External Tensor Product

Suppose that V is a group representation of G, and W is a group representation of H. Then the vector space tensor product V tensor W is a group representation of the group direct product G×H. An element (g,h) of G×H acts on a basis element v tensor w by

 (g,h)(v tensor w)=gv tensor hw.

To distinguish from the representation tensor product, the external tensor product is denoted V□AdjustmentBox[x, BoxMargins -> {{-0.65, 0.13913}, {-0.5, 0.5}}, BoxBaselineShift -> -0.1]W, although the only possible confusion would occur when G=H.

When V and W are irreducible representations of G and H respectively, then so is the external tensor product. In fact, all irreducible representations of G×H arise as external direct products of irreducible representations.

See also

Group, Group Representation, Irreducible Representation, Representation Tensor Product, Vector Space, Vector Space Tensor Product

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "External Tensor Product." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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