Projective representation

In the field of representation theory, a projective representation of a group G on a finite-dimensional vector space V over a field K is defined as a group homomorphism P: G \to \mathrm{PGL}(V), where \mathrm{PGL}(V) denotes the projective general linear group, obtained as the quotient of the general linear group \mathrm{GL}(V) by the subgroup of scalar multiples of the identity matrix.^[1] Equivalently, it consists of a map \rho: G \to \mathrm{GL}(V) together with a function \alpha: G \times G \to K^\times (a 2-cocycle) satisfying \rho(g) \rho(h) = \alpha(g,h) \rho(gh) for all g, h \in G, and \rho(e) = I for the identity element e \in G.^[2] Projective representations generalize ordinary linear representations, in which the cocycle \alpha is trivial (i.e., \alpha(g,h) = 1 for all g,h), and play a fundamental role in algebraic structures such as central extensions and twisted group algebras K_\alpha G.^[2] The cohomology class of \alpha lies in the second cohomology group H^2(G, K^\times), which measures the extent to which a projective representation fails to lift to a linear one; for K = \mathbb{C}, this is the Schur multiplier of G.^[1] Every projective representation lifts to an ordinary linear representation of a representation group—a finite central extension \tilde{G} of G with kernel isomorphic to the Schur multiplier—allowing the study of projective representations through linear ones.^[1] For example, the Schur multiplier of the symmetric group S_n (with n \geq 4) is \mathbb{Z}/2\mathbb{Z}, yielding two non-isomorphic representation groups except for n=6.^[1] The concept was introduced by Issai Schur in 1904 as part of his foundational work on the representations of finite groups, including studies of symmetric and alternating groups, and further developed in his papers through 1911.^[1] Key results include Clifford's theorem (1937), which extends irreducible linear representations of normal subgroups to projective ones, and Mackey's refinement (1958) for general projective cases.^[2] These ideas have applications in the classification of finite simple groups, where Schur multipliers inform the structure of covering groups.^[2] In quantum mechanics, projective representations are crucial because physical states are elements of the projective Hilbert space (rays modulo phase factors), so symmetry groups act via projective unitary representations rather than ordinary ones.^[3] A prominent example is the rotation group \mathrm{SO}(3), whose projective representations correspond to half-integer spin systems, such as spin-1/2 particles transforming under the double cover \mathrm{SU}(2); these yield dimensions $2j+1 for half-integer j.^[3] Bargmann's theorem (1954) establishes that continuous Lie groups admitting projective unitary representations possess central extensions by \mathrm{U}(1) to which these lift as ordinary unitary representations, facilitating the classification of particle representations under the Poincaré group.

Definitions and Foundations

Linear and Projective Representations

A linear representation of a group G over a field K is a homomorphism \rho: G \to \mathrm{GL}(V), where V is a vector space over K and \mathrm{GL}(V) is the general linear group consisting of all invertible linear transformations of V.^[4] This homomorphism preserves the group operation exactly, satisfying \rho(gh) = \rho(g)\rho(h) for all g, h \in G.^[4] The dimension of V is called the degree of the representation.^[4] In contrast, a projective representation of G on V is a homomorphism P: G \to \mathrm{PGL}(V), where \mathrm{PGL}(V) = \mathrm{GL}(V)/K^\times is the projective general linear group, obtained by quotienting \mathrm{GL}(V) by the scalar multiples of the identity (with K^\times the multiplicative group of nonzero elements of K).^[5] Equivalently, it can be described via operators in \mathrm{GL}(V) satisfying P(g)P(h) = c(g,h) P(gh) for some scalar c(g,h) \in K^\times, where the projective equivalence ignores these phase factors.^[5] Thus, projective representations arise naturally as linear representations modulo scalars, capturing actions up to overall scaling.^[5] The study of projective representations originated in the early 20th century with Issai Schur's work on the irreducible representations of symmetric and alternating groups, where projective aspects emerged as essential for completeness during the period 1904–1911.^[6] Schur's investigations, including the construction of representation groups for these permutation groups, highlighted how projective representations extend the linear framework to address certain irreducibility phenomena.^[6] Projective representations of G can often be lifted to linear representations of a central extension of G, thereby reducing the analysis to standard linear cases.^[5] The obstruction to such a direct lift from projective to linear on G itself is measured by elements of the second cohomology group H^2(G, K^\times).^[5]

