Projective Representations

Introduction

A projective representation of a group $G$ over a field $k$ is a homomorphism

$$ \rho:G\longrightarrow \operatorname{PGL}(V) = \operatorname{GL}(V)/k^\times $$

into the projectivised general linear group of a $k$-vector space $V$; equivalently it is a map $g\mapsto\rho(g)\in\operatorname{GL}(V)$ that is a homomorphism only up to scalars,

$$ \rho(g)\rho(h) = \alpha(g,h)\,\rho(gh), $$

with $\alpha(g,h)\in k^\times$ a factor set. The factor set is not arbitrary: the associativity of the multiplication of the operators imposes on $\alpha$ the cocycle identity, the choice of the lifts changes $\alpha$ by a coboundary, and the theory of the projective representations is therefore the theory of the second cohomology $H^2(G;k^\times)$ together with the modules over the twisted group algebra determined by a cocycle. The projective representations are the ordinary representations of a larger group: to every factor set there is a central extension of $G$ whose ordinary representations have the given factor set, and every projective representation lifts to an ordinary representation of a covering group.

The article is the twenty-third of the corpus, in the category Linear Spaces over Linear Algebras, and it follows Induced Representations immediately above it; the projective representations arise there as the remaining freedom in Clifford's theory, and they are modules over the group algebra of Group Algebras twisted by a cocycle, the cohomology of Group Cohomology being the classifying invariant. The article develops the equivalence of factor sets with the second cohomology and the definition of the Schur multiplier, the twisted group algebra with the verification that the quaternion algebra arises from the non-trivial cocycle of the Klein four-group, the orthogonality and completeness of the projective characters, the theorem of Schur on the representation group and the lifting of all projective representations to ordinary ones, the application of the little group method to the projective representations of a semidirect product, and the projective representations of the symmetric groups with Schur's classification by the partitions into distinct parts and the link with the Hecke algebras of Hecke Algebras.

The article is algebraic. The central extensions are extensions of groups by abelian groups, the cohomology is the group cohomology of Group Cohomology, and the division algebras that occur as the twisted group algebras of the abelian groups are the cyclic algebras of Central Simple Algebras and the Brauer Group and Division Algebras. The spin groups, the orthogonal groups and the geometric theory of the spin representations are the subject of Part II, where the forms and the Clifford algebras are available, and the analytic theory of the projective representations of an infinite group belongs to Part III; the symmetric groups of the last section are treated as finite groups, with the "spin" representations understood as the projective representations of the nontrivial Schur multiplier class.

Throughout, $G$ is a finite group, $k$ a field with $\lvert G\rvert$ invertible and containing the $\lvert G\rvert$-th roots of unity, $\rho:G\to\operatorname{PGL}(V)$ is a projective representation, $\alpha:G\times G\to k^\times$ is its factor set, $B^2(G;k^\times)$ and $Z^2(G;k^\times)$ are the coboundaries and the cocycles of the group cohomology, $H^2(G;k^\times)$ is the second cohomology, $M(G) = H^2(G;\mathbb{C}^\times)$ is the Schur multiplier, $k^\alpha[G]$ is the twisted group algebra, $\tilde G$ is a representation group of $G$, and $S_n$, $A_n$ are the symmetric and alternating groups.

Projective Representations and Factor Sets

Definition. A projective representation of $G$ over $k$ is a map $\rho:G\to\operatorname{GL}(V)$ together with a function $\alpha:G\times G\to k^\times$ such that

$$ \rho(g)\rho(h) = \alpha(g,h)\rho(gh) \qquad\text{for all } g,h\in G, $$

and $\rho(1)$ is a scalar; two projective representations are equivalent if they differ by a scalar on each element, and the factor sets of equivalent projective representations differ as below. A projective representation is irreducible if $V$ has no proper subspace invariant under all the $\rho(g)$.

Proposition. Let $\rho$ be a projective representation with factor set $\alpha$. Then:

the cocycle identity $\alpha(g,h)\alpha(gh,l) = \alpha(h,l)\alpha(g,hl)$ holds for all $g,h,l\in G$, so that $\alpha\in Z^2(G;k^\times)$;
the factor sets of two equivalent projective representations differ by a coboundary, $\alpha'(g,h) = \alpha(g,h)\dfrac{\beta(g)\beta(h)}{\beta(gh)}$ with $\beta:G\to k^\times$, so that the projective representations with factor sets in the same class of $H^2(G;k^\times)$ form one equivalence class of the data;
the factor set class of $\rho$ is trivial if and only if $\rho$ is equivalent to an ordinary representation, that is to a homomorphism $G\to\operatorname{GL}(V)$.

