Involutions of a Group Ring

Introduction

The group ring $K[G]$ carries an involution built from two reversals, the inversion $g \mapsto g^{-1}$ of the group and an involution $\sigma$ of the coefficient ring $K$. The map

$$ \Bigl(\sum_g a_g\,g\Bigr)^{*} = \sum_g \sigma(a_g)\,g^{-1} $$

is anti-multiplicative because the two reversals compensate: the inverse of a product is the product of the inverses in the opposite order, and $\sigma$ is anti-multiplicative. This standard involution is the one used throughout the representation theory of a finite group, and it is only the first of a family: any anti-automorphism of $G$ of order two, composed with $\sigma$ on the coefficients, gives an involution of $K[G]$, and the standard one is the case of inversion. When $G$ is abelian, inversion is an automorphism, so the standard involution is an automorphism of $K[G]$ and the distinction between the two kinds of involution disappears.

This article fixes the standard involution, derives the involutions induced by an anti-automorphism of the group, records the compatibility with the coefficients and the passage to a twisted group ring, and computes the symmetric and skew elements. It assumes Rings and Involutive Rings for the involution of a ring, Groups for the group and its anti-automorphisms, and Group Algebras for the group ring and its augmentation; the classification of all the involutions of $K[G]$ by a cocycle is stated in the form in which the twisted group ring carries it, and the applications to representation theory are not touched. Throughout, $G$ is a group, $K$ is a commutative ring with $1 \neq 0$ and $\sigma$ is an involution of $K$ (possibly the identity), $K[G]$ is the group ring, and $\varepsilon$ is its augmentation; the general element is $\sum_g a_g g$ with $a_g \in K$ and finite support.

The Standard Involution

Definition. The standard involution of $K[G]$ is the map

$$ \Bigl(\sum_g a_g\,g\Bigr)^{*} = \sum_g \sigma(a_g)\,g^{-1}, $$

extended linearly.

Theorem. The standard involution is an involution of $K[G]$; it restricts to $\sigma$ on the copy of $K$ inside $K[G]$, and it commutes with the augmentation, $\varepsilon(x^{*}) = \varepsilon(x)$.

Proof. Additivity is entrywise, and $1^{*} = 1$ because $1 = 1\cdot e$ and $e^{-1} = e$. For the anti-multiplicativity, let $x = \sum_g a_g g$ and $y = \sum_h b_h h$; then $xy = \sum_{g,h} a_g b_h\,gh$ and

$$ (xy)^{*} = \sum_{g,h}\sigma(a_gb_h)\,(gh)^{-1} = \sum_{g,h}\sigma(b_h)\sigma(a_g)\,h^{-1}g^{-1} = y^{*}x^{*}, $$

using $(gh)^{-1} = h^{-1}g^{-1}$ and $\sigma(a_gb_h) = \sigma(b_h)\sigma(a_g)$; the reindexing of a finite-support sum is legitimate. The square is the identity because $(g^{-1})^{-1} = g$ and $\sigma^2 = \mathrm{id}$, and the restriction to $K$ is $\sigma$ because $g = e$ gives $a_e e \mapsto \sigma(a_e)e$. The augmentation of $x^{*}$ is the sum of the $\sigma(a_g)$ over those $g$ with $g^{-1}$ ranging over the support, which is the sum of the $a_g$ because $\sigma(1) = 1$ and $g \mapsto g^{-1}$ is a bijection of $G$.

Corollary (the two cases of the group). The standard involution is an automorphism of $K[G]$ exactly when inversion is an automorphism of $G$, that is, when $G$ is abelian; for a nonabelian $G$ it is a genuine anti-automorphism. In the abelian case $K[G]$ is commutative, and the standard involution is an automorphism of order dividing two whose fixed ring is the subring of the elements with $a_g = \sigma(a_{g^{-1}})$.

Proof. The standard involution reverses products, so it is multiplicative exactly when the product is commutative, and the computation $(gh)^{-1} = h^{-1}g^{-1}$ shows that inversion is an automorphism exactly when $gh = hg$ for all $g, h$. The fixed elements are those with $\sigma(a_g) = a_{g^{-1}}$ by comparison of coefficients.

Involutions from an Anti-Automorphism of the Group

Definition. Let $\iota$ be an anti-automorphism of $G$ of order two, $\iota(gh) = \iota(h)\iota(g)$ and $\iota^2 = \mathrm{id}$. The involution of $K[G]$ induced by $\iota$ is

$$ \Bigl(\sum_g a_g\,g\Bigr)^{\iota} = \sum_g \sigma(a_g)\,\iota(g). $$

Proposition. The map $x \mapsto x^{\iota}$ is an involution of $K[G]$, and the standard involution is the case $\iota(g) = g^{-1}$.

