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


Authors: Hans-Christian Herbig, Daniel Herden and Christopher Seaton
Journal: Proc. Amer. Math. Soc. 143 (2015), 4583-4596
MSC (2010): Primary 05A15; Secondary 11B68, 13A50, 53D20
DOI: https://doi.org/10.1090/proc/12806
Published electronically: July 29, 2015
MathSciNet review: 3391019
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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\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.


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Hans-Christian Herbig
Affiliation: Departamento de Matemática Aplicada, Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos 149, Centro de Tecnologia - Bloco C, CEP: 21941-909, Rio de Janeiro, Brazil
Email: herbig@labma.ufrj.br

Daniel Herden
Affiliation: Department of Mathematics, Baylor University, One Bear Place #97328, Waco, Texas 76798-7328
Email: Daniel_Herden@baylor.edu

Christopher Seaton
Affiliation: Department of Mathematics and Computer Science, Rhodes College, 2000 N. Parkway, Memphis, Tennessee 38112
Email: seatonc@rhodes.edu

DOI: https://doi.org/10.1090/proc/12806
Received by editor(s): April 3, 2014
Published electronically: July 29, 2015
Additional Notes: The first and second author were supported by the grant GA CR P201/12/G028. The third author was supported by a Rhodes College Faculty Development Grant as well as the E.C. Ellett Professorship in Mathematics.
Communicated by: Harm Derksen
Article copyright: © Copyright 2015 American Mathematical Society