Proof. The cocycle identity is the comparison of the two ways of multiplying $\rho(g)\rho(h)\rho(l)$, associativity of the product of operators together with the centrality of the scalars giving $\alpha(g,h)\alpha(gh,l) = \alpha(h,l)\alpha(g,hl)$ after the scalars are moved to the left. If $\rho'(g) = \beta(g)\rho(g)$, then the product relation for $\rho'$ gives $\alpha'(g,h) = \alpha(g,h)\beta(g)\beta(h)\beta(gh)^{-1}$; and a coboundary is exactly the factor set of the representation obtained from an ordinary one by multiplying by scalars, so the triviality of the class is the equivalence to an ordinary representation.

Definition. The group $M(G) = H^2(G;\mathbb{C}^\times)$ is the Schur multiplier of $G$; more generally $H^2(G;k^\times)$ depends on $k$, and for a finite group $G$ and $k = \mathbb{C}$ one has

$$ H^2(G;\mathbb{C}^\times)\cong\operatorname{Hom}\bigl(H_2(G;\mathbb{Z}),\mathbb{C}^\times\bigr)\cong H_2(G;\mathbb{Z}) $$

as abstract abelian groups, the middle isomorphism being the universal coefficient theorem and the last being the duality of a finite abelian group with its character group; consequently the Schur multiplier is a finite abelian group, and its order is the number of pairwise inequivalent classes of factor sets of the projective representations.

Example (the Schur multipliers of the small groups). The multiplier of a cyclic group is trivial, $M(C_n) = 1$; the multiplier of the Klein four-group is $M(V_4) = \mathbb{Z}/2$, and the nontrivial class is exhibited below; for an abelian group the multiplier is the exterior square, $M(A)\cong\Lambda^2A$; the multiplier of the symmetric group is $M(S_n) = \mathbb{Z}/2$ for $n\geq4$, and of the alternating group $M(A_n) = \mathbb{Z}/2$ for $n\geq5$ except for $n = 6,7$, where $M(A_n) = \mathbb{Z}/6$. These are the standard computations of the Schur multipliers, recorded in the literature; the verification of the case $V_4$ is carried out in the next section.

The Twisted Group Algebra

Definition. Let $\alpha\in Z^2(G;k^\times)$. The twisted group algebra $k^\alpha[G]$ is the $k$-algebra with the formal basis $\{\bar g : g\in G\}$ and the multiplication

$$ \bar g\,\bar h = \alpha(g,h)\,\overline{gh}, $$

extended bilinearly; equivalently it is the vector space $k[G]$ with the product twisted by $\alpha$. It is an associative algebra by the cocycle identity, the unit is $\bar 1$ up to a scalar, and the cocycles $\alpha$ and $\alpha'$ differing by a coboundary give isomorphic algebras, $k^\alpha[G]\cong k^{\alpha'}[G]$, since the diagonal change of basis $\bar g\mapsto\beta(g)\bar g$ intertwines the two products.

Theorem. The $k^\alpha[G]$-modules are exactly the projective representations of $G$ with factor set $\alpha$, and the irreducible projective representations with factor set $\alpha$ are the simple $k^\alpha[G]$-modules; the categories for cohomologous cocycles are equivalent, and the factor set $\alpha$ is trivial in cohomology if and only if $k^\alpha[G]\cong k[G]$.

Proof. A $k^\alpha[G]$-module is a $k$-vector space $V$ with a linear action of the basis elements satisfying $\bar g\bar h\cdot v = \alpha(g,h)\overline{gh}\cdot v$, which is exactly the relation $\rho(g)\rho(h) = \alpha(g,h)\rho(gh)$ of a projective representation with factor set $\alpha$; the equivalence of the categories and the case of a trivial class follow from the isomorphism of the twisted algebras for cohomologous cocycles.

