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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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up vote 6 down vote accepted

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 $R=\sum_{0\leq i\leq k}is^{n-1}_{k-i}$. Here $s^{a}_{b}=\dim \mathrm{Sym}^bU$ where $\dim U=a$ which equals $\binom{a+b-1}{b}$.

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Thanks a lot! Could you please tell me where I can read about these things? –  Brian Sep 27 '10 at 4:38
    
@Brian, try a text on multilinear algebra, for example "finite dimensional multilinear algebra" by M. Marcus discusses such topics in chapter 2. –  Gjergji Zaimi Sep 27 '10 at 5:11
    
@Gjergji: Thanks a lot! –  Brian Sep 27 '10 at 5:34
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More precisely, $$\det T_k=(\det T)^N,\qquad N=C_{n-1}^{k-1}.$$ $N$ is a binomial coefficient. –  Denis Serre Sep 27 '10 at 5:46
    
Actually, a density argument would also work (assume that everything is diagonalizable). –  Brian Dec 8 '10 at 1:22

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