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
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