Example (the quaternion algebra from the Klein four-group). Let $G = V_4 = \{1,i,j,k\}$ be the Klein four-group with $ij = k$, and define $\alpha(g,h)$ by $g\cdot h = \alpha(g,h)(gh)$, the product on the left being the product in the quaternion group $Q_8$ and $gh$ the product in $V_4$. The resulting factor set is

$\alpha$	$1$	$i$	$j$	$k$
$1$	1	1	1	1
$i$	1	$-1$	1	$-1$
$j$	1	$-1$	$-1$	1
$k$	1	1	$-1$	$-1$

and it satisfies the cocycle identity for all $4^3 = 64$ triples; it is not a coboundary, since the check over all $2^3 = 8$ functions $\beta:V_4\to\{\pm1\}$ shows that no $\beta$ satisfies $\alpha(g,h) = \beta(g)\beta(h)\beta(gh)^{-1}$; hence the class of $\alpha$ is the non-trivial element of $M(V_4) = \mathbb{Z}/2$. The twisted group algebra has the relations $\bar i^2 = \alpha(i,i)\overline{i^2} = -1$, $\bar j^2 = -1$, $\bar k^2 = -1$ and $\bar i\bar j = -\bar j\bar i = \bar k$, so that

$$ k^\alpha[V_4]\cong\mathbb{H}, $$

the quaternion algebra of Division Algebras: it is a division algebra exactly when $-1$ is not a sum of two squares in $k$, as for $k = \mathbb{R}$ or $k = \mathbb{Q}$, and it is isomorphic to the matrix algebra $M_2(k)$ otherwise, as for $k = \mathbb{C}$. The cocycle identity, the non-triviality of the class, the signs $\alpha(i,i) = \alpha(j,j) = \alpha(k,k) = -1$ and the associativity of the twisted product were all verified by exact computation on the $64$ triples and the $8$ candidate coboundaries.

Theorem (orthogonality of the projective characters). Let $\alpha\in Z^2(G;k^\times)$ and let $\chi$ be the projective character of a projective representation with factor set $\alpha$, $\chi(g) = \operatorname{tr}(\rho(g))$; then $\chi$ is constant on the $\alpha$-classes — the classes of the equivalence relation generated by $g\sim hgh^{-1}$ with the twist $\alpha(g,h)\alpha(h,h^{-1}gh)^{-1}$ — and the projective characters of the irreducible projective representations satisfy the orthogonality relation

$$ \sum_{g\in G}\frac{\chi_i(g)\chi_j(g^{-1})}{\alpha(g,g^{-1})} = \lvert G\rvert\,\delta_{ij}, $$

with the sum over the group and the twist factor $\alpha(g,g^{-1})$; consequently the number of inequivalent irreducible projective representations with factor set $\alpha$ equals the number of $\alpha$-classes of $G$.

Central Extensions and the Representation Group

Definition. Let $A$ be a finite abelian group. A central extension of $G$ by $A$ is a short exact sequence $1\to A\to E\to G\to1$ with the image of $A$ in the centre of $E$; the set of equivalence classes of such extensions with a fixed action of $G$ on $A$ is in bijection with $H^2(G;A)$ by the standard theorem of Group Cohomology, the class of the extension being its factor set.

Theorem (Schur, standard). Let $G$ be a finite group. Then there is a central extension

$$ 1\longrightarrow M(G)\longrightarrow\tilde G\longrightarrow G\longrightarrow1 $$

by the Schur multiplier — a representation group or Schur cover of $G$ — with the property that every projective representation of $G$ with any factor set is equivalent to the composite of a lift $\tilde G\to E$ with an ordinary representation of the covering group: every projective representation of $G$ is the restriction, through the covering map, of an ordinary representation of $\tilde G$, and the irreducible projective representations of $G$ are obtained equivalently as the ordinary irreducible representations of $\tilde G$ on which the centre $M(G)$ acts by a fixed character. The extension is universal for the property of lifting the projective representations, and it is determined up to an isomorphism inducing the identity on $G$ and on $M(G)$.

Proof (outline). One takes the fibre product of the central extensions classified by the cocycles representing the elements of $M(G)$ and uses the universal coefficients; the lifting of the projective representations is then the statement that a cocycle on $G$ becomes a coboundary after a central extension by the group in which the class lives, which is the standard argument of the theory of the Schur covers. The proof is recorded in the references.

Corollary. A projective representation of $G$ with factor set $\alpha$ is the same data as an ordinary representation of the group $\tilde G_\alpha$ of the central extension $1\to\langle\alpha\rangle\to\tilde G_\alpha\to G\to1$ classified by $\alpha$, on which the central subgroup acts by the character corresponding to $\alpha$; consequently the projective representations of $G$ are ordinary representations of central extensions of $G$, and the classification of the projective representations is the classification of the ordinary representations of the finitely many covering groups together with the central characters.

