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 $\det T_k$?
We have that $\det T_k$ is a fixed (depending on $n=\dim V$ and $k$ only) power of $\det T$. To see this, as well as getting the power, one can for instance note that $\mathrm{SL}(V)$ is the commutator subgroup of $\mathrm{GL}(V)$ (except for extremely small finite fields but we can always increase the size of the field) and hence if $\det T=1$ then $\det T_k=1$. We can then write any $T\in\mathrm{GL}(V)$ in the form $DS$, where $S\in\mathrm{SL}(V)$ and $D$ a diagonal matrix with diagonal entries $(t,1,1,\ldots,1)$. Then $\det((DS)_k)=\det(D_k)\cdot1$ so it suffices to compute $\det(D_k)$ but in the standard basis of $\mathrm{Sym}^kV$, given a basis $e_1,\ldots,e_n$ of $V$, $D_k$ is a diagonal entries and its determinant is $t^R$, where $R=\sum_{0\leq i\leq k}is^{n1}_{ki}$. Here $s^{a}_{b}=\dim \mathrm{Sym}^bU$ where $\dim U=a$ which equals $\binom{a+b1}{b}$.

$\begingroup$ Thanks a lot! Could you please tell me where I can read about these things? $\endgroup$ – Brian Sep 27 '10 at 4:38

$\begingroup$ @Brian, try a text on multilinear algebra, for example "finite dimensional multilinear algebra" by M. Marcus discusses such topics in chapter 2. $\endgroup$ – Gjergji Zaimi Sep 27 '10 at 5:11

1$\begingroup$ More precisely, $$\det T_k=(\det T)^N,\qquad N=C_{n1}^{k1}.$$ $N$ is a binomial coefficient. $\endgroup$ – Denis Serre Sep 27 '10 at 5:46

$\begingroup$ Actually, a density argument would also work (assume that everything is diagonalizable). $\endgroup$ – Brian Dec 8 '10 at 1:22

2$\begingroup$ There is a typo in Denis Serre's formula : one has $N=C_{n+k1}^{n}$. $\endgroup$ – Olivier Benoist Nov 29 '13 at 16:19