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Let V $V$ be a vector space over some field $k$ and $T \in \mathrm{GL}(V)$. Then, we can view $T\in \mathrm{GL}(\mathrm{Sym}^k(V))$ where $\mathrm{Sym}^k(V)$ denotes the $k^\mathrm{th}$ symmetric power of $V$ and denote it $T_k$. Knowing $\det T$, is there a general formula for $\det T_k$?

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Let V be a vector space over some field $k$ and $T \in \mathrm{GL}(V)$. mathrm{GL}(V)$. Then, we can view $T\in \mathrm{GL}(\mathrm{Sym}^k(V))$ where $\mathrm{Sym}^k(V)$ denotes the $k^\mathrm{th}$ symmetric power of$V$and denote it $T_k$. T_k$. Knowing $\det T$, is there a general formula for $\det T_k$?

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Let V be a vector space over some field $k$ and $T \in \mathrm{GL}(V)$. Then, we can view $T\in \mathrm{GL}(\mathrm{Sym}^k(V))$ where $\mathrm{Sym}^k(V)$ denotes the $k^\mathrm{th}$ symmetric power of $V$ and denote it $T_k$. Knowing $\det T$, is there a general formula for $\det T_k$.?

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