Invariant of a binary form: Difference between revisions

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{{main|Glossary of invariant theory}}

A binary form (of degree ''n'') is a homogeneous polynomial &Sigma;<math>\sum_{{su|b=''i''=0|p=''}^n \binom{n''}{i} (a_{{su|p=''n''|b=''-i''}})''a''_{''n''&minus;''i''}''x''^{''^{n''&minus;''-i''}''}y''^{''^i''} = ''a''_{''a_nx^n''}''x''^''n'' + (\binom{n}{su|p=''n''|b=1}})''a''_{'' a_{n''&minus;-1}''}x''^{''^{n''&minus;-1}''}y'' + ...\cdots + ''a''₀''y''<sup>''a_0y^n''</supmath>. The group ''SL''<submath>2SL_2(\mathbb{C})</submath>('''C''') acts on these forms by taking ''<math>x''</math> to ''<math>ax''&nbsp; +&nbsp;'' by''</math> and ''<math>y''</math> to ''<math>cx''&nbsp; +&nbsp;'' dy''</math>. This induces an action on the space spanned by ''a''<sub>0</submath>a_0, ...\ldots, ''a''<sub>''n''a_n</submath> and on the polynomials in these variables. An '''invariant''' is a polynomial in these ''<math>n''&nbsp; +&nbsp; 1</math> variables ''a''<sub>0</submath>a_0, ...\ldots, ''a''<sub>''n''a_n</submath> that is invariant under this action. More generally a '''covariant''' is a polynomial in ''a''<submath>0a_0, \ldots, a_n</submath>, ..., ''a''<submath>''n''x</submath>, ''x'', ''<math>y''</math> that is invariant, so an invariant is a special case of a covariant where the variables ''<math>x''</math> and ''<math>y''</math> do not occur. More generally still, a '''simultaneous invariant''' is a polynomial in the coefficients of several different forms in ''<math>x''</math> and&nbsp;'' <math>y''</math>.

In terms of [[representation theory]], given any representation ''<math>V''</math> of the group ''SL''<sub>2</submath>SL_2('''\mathbb{C'''})</math> one can ask for the ring of invariant polynomials on ''<math>V''</math>. Invariants of a binary form of degree ''<math>n''</math> correspond to taking ''<math>V''</math> to be the <math>(''n''&nbsp; +&nbsp; 1)</math>-dimensional irreducible representation, and covariants correspond to taking ''<math>V''</math> to be the sum of the irreducible representations of dimensions 2 and&nbsp;'' <math>n''&nbsp; +&nbsp; 1</math>.

The invariants of a binary form form a [[graded algebra]], and {{harvtxt|Gordan|1868}} proved that this algebra is finitely generated if the base field is the complex numbers.

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===Covariants of a binary linear form===

For linear forms ''ax''<math>F_1(x,y) = Ax + ''by''By</math> the only invariants are constants. The algebra of covariants is generated by the form itself of degree 1 and order 1.

===Covariants of a binary quadric===

The algebra of invariants of the quadratic form ''ax''<supmath>F_2(x,y) = Ax^2</sup> + 2''bxy''2Bxy + ''cy''<sup>Cy^2</supmath> is a polynomial algebra in 1 variable generated by the discriminant ''b''<supmath>B^2 - AC</supmath> &minus; ''ac'' of degree 2. The algebra of covariants is a polynomial algebra in 2 variables generated by the discriminant together with the form ''f'' itself (of degree 1 and order 2). {{harv|Schur|1968|loc=II.8}} {{harv|Hilbert|1993|loc=XVI, XX}}

===Covariants of a binary cubic===

The algebra of invariants of the cubic form ''ax''<supmath>F_3(x,y) = Ax^3</sup> + 3''bx''²''y''3Bx^2y + 3''cxy''^{3Cxy^2} + ''dy''<sup>Dy^3</supmath> is a polynomial algebra in 1 variable generated by the discriminant ''D''<math>\Delta = 3''b''^{3B^2C^2}''c''² + 6''abcd''6ABCD- &minus;4B^3D 4''b''³''d''- &minus;4C^3A 4''c''³''a''- &minus; ''a''²''d''^{A^2D^2</supmath> of degree 4. The algebra of covariants is generated by the discriminant, the form itself (degree 1, order 3), the Hessian ''<math>H''</math> (degree 2, order 2) and a covariant ''<math>T''</math> of degree 3 and order 3. They are related by the [[Syzygy (mathematics)|syzygy]] 4''H''<supmath>4H^3}=''Df''^{^2}-''T''<sup>^2</supmath> of degree 6 and order 6. {{harv|Schur|1968|loc=II.8}} {{harv|Hilbert|1993|loc=XVII, XX}}