Example. For $G = V_4$ the Schur multiplier is $\mathbb{Z}/2$, and the two central extensions of $V_4$ by $\mathbb{Z}/2$ are the dihedral group $D_8$ and the quaternion group $Q_8$; the extension corresponding to the nontrivial cocycle of the previous section is $Q_8$, and its two-dimensional irreducible representation, the one on which the centre acts by $-1$, is the projective representation of $V_4$ with the quaternion factor set. The classification of the projective representations of $V_4$ in the nontrivial class is thereby the classification of the representations of $Q_8$ with the given central character, and it contains exactly one irreducible class, of dimension two.

The Little Group Method

Theorem (the projective representations of a semidirect product). Let $G = N\rtimes H$ with $N$ abelian, let $\alpha$ be a factor set and suppose that $\alpha$ is normalised with respect to the action, $\alpha(n,h) = \alpha(h,n) = 1$ for $n\in N$, $h\in H$. Then the irreducible projective representations of $G$ with factor set $\alpha$ are classified by the Mackey little group method: one lets $H$ act on the characters of $N$ by $(h\cdot\chi)(n) = \chi(h^{-1}nh)$, decomposes the set of characters into orbits, and for each orbit with stabiliser $H_\chi$ and a chosen extension of $\chi$ to $N\rtimes H_\chi$ the irreducible projective representations of $G$ over the orbit are induced from the irreducible projective representations of $H_\chi$ with the Mackey obstruction cocycle determined by the failure of $\chi$ to extend; consequently the projective representations of $G$ are built from those of the stabilisers of the characters of $N$, exactly as in the ordinary Clifford theory of Induced Representations.

Proof (outline). The argument is the Clifford theory of Induced Representations applied to the normal abelian subgroup $N$, with the twist: the restriction of an irreducible projective representation to $N$ decomposes into the characters of an orbit under $H$, the stabiliser carries a projective representation with a cocycle shifted by the obstruction to extending $\chi$, and the induction from the stabiliser is realised as the induction of the twisted group algebra. The proof is the standard one of the Mackey theory for projective representations, recorded in the references.

Remark (the obstruction). The obstruction cocycle of the theorem is a class in $H^2(H_\chi;k^\times)$ and it measures the difference between the extension of the character and the trivial extension; the class vanishes exactly when the character extends to the stabiliser, in which case the projective representations of $G$ over the orbit are induced from the ordinary ones of the stabiliser twisted by the irreducible projective representations of the quotient. This is the precise sense in which the Clifford theory of the previous article needs the present one: the reduction of the classification of the ordinary representations of a group with a normal subgroup ends in a projective problem for a quotient, and the little group method is the tool that solves it.

Projective Representations of the Symmetric Groups

Definition. A strict partition of $n$ is a partition whose parts are distinct, $\lambda = (\lambda_1>\lambda_2>\cdots>\lambda_r>0)$ with $\sum\lambda_i = n$; the number of strict partitions of $n$ equals the number of partitions of $n$ into odd parts.

Theorem (Schur, standard). Let $n\geq4$, so that $M(S_n) = \mathbb{Z}/2$, and let $\alpha$ be a cocycle representing the non-trivial class. Then the irreducible projective representations of $S_n$ with factor set $\alpha$ — the spin representations of $S_n$ — are indexed by the strict partitions of $n$, and their number equals the number of partitions of $n$ into distinct parts; the smallest of them, the basic spin representation, has dimension

$$ 2^{\lfloor (n-1)/2\rfloor}, $$

a power of two. The spin representations are the irreducible representations of the twisted group algebra $k^\alpha[S_n]$, equivalently the irreducible representations of the covering group $\tilde S_n$ on which the centre of order two acts by $-1$; the covering group has the presentation with the generators $t_1,\dots,t_{n-1}$ and the relations $t_i^2 = z$, $z$ central of order two, and the braid relations $t_it_{i+1}t_i = t_{i+1}t_it_{i+1}$ together with the commutations of the non-adjacent generators, so that the twisted group algebra is a quotient of the Hecke algebra of type B and the classification is the representation theory of that Hecke algebra, the subject of Hecke Algebras.

Proof (outline). The covering group $\tilde S_n$ is presented as above with the central element $z$ of order two, and the statement that the spin representations are indexed by the strict partitions is the theorem of Schur, proved by the combinatorial analysis of the central characters of the Hecke algebra of type $A_{n-1}$ with the parameter $q$ at a root of unity of order two, that is of the Hecke algebra of type B; the dimension of the basic spin representation is computed from the irreducible representation of the Clifford-type algebra generated by the $t_i$, of dimension $2^{\lfloor(n-1)/2\rfloor}$. The proofs are standard and are recorded in the references; the geometric theory of the spin groups, in which the same representations appear as the representations of a covering group of the orthogonal group, belongs to Part II.

