Representation theory of finite groups
Representation theory of finite groups is a branch of mathematics that examines how finite groups act on finite-dimensional vector spaces via linear transformations, typically over the complex numbers, by means of homomorphisms from the group to the general linear group GL(V). This framework translates abstract group symmetries into concrete linear algebraic structures, enabling the decomposition of representations into irreducible components and the use of characters—trace functions on these representations—to analyze group properties such as conjugacy classes and subgroup relations.[1][2] The theory originated in the late 19th century, with early roots in the study of characters for abelian groups dating back to Gauss in the early 1800s for applications in number theory. A pivotal milestone came in 1896 when Ferdinand Georg Frobenius, prompted by Richard Dedekind's inquiries into group determinants, extended characters to non-abelian finite groups and proved their orthogonality properties, laying the foundation for modern character theory. Subsequent developments by Issai Schur in the early 20th century refined the theory through index theory and integral representations, while Richard Brauer's work in the 1930s advanced modular representations in positive characteristic, including key results on decomposition matrices and blocks.[3][4] Central concepts include the group algebra kG, where representations correspond to modules over this algebra, and Maschke's theorem, which asserts semisimplicity—every representation decomposes as a direct sum of irreducibles—when the field characteristic does not divide the group order. The number of irreducible representations equals the number of conjugacy classes, and the sum of the squares of their dimensions equals the group order, with character degrees dividing the group order by Frobenius's divisibility theorem. Induction and restriction functors relate representations of subgroups to the full group, governed by Frobenius reciprocity, while blocks and defect groups classify modular representations via Brauer's theorems.[5][1] Beyond pure mathematics, the theory finds applications in physics for modeling symmetries in quantum mechanics and particle interactions, in chemistry for molecular vibrations, and in number theory through connections to Galois representations and the Langlands program. It also informs combinatorics via symmetric functions and topology through equivariant cohomology, underscoring its interdisciplinary impact.[2][5]Basic definitions
Linear representations
In representation theory, a linear representation of a finite group G is a group homomorphism \rho: G \to \mathrm{GL}(V), where V is a finite-dimensional vector space over the complex numbers \mathbb{C}.[6] This homomorphism assigns to each group element g \in G an invertible linear transformation \rho(g) \in \mathrm{GL}(V), preserving the group operation via \rho(gh) = \rho(g) \rho(h) for all g, h \in G, and the identity element e \in G maps to the identity transformation \rho(e) = \mathrm{Id}_V.[2] The dimension of the representation, denoted \dim \rho or simply n if clear from context, is the dimension of V, which equals the size of the square matrices representing the transformations in a chosen basis.[7] By selecting an ordered basis for V, the representation \rho can be expressed as a matrix representation, where each \rho(g) corresponds to an n \times n invertible complex matrix, and the homomorphism property ensures that the matrix product corresponds to the group multiplication.[8] This matrix form facilitates computations, as group actions become matrix multiplications, though equivalence under change of basis means different choices yield similar matrices.[6] A subspace W \subseteq V is invariant under \rho if \rho(g) w \in W for all w \in W and g \in G.[2] The representation \rho is reducible if it admits a proper nontrivial invariant subspace (i.e., $0 \subsetneq W \subsetneq V), allowing V to decompose into smaller subspaces preserving the action; otherwise, it is irreducible, meaning no such decomposition exists beyond the trivial ones.[7] This distinction captures whether the group's action on V can be "broken down" into independent components.[8] A fundamental example is the trivial representation, where V = \mathbb{C} (so \dim \rho = 1) and \rho(g) = 1 (the identity scalar) for every g \in G, making every vector fixed by the group action.[9] This one-dimensional representation always exists and is irreducible.[10]Representations over group algebras
