The Laurent coefficients of the Hilbert series of a Gorenstein algebra

Abstract

By a theorem of R. Stanley, a graded Cohen-Macaulay domain $A$ is Gorenstein if and only if its Hilbert series satisfies the functional equation \[ \operatorname{Hilb}_A(t^{-1})=(-1)^d t^{-a}\operatorname{Hilb}_A(t), \] where $d$ is the Krull dimension and $a$ is the a-invariant of $A$. We reformulate this functional equation in terms of an infinite system of linear constraints on the Laurent coefficients of ${\mathrm{Hilb}}_A(t)$ at $t=1$. The main idea consists of examining the graded algebra $\mathcal{F}={⨁}_{r\in \mathbb{Z}}{\mathcal{F}}_r$ of formal power series in the variable $x$ that fulfill the condition $\phi (x/(x-1))=(1-x{)}^r\phi (x)$. As a byproduct, we derive quadratic and cubic relations for the Bernoulli numbers. The cubic relations have a natural interpretation in terms of coefficients of the Euler polynomials. For the special case of degree $r=-(a+d)=0$, these results have been investigated previously by the authors and involved merely even Euler polynomials. A link to the work of H. W. Gould and L. Carlitz on power sums of symmetric number triangles is established.

31 pages. From v1: Improved exposition, simplified proofs of Proposition 4.9 and Theorem 5.3, and corrected some errors. From v2: Changed the title to more clearly reflect the contents of the paper; replaced the example at the end of Section 2 with a more relevant example; minor corrections and updates

References in corpus (3)

• The Hilbert series of a linear symplectic circle quotient
• On compositions with $x^2/(1-x)$
• The Hilbert series and $a$-invariant of circle invariants

Cited by in corpus (4)

• Futaki Invariants and Reflexive Polygons
• Some Open Questions in Quiver Gauge Theory
• Hilbert series of symplectic quotients by the 2-torus
• Multigraded Hilbert series of invariants, covariants, and symplectic quotients for some rank $1$ Lie groups