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Questions tagged [symmetric-algebra]

A quotient of the tensor algebra by the ideal generated by symmetric tensors.

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$G$ is Lie group and $V$ is a representations of $G$,prove representations $V \otimes V \cong S^2(V) \oplus \Lambda^2(V)$

Let $G$ a Lie group and let $V$ a representations of $G$. Then we have the following representations are isomorphic: \begin{align} V \otimes V \cong S^2(V) \oplus \Lambda^2(V) \end{align} I have no ...
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Proving Symmetric Difference of A and B

Let A and B be sets. Define the symmetric difference of A and B as A∆B= (A ∪ B) − (A ∩ B). (a) Prove that A∆B = (A − B) ∪ (B − A) I tried to start this but am getting really lost. if someone could ...
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Definition of symmetric power of a linear represenation

I'm reading Kowalski's Representation theory, and there's a part about the symmetric and antisymmetric powers of a representation, and I'd like to ask a question about those. So there's a proposition ...
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The symmetric algebra of a vector space is generated by powers

Let $V$ be a finite-dimensional vector space over a field $k \supseteq \mathbb Q$ (characterstic $0$). In Helgason's Groups and Geometric Analysis it is mentioned that the symmetric algebra $S(V)$ is ...