Poincare_Series

Let V d be the complex vector space of binary forms of degree d with the canonical action of the special linear group SL 2 =SL 2 (ℂ), and let V d =V d 1 ⊕⋯⊕V d n . The action of SL 2 is extended to an action on the coordinate algebras ℂ[V d ] and ℂ[V d ⊕ℂ 2 ]. The algebras ℐ d =ℂ[V d ] SL 2 and 𝒞 d =ℂ[V d ⊕ℂ 2 ] SL 2 are known, respectively, as the algebras of joint invariants and joint covariants of n binary forms of degrees d 1 ,...,d n . They are among the most intensively studied objects in classical invariant theory of the 19th century. The algebra ℂ[V d ⊕ℂ 2 ] is ℤ n+1 -multigraded in a natural way assuming that the elements of V d 1 ,...,V d n and ℂ 2 are of degree (1,0,...,0,0),...,(0,0,...,1,0),(0,0,...,0,1), respectively. The algebras ℐ d and 𝒞 d are graded subalgebras of ℂ[V d ⊕ℂ 2 ]. In the paper under review the author establishes formulas for the Poincaré (or the Hilbert) series 𝒫(𝒞 d ,z 1 ,...,z n ,t) and 𝒫(ℐ d ,z 1 ,...,z n ,t) which count the dimensions of the multihomogeneous components of the algebras. First he presents an analogue of the classical Cayley-Sylvester formula which expresses the dimensions of the multihomogeneous components in terms of the number of solutions in nonnegative integers of a system of linear equations. Then the author gives an analogue of the Springer-Brion formula to express the Poincaré sereis as a formal power series. To compute the series the author uses the MacMahon partition analysis (or Ω-calculus). The author has also developed a special Maple package (available online) for explicit computations.

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