In representation theory, the group algebra \mathbb{C}[G] of a finite group G over the complex numbers is the vector space consisting of all formal linear combinations \sum_{g \in G} a_g g, where a_g \in \mathbb{C} and only finitely many coefficients are nonzero, equipped with the convolution product defined by \left( \sum_{g \in G} a_g g \right) \left( \sum_{h \in G} b_h h \right) = \sum_{k \in G} c_k k with c_k = \sum_{gh = k} a_g b_h.[11] This makes \mathbb{C}[G] into an associative unital algebra of dimension |G|, where the unit is the identity element e of G.[5] A linear representation \rho: G \to \mathrm{GL}(V) of G on a complex vector space V is equivalent to a left \mathbb{C}[G]-module structure on V, where the action is given by \left( \sum_{g \in G} a_g g \right) \cdot v = \sum_{g \in G} a_g \rho(g) v for v \in V.[11] Conversely, any left \mathbb{C}[G]-module M defines a representation by restricting the action to the basis elements g \in G. This equivalence shifts the study of representations from homomorphisms into matrix groups to modules over the algebra \mathbb{C}[G], enabling the application of tools from ring and module theory.[5] Since G is finite, \mathbb{C}[G] is a finite-dimensional algebra over \mathbb{C}, and thus Artinian as a ring, meaning it satisfies the descending chain condition on left ideals.[5] This property ensures that \mathbb{C}[G]-modules of finite length decompose in controlled ways, providing a foundational framework for decomposing representations into simpler components and analyzing their structure via the algebra's ideals and quotients.[11]Homomorphisms and equivalence of representations
In representation theory, a homomorphism between two representations of a finite group G, denoted (\rho, V) and (\sigma, W) where \rho: G \to \mathrm{GL}(V) and \sigma: G \to \mathrm{GL}(W) over a field K (typically \mathbb{C}), is a linear map \phi: V \to W satisfying \phi \circ \rho(g) = \sigma(g) \circ \phi for all g \in G.[12] Such maps preserve the group action and are also known as intertwining operators between the representations.[12] The set of all such homomorphisms forms a vector space \mathrm{Hom}_G(V, W), which consists of the G-equivariant linear maps from V to W.[12] This space is naturally a subspace of \mathrm{Hom}_K(V, W) invariant under the induced G-action, and its dimension provides a measure of the structural similarity between the representations.[12] When V = W, \mathrm{Hom}_G(V, V) is the space of endomorphisms of the representation. Two representations (\rho, V) and (\sigma, W) are equivalent if there exists an invertible homomorphism \phi: V \to W, meaning \phi is a linear isomorphism that intertwines the actions of G. Equivalence implies that the representations are isomorphic as modules over the group algebra K[G], preserving all representation-theoretic properties such as dimensions and characters. Over \mathbb{C}, every finite-dimensional representation admits a unique (up to isomorphism) isotypic decomposition into a direct sum of isotypic components, where each component is a direct sum of copies of a single irreducible representation.[2] These components serve as the building blocks for understanding the multiplicity of each irreducible in the overall decomposition.[2]Introductory examples
Permutation representations
A permutation representation of a finite group G is constructed from a left action of G on a finite set X. This action induces a representation \rho: G \to \mathrm{GL}(\mathbb{C}^X) on the complex vector space \mathbb{C}^X of all functions f: X \to \mathbb{C}, defined by (\rho(g) f)(x) = f(g^{-1} x) for g \in G, f \in \mathbb{C}^X, and x \in X. In the standard basis \{e_y \mid y \in X\} where e_y(z) = \delta_{y z}, the matrices \rho(g) are permutation matrices corresponding to the action on X.[10] The permutation representation on \mathbb{C}^X decomposes as a direct sum \bigoplus_{\mathcal{O}} V_{\mathcal{O}}, where the sum is over the orbits \mathcal{O} of the G-action on X, and each V_{\mathcal{O}} is the subspace of functions supported on \mathcal{O}. Each such V_{\mathcal{O}} is a transitive permutation representation, meaning the action restricts to a single orbit on the support of \mathcal{O}. These transitive components play the role of indecomposables within the category of permutation representations: a permutation representation decomposes uniquely (up to isomorphism) into a direct sum of transitive ones, and no transitive permutation representation admits a further nontrivial direct sum decomposition into permutation subrepresentations.