Symbolic method
Encyclopedia
In mathematics
, the symbolic method in invariant theory
is an algorithm
developed by Arthur Cayley
, , , and in the 19th century for computing invariant
s of algebraic forms. It is based on treating the form as if it were a power of a degree one form.
is a binary quadratic form with an invariant given by the discriminant
The symbolic representation of the discriminant is
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
Squaring this we get
Next we pretend that
so that
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
is a binary form of higher degree, then one introduces new variables a1, a2, b1, b2, c1, c2, with the properties
What this means is that the following two vector spaces are naturally isomorphic:
The isomorphism is given by mapping aa, bb, .... to Aj. This mapping does not preserve products of polynomials.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the symbolic method in invariant theory
Invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
is an algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
developed by Arthur Cayley
Arthur Cayley
Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
, , , and in the 19th century for computing invariant
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
s of algebraic forms. It is based on treating the form as if it were a power of a degree one form.
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 . Suppose thatis a binary quadratic form with an invariant given by the discriminant
The symbolic representation of the discriminant is
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
Squaring this we get
Next we pretend that
so that
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
Higher degrees
More generally ifis a binary form of higher degree, then one introduces new variables a1, a2, b1, b2, c1, c2, with the properties
What this means is that the following two vector spaces are naturally isomorphic:
- The vector space of homogeneous polynomials in A0,...An 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 aa, bb, .... to Aj. 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