Hilbert polynomials of the algebras of $SL_ 2$-invariants

Abstract

We consider one of the fundamental problems of classical invariant theory, the research of Hilbert polynomials for an algebra of invariants of Lie group $SL_2$. Form of the Hilbert polynomials gives us important information about the structure of the algebra. Besides, the coefficients and the degree of the Hilbert polynomial play an important role in algebraic geometry. It is well known that the Hilbert function of the algebra $SL_n$-invariants is quasi-polynomial. The Cayley-Sylvester formula for calculation of values of the Hilbert function for algebra of covariants of binary $d$-form $\mathcal{C}_{d}= \mathbb{C}[V_d\oplus \mathbb{C}^2]_{SL_2}$ (here $V_d$ is the $d+1$-dimensional space of binary forms of degree $d$) was obtained by Sylvester. Then it was generalized to the algebra of joint invariants for $n$ binary forms. But the Cayley-Sylvester formula is not expressed in terms of polynomials.

In our article we consider the problem of computing the Hilbert polynomials for the algebras of joint invariants and joint covariants of $n$ linear forms and $n$ quadratic forms. We express the Hilbert polynomials $\mathcal{H} \mathcal{I}^{(n)}_1,i)=\dim(\mathcal{C}^{(n)}_1)_i, \mathcal{H}(\mathcal{C}^{(n)}_1,i)=\dim(\mathcal{C}^{(n)}_1)_i,$ $\mathcal{H}(\mathcal{I}^{(n)}_2,i)=\dim(\mathcal{I}^{(n)}_2)_i, \mathcal{H}(\mathcal{C}^{(n)}_2,i)=\dim(\mathcal{C}^{(n)}_2)_i$ of those algebras in terms of quasi-polynomial. We also present them in the form of Narayana numbers and generalized hypergeometric series.