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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:

$$g(X_{ij})=g_{ik}X_{kj}$$

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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2 Answers 2

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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 12.3.1.1 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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