Proof. The same computation as for the standard involution, with $\iota(gh) = \iota(h)\iota(g)$ in place of $(gh)^{-1} = h^{-1}g^{-1}$, gives anti-multiplicativity; the order-two condition is $\iota^2 = \mathrm{id}$; additivity, the unit and the coefficient restriction are as before. Inversion is the anti-automorphism $g \mapsto g^{-1}$, which has order two and is an automorphism exactly when $G$ is abelian.

Proposition (twisting by a cocycle). Let $\iota$ be an anti-automorphism of $G$ of order two and let $u : G \to K^{\times}$ be a map with

$$ u_e = 1, \qquad u_{gh} = u_h\,u_g, \qquad u_g\,u_{\iota(g)} = 1 \quad \text{for all } g, h \in G . $$

Then

$$ \Bigl(\sum_g a_g\,g\Bigr)^{*} = \sum_g \sigma(a_g)\,u_g\,\iota(g) $$

is an involution of $K[G]$ restricting to $\sigma$ on the coefficients; the involution induced by $\iota$ is the case $u = 1$, and the standard involution is the case $u = 1$ and $\iota(g) = g^{-1}$.

Proof. Additivity and $1^{*} = 1$ are immediate, the restriction to $K$ is $\sigma$, and $\sigma^2 = \mathrm{id}$ with $\iota^2 = \mathrm{id}$ and $u_g u_{\iota(g)} = 1$ give $(x^{*})^{*} = x$. For the anti-multiplicativity, it suffices to check the products of group elements: $(g h)^{*} = \sigma(1)u_{gh}\iota(gh) = u_{gh}\iota(h)\iota(g)$, while $h^{*}g^{*} = u_h\iota(h)\,u_g\iota(g) = u_hu_g\,\iota(h)\iota(g)$, and the two agree by $u_{gh} = u_hu_g$ (the coefficients $u_g$ are central scalars). The case $u = 1$ is the involution induced by $\iota$.

Remark. The map $u$ is a cocycle, an anti-homomorphism of $G$ into the units of $K$ that is inverted by $\iota$; when $K^{\times}$ is abelian it is a homomorphism and its class in $H^1(G;K^{\times})$ is an invariant of the involution. That every involution of $K[G]$ that restricts to $\sigma$ on the coefficients arises from a pair $(\iota, u)$ as above is the classification theorem of Hertweck; it is quoted here, not used, and the family above is the part the article works with.

The Twisted Group Ring

Let $\alpha : G \times G \to K^{\times}$ be a two-cocycle in the sense of group cohomology, and let $K^{\alpha}[G]$ be the twisted group ring with basis $\bar g$ and product

$$ \bar g\,\bar h = \alpha(g,h)\,\overline{gh}, \qquad \alpha(g,h)\alpha(gh,k) = \alpha(h,k)\alpha(g,hk). $$

Proposition. The assignment $\bar g^{*} = \overline{g^{-1}}$ extends to an involution of $K^{\alpha}[G]$ with coefficient involution $\sigma$ if and only if

$$ \alpha(g,h) = \alpha(h^{-1},g^{-1}) \qquad \text{for all } g, h \in G . $$

Proof. By linearity it suffices to check the products of basis elements. On the one hand $(\bar g\bar h)^{*} = \alpha(g,h)\,\overline{gh}^{*} = \alpha(g,h)\overline{(gh)^{-1}}$. On the other hand $\bar h^{*}\bar g^{*} = \overline{h^{-1}}\,\overline{g^{-1}} = \alpha(h^{-1},g^{-1})\overline{h^{-1}g^{-1}} = \alpha(h^{-1},g^{-1})\overline{(gh)^{-1}}$. The two agree for all $g, h$ exactly under the stated identity. The square is the identity because $\overline{g^{-1}}^{*} = \bar g$ and $\sigma^2 = \mathrm{id}$, and additivity and the unit are immediate.

Corollary. For the untwisted group ring, where $\alpha = 1$, the condition is automatic and the standard involution exists for every $G$; the twist obstructs the involution exactly by the difference between $\alpha(g,h)$ and $\alpha(h^{-1},g^{-1})$, and the obstruction vanishes when $\alpha$ is symmetric in this sense. The twisted group ring is the algebra of a projective representation of $G$ of cocycle $\alpha$, and the involution it carries is the one that makes that algebra involutive.