[13][14] The orbit-stabilizer theorem relates the structure of these representations to subgroups of G. For a transitive action on a set \mathcal{O} with |\mathcal{O}| = n, fix x \in \mathcal{O} and let H = \mathrm{Stab}_G(x) be its stabilizer; then n = |G|/|H|, and the dimension of the corresponding representation space is n. Every transitive G-set is G-equivariantly isomorphic to the set of left cosets G/H with the natural action g \cdot (kH) = (gk)H, yielding the transitive permutation representation on \mathbb{C}^{G/H}. Thus, all transitive permutation representations arise as coset actions for some subgroup H \leq G.[13] A concrete example is the cyclic group C_m = \langle r \rangle of order m acting on itself by left multiplication, which is a transitive action yielding the regular representation on \mathbb{C}^{C_m}. This decomposes as the direct sum of the trivial representation (spanned by constant functions) and the sum of the remaining m-1 one-dimensional irreducible representations (corresponding to the non-trivial characters of C_m). The regular representation is a special case of a transitive permutation representation, arising when H = \{e\} is the trivial subgroup.[15]Regular representations
The left regular representation of a finite group G is the homomorphism \lambda: G \to \mathrm{GL}(\mathbb{C}[G]) defined by \lambda(g) \left( \sum_{h \in G} a_h h \right) = \sum_{h \in G} a_h (g h), where \mathbb{C}[G] is the group algebra with basis \{h \mid h \in G\} and the action corresponds to left multiplication on the basis elements.[16] This representation arises naturally from the action of G on itself by left multiplication, viewing \mathbb{C}[G] as the space of functions on G.[7] The right regular representation \rho: G \to \mathrm{GL}(\mathbb{C}[G]) is defined analogously by \rho(g) \left( \sum_{h \in G} a_h h \right) = \sum_{h \in G} a_h (h g^{-1}), ensuring it is a group homomorphism.[7] The left and right regular representations commute, as the left action \lambda(g) and right action \rho(k) satisfy \lambda(g) \rho(k) = \rho(k) \lambda(g) for all g, k \in G.[7] Moreover, the right regular representation is equivalent to the left regular representation composed with the inversion automorphism g \mapsto g^{-1}, which preserves traces and thus yields the same character.[17] Both the left and right regular representations have dimension |G|, as they act on the |G|-dimensional vector space \mathbb{C}[G].[16] Over the complex numbers, the regular representation decomposes as a direct sum of all irreducible representations of G, where each irreducible representation \rho appears with multiplicity equal to its dimension \dim \rho: \mathbb{C}[G] \cong \bigoplus_{\rho \in \mathrm{Irr}(G)} (\dim \rho) \cdot \rho. This decomposition highlights the regular representation's role in spanning the full representation theory of G.[2][17] The character \chi_{\mathrm{reg}} of the regular representation is given by \chi_{\mathrm{reg}}(g) = \begin{cases} |G| & \text{if } g = e, \\ 0 & \text{if } g \neq e, \end{cases} where e is the identity element; this follows from the action permuting the basis with a single fixed point only at the identity.[7][16] This character formula is central to computing multiplicities in the decomposition using inner products with irreducible characters.[17]Irreducibility
Definition of irreducible representations
In the context of representation theory of finite groups, an irreducible representation of a finite group G over the complex numbers \mathbb{C} is a representation \rho: G \to \mathrm{GL}(V) on a finite-dimensional vector space V \neq \{0\} such that the only G-invariant subspaces of V are \{0\} and V itself.[18] This means there are no proper nontrivial subspaces W \subset V satisfying \rho(g)W = W for all g \in G.[19] Equivalently, irreducible representations correspond to simple modules over the complex group algebra \mathbb{C}[G], where the category of finite-dimensional representations of G is isomorphic to the category of finite-dimensional \mathbb{C}[G]-modules, and simplicity means no proper nontrivial submodules.[15] A fundamental result states that every irreducible representation of G over \mathbb{C} is determined up to isomorphism by its character, the function \chi_\rho(g) = \mathrm{tr}(\rho(g)).[20] Moreover, the number of isomorphism classes of irreducible representations of G equals the number of conjugacy classes of G.[15]Schur's lemma
