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 M\"{o}bius 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.