Poincare series for the algebras of joint invariants and covariants of n quadratic forms

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Poincare series for the algebras of joint invariants and covariants of n quadratic forms

Journal Title: Карпатські математичні публікації - Year 2017, Vol 9, Issue 1

Abstract

We consider one of the fundamental objects of classical invariant theory, namely the Poincare series for an algebra of invariants of Lie group $SL_2$. The first two terms of the Laurent series expansion of Poincar\'e series at the point $z = 1$ give us an important information about the structure of the algebra $\mathcal{I}_{d}.$ It was derived by Hilbert for the algebra ${\mathcal{I}_{d}=\mathbb{C}[V_d]^{\,SL_2}}$ of invariants for binary $d-$form (by $V_d$ we denote the vector space over $\mathbb{C}$ consisting of all binary forms homogeneous of degree $d$). Springer got this result, using explicit formula for the Poincar\'e series of this algebra. We consider this problem for the algebra of joint invariants ${\mathcal{I}_{2n}{=}\mathbb{C} \underbrace{V_2 {\oplus} V_2 {\oplus} \cdots {\oplus} V_2}_{\text{$n$ times}} ]^{SL_2}}$ and the algebra of joint covariants ${\mathcal{C}_{2n}{=}\mathbb{C}[\underbrace{V_2 {\oplus} V_2 {\oplus} \cdots {\oplus} V_2}_{\text{$n$ times}} \oplus}\mathbb{C}^2 ]^{SL_2}}$ of $n$ quadratic forms. We express the Poincar\'e series $\mathcal{P} \mathcal{C}_{2n},z)=\sum_{j=0}^{\infty }\dim(\mathcal{C}_{2n})_{j}\, z^j$ and $\mathcal{P}(\mathcal{I}_{2n},z)=\sum_{j=0}^{\infty }\dim(\mathcal{I}_{2n})_{j}\, z^j$ of these algebras in terms of Narayana polynomials. Also, for these algebras we calculate the degrees and asymptotic behavious of the degrees, using their Poincare series.

Authors and Affiliations

N. Ilash

Keywords

EP ID EP325055
DOI 10.15330/cmp.9.1.57-62
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