On compositions with 𝑥²/(1-𝑥)

Abstract

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 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 → 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.

Symplectic reduction at zero angular momentum

“…In fact, we observe in each case that d = −a, and hence that the quotient ring is graded Gorenstein; see [6,Section 3.7.3]. When k = 2, these computations were considered in [15,Section 8.3.1], and the relation between the coefficients of the Laurent expansion of the Hilbert series at t = 1 considered there and the graded Gorenstein condition is explained by [11,Corollary 1.8]. Note that the cases corresponding to n = 1 are orbifolds in a strict sense: the moment map is 0, and the group O 1 = Z 2 is finite.…”