===Covariants of a binary quartic===

The algebra of invariants of a quartic form is generated by invariants ''<math>i''</math>, ''<math>j''</math> of degrees 2, 3:
<math display="block">

\begin{aligned}

F_4(x, y)&=A x^4+4 B x^3 y+6 C x^2 y^2+4 D x y^3+E y^4\\

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j_{F_4}&=A C E+2 B C D-C^3-B^2 E-A D^2

\end{aligned}

</math>

</math>This ring is naturally isomorphic to the ring of modular forms of level 1, with the two generators corresponding to the Eisenstein series ''E''₄ and ''E''₆. The algebra of covariants is generated by these two invariants together with the form ''f'' of degree 1 and order 4, the Hessian ''H'' of degree 2 and order 4, and a covariant ''T'' of degree 3 and order 6. They are related by a syzygy {{math|1=''jf''³ – ''Hf''²''i'' + 4''H''³ + ''T''² = 0}} of degree 6 and order 12. {{harv|Schur|1968|loc=II.8}} {{harv|Hilbert|1993|loc=XVIII, XXII}}

</math>This ring is naturally isomorphic to the ring of modular forms of level 1, with the two generators corresponding to the Eisenstein series ''E''<submath>4E_4</submath> and ''E''<submath>6E_6</submath>. The algebra of covariants is generated by these two invariants together with the form ''<math>f''</math> of degree 1 and order 4, the Hessian ''<math>H''</math> of degree 2 and order 4, and a covariant ''<math>T''</math> of degree 3 and order 6. They are related by a syzygy {{<math|1=''>jf''^{^3} –- ''Hf''²''i''^2i + 4''H''^{4H^3} + ''T''^{^2} = 0}}</math> of degree 6 and order 12. {{harv|Schur|1968|loc=II.8}} {{harv|Hilbert|1993|loc=XVIII, XXII}}

===Covariants of a binary quintic===

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|+ (basic invariant, basic covariant)

! Degree of forms !! 1 !! 2 !! 3 !! 4 !! 5

!n

|-

| 1

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|| (5, 20)

|| (23, 94)

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| 2

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|| (6, 18)

|| (29, 92)

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|-

| 3

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|| (20, 63)

||(8, 28)

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|-

| 4

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||

|| (8, 28)

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Multiple forms:

* Covariants of several linear forms: The ring of invariants of ''<math>n''</math> linear forms is generated by ''<math>n''(''n''–1-1)/2</math> invariants of degree 2. The ring of covariants of ''<math>n''</math> linear forms is essentially the same as the ring of invariants of ''<math>n''+1</math> linear forms.

* Covariants of several linear forms: The ring of invariants of ''n'' linear forms is generated by ''n''(''n''–1)/2 invariants of degree 2. The ring of covariants of ''n'' linear forms is essentially the same as the ring of invariants of ''n''+1 linear forms.

* Covariants of several linear and quadratic forms:

** The ring of invariants of a sum of ''<math>m''</math> linear forms and ''<math>n''</math> quadratic forms is generated by ''<math>m''(''m''–1-1)/2 + ''n''(''n''+1)/2</math> generators in degree 2, ''<math>nm''(''m''+1)/2 + ''n''(''n''–1-1)(''n''–2-2)/6</math> in degree 3, and ''<math>m''(''m''+1)''n''(''n''–1-1)/4</math> in degree 4.

** For the number of generators of the ring of covariants, change ''<math>m''</math> to ''<math>m''+1</math>.

* Covariants of many cubics or quartics: See {{harvtxt|Young|1898}}.

==See also==