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induced representation (Definition)

Let $ G$ be a group, $ H \subset G$ a subgroup, and $ V$ a representation of $ H$, considered as a $ \mathbb{Z}[H]$-module. The induced representation of $ \rho$ on $ G$, denoted $ \operatorname{Ind}_H^G(V)$, is the $ \mathbb{Z}[G]$-module whose underlying vector space is the direct sum

$\displaystyle \bigoplus_{\sigma \in G/H} \sigma V $
of formal translates of $ V$ by left cosets $ \sigma$ in $ G/H$, and whose multiplication operation is defined by choosing a set $ \{g_\sigma\}_{\sigma \in G/H}$ of coset representatives and setting
$\displaystyle g(\sigma v) := \tau (h v) $
where $ \tau$ is the unique left coset of $ G/H$ containing $ g \cdot g_\sigma$ (i.e., such that $ g \cdot g_\sigma = g_\tau \cdot h$ for some $ h \in H$).

One easily verifies that the representation $ \operatorname{Ind}_H^G(V)$ is independent of the choice of coset representatives $ \{g_\sigma\}$.



"induced representation" is owned by djao.
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example of induced representation (Example) by rspuzio
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Cross-references: independent, coset, operation, multiplication, left cosets, translates, direct sum, vector space, representation, subgroup, group
There are 3 references to this entry.

This is version 1 of induced representation, born on 2002-02-05.
Object id is 1823, canonical name is InducedRepresentation.
Accessed 3588 times total.

Classification:
AMS MSC: 20C99 (Group theory and generalizations :: Representation theory of groups :: Miscellaneous)
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