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\\
i_{F_4}&=A E-4 B D+3 C^2 \\
j_{F_4}&=A C E+2 B C D-C^3-B^2 E-A D^2
\end{aligned}
</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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==Invariants of several binary forms==

The covariants of a binary form are essentially the same as joint invariants of a binary form and a binary linear form. More generally, on can ask for the joint invariants (and covariants) of any collection of binary forms. Some cases that have been studied are listed below.

{| class="wikitable"

|+ (basic invariant, basic covariant)

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

|-

| 1

| (1, 3)

|| (2, 5)

|| (4, 13)

|| (5, 20)

|| (23, 94)

|-

| 2

|

|| (3, 6)

|| (5, 15)

|| (6, 18)

|| (29, 92)

|-

| 3

|

||

||

|| (20, 63)

||(8, 28)

|-

| 4

|

||

||

||

||

|}

Notes:

===Covariants of two linear forms===

There are 1 basic invariant and 3 basic covariants.

===Covariants of a linear form and a quadratic===

There are 2 basic invariants and 5 basic covariants.

===Covariants of a linear form and a cubic===

There are 4 basic invariants (essentially the covariants of a cubic) and 13 basic covariants.

===Covariants of a linear form and a quartic===

There are 5 basic invariants (essentially the basic covariants of a quartic) and 20 basic covariants.

===Covariants of a linear form and a quintic===

There are 23 basic invariants (essentially the basic covariants of a quintic) and 94 basic covariants.

===Covariants of a linear form and a quantic===

===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 two quadratics===

There are 3 basic invariants and 6 basic covariants.

===Covariants of two quadratics and a linear form===

===Covariants of several linear and quadratic forms===

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

For the number of generators of the ring of covariants, change ''m'' to ''m''+1.

===Covariants of a quadratic and a cubic===

There are 5 basic invariants and 15 basic covariants

===Covariants of a quadratic and a quartic===

There are 6 basic invariants and 18 basic covariants

===Covariants of a quadratic and a quintic===

There are 29 basic invariants and 92 basic covariants

===Covariants of a cubic and a quartic===

There are 20 basic invariants and 63 basic covariants

===Covariants of two quartics===

There are 8 basic invariants (3 of degree 2, 4 of degree 3, and 1 of degree 4) and 28 basic covariants. (Gordan gave 30 covariants, but Sylvester showed that two of these are reducible.)

* The basic invariants of a linear form are essentially the same as its basic covariants.

===Covariants of many cubics or quartics===

* For two quartics, there are 8 basic invariants (3 of degree 2, 4 of degree 3, and 1 of degree 4) and 28 basic covariants. (Gordan gave 30 covariants, but Sylvester showed that two of these are reducible.)

Multiple forms:

The numbers of generators of invariants or covariants were given by {{harvtxt|Young|1898}}.

* Covariants of several linear forms: The ring of invariants of <math>n</math> linear forms is generated by <math>n(n-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 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)/2 + n(n+1)/2</math> generators in degree 2, <math>nm(m+1)/2 + n(n-1)(n-2)/6</math> in degree 3, and <math>m(m+1)n(n-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==