Description via Cocycles and Group Cohomology

A projective representation \rho: G \to \mathrm{GL}(V) of a group G on a vector space V over a field F can be formalized using a 2-cocycle c: G \times G \to F^\times, where F^\times denotes the multiplicative group of the field. The function c satisfies the 2-cocycle condition

c(g,h) \, c(gh,k) = c(g,hk) \, c(h,k)

for all g, h, k \in G, and the representation obeys

\rho(g) \rho(h) = c(g,h) \, \rho(gh)

for all g, h \in G. This scalar factor c(g,h) accounts for the phase ambiguity inherent in projective representations.^[7] In the framework of group cohomology, with F^\times viewed as a trivial G-module, the equivalence classes of projective representations correspond bijectively to elements of the second cohomology group H^2(G, F^\times) = Z^2(G, F^\times) / B^2(G, F^\times), where Z^2(G, F^\times) is the group of 2-cocycles and B^2(G, F^\times) is the subgroup of 2-coboundaries. The Schur multiplier of G is precisely this group H^2(G, F^\times), which classifies the possible scalar factors up to equivalence. For example, over \mathbb{C}, the Schur multiplier of the symmetric group S_n for n \geq 4 is \mathbb{Z}/2\mathbb{Z}, yielding one nontrivial cohomology class, while for the alternating group A_n, it is \mathbb{Z}/2\mathbb{Z} for n=4,5 and n \geq 8, and \mathbb{Z}/6\mathbb{Z} for n=6,7.^[8] Two 2-cocycles c, c': G \times G \to F^\times define equivalent projective representations if they are cohomologous, meaning there exists a 1-cochain b: G \to F^\times such that

c'(g,h) = \frac{b(g) \, b(h)}{b(gh)} \, c(g,h)

for all g, h \in G; here, \frac{b(g) \, b(h)}{b(gh)} is the coboundary \delta b. This equivalence relation identifies cocycles within the same cohomology class \in H^2(G, F^\times).^[5] The cohomology class $$ provides an obstruction to lifting the projective representation to an ordinary linear representation of G: such a lift exists if and only if = 0 in H^2(G, F^\times), in which case the cocycle is a coboundary and the scalars can be absorbed into a genuine representation. Linear representations arise as the special case where c(g,h) = 1 for all g,h \in G.^[7]

Examples for Finite Groups

Representations of Finite Abelian Groups

For finite abelian groups G, the classification of projective representations over \mathbb{C} leverages the explicit structure of the second cohomology group H^2(G, \mathbb{C}^*). This group is isomorphic to \operatorname{Hom}(\wedge^2 G, \mathbb{C}^*), where \wedge^2 G denotes the exterior square of G. The isomorphism arises because the Schur multiplier M(G) = H_2(G, \mathbb{Z}) \cong \wedge^2 G for abelian G, and by the universal coefficient theorem, H^2(G, \mathbb{C}^*) \cong \operatorname{Hom}(H_2(G, \mathbb{Z}), \mathbb{C}^*) since the \operatorname{Ext} term vanishes for the divisible module \mathbb{C}^*.^[9] This identification allows cocycles to be described as alternating bilinear forms on G with values in \mathbb{C}^*, facilitating explicit computations of projective structures. A simple illustration occurs for G = \mathbb{Z}/2\mathbb{Z}. Here, \wedge^2 G = 0 since G is cyclic of prime order, yielding H^2(G, \mathbb{C}^*) = 1, the trivial group. Consequently, every projective representation of G is equivalent to an ordinary linear representation. The irreducible projective representations coincide with the one-dimensional linear characters: the trivial representation, where the generator acts as the identity, and the sign representation, where it acts by multiplication by -1. For a case exhibiting nontrivial projective structure, consider G = \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}, the Klein four-group. The exterior square \wedge^2 G \cong \mathbb{Z}/2\mathbb{Z}, so H^2(G, \mathbb{C}^*) \cong \mathbb{Z}/2\mathbb{Z}. The nontrivial cohomology class corresponds to central extensions of G by \mathbb{Z}/2\mathbb{Z}, such as the dihedral group D_4 of order 8. This group admits a faithful irreducible representation on \mathbb{C}^2 given by matrices

