Poincare series for the algebras of joint invariants and covariants of n quadratic forms
We consider one of the fundamental problems of classical invariant theory - the research of Poincare series for an algebra of invariants of Lie group SL_2. The first two terms of the Laurent series expansion of Poincare series at the point z = 1 give us important information about the structure of the algebra I_d. It was derived by Hilbert for the algebra I_d=C[V_d]^ SL_2 of invariants for binary d-form (by V_d denote the vector space over C consisting of all binary forms homogeneous of degree d). Springer got this result, using explicit formula for the Poincare series of this algebra. We consider this problem for the algebra of joint invariants I_2n=C[V_2 V_2 … V_2_n times ]^SL_2 and the algebra of joint covariants C_2n=C[V_2 V_2 … V_2_n times C^2 ]^SL_2 of n quadratic forms. We express the Poincare series P(C_2n,z)=_j=0^∞ (C_2n)_j z^j and P(I_2n,z)=_j=0^∞ (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. , number=1, journal=Carpathian Mathematical Publications, author=Ilash, N.B., year=2017, month=Jun., pages=57–62