Example (the rational group algebra of the quaternion group). For $G = Q_8 = \{\pm 1, \pm i, \pm j, \pm k\}$ and $K = \mathbb{Q}$ with $\sigma = \mathrm{id}$, the standard involution sends $g$ to $g^{-1}$, which is $-g$ for the six elements $\pm i, \pm j, \pm k$ and fixes $\pm 1$. The group algebra decomposes as

$$ \mathbb{Q}[Q_8] \cong \mathbb{Q}^4 \times \mathbb{H}(\mathbb{Q}), $$

with four one-dimensional factors and the quaternion algebra; under this decomposition the standard involution is the identity on the four copies of $\mathbb{Q}$ and the quaternion conjugation on $\mathbb{H}(\mathbb{Q})$, so its fixed part is $4+1 = 5$-dimensional, spanned by the four coordinate units together with the real line of the quaternion factor, and its skew part is the $3$-dimensional space of pure quaternions.

Compatibility with the Coefficients

Proposition. An involution of $K[G]$ that restricts to the involution $\sigma$ of $K$ is determined by its values on the group elements: the ring is generated as a $K$-module by $G$, so the involution is fixed by $\sigma$ and by the images of the group elements. The involutions of the previous section are exactly those of the form $g^{*} = u_g\iota(g)$; that these exhaust the involutions restricting to $\sigma$ is Hertweck's theorem, quoted rather than proved here.

Proof. The generation statement is the definition of the group ring; the form of the involutions of the previous section is its construction, and the exhaustiveness is the deferred classification.

Corollary (elements fixed by the standard involution). For the standard involution the fixed elements are $\{x : a_g = \sigma(a_{g^{-1}})\ \text{for all}\ g\}$ and the skew elements are $\{x : a_g = -\sigma(a_{g^{-1}})\}$; both are additive subgroups, and they are the eigenspaces when $2$ is invertible in $K$. For a finite group the symmetric part is a $K^\sigma$-module of rank at least the number of conjugacy classes of $G$ that are stable under inversion.

Proof. The fixed-element condition is the comparison of coefficients in $x^{*} = x$; the eigenspace statement is the additive decomposition of Involutive Rings. The rank statement follows because a stable conjugacy class sum is fixed under $\sigma = \mathrm{id}$ and the class sums are linearly independent.

Summary

The group ring $K[G]$ carries the standard involution $(\sum a_g g)^{*} = \sum \sigma(a_g)g^{-1}$, anti-multiplicative because the inversion of $G$ and the involution $\sigma$ of $K$ reverse in tandem, with $1^{*} = 1$ and $\varepsilon(x^{*}) = \varepsilon(x)$. It restricts to $\sigma$ on the coefficients, it is an automorphism exactly when $G$ is abelian, and its fixed elements are those with $a_g = \sigma(a_{g^{-1}})$. More generally an anti-automorphism $\iota$ of $G$ of order two gives the involution $(\sum a_g g)^{\iota} = \sum \sigma(a_g)\iota(g)$, and the standard involution is the case of inversion; every involution of $K[G]$ fixing the coefficients is of the form $g^{*} = u_g\iota(g)$ with $u$ a cocycle, the invariant of the involution being the class of $u$ in $H^2(G;K^{\times})$.

On a twisted group ring $K^{\alpha}[G]$, with $\bar g\bar h = \alpha(g,h)\overline{gh}$, the assignment $\bar g^{*} = \overline{g^{-1}}$ extends to an involution if and only if $\alpha(g,h) = \alpha(h^{-1},g^{-1})$, a condition automatic for the untwisted ring. The symmetric and skew elements are the eigenspaces of the standard involution, and for a finite group the symmetric part contains the inversion-stable class sums.

Summary of Notation

Symbol	Meaning
$G$, $K$, $\sigma$	Group, commutative coefficient ring, involution of $K$
$K[G]$	Group ring; $\varepsilon$ its augmentation
$x^{*} = \sum \sigma(a_g)g^{-1}$	Standard involution, from inversion
$x^{\iota} = \sum \sigma(a_g)\iota(g)$	Involution induced by an anti-automorphism $\iota$ of order two
$g^{*} = u_g\iota(g)$	General involution fixing the coefficients; $u$ a cocycle
$\alpha$, $K^{\alpha}[G]$	Two-cocycle and twisted group ring; $\bar g\bar h = \alpha(g,h)\overline{gh}$
$\alpha(g,h) = \alpha(h^{-1},g^{-1})$	Existence of the involution on the twisted group ring
$a_g = \sigma(a_{g^{-1}})$	Fixed elements of the standard involution
$\mathbb{Q}[Q_8] \cong \mathbb{Q}^4 \times \mathbb{H}(\mathbb{Q})$	Example; conjugation on the quaternion factor