\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},

where the generators of G map to \pm \sigma_x, \pm \sigma_z (up to the central scalars \pm 1), and the product generator maps to \pm i \sigma_y with \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}. This yields a two-dimensional irreducible projective representation of G, where the cocycle takes values in \{ \pm 1 \} to account for the phase factors. In contrast to linear representations of abelian groups, which are completely reducible into one-dimensional irreducibles, such projective irreducibles can have higher dimension determined by the order of the extension kernel.^[10] More generally, for a finite abelian G, while all irreducible linear representations over \mathbb{C} are one-dimensional (corresponding to characters of G), irreducible projective representations associated to nontrivial cocycles in H^2(G, \mathbb{C}^*) can have dimensions greater than one. These dimensions divide |G| and reflect the structure of central extensions classified by the cohomology; for elementary abelian p-groups of rank r, the maximal dimension is p^{\lfloor r/2 \rfloor} for odd p. Higher-dimensional projective representations thus arise naturally from nontrivial elements of \operatorname{Hom}(\wedge^2 G, \mathbb{C}^*), enabling constructions beyond the linear case.^[11] Projective representations of finite abelian groups connect to an extended character theory, where a projective character is the trace function \chi: G \to \mathbb{C} of a projective representation. Unlike ordinary characters, these satisfy modified orthogonality relations incorporating the cocycle, such as \sum_{g \in G} \overline{\omega(g)} \chi_1(g) \chi_2(g^{-1}) = |G| \delta_{\rho_1, \rho_2} for projective representations \rho_1, \rho_2 with multiplier \omega \in Z^2(G, \mathbb{C}^*). This framework generalizes classical character theory to accommodate the cohomological data.^[12]

Discrete Fourier Transform Example

A concrete example of a projective representation arises for the finite abelian group G = \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}, where p is prime, realized on the Hilbert space L^2(\mathbb{Z}/p\mathbb{Z}) of dimension p using operators inspired by the discrete Fourier transform. Identify elements of G as pairs (a, b) with a, b \in \mathbb{Z}/p\mathbb{Z}. The translation operator T_a acts on the standard orthonormal basis \{ |x\rangle \mid x \in \mathbb{Z}/p\mathbb{Z} \} by T_a |x\rangle = |x + a\rangle, implementing shifts in the "position" basis. The boost operator S_b acts by multiplication in the Fourier domain, S_b |x\rangle = \omega^{b x} |x\rangle, where \omega = e^{2\pi i / p} is a primitive p-th root of unity.^[13] These operators satisfy the projective relation T_a S_b = \omega^{-a b} S_b T_a, defining a projective unitary representation \rho: G \to \mathrm{PU}(p) via \rho(a, b) = T_a S_b. The phase factor \omega^{-a b} corresponds to the 2-cocycle c((a, 0), (0, b)) = \omega^{-a b} (extended bilinearly to general elements), which is nontrivial as a p-th root of unity. This representation is irreducible, as the operators generate the full matrix algebra M_p(\mathbb{C}) through their Weyl-Heisenberg structure.^[13]^[14] The projective representation lifts to an ordinary linear representation of the central extension known as the Heisenberg group over \mathbb{Z}/p\mathbb{Z}, a nilpotent group of order p^3 and dimension 3 (as a vector space over \mathbb{F}_p) with multiplication ((a, b, z), (a', b', z')) = (a + a', b + b', z + z' + a b'). The map sends (a, b) \mapsto T_a S_b e^{i \theta z} for central elements, preserving the group law exactly.^[13] The cohomology class \in H^2(G, \mathbb{U}(1)) \cong \mathbb{Z}/p\mathbb{Z} generates the group, confirming the cocycle is essentially nontrivial and classifies this extension up to equivalence. This example illustrates how discrete Fourier methods yield faithful projective structures for finite abelian groups beyond one dimension.^[14]

