The Hilbert series and a-invariant of circle invariants |
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Authors: | L Emily Cowie Hans-Christian Herbig Daniel Herden Christopher Seaton |
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Institution: | 1. Department of Mathematics, 303 Lockett Hall, Louisiana State University, Baton Rouge, LA 70803, USA;2. Departamento de Matemática Aplicada, Av. Athos da Silveira Ramos 149, Centro de Tecnologia – Bloco C, CEP 21941-909, Rio de Janeiro, Brazil;3. Department of Mathematics, Baylor University, One Bear Place #97328, Waco, TX 76798-7328, USA;4. Department of Mathematics and Computer Science, Rhodes College, 2000 N. Parkway, Memphis, TN 38112, USA |
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Abstract: | Let V be a finite-dimensional representation of the complex circle determined by a weight vector . We study the Hilbert series of the graded algebra of polynomial -invariants in terms of the weight vector a of the -action. In particular, we give explicit formulas for as well as the first four coefficients of the Laurent expansion of at . The naive formulas for these coefficients have removable singularities when weights pairwise coincide. Identifying these cancelations, the Laurent coefficients are expressed using partial Schur polynomials that are independently symmetric in two sets of variables. We similarly give an explicit formula for the a-invariant of in the case that this algebra is Gorenstein. As an application, we give methods to identify weight vectors with Gorenstein and non-Gorenstein invariant algebras. |
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Keywords: | Primary 13A50 secondary 13H10 05E05 |
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