Schur's lemma is a fundamental result in the representation theory of finite groups over the complex numbers. It asserts that if \rho: G \to \mathrm{GL}(V) is an irreducible representation of a finite group G on a finite-dimensional complex vector space V, then every G-equivariant endomorphism T: V \to V (i.e., T \rho(g) = \rho(g) T for all g \in G) is a scalar multiple of the identity operator. In other words, the space of intertwiners is \mathrm{Hom}_G(V, V) = \mathbb{C} \cdot \mathrm{Id}_V.[21] To prove this, suppose T \in \mathrm{Hom}_G(V, V) is nonzero. Then the image \mathrm{im}(T) is a nonzero G-invariant subspace of V. By irreducibility, \mathrm{im}(T) = V, so T is surjective. The kernel \ker(T) is also G-invariant; since \dim V < \infty, surjectivity implies that \dim \ker(T) = 0, so T is injective and hence invertible. Thus, every nonzero element of \mathrm{End}_G(V) is invertible, making \mathrm{End}_G(V) a division algebra over \mathbb{C}. As the only finite-dimensional division algebra over \mathbb{C} is \mathbb{C} itself (up to isomorphism), and \mathbb{C} acts faithfully by scalars on V, it follows that all such endomorphisms are scalar multiples of the identity.[19] The lemma extends naturally to intertwiners between distinct irreducibles: if V and W are inequivalent irreducible representations of G, then \mathrm{Hom}_G(V, W) = \{0\}. To see this, any nonzero T: V \to W would have \mathrm{im}(T) = W by irreducibility of W, making T surjective and thus an isomorphism (since \dim V = \dim W), which contradicts the inequivalence of V and W.[19] Over fields other than \mathbb{C}, such as \mathbb{R}, the endomorphism algebra \mathrm{End}_G(V) for an irreducible representation may be isomorphic to \mathbb{R}, \mathbb{C}, or the quaternions \mathbb{H}, reflecting different types of irreducibility, though the complex case remains the primary setting for finite group representations.[5]Character theory
Definition and basic properties of characters
In representation theory of finite groups, a character associated to a representation \rho: G \to \mathrm{GL}(V) over the complex numbers, where V is a finite-dimensional vector space, 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, with \operatorname{tr} denoting the trace of the linear operator \rho(g).[5][9] This concept was introduced by Georg Frobenius in 1896 as a tool to study the structure of finite groups through their linear representations.[3] Characters are class functions, meaning \chi_\rho(hgh^{-1}) = \chi_\rho(g) for all g, h \in G, since the trace is invariant under similarity transformations.[5][22] Basic properties of characters include the evaluation at the identity element \chi_\rho(e) = \dim V, which is the degree of the representation.[5][22] In general, \chi_\rho(gh) does not simplify directly in terms of \chi_\rho(g) and \chi_\rho(h), reflecting the non-commutative nature of the group.[5] However, for the inverse, \chi_\rho(g^{-1}) = \overline{\chi_\rho(g)}, the complex conjugate, because the eigenvalues of \rho(g) are roots of unity and the representation can be chosen unitary.[5] Characters exhibit additivity under direct sums of representations: if V \oplus W carries the representation \rho_V \oplus \rho_W, then \chi_{V \oplus W} = \chi_V + \chi_W.[5][9][22] Moreover, characters are invariant under equivalence of representations, so isomorphic representations yield the same character function.[5][22] As an example, the character of the regular representation of G equals |G| at the identity and zero elsewhere.[5]Orthogonality relations