Projective Representations of Lie Groups

Projective Representations of SO(3) and Spin

The second cohomology group H^2(\mathrm{SO}(3), \mathrm{U}(1)) is isomorphic to \mathbb{Z}/2\mathbb{Z}, indicating that projective representations of the rotation group SO(3) are classified into two equivalence classes: the ordinary (single-valued) representations and the projective (double-valued) ones corresponding to half-integer spin.^[15] The universal covering group of SO(3) is the special unitary group SU(2), which provides a 2:1 homomorphism \mathrm{SU}(2) \to \mathrm{SO}(3) that is a central extension by \mathbb{Z}/2\mathbb{Z}. The irreducible linear representations of SU(2), labeled by spin quantum numbers j = 0, 1/2, 1, 3/2, \dots, descend to representations of SO(3) via this covering map; those with integer j yield ordinary irreducible representations of SO(3), while those with half-integer j yield irreducible projective representations, differing by a sign under 360-degree rotations. A canonical example is the spin-1/2 projective representation on \mathbb{C}^2, realized using the Pauli matrices \sigma_x, \sigma_y, \sigma_z. For a rotation R by angle \theta around unit vector \mathbf{n}, the representation is given by

\rho(R) = e^{-i (\theta/2) \mathbf{n} \cdot \boldsymbol{\sigma}},

where \boldsymbol{\sigma} = (\sigma_x, \sigma_y, \sigma_z). The composition of two such rotations satisfies \rho(R) \rho(S) = c(R,S) \rho(RS) with cocycle c(R,S) = \pm 1, reflecting the double-valued nature. In quantum mechanics, half-integer spin representations describe fermions, such as electrons, where the projective nature enforces antisymmetric wave functions under 360-degree rotations, distinguishing them from bosons, which transform under integer-spin ordinary representations of SO(3).^[16] By Bargmann's theorem, all finite-dimensional unitary projective representations of SO(3) are equivalent to those obtained by projecting the finite-dimensional irreducible unitary representations of SU(2).

Universal Covers and Central Extensions

For a connected Lie group G, its universal covering group \hat{G} is a central extension of G by the fundamental group \pi_1(G), with the covering map \hat{G} \to G having a discrete central kernel isomorphic to \pi_1(G). Linear representations of \hat{G} descend to projective representations of G provided that the kernel acts by scalar multiples on the representation space, reflecting the phase factors inherent in projective actions. This mechanism systematically generates projective representations from ordinary ones on the cover, as the deck transformations introduce the necessary multipliers.^[17] More generally, projective representations of G can be realized as linear representations of a central extension H of G by the circle group U(1). Such an extension is constructed as H = \hat{G} \times_c U(1), where c: \hat{G} \times \hat{G} \to U(1) is a 2-cocycle representing a class in the second cohomology group H^2(\hat{G}, U(1)), and the group law on H incorporates the cocycle via (g_1, z_1)(g_2, z_2) = (g_1 g_2, z_1 z_2 c(g_1, g_2)). The cohomology class $$ classifies the possible extensions, and unitary representations of H project to projective unitary representations of G. A concrete instance of this arises with the double cover SU(2) \to SO(3), where linear spin representations on the cover yield projective representations on the orthogonal group.^[17] An illustrative example beyond the rotation group occurs with G = \mathrm{SL}(2, \mathbb{R}), whose universal cover \widetilde{\mathrm{SL}}(2, \mathbb{R}) is infinite-sheeted due to \pi_1(\mathrm{SL}(2, \mathbb{R})) \cong \mathbb{Z}. Linear representations of this cover descend to projective representations of \mathrm{SL}(2, \mathbb{R}), which is the double cover of the Lorentz group \mathrm{SO}(2,1) and thus relevant for transformations in two-dimensional Minkowski space.^[18] In general, a projective unitary representation of G lifts to a linear unitary representation of the central extension if the associated cocycle is cohomologous to a coboundary in the cover, meaning it arises from a 1-cochain on \hat{G}. For simply connected \hat{G}, the second Lie algebra cohomology H^2(\mathfrak{g}, \mathbb{R}) is often trivial—particularly for semisimple Lie algebras—facilitating such lifts by ensuring that smooth central extensions are topologically trivial.^[17]

