Bivariate Poincaré Series for the Algebra of Covariants of a Binary Form

Abstract

A formula for computation of the bivariate Poincaré series for the algebra of covariants of binary -form is found.

1. Introduction

Let be the complex vector space of binary forms of degree endowed with the natural action of the special linear group . Consider the corresponding action of the group on the coordinate rings and . Denote by and by the subalgebras of -invariant polynomial functions. In the language of classical invariant theory, the algebras and are called the algebra of invariants and the algebra of covariants for the binary form of degree , respectively. The algebra is a finitely generated bigraded algebra: where each subspace of covariants of degree and order is finite dimensional. The formal power series , is called the bivariate Poincaré series of the algebra of covariants . It is clear that the series is the Poincaré series of the algebra and the series is the Poincaré series of the algebra with respect to the usual grading of the algebras under degree. The finitely generation of the algebra of covariants implies that its bivariate Poincaré series is the power series expansion of a rational function of two variables . We consider here the problem of computing efficiently this rational function.

Calculating the Poincaré series of the algebras of invariants and covariants was an important object of research in invariant theory in the 19th century. For the cases , the series were calculated by Sylvester, see in [1, 2] the big tables of , named them as generating functions for covariants, reduced form. All those calculations are correct up to .

Relatively recently, Springer [3] found an explicit formula for computing the Poincaré series of the algebra of invariants . In the paper we have proved a Cayley-Sylvester-type formula for calculating of and a Springer-type formula for calculation of . By using the formula, the bivariate Poincaré series is calculated for .

2. Cayley-Sylvester-Type Formula for

To begin, we give a proof of the Cayley-Sylvester-type formula for the dimension of the graded component .

Let and be standard irreducible representation of the Lie algebra . The basis elements ,, of the algebra act on by the derivations : The action of is extended to an action on the symmetrical algebra in the natural way.

Let be the maximal unipotent subalgebra of . The algebra , defined by is called the algebra of semi-invariants of the binary form of degree . For any element , a natural number is called the order of the element if the number is the smallest natural number such that It is clear that any semi-invariant of order is the highest weight vector for an irreducible -module of the dimension in .

The classical theorem [4] of Roberts implies an isomorphism of the algebra of covariants and the algebra of semi-invariants. Furthermore, the order is preserved through the isomorphism. Thus, it is enough to compute the Poincaré series of the algebra .

The algebra is -graded and each is a completely reducible representation of the Lie algebra . Thus, the following decomposition holds here is the multiplicity of the representation in the decomposition of . On the other hand, the multiplicity of the representation is equal to the number of linearly independent homogeneous semi-invariants of degree and order for the binary -form. This argument proves the following.

Lemma 2.1.

The set of weights (eigenvalues of the operator ) of a representation denote by , in particular, .

A formal sum is called the character of a representation , here denotes the multiplicity of the weight . Since, the multiplicity of any weight of the irreducible representation is equal to 1, we have

The character of the representation equals (see [5]), where is the complete symmetrical function

By replacing with , , we obtain the specialized expression for : here is the number nonnegative integer solutions of the equation on the assumption that . In particular, the coefficient of (the multiplicity of zero weight ) is equal to , and the coefficient of is equal to .

On the other hand, the decomposition (*) implies the equality for the characters: We can summarize what we have shown so far in the following.

Theorem 2.2.

Proof. The weight appears once in any representation , for . Therefore Similarly, Thus, By using Lemma 2.1 we obtain

For another proof of the formula see [3].

Note that the original Cayley-Sylvester formula is Also, in [6] we proved that Here are the components of standard grading of the algebras , under degree.

3. Calculation of

It is well known that the number of nonnegative integer solutions of the following system is given by the coefficient of of the generating function We will use the notation to denote the coefficient of in the series expansion of . Thus It is clear that

Similarly, the number of nonnegative integer solutions of the following system

equals

Therefore, Thus, the following statement holds.

Theorem 3.1. The number of linearly independent covariants of degree and order for the binary d- form is given by the formula
It is clear that By using the decomposition where is the -binomial coefficient one obtains the well-known formula for instance, see [3].

4. Explicit Formula for

Let us prove Springer-type formula for the bivariate Poincaré series of the algebra covariants of the binary -form. Consider the -algebra of formal power series. For an integer define the -linear function in the following way

The main idea of the ensuing calculations is that the Poincaré series can be expressed in terms of function . The following simple but important statement holds.

Lemma 4.1.

Proof. Theorem 2.2 implies that . Then

Let be a -linear function defined by

for . Note that and , . Also, put for . It is clear that if , .

In important special cases, calculating the functions can be reduced to calculating the functions . The following statements hold.

Lemma 4.2. (i) For holds .
(ii) For and for holds

Proof. (i) The statement follows from the linearity of the function and from the following simple observation: for and .
(ii) Let . Then for we have

Now we can present Springer-type formula for calculating of the bivariate Poincaré series .

Theorem 4.3. here is -shifted factorial.

Proof. Consider the partial fraction decomposition of the rational function : It is easy to see, that Using the above Lemmas we obtain

Corollary 4.4. A denominator of the bivariate Poincaré series , can be written in the form where are the degrees of elements of homogeneous system of parameters for the algebra of invariants .

Proof. The formula of Theorem 4.3 implies that the bivariate Poincaré series has the form for some polynomials . Thus, the Poincaré series for the algebra of invariants has the form The algebra of invariants is Cohen-Macaulay, and its transcendence degree for equals , see [3]. Therefore it has a homogeneous system of parameters and the denominator of its Poincaré series can be written in the following way , where are the degrees of elements of this homogeneous system of parameters.

5. Examples

For direct computations we use the following technical lemma.

Lemma 5.1. For one has here , and are natural numbers.

Proof. Taking into account Lemma 4.2 we get In a similar fashion we prove the general case.

By using Lemma 5.1 the bivariate Poincaré series for are found. All these results agree with Sylvester's calculations up to , see [1, 2].

Below is the list of several series:

References

1. J. J. Sylvester and F. Franklin, “Tables of the generating functions and groundforms for the binary quantics of the first ten orders,” American Journal of Mathematics, vol. 2, no. 3, pp. 223–251, 1879. View at: Publisher Site | Google Scholar
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3. T. A. Springer, “On the invariant theory of SU(2),” Indagationes Mathematicae, vol. 42, no. 3, pp. 339–345, 1980. View at: Google Scholar | Zentralblatt MATH
4. M. Roberts, “The covariants of a binary quantic of the n-th degree,” The Quarterly Journal of Mathematics, vol. 4, pp. 168–178, 1861. View at: Google Scholar
5. W. Fulton and J. Harris, Representation Theory, vol. 129 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1991.
6. L. Bedratyuk, “The Poincaré series for the algebra of covariants of a binary form,” International Journal of Algebra, vol. 4, no. 25, pp. 1201–1207, 2010. View at: Google Scholar

Copyright

Copyright © 2011 Leonid Bedratyuk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.