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Let $k$ be a field and let $A$ be the polynomial ring over $k$ in $3n$ variables: $A = k[X_{ij} \vert i=1,2,3 \quad j=1,2,\cdots,n]$.

${\rm SO}_3(k)$ acts on $A$ in the following way: Given $g \in {\rm SO}_3(k)$, we define:


with respect to the summation convention.

Can the ring of invariants of this action be expressed in terms of generators and relations? I get the feeling that this is a standard exercise in invariant theory, but am not sure where to look.

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up vote 5 down vote accepted

This is addressed by the classical invariant theory, but the answer is more complicated than for general linear or orthogonal groups (in particular, not all minimal generators are quadratic). Let $k$ be a field of characteristic 0. The group $G=SO_m$ acts on $m\times n$ matrices by the left multiplication and this induces a $G$-action on $A=k[X_{ij}].$ Let us view the variables as the entries of the $m\times n$ generic matrix over $k.$ Then the algebra of invariants $A^G$ is generated by:

1 Scalar products of the columns of the matrix $X.$

2 Order $m$ minors of the matrix $X.$

This is the First Fundamental Theorem (FFT) of classical invariant theory for $SO_m.$ In fact, the elements of the first type generate $O_m$-invariants and the elements of the second type generate $SL_m$-invariants ($SO_m=O_m\cap SL_m$).

Moreover, all relations between these generators are also known (the Second Fundamental Theorem, SFT) and there is a good description of a standard monomial basis of $A^G.$ If I am not mistaken, the last part is due to Laskshmibai and coauthors. A comprehensive modern reference is

Laskshmibai and Raghavan, Standard monomial theory. Invariant theoretic approach. Encyclopaedia of Mathematical Sciences, vol 137 (Invariant Theory and Algebraic Transformation Groups VIII), Springer.
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The theorem remains true unless $\text{char} k=2.$ But already for orthogonal groups in char 2, there is a drastic difference (not all invariants are generated by the quadratic ones). – Victor Protsak Aug 13 '10 at 2:42
Thankyou. The result I need is the one given in H. Weyl, the classical groups II.17 - that the relations between determinants and inner products are generated by relations of two types: J2, J3 (which I believe imply J1). It is not clear to me what the required hypotheses on the ground field k are, either in Weyl or Laskshmibai (L&R). Also, it is not obvious that the relations (i-iii) in Prop of L&R are the same as Weyl's J2,J3. For the rest of my paper I only need that char(k) is not 2, so it would be ideal if as you suggest the above result held for all fields of char not 2. – tkf Feb 14 '11 at 20:14

H. Weyl, The classical groups chapter V

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