Finite-Dimensional Unitary Projective Representations

A finite-dimensional unitary projective representation of a Lie group G is a map \rho: G \to \mathrm{U}(H), where H is a finite-dimensional complex Hilbert space and \mathrm{U}(H) is the unitary group on H, such that \rho(g)^*\rho(g) = I for all g \in G (unitarity up to phase is automatic since phases are in \mathrm{U}(1)), and \rho(gh) = c(g,h) \rho(g) \rho(h) for all g, h \in G, where c: G \times G \to \mathrm{U}(1) is a measurable cocycle satisfying the 2-cocycle condition.^[19] Such representations arise naturally in quantum mechanics, where observables transform linearly but states transform projectively under symmetries.^[20] For an irreducible finite-dimensional unitary projective representation over \mathbb{C}, Schur's lemma extends to assert that the algebra of intertwiners consists precisely of scalar multiples by elements of \mathrm{U}(1).^[21] Moreover, one can select a gauge equivalent representative \tilde{\rho} such that \det \tilde{\rho}(g) = 1 for all g \in G, meaning the image lies in the special linear group \mathrm{SL}(H); this is achieved by adjusting the cocycle by a suitable 1-coboundary to compensate for the determinant phase, which lies in \mathrm{U}(1) due to unitarity.^[19] This normalization simplifies analysis and ensures the representation factors through a projective special unitary group. In finite dimensions, every unitary projective representation of G is equivalent to a genuine linear unitary representation of a central extension \tilde{G} of G by \mathrm{U}(1), obtained via de-projectivization: specifically, the cocycle c defines a central extension $1 \to \mathrm{U}(1) \to \tilde{G} \to G \to 1, and there exists a unitary representation \tilde{\rho}: \tilde{G} \to \mathrm{U}(H) lifting \rho.^[19] For a concrete gauge fixing, one may choose representatives such that \operatorname{Tr}(\rho(g)) is real and fixed (e.g., via a "trace-zero" condition adjusted for the identity), ensuring uniqueness up to equivalence. This process is always possible in finite dimensions, contrasting with infinite-dimensional cases where obstructions may arise. For compact Lie groups, the classification of finite-dimensional irreducible unitary projective representations follows from the Peter-Weyl theorem and covering group theory: all such irreducibles arise as linear irreducible unitary representations of the universal cover \tilde{G} of G, restricted to those that are non-trivial on the kernel of the covering map (or tensor products with one-dimensional representations of the center if needed to match the cocycle class).^[20] The irreducibles of \tilde{G} are finite-dimensional and completely classified by highest weights, and projecting down yields the projective irreducibles of G. A canonical example is the rotation group \mathrm{SO}(3), whose universal cover is \mathrm{SU}(2); the finite-dimensional irreducible representations of \mathrm{SU}(2) (labeled by half-integers j = 0, 1/2, 1, \dots, of dimension $2j+1) project to linear representations of \mathrm{SO}(3) precisely when j is integer, while half-integer j yield irreducible projective representations of \mathrm{SO}(3), essential for describing spin in quantum mechanics.^[22]

