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Example 1.
Define the standard representation and the adjoint representation for . Then form the tensor product representation. First, set up the representation spaces.
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V1 >
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Define the standard representation.
V2 >
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V2 >
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| (2.1) |
sl2 >
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Define the adjoint representation using the Adjoint command.
sl2 >
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We will need a 6-dimensional vector space to represent the tensor product of rho1 and rho2.
sl2 >
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W1 >
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Use the Query command to verify that rho1 is a representation.
sl2 >
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Example 2.
Compute the representation of rho1 (the standard representation of sl2) on the 3rd symmetric product of . First, use the GenerateSymmetricTensors command to generate a basis T1 for .
sl2 >
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V1 >
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| (2.3) |
We will need a - dimensional representation space.
V1 >
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W2 >
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Example 3.
Compute the representation of rho1 (the standard representation of sl2) on the 2nd exterior product of the 3rd symmetric product
sl2 >
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W2 >
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| (2.4) |
We will need a 6-dimensional representation space.
W2 >
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W3 >
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Use the Invariants command to calculate the invariants of this representation.