The orthogonality relations constitute a cornerstone of character theory for finite groups, providing analytic tools to classify irreducible representations and decompose arbitrary representations. These relations arise from the structure of the group algebra over the complex numbers and the properties of irreducible characters. Central to this framework is the space of class functions on a finite group G, which are functions constant on conjugacy classes, forming a vector space of dimension equal to the number of conjugacy classes of G.[23] To quantify orthogonality, an inner product is defined on the space of class functions. For class functions \chi and \psi on G, the inner product is given by \langle \chi, \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \chi(g) \overline{\psi(g)}, where \overline{\psi(g)} denotes the complex conjugate of \psi(g), and |G| is the order of G. This Hermitian inner product induces a positive definite norm on the space, enabling the notion of orthonormal bases.[23][5] The row orthogonality theorem asserts that the irreducible characters of G are orthonormal with respect to this inner product. Specifically, if \chi_i and \chi_j are irreducible characters of G, then \langle \chi_i, \chi_j \rangle = \delta_{ij}, where \delta_{ij} is the Kronecker delta (equal to 1 if i = j and 0 otherwise). To establish this, consider the irreducible representations \rho_i and \rho_j affording \chi_i and \chi_j, respectively. The inner product \langle \chi_i, \chi_j \rangle equals the multiplicity of \rho_i in \rho_j \otimes \overline{\rho_j}, or equivalently, \dim \Hom_G(V_i, V_j), where V_i and V_j are the representation spaces. By Schur's lemma, this dimension is 1 if i = j and 0 otherwise, yielding the orthogonality.[23][5] The column orthogonality theorem provides relations among character values at fixed elements. For elements g, h \in G, the sum over irreducible characters satisfies \sum_{\chi \in \Irr(G)} \chi(g) \overline{\chi(h)} = \frac{|G|}{|C_G(g)|} \delta_{cl(g), cl(h)}, where \Irr(G) is the set of irreducible characters of G, C_G(g) is the centralizer of g in G, and \delta_{cl(g), cl(h)} is 1 if g and h are conjugate (i.e., in the same conjugacy class) and 0 otherwise. This follows from viewing the character table as a matrix whose columns are orthogonal with weights given by centralizer orders, derived from the row orthogonality and the fact that sums over conjugacy classes preserve the structure.[23][5] These orthogonality relations imply the linear independence of the irreducible characters over \mathbb{C}, as the orthonormality ensures no nontrivial linear relation among them. Moreover, they form a complete orthonormal basis for the space of class functions: any class function \phi on G expands uniquely as \phi = \sum_{\chi \in \Irr(G)} \langle \phi, \chi \rangle \chi, with the number of irreducible characters equaling the number of conjugacy classes. This completeness enables the projection formula for decomposition multiplicities and underpins the classification of representations via characters.[23][5]Character tables
A character table of a finite group G is a square array whose rows are indexed by the irreducible characters of G and whose columns are indexed by the conjugacy classes of G, with the entry in row i and column j given by the value of the i-th irreducible character \chi_i on a representative of the j-th conjugacy class, denoted \chi_i(\mathrm{cl}(g_j)).[24] The degrees of the irreducible representations appear in the first column as \chi_i(1), and the number of rows (and columns) equals the number of conjugacy classes, which by the fundamental theorem of character theory equals the number of irreducible representations up to isomorphism.[24] Character tables are computed using the orthogonality relations of characters, which provide a system of linear equations allowing the values to be solved for once some are known, such as the degrees from dimensions of representations.[24] Specifically, the first orthogonality relation states that the irreducible characters form an orthonormal basis for the space of class functions with respect to the inner product \langle \chi, \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \chi(g) \overline{\psi(g)}, enabling the determination of unknown character values through matrix inversion or direct solving of the orthogonality equations.[24] A concrete example is the character table of the symmetric group S_3, which has order 6 and three conjugacy classes: the identity (size 1), transpositions like (1\,2) (size 3), and 3-cycles like (1\,2\,3) (size 2).[25] S_3 has three irreducible representations: the trivial representation of degree 1, the sign representation of degree 1, and a 2-dimensional irreducible representation.[25] The character table is as follows:| Irreducible character | \{e\} | \{(1\,2)\} | \{(1\,2\,3)\} |
|---|---|---|---|
| Trivial (\chi_1) | 1 | 1 | 1 |
| Sign (\chi_2) | 1 | -1 | 1 |
| 2-dimensional (\chi_3) | 2 | 0 | -1 |