nLab extended representation

Idea

Given a subgroup inclusion $H \xhookrightarrow{\iota} G$ there is the evident functor $\iota^\ast \,\colon\, Rep(G) \xrightarrow{\;} Rep(H)$ sending any linear representation of $G$ to its restricted representation of $H$. The adjoints to this construction are given by forming left/right induced representations.

But one may also ask, for any linear representation $\rho \in Rep(H)$ of the subgroup, to extend it to a representation of $G$, namely to find any $\widehat{\rho} \in Rep(G)$ such that $\rho \,=\, \iota^\ast\big(\widehat{\rho}\big)$.

Specifically, for $R$ the ground field (or ground ring) and $GL_n(R)$ denoting its general linear groups ($n \in \mathbb{N}$), then the given representation is equivalently a group homomorphism $H \xrightarrow{\rho} GL_n(R)$ and an extended representation of $\rho$ is an extension of this morphism along the group inclusion:

Specifically when $H \xrightarrow{\iota} G$ is a normal subgroup inclusion and $\rho$ is an irreducible representation, there is a classification of extended irreps, — hence of irreps of $G$ relative to the given irrep of $H$ — either on the nose or at least as projective representations (Isaacs 1976)

Properties

Extensions along normal subgroup inclusions

Consider

• $N \xhookrightarrow{\iota} G$ a normal subgroup inclusion,

• $\rho \in Rep(N)$ an $N$-representation.

It is immediate but important that:

Proposition

For $\rho$ to admit any extension to $G$ it is necessary that the isomorphism class $[\rho]$ of $\rho$ is invariant under the conjugation action

$$\begin{array}{ccc} Rep(H) &\overset{(-)^g}{\longrightarrow}& Rep(H) \\ \rho &\mapsto& \rho\big(g(-)g^{-1}\big) \end{array}$$

by all elements $g \in G$:

(1)$$\underset{ g \in G }{\forall} \;\;\; [\rho] = \big[\rho^g\big] \,.$$

Proof

Consider $\widehat{\rho}$ any extension and $g \in G$ any element. Then the fact that $\widehat{\rho}$ is a $G$-representation extending $\rho$ implies for $n \in N$ that

$$\begin{array}{ccl} \widehat{\rho}(g) \circ \rho(n) \circ \widehat{\rho}(g^{-1}) &=& \widehat{\rho}(g) \circ \widehat{\rho}(n) \circ \widehat{\rho}(g^{-1}) \\ &=& \widehat{\rho}\big(g n g^{-1}\big) \\ &=& \rho\big(g n g^{-1}\big) \\ &=& \rho^g(n) \,. \end{array}$$

But this says equivalently that $\widehat{\rho}(g)$ is an intertwiner which constitutes an isomorphism of representations

$$\rho \xrightarrow[\sim]{ \widehat{\rho}(g) } \rho^g \,,$$

and hence that $[\rho] = \big[\rho^g\big]$.

Now assume in addition that

• the ground field is algebraically closed field, like the complex numbers $\mathbb{C}$,

• $\rho$ is an irreducible representation,

• $\rho$ is $G$-invariant (1)

Proposition

There always exists a projective extension of $\rho$ in the sense of a projective representation $\widetilde{\rho} \in PRep(G)$ satisfying, for $n \in N$ and $g \in G$,

1. $\widehat{\rho}(n) \,=\, \rho(n)$,

2. $\widehat{\rho}(g n) = \widehat{\rho}(g) \circ \widehat{\rho}(n)$,

3. $\widehat{\rho}(n g) = \widehat{\rho}(n) \circ \widehat{\rho}(g)$,

and any pair $\widehat{\rho}, \widehat{\rho}'$ of such differ only by a function on $G$ with values in the group of units $k^{\times}$ and depending only on cosets,

(2)$$\mu \,\colon\, G \xrightarrow{\;} G/N \xrightarrow{\;} k^\times \,,$$

in that

$$\widetilde{\rho}'(-) \;=\; \widetilde{\rho}(-) \cdot \mu(-) \,.$$

(Isaacs 1976 Thm. 11.2)

Proposition

Given any projective extension $\widetilde{\rho}$ according to Prop. , the corresponding 2-cocycle in the group cohomology of $G$,

$$\widetilde{\alpha} \,\colon\, G \times G \longrightarrow k^{\times} \,, \;\;\;\;\;\;\;\;\;\; \text{s.t.} \;\;\;\;\;\;\;\;\;\; \underset{g_i \in G}{\forall} \;\; \widetilde{\rho}(g_1) \circ \widetilde{\rho}(g_1) \;=\; \widetilde{\rho}(g_1 g_2) \cdot \widetilde{\alpha}(g_1, g_2) \,,$$

comes from a 2-cocycle $\alpha$ on the quotient group

$$\widetilde{\alpha} \;\colon\; G \times G \twoheadrightarrow (G/N) \times (G/N) \xrightarrow{ \alpha } k^{\times}$$

whose group cohomology class

$$[\alpha] \,\in\, H^2\big(G/N;\, k^\times\big)$$

depends only on $[\rho]$ (is independent of the choice of $\widetilde{\rho}$).

Finally, $\rho$ admits an actual extension $\widehat{\rho}$ iff that class is trivial, $[\alpha] = 0$.

(Isaacs 1976 Thm. 11.7)

Remark

It follows moreover that if an extension $\widehat{\rho}$ exists, then this is unique up to multiplication with (the pullback to $G$ of) a group character $\mu$ on $G/N$ (2).

Related concepts

• extension

• restricted representation

• induced representation

References

Monographs:

• I. Martin Isaacs: around Thm. 11.7 in: Character theory of finite groups, Academic Press, New York (1976) [ISBN:978-0-8218-4229-4]

• Gabriel Navarro, sections 5 & 6 of: Character Theory and the McKay Conjecture, Cambridge University Press (2018) [doi:10.1017/9781108552790]

Further discussion:

• Peter Schmid: Extending irreducible representations of normal subgroups, Journal of Algebra 94 1 (1985) 30-51 [doi:10.1016/0021-8693(85)90203-0]

• G. Karpilovsky: On extension of characters from normal subgroups, Proceedings of the Edinburgh Mathematical Society 27 (1984) 7-9 [doi:10.1017/S0013091500022057, pdf]

• Qiang Huang: Extension of irreducible characters from normal subgroups, Linear and Multilinear Algebra 27 2 (1990) [doi;10.1080/03081089008818001]

• Robert B. Howlett : Extending characters from normal subgroups, in Topics in Algebra, Lecture Notes in Mathematics 697 (2006) 1–7 [doi:10.1007/BFb0103119]