How to compute the ring of invariants of SO_3(k) acting on a polynomial ring - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T11:55:41Z http://mathoverflow.net/feeds/question/35420 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/35420/how-to-compute-the-ring-of-invariants-of-so-3k-acting-on-a-polynomial-ring How to compute the ring of invariants of SO_3(k) acting on a polynomial ring tkf 2010-08-13T01:35:29Z 2010-08-13T02:31:12Z <p>Let $k$ be a field and let $A$ be the polynomial ring over $k$ in $3n$ variables: $A = k[X_{ij} \vert i=1,2,3 \quad j=1,2,\cdots,n]$.</p> <p>${\rm SO}_3(k)$ acts on $A$ in the following way: Given $g \in {\rm SO}_3(k)$, we define:</p> <p>$$g(X_{ij})=g_{ik}X_{kj}$$ </p> <p>with respect to the summation convention.</p> <p>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. </p> http://mathoverflow.net/questions/35420/how-to-compute-the-ring-of-invariants-of-so-3k-acting-on-a-polynomial-ring/35423#35423 Answer by Richard Borcherds for How to compute the ring of invariants of SO_3(k) acting on a polynomial ring Richard Borcherds 2010-08-13T02:07:35Z 2010-08-13T02:07:35Z <p>H. Weyl, <a href="http://books.google.com/books?isbn=978-0-691-05756-9" rel="nofollow">The classical groups</a> chapter V</p> http://mathoverflow.net/questions/35420/how-to-compute-the-ring-of-invariants-of-so-3k-acting-on-a-polynomial-ring/35426#35426 Answer by Victor Protsak for How to compute the ring of invariants of SO_3(k) acting on a polynomial ring Victor Protsak 2010-08-13T02:31:12Z 2010-08-13T02:31:12Z <p>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:</p> <p>1 Scalar products of the columns of the matrix $X.$</p> <p>2 Order $m$ minors of the matrix $X.$</p> <p>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 ($SO_m=O_m\cap SL_m$).</p> <p>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</p> <blockquote> Laskshmibai and Raghavan, <em>Standard monomial theory. Invariant theoretic approach.</em> Encyclopaedia of Mathematical Sciences, vol 137 (Invariant Theory and Algebraic Transformation Groups VIII), Springer. </blockquote>