Invariant of a binary form: Difference between revisions

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A binary form (of degree ''n'') is a homogeneous polynomial Σ{{su|b=''i''=0|p=''n''}} ({{su|p=''n''|b=''i''}})''a''_{''n''−''i''}''x''^{''n''−''i''}''y''^''i'' = ''a''_''n''''x''^''n'' + ({{su|p=''n''|b=1}})''a''_{''n''−1}''x''^{''n''−1}''y'' + ... + ''a''₀''y''^''n''. The group ''SL''₂('''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''₀, ..., ''a''_''n'' and on the polynomials in these variables. An '''invariant''' is a polynomial in these ''n'' + 1 variables ''a''₀, ..., ''a''_''n'' that is invariant under this action. More generally a '''covariant''' is a polynomial in ''a''₀, ..., ''a''_''n'', ''x'', ''y'' that is invariant, so an invariant is a special case of a covariant where the variables ''x'' and ''y'' do not occur. More generally still, a '''simultaneous invariant''' is a polynomial in the coefficients of several different forms in ''x'' and ''y''.

In terms of representation theory, given nayany representation ''V'' of the group ''SL''₂('''C''') one can ask for the ring of invaraintinvariant 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.