojective K-theoretic Hall algebras and providing finite-length monoidal abelian envelopes for combinatorial problems. These constructions enhance understanding of stability conditions and tilting in non-abelian settings. Emerging connections link quiver representations to , particularly through , where quiver representations model filtrations in high-dimensional data. This intersection applies representation-theoretic invariants to detect persistent topological features in datasets, aiding robust feature extraction in applications. In the , Victor Ginzburg introduced Calabi-Yau completions for algebras, extending to Lie superalgebras by equipping their enveloping algebras with differential graded structures that mimic Calabi-Yau geometry. These completions, defined via superpotentials, yield 3-Calabi-Yau categories whose derived equivalences preserve homological properties, facilitating Koszul duality for representations of supergroups and connections to mirror symmetry.

Generalizations

Representations in abelian categories

In an abelian category \mathcal{C}, a representation of a group G is given by an object M \in \mathcal{C} equipped with a group homomorphism \rho: G \to \Aut_{\mathcal{C}}(M), where \Aut_{\mathcal{C}}(M) denotes the group of automorphisms of M in \mathcal{C}; this action must preserve the abelian structure, meaning it is compatible with the kernels and cokernels inherent to \mathcal{C}. More generally, the category \Rep(G, \mathcal{C}) of such representations has objects as pairs (M, \rho) and morphisms as \mathcal{C}-morphisms intertwining the actions. For finite groups, the functor \rho often preserves finite products when \mathcal{C} admits them, ensuring the representation respects direct sums. This framework generalizes classical representations beyond vector spaces, allowing actions on objects like modules or sheaves while leveraging exact sequences for subrepresentations. Irreducible representations in \Rep(G, \mathcal{C}) correspond to simple objects, those with no proper subobjects that are themselves subrepresentations (i.e., the only subrepresentations are 0 and the object itself). Complete reducibility holds if every object in \Rep(G, \mathcal{C}) decomposes as a of objects, making the semisimple. This property fails in general abelian categories but occurs under suitable conditions, such as when \mathcal{C} is the of vector spaces over a k of not dividing |G| for finite G, by Maschke's theorem, which implies every representation is semisimple. In , for \mathcal{C} the of abelian groups, representations of G are \mathbb{Z}G-s, where torsion elements can prevent complete reducibility; for example, the augmentation module \mathbb{Z} over \mathbb{Z}G for finite G has a short $0 \to I_G \to \mathbb{Z}G \to \mathbb{Z} \to 0 that does not split, exhibiting non-semisimplicity due to torsion issues. The K_0(\Rep(G, \mathcal{C})) is the on isomorphism classes of objects in \Rep(G, \mathcal{C}), modulo relations [A] + [C] = [B] from short exact sequences $0 \to A \to B \to C \to 0; it captures formal differences of representations and, in semisimple cases, is freely generated by classes of irreducibles. The rank function on K_0(\Rep(G, \mathcal{C})), when defined (e.g., via a fiber functor to vector spaces), yields an that measures virtual dimensions of representations, aiding in decomposition studies. Tilting modules provide a bridge between different abelian categories in representation theory: a tilting module T over a (such as the group algebra kG) is a finitely generated with \Ext^1(T, T) = 0, \pd(T) < \infty, and \id({}^\vee T) < \infty (where {}^\vee T = \Hom_k(T, k)), inducing an equivalence between derived categories of s over the \End(T) and the original category. This construction, seminal in relating representations across s, facilitates transfers of homological properties like global dimension.

