Hilbert polynomials of the algebras of SL2 -invariants
Journal Title: Карпатські математичні публікації - Year 2018, Vol 10, Issue 2
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 SL2. 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 SLn-invariants is quasi-polynomial. The Cayley-Sylvester formula for calculation of values of the Hilbert function for algebra of covariants of binary d-form Cd=C[Vd⊕C2]SL2 (here Vd 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 HI(n)1,i)=dim(C(n)1)i,H(C(n)1,i)=dim(C(n)1)i, H(I(n)2,i)=dim(I(n)2)i,H(C(n)2,i)=dim(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.
Authors and Affiliations
N. Ilash
Keywords
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- EP ID EP533525
- DOI 10.15330/cmp.10.2.303-312
- Views 65
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