A basic problem in invariant theory is to find the invariants for the action of on binary forms of a given degree. For example, an invariant of the quadratic binary form is the discriminant .
The purpose of this post is to do all the tedious book-keeping needed to explain, in modern language, the sense in which is invariant, keeping careful track of all the various actions. We will also prove that is, in a sense to be made precise, the unique invariant of .
Actions on and on
Throughout let . Let , thought of as a set of column vectors, acted on by in the natural way:
Let be the dual space to , thought of as a set of row vectors. Then acts on by
More briefly, we may write , for . Note that the inverse is essential to make the action well-defined.
Action on binary -forms
We now show how the action of on induces
an action on the space of binary -forms. By definition, a binary -form is a function of the form
for some . (The binomial coefficients could be omitted.) Observe that if then
is a binary -form. Conversely, the binary -form
is equal to where , form the standard basis for . We may therefore identify the vector space of binary -forms with . By functoriality of , the action of is given by
on the elements spanning , and so by for a general .
Example. In the quadratic case, if
and . Since
the new coefficients are related to the old by the matrix .
Invariants: modern definition
The actions of on and induce an action on , defined by . Consider the map defined by . By definition of the actions, we have
More generally, we may define by , and again it follows that for all and . This shows that is an invariant of of weight , in the sense defined below.
Definition. Let be a representation . An invariant of of weight is a function such that for all and .
Because of the presence of in , it is also common to call the function above a covariant. Thinking of as `evaluate ‘, explains the comment towards the end of I.3 in Hilbert’s Theory of algebraic invariants: `The simplest example of a covariant is the form itself.’
Invariants and covariants: classical definition
Let . Set and and substitute for and in to get
where , and . A covariant of of weight is a function of and which is multiplied by when are replaced with respectively. (To test this one must, of course, rewrite the expression in terms of .) An invariant of of weight is a covariant of weight that is independent of and .
For example, the function is an invariant of weight , since
As in Hilbert’s remark, the function is a covariant of weight .
This definition is equivalent to the definition by actions above. Define . Observe that if is the matrix above then
and, by definition of the coefficients , we have
Thus . So a covariant of weight is the same as a function of that is unchanged when are replaced with . This is equivalent to the classical definition just given. It is interesting to note how the classical definition suppresses the inverse in the action by `starting’ with and .
Exercise. More generally, let . Define , and suppose that . Show that and hence that the row vectors of coefficients and are related by
For example, when we get
This agrees with the example earlier, noting that we now have , whereas in the earlier example, we had . As a further check, one can observe that if and then and correspondingly
Invariants of the binary quadratic form
Now suppose that is an invariant of weight . Assume also that is polynomial of degree , so is given by a polynomial of degree in the coefficients of . (By working in , and using that preserves the -degree, it is easy to show that any polynomial invariant is a sum of such homogeneous polynomial invariants.) So is an element of
with the property that for all . In characteristic zero, although not in prime characteristic, symmetric powers commute with duality. So finally, noting that (one last inversion), we obtain an element
spanning a -dimensional subrepresentation isomorphic to . The coefficients in the matrices of acting on and on are of degree two in . Hence acts with degree on the left-hand side, and with degree on the right-hand side. It follows that there are no non-zero invariants unless . In this case, identifying with , the number of linearly independent invariants is the multiplicity
By the Cayley–Sylvester formula, this multiplicity is one when is even, and zero when is odd. (For an symmetric group proof of the Cayley–Sylvester formula see Corollary 2.12 in this paper of Eugenio Giannelli.) Since powers of the discriminant give an invariant in each even degree, the discriminant generates the ring of invariants.