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Hilbert series associated to symplectic quotients by ${SU}_{2}$

Departamento de Matemática Aplicada, Universidade Federal do Rio de Janeiro, Avenedo Athos da Silveira Ramos 149, Centro de Tecnologia — Bloco C, CEP 21941-909 Rio de Janeiro, Brazil

E-mail Address: herbighc@gmail.com

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Daniel Herden

Department of Mathematics, Baylor University, One Bear Place #97328, Waco, TX 76798-7328, USA

E-mail Address: daniel_herden@baylor.edu

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, and

Christopher Seaton

Department of Mathematics and Computer Science, Rhodes College, 2000 North Parkway, Memphis, TN 38112, USA

E-mail Address: seatonc@rhodes.edu

Corresponding author.

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https://doi.org/10.1142/S0218196720500435Cited by:3 (Source: Crossref)

Abstract

We compute the Hilbert series of the graded algebra of real regular functions on the symplectic quotient associated to an ${SU}_{2}$ -module and give an explicit expression for the first nonzero coefficient of the Laurent expansion of the Hilbert series at $t = 1$ . Our expression for the Hilbert series indicates an algorithm to compute it, and we give the output of this algorithm for all representations of dimension at most $10$ . Along the way, we compute the Hilbert series of the module of covariants of an arbitrary ${SL}_{2}$ - or ${SU}_{2}$ -module as well as its first three Laurent coefficients.

Communicated by J. McCullough

Keywords:

AMSC: 53D20, 13A50, 14L30, 05E05

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