Infinite-Dimensional Unitary Projective Representations: The Heisenberg Case

The Heisenberg group H_3(\mathbb{R}) is defined as the group of $3 \times 3 upper triangular matrices with ones on the diagonal, parametrized by elements (x, y, z) \in \mathbb{R}^3 under the multiplication rule (x_1, y_1, z_1) \cdot (x_2, y_2, z_2) = (x_1 + x_2, y_1 + y_2, z_1 + z_2 + x_1 y_2).^[23] This structure makes H_3(\mathbb{R}) a central extension of the abelian group \mathbb{R}^2 (identified with the quotient by the center Z = \{ (0,0,z) \} \cong \mathbb{R}) by \mathbb{R}, where the commutator [(x,y,z), (x',y',z')] = (0,0, x y' - y x') lies in the center.^[23] A key example of an infinite-dimensional unitary projective representation arises from the Schrödinger representation of H_3(\mathbb{R}) on the Hilbert space L^2(\mathbb{R}, dx).^[23] This representation is realized via unitary operators consisting of translations T_a f(x) = f(x - a) for a \in \mathbb{R} and modulations M_b f(x) = e^{i b x} f(x) for b \in \mathbb{R}, extended to the full group action by \pi(x,y,z) f(t) = e^{i (z + x y / 2)} M_y T_x f(t).^[23] Projecting to the quotient \mathbb{R}^2, these induce a projective representation where the operators satisfy the Weyl relation T_a M_b = e^{i a b} M_b T_a, corresponding to a nontrivial 2-cocycle \omega((a,0),(0,b)) = e^{i a b} on \mathbb{R}^2 with values in U(1).^[23] The Stone–von Neumann theorem asserts that every infinite-dimensional irreducible unitary representation of H_3(\mathbb{R}) with nontrivial central character (i.e., where the center acts by a non-identity character) is unitarily equivalent to this Schrödinger representation.^[23] Unitarity is preserved on L^2(\mathbb{R}, dx) with respect to the Lebesgue measure, as both T_a and M_b are isometric isomorphisms.^[23] However, the induced cocycle on the quotient \mathbb{R}^2 is cohomologically nontrivial, lying in H^2(\mathbb{R}^2, U(1)) \cong \mathbb{R}, which prevents it from lifting to a true representation of \mathbb{R}^2 without the central extension.^[23] This construction provides the foundational infinite-dimensional unitary projective representation for the Heisenberg group and serves as a basis for phase space formulations in quantum mechanics.^[23] It extends discrete analogs, such as the finite discrete Fourier transform, to the continuous setting.^[23]

Infinite-Dimensional Unitary Projective Representations: Bargmann's Theorem

Bargmann's theorem provides a fundamental criterion for lifting projective unitary representations of connected Lie groups to linear unitary representations, particularly in infinite-dimensional Hilbert spaces relevant to quantum mechanics. Specifically, for a connected Lie group G with Lie algebra \mathfrak{g}, if the second continuous cohomology group H^2(\mathfrak{g}, \mathbb{R}) = 0, then every strongly continuous projective unitary representation of G on a separable Hilbert space lifts to a linear unitary representation of the universal covering group \hat{G}. This result ensures that under the vanishing cohomology condition, the phase factors defining the projective representation can be absorbed into a true representation of the covering group, avoiding the need for additional central extensions. The proof relies on the exponential map and analytic continuation properties of the representations. A projective representation is described by a continuous 2-cocycle with values in U(1), which corresponds to a Lie algebra cocycle via the infinitesimal representation. Since H^2(\mathfrak{g}, \mathbb{R}) = 0, this cocycle is a coboundary, allowing the construction of a linear representation on the Lie algebra level. Strongly continuous extension to the group level then follows using the universal cover \hat{G}, where the cocycle trivializes globally due to the simply connected nature of \hat{G}. This approach highlights the role of continuous cohomology in de-projectivizing representations without introducing non-trivial phases. When H^2(\mathfrak{g}, \mathbb{R}) \neq 0, such as for certain nilpotent Lie algebras underlying groups like the Heisenberg group, lifts generally require non-trivial central extensions, potentially infinite-dimensional in the infinite-dimensional setting, and cannot be achieved solely via the universal cover. For instance, the Heisenberg case illustrates this non-vanishing cohomology, necessitating a central extension to realize projective representations linearly. The theorem extends naturally to infinite-dimensional contexts for strongly continuous representations, confirming that quantum symmetries of simply connected Lie groups with vanishing H^2 admit linear lifts on Hilbert spaces, a key insight for applications in quantum field theory. Historically, Bargmann's 1954 work built on Wigner's earlier investigations into ray representations in quantum mechanics, providing a rigorous framework for when projective symmetries can be linearized, with later extensions to Banach spaces and infinite-dimensional Lie groups appearing in post-2000 literature.