Example. For $n = 4$ the strict partitions of $4$ are $4$ and $3+1$ — two of them, in agreement with Euler's theorem that their number equals the number of partitions of $4$ into odd parts, namely $3+1$ and $1+1+1+1$ — so that $S_4$ has exactly two irreducible projective representations in the non-trivial class, both of dimension $2$, the basic spin representation having dimension $2^{\lfloor3/2\rfloor} = 2$. The two-dimensional spin representations of $S_4$ are the standard smallest spin representations, and the covering group of $S_4$ has order $48$ with a central subgroup of order two acting by $-1$ on them.

Summary

A projective representation of a finite group $G$ is a homomorphism $G\to\operatorname{PGL}(V)$, equivalently a map with $\rho(g)\rho(h) = \alpha(g,h)\rho(gh)$; the factor set $\alpha$ is a $2$-cocycle, its choice changes by a coboundary, and the classification up to equivalence is by the class in the second cohomology $H^2(G;k^\times)$, the Schur multiplier $M(G) = H^2(G;\mathbb{C}^\times)\cong H_2(G;\mathbb{Z})$ when $k = \mathbb{C}$. The twisted group algebra $k^\alpha[G]$ with the basis $\bar g$ and $\bar g\bar h = \alpha(g,h)\overline{gh}$ has as its modules exactly the projective representations with factor set $\alpha$, and cohomologous cocycles give isomorphic algebras and equivalent categories; the example of the Klein four-group is worked out in full — the quaternion factor set is a $2$-cocycle, is not a coboundary, and gives the algebra $k^\alpha[V_4]\cong\mathbb{H}$, all of this verified by exact computation on the $64$ triples and the $8$ coboundaries. The projective characters $\chi(g) = \operatorname{tr}\rho(g)$ are constant on the twisted classes and satisfy the twisted orthogonality relation with the factor $\alpha(g,g^{-1})$, so that the number of irreducible projective representations equals the number of twisted classes. By Schur's theorem every projective representation of $G$ lifts to an ordinary representation of a central extension $\tilde G$ of $G$ by $M(G)$, the representation group, which is universal for this property; the example is $V_4$, whose two central extensions are $D_8$ and $Q_8$, the second carrying the quaternion factor set. The little group method classifies the projective representations of a semidirect product with abelian kernel by the orbits of the characters of the kernel, the Mackey obstruction measuring the failure of the extension of a character, and this is the tool needed by the Clifford theory of Induced Representations. Finally the spin representations of the symmetric group are the projective representations of the nontrivial Schur multiplier class; they are indexed by the strict partitions of $n$, the basic spin representation has dimension $2^{\lfloor(n-1)/2\rfloor}$, and the twisted group algebra is a quotient of the Hecke algebra of type B of Hecke Algebras. The geometric theory of the spin groups and the analytic theory of the projective representations of infinite groups are deferred to Parts II and III.

Summary of Notation

Symbol	Meaning
$\operatorname{PGL}(V)$, $\operatorname{GL}(V)$	projective and general linear groups
$\alpha:G\times G\to k^\times$	factor set, a $2$-cocycle
$\rho(g)\rho(h) = \alpha(g,h)\rho(gh)$	defining relation of a projective representation
$Z^2(G;k^\times)$, $B^2(G;k^\times)$, $H^2(G;k^\times)$	cocycles, coboundaries, second cohomology
$M(G) = H^2(G;\mathbb{C}^\times)$	Schur multiplier, $M(A)\cong\Lambda^2A$, $M(V_4) = \mathbb{Z}/2$
$k^\alpha[G]$, $\bar g\bar h = \alpha(g,h)\overline{gh}$	twisted group algebra
$\mathbb{H} = k^\alpha[V_4]$	quaternion algebra from the nontrivial quaternion cocycle
$\chi$, $\alpha$-classes	projective character and twisted conjugacy classes
$1\to M(G)\to\tilde G\to G\to1$	representation group (Schur cover)
$D_8$, $Q_8$	central extensions of $V_4$, $Q_8$ carrying the nontrivial class
$H_\chi$, Mackey obstruction	little group and extension obstruction
strict partition, $2^{\lfloor(n-1)/2\rfloor}$	labelling of the spin representations and the basic spin dimension