In mathematics, the symbolic method in

invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...

is an

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

developed by

Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, Cayley enjoyed solving complex maths problems ...

Siegfried Heinrich Aronhold Siegfried Heinrich Aronhold (16 July 1819 – 13 March 1884) was a German mathematician who worked on invariant theory and introduced the symbolic method. He was born in Angerburg, East Prussia, and died, aged 64, in Berlin Berlin ( , ) is ...

Alfred Clebsch Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Berlin. ...

, and

Paul Gordan __NOTOC__ Paul Albert Gordan (27 April 1837 – 21 December 1912) was a Jewish-German mathematician, a student of Carl Jacobi at the University of Königsberg before obtaining his PhD at the University of Breslau (1862),. and a professor a ...

in the 19th century for computing invariants of

algebraic form In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...

s. It is based on treating the form as if it were a power of a degree one form, which corresponds to embedding a symmetric power of a vector space into the symmetric elements of a

tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...

of copies of it.

Symbolic notation

The symbolic method uses a compact, but rather confusing and mysterious notation for invariants, depending on the introduction of new symbols ''a'', ''b'', ''c'', ... (from which the symbolic method gets its name) with apparently contradictory properties.

Example: the discriminant of a binary quadratic form

These symbols can be explained by the following example from Gordan. Suppose that :

\displaystyle  f(x) = A_0x_1^2+2A_1x_1x_2+A_2x_2^2

is a binary quadratic form with an invariant given by the discriminant :

\displaystyle \Delta=A_0A_2-A_1^2.

The symbolic representation of the discriminant is :

\displaystyle 2\Delta=(ab)^2

where ''a'' and ''b'' are the symbols. The meaning of the expression (''ab'')² is as follows. First of all, (''ab'') is a shorthand form for the determinant of a matrix whose rows are ''a''₁, ''a''₂ and ''b''₁, ''b''₂, so :

\displaystyle (ab)=a_1b_2-a_2b_1.

Squaring this we get :

\displaystyle (ab)^2=a_1^2b_2^2-2a_1a_2b_1b_2+a_2^2b_1^2.

Next we pretend that :

\displaystyle f(x)=(a_1x_1+a_2x_2)^2=(b_1x_1+b_2x_2)^2

so that :

\displaystyle  A_i=a_1^a_2^= b_1^b_2^

and we ignore the fact that this does not seem to make sense if ''f'' is not a power of a linear form. Substituting these values gives :

\displaystyle (ab)^2= A_2A_0-2A_1A_1+A_0A_2 = 2\Delta.

Higher degrees

More generally if :

\displaystyle  f(x) = A_0x_1^n+\binomA_1x_1^x_2+\cdots+A_nx_2^n

is a binary form of higher degree, then one introduces new variables ''a''₁, ''a''₂, ''b''₁, ''b''₂, ''c''₁, ''c''₂, with the properties :

f(x)=(a_1x_1+a_2x_2)^n=(b_1x_1+b_2x_2)^n=(c_1x_1+c_2x_2)^n=\cdots.

What this means is that the following two vector spaces are naturally isomorphic: *The vector space of homogeneous polynomials in ''A''₀,...''A''_''n'' of degree ''m'' *The vector space of polynomials in 2''m'' variables ''a''₁, ''a''₂, ''b''₁, ''b''₂, ''c''₁, ''c''₂, ... that have degree ''n'' in each of the ''m'' pairs of variables (''a''₁, ''a''₂), (''b''₁, ''b''₂), (''c''₁, ''c''₂), ... and are symmetric under permutations of the ''m'' symbols ''a'', ''b'', ...., The isomorphism is given by mapping ''a'a'', ''b'b'', .... to ''A''_''j''. This mapping does not preserve products of polynomials.

More variables

The extension to a form ''f'' in more than two variables ''x''₁, ''x''₂, ''x''₃,... is similar: one introduces symbols ''a''₁, ''a''₂, ''a''₃ and so on with the properties :

f(x)=(a_1x_1+a_2x_2+a_3x_3+\cdots)^n=(b_1x_1+b_2x_2+b_3x_3+\cdots)^n=(c_1x_1+c_2x_2+c_3x_3+\cdots)^n=\cdots.

Symmetric products

The rather mysterious formalism of the symbolic method corresponds to embedding a symmetric product S^''n''(''V'') of a vector space ''V'' into a tensor product of ''n'' copies of ''V'', as the elements preserved by the action of the symmetric group. In fact this is done twice, because the invariants of degree ''n'' of a quantic of degree ''m'' are the invariant elements of S^''n''S^''m''(''V''), which gets embedded into a tensor product of ''mn'' copies of ''V'', as the elements invariant under a wreath product of the two symmetric groups. The brackets of the symbolic method are really invariant linear forms on this tensor product, which give invariants of S^''n''S^''m''(''V'') by restriction.

References

* Footnotes