Symbolic method

Last updated October 26, 2023

In mathematics, the symbolic method in invariant theory is an algorithm developed by Arthur Cayley,^[1] Siegfried Heinrich Aronhold,^[2] Alfred Clebsch,^[3] and Paul Gordan ^[4] in the 19th century for computing invariants of algebraic forms. 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 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.^[5] Suppose that

\displaystyle f(x)=A_{0}x_{1}^{2}+2A_{1}x_{1}x_{2}+A_{2}x_{2}^{2}

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

\displaystyle \Delta =A_{0}A_{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 a₁, a₂ and b₁, b₂, so

\displaystyle (ab)=a_{1}b_{2}-a_{2}b_{1}.

Squaring this we get

\displaystyle (ab)^{2}=a_{1}^{2}b_{2}^{2}-2a_{1}a_{2}b_{1}b_{2}+a_{2}^{2}b_{1}^{2}.

Next we pretend that

\displaystyle f(x)=(a_{1}x_{1}+a_{2}x_{2})^{2}=(b_{1}x_{1}+b_{2}x_{2})^{2}

so that

\displaystyle A_{i}=a_{1}^{2-i}a_{2}^{i}=b_{1}^{2-i}b_{2}^{i}

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_{2}A_{0}-2A_{1}A_{1}+A_{0}A_{2}=2\Delta .

Higher degrees

More generally if

\displaystyle f(x)=A_{0}x_{1}^{n}+{\binom {n}{1}}A_{1}x_{1}^{n-1}x_{2}+\cdots +A_{n}x_{2}^{n}

is a binary form of higher degree, then one introduces new variables a₁, a₂, b₁, b₂, c₁, c₂, with the properties

f(x)=(a_{1}x_{1}+a_{2}x_{2})^{n}=(b_{1}x_{1}+b_{2}x_{2})^{n}=(c_{1}x_{1}+c_{2}x_{2})^{n}=\cdots .

What this means is that the following two vector spaces are naturally isomorphic:

The vector space of homogeneous polynomials in A₀,...A_n of degree m
The vector space of polynomials in 2m variables a₁, a₂, b₁, b₂, c₁, c₂, ... that have degree n in each of the m pairs of variables (a₁, a₂), (b₁, b₂), (c₁, c₂), ... and are symmetric under permutations of the m symbols a, b, ....,

The isomorphism is given by mapping a^n−j
₁a^j
₂, b^n−j
₁b^j
₂, .... to A_j. This mapping does not preserve products of polynomials.

More variables

The extension to a form f in more than two variables x₁, x₂, x₃,... is similar: one introduces symbols a₁, a₂, a₃ and so on with the properties

f(x)=(a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}+\cdots )^{n}=(b_{1}x_{1}+b_{2}x_{2}+b_{3}x_{3}+\cdots )^{n}=(c_{1}x_{1}+c_{2}x_{2}+c_{3}x_{3}+\cdots )^{n}=\cdots .

Symmetric products

The rather mysterious formalism of the symbolic method corresponds to embedding a symmetric product Sⁿ(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 SⁿS^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 SⁿS^m(V) by restriction.

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References

Gordan, Paul (1987) [1887]. Kerschensteiner, Georg (ed.). Vorlesungen über Invariantentheorie (2nd ed.). New York York: AMS Chelsea Publishing. ISBN 9780828403283. MR 0917266.

Footnotes

↑ Cayley, Arthur (1846). "On linear transformations". Cambridge and Dublin Mathematical Journal: 104–122.
↑ Aronhold, Siegfried Heinrich (1858). "Theorie der homogenen Functionen dritten Grades von drei Veränderlichen". Journal für die reine und angewandte Mathematik (in German). 1858 (55): 97–191. doi:10.1515/crll.1858.55.97. ISSN 0075-4102. S2CID 122247157.
↑ Clebsch, A. (1861). "Ueber symbolische Darstellung algebraischer Formen". Journal für die Reine und Angewandte Mathematik (in German). 1861 (59): 1–62. doi:10.1515/crll.1861.59.1. ISSN 0075-4102. S2CID 119389672.
↑ Gordan 1887.
↑ Gordan 1887, v. 2, p.g. 1-3.