DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 2026-08-29

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The induced map $Λ^{k} T$ on exterior powers

Definition

Let $T : V \to W$ be linear. Iterating the induced map of Module homomorphisms induce tensor-product homomorphisms functorially gives the $k$ -fold tensor power

$T^{\otimes k} : V^{\otimes k} ⟶ W^{\otimes k}, T^{\otimes k} (v_{1} \otimes \dots \otimes v_{k}) = T (v_{1}) \otimes \dots \otimes T (v_{k}),$

with $T^{\otimes 0} = {id}_{F}$ and $T^{\otimes 1} = T$ . A pure tensor with $v_{i} = v_{j}$ maps to the pure tensor with $T (v_{i}) = T (v_{j})$ , so $T^{\otimes k} (W_{k} (V)) \subseteq W_{k} (W)$ for the relation subspaces of The $k$ th exterior power as the tensor-power quotient by repeated-vector relations. Hence $T^{\otimes k}$ descends to the quotient, defining the $k$ th exterior power

$Λ^{k} T : Λ^{k} V ⟶ Λ^{k} W, Λ^{k} T (v_{1} \land \dots \land v_{k}) = T (v_{1}) \land \dots \land T (v_{k}) .$

For $k = 0$ this is $Λ^{0} T = {id}_{F}$ , and for $k = 1$ it is $T$ itself under $Λ^{1} V = V$ .

Depends on

Dependency tree · 2 levels

Sources

Keith Conrad, Exterior Powers

The induced map ΛkTΛkT on exterior powers

Definition

Depends on

Dependency tree · 2 levels

Sources

The induced map $Λ^{k} T$ on exterior powers