Set-theoretic and combinatorial representations

A permutation representation of a finite group G arises from a of G on a X, which induces a \rho: G \to S_{|X|}, where S_{|X|} is the on |X| letters. This action permutes the elements of X, and choosing a basis for the \mathbb{C}^X consisting of the characteristic functions of singletons in X linearizes the representation, yielding an equivalent linear representation in \mathrm{GL}(|X|, \mathbb{C}) via matrices. The polynomial provides a that encodes the cycle structures of in such representations, facilitating the enumeration of orbits under group actions. For a group G acting on a set of n, the is defined as Z(G; s_1, s_2, \dots) = \frac{1}{|G|} \sum_{g \in G} \prod_{k=1}^n s_k^{c_k(g)}, where c_k(g) denotes the number of cycles of length k in the g. This polynomial, introduced by Pólya, allows substitution of variables (e.g., s_k = x_1^k + x_2^k + \cdots) to count colorings or labelings fixed by the action, yielding the number of distinct orbits. Burnside's lemma offers a foundational tool for orbit counting in permutation representations, stating that the number of orbits of G on X is |\mathrm{Orbits}| = \frac{1}{|G|} \sum_{g \in G} \mathrm{fix}(g), where \mathrm{fix}(g) is the number of points in X fixed by g. This average number of fixed points directly applies to combinatorial enumeration problems, such as counting distinct configurations up to . Combinatorial generalize permutation representations by viewing them as functors F: \mathbf{Set}_\mathrm{fin} \to \mathbf{Set}_\mathrm{fin}, where \mathbf{Set}_\mathrm{fin} is the category of finite sets and bijections, assigning to each finite set I a set F[I] of structures on I, with transport along bijections preserving the action. When equipped with a group action, species encode equivariant combinatorial objects, enabling the study of labeled structures under symmetries via exponential generating functions \sum_{n \geq 0} |F| \frac{x^n}{n!}. A classic example is counting necklaces with n beads and k colors under the D_n of order $2n, which includes rotations and reflections. Applying to the action on k^n colorings yields the number of distinct necklaces as the average number of fixed colorings over the $2n group elements; for instance, rotations fix colorings periodic with the cycle lengths, while reflections fix those invariant under flips. Another key example involves the symmetric group S_n acting on the set of standard Young tableaux of a fixed shape \lambda \vdash n, where a standard Young tableau fills the boxes of the Ferrers diagram of \lambda with $1 to n increasingly across rows and down columns. This permutation representation, via row and column symmetrizers, underlies the irreducible representations of S_n, with the dimension given by the hook-length formula.

Representations of categories and functors

In representation theory, a representation of a small category \mathcal{C} over k is defined as a functor \rho: \mathcal{C} \to \Vect_k, where \Vect_k denotes the category of finite-dimensional vector spaces over k. This assigns to each object in \mathcal{C} a vector space and to each morphism a linear map, preserving composition and identities. More generally, representations can target any \mathcal{D}, yielding a functor \rho: \mathcal{C} \to \mathcal{D}. Such functorial representations extend classical notions by encoding the structure of \mathcal{C} through categorical actions on modules or sheaves. A prominent example arises with quivers, which are finite directed multigraphs possibly equipped with relations forming an in the path algebra. A representation of a quiver Q = (Q_0, Q_1) over k assigns to each i \in Q_0 a finite-dimensional V_i and to each a: i \to j in Q_1 a V_a: V_i \to V_j, such that the assignments respect relations and composition of paths corresponds to composition of maps. Equivalently, quiver representations are precisely the finite-dimensional over the path algebra kQ. Representations of quivers with relations model modules over finite-dimensional algebras, facilitating the study of their module categories. Gabriel's theorem provides a complete of quivers admitting only finitely many classes of indecomposable finite-dimensional . Specifically, a connected Q over an has finite representation type if and only if its underlying undirected is a of type A_n, D_n, or E_{6,7,8}; in these cases, the indecomposables are in with the positive roots of the corresponding via dimension vectors. For quivers without oriented cycles, the is hereditary, and Gabriel's criterion ensures fails in general unless the quiver is trivial, but finite type imposes a rigid structure on decompositions. Auslander-Reiten theory builds on this by analyzing the category through almost split sequences, which are short exact sequences $0 \to A \to B \to C \to 0 that do not split, where A and C are indecomposable, and every non-isomorphism from an indecomposable to C (or from A) factors uniquely through B. These sequences exist for non-projective (or non-injective) indecomposables in artinian rings and form the Auslander-Reiten , a whose vertices are indecomposables and arrows represent irreducible morphisms, enabling the translation of representations into combinatorial data for classification. Developed by Auslander and Reiten in the , this framework reveals the connectivity and structure of indecomposables, particularly for hereditary algebras like path algebras of acyclic s. Higher-dimensional generalizations involve 2-representations of 2-categories, defined as functors from a 2-category \mathcal{C} to the 2-category \mathbf{2}\mathbf{-}\mathbf{Cat} of (finitary) categories, preserving 1- and 2-morphisms up to natural isomorphism. Simple transitive 2-representations classify building blocks analogous to simples in 1-dimensional theory, with applications in categorification where tensor products of representations lift to functors between categories. In knot theory, 2-representations of 2-quantum groups, as introduced by Khovanov and Lauda, categorify quantum knot invariants like the Jones polynomial via Khovanov homology, where link diagrams yield 2-functors acting on graded categories to produce homological invariants. These structures connect algebraic representations to topological invariants, with Rouquier's fibrations providing explicit 2-representations for sl_n quantum groups.