On compositions with $x^2/(1-x)$

Abstract

In the past, empirical evidence has been presented that Hilbert series of symplectic quotients of unitary representations obey a certain universal system of infinitely many constraints. Formal series with this property have been called \emph{symplectic}. Here we show that a formal power series is symplectic if and only if it is a formal composite with the formal power series $x^2/(1-x)$. Hence the set of symplectic power series forms a subalgebra of the algebra of formal power series. The subalgebra property is translated into an identity for the coefficients of the even Euler polynomials, which can be interpreted as a cubic identity for the Bernoulli numbers. Furthermore we show that a rational power series is symplectic if and only if it is invariant under the idempotent Möbius transformation $x\mapsto x/(x-1)$. It follows that the Hilbert series of a graded Cohen-Macaulay algebra $A$ is symplectic if and only if $A$ is Gorenstein with its a-invariant and its Krull dimension adding up to zero. It is shown that this is the case for algebras of regular functions on symplectic quotients of unitary representations of tori.

15 pages

References in corpus (2)

• The Hilbert series of a linear symplectic circle quotient
• On Orbifold Criteria for Symplectic Toric Quotients

Cited by in corpus (7)

• Symplectic quotients have symplectic singularities
• Symplectic reduction at zero angular momentum
• The Laurent coefficients of the Hilbert series of a Gorenstein algebra
• Hilbert series associated to symplectic quotients by ${\mathrm{SU}}_2$
• The Hilbert series of ${\mathrm{SL}}_2$-invariants
• Hilbert series of symplectic quotients by the 2-torus
• Multigraded Hilbert series of invariants, covariants, and symplectic quotients for some rank $1$ Lie groups