Content deleted Content added
→Terminology: covariants |
→Terminology: representation theory |
||
Line 4:
A binary form (of degree ''n'') is a homogeneous polynomial Σ{{su|b=''i''=0|p=''n''}} ({{su|p=''n''|b=''i''}})''a''<sub>''n''−''i''</sub>''x''<sup>''n''−''i''</sup>''y''<sup>''i''</sup> = ''a''<sub>''n''</sub>''x''<sup>''n''</sup> + ({{su|p=''n''|b=1}})''a''<sub>''n''−1</sub>''x''<sup>''n''−1</sup>''y'' + ... + ''a''<sub>0</sub>''y''<sup>''n''</sup>. The group ''SL''<sub>2</sub>('''C''') acts on these forms by taking ''x'' to ''ax''+''by'' and ''y'' to ''cx''+''dy''. This induces an action on the space spanned by ''a''<sub>0</sub>, ..., ''a''<sub>''n''</sub> and on the polynomials in these variables. An '''invariant''' is a polynomial in these ''n''+1 variables ''a''<sub>0</sub>, ..., ''a''<sub>''n''</sub> that is invariant under this action. More generally a '''covariant''' is a polynomial in ''a''<sub>0</sub>, ..., ''a''<sub>''n''</sub>, ''x'', ''y'' that is invariant, so an invariant is a special case of a covariant where the variables ''x'' and ''y'' do not occur.
In terms of representation theory, given nay representation ''V'' of the group ''SL''<sub>2</sub>('''C''') one can ask for the ring of invaraint polynomials on ''V''. Invariants of a binary form of degree ''n'' correspond to taking ''V'' to be the ''n''+1 dimensional irreducible representation, and covariants correspond to taking ''V'' to be the sum of the irreducible representations of dimensions 2 and ''n''+1.
The invariants of a binary form are a graded algebra, and {{harvtxt|Gordan|1868}} proved that this algebra is finitely generated if the base field is the complex numbers.
| |||