{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T19:22:05Z","timestamp":1773256925239,"version":"3.50.1"},"reference-count":41,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2021,1,1]],"date-time":"2021-01-01T00:00:00Z","timestamp":1609459200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,5,21]]},"abstract":"Abstract<\/jats:title>\n Let \n \n \n \n F<\/m:mi>\n <\/m:math>\n F<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> be a finite group and \n \n \n \n X<\/m:mi>\n <\/m:math>\n X<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> be a complex quasi-projective \n \n \n \n F<\/m:mi>\n <\/m:math>\n F<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-variety. For \n \n \n \n r<\/m:mi>\n \u2208<\/m:mo>\n N<\/m:mi>\n <\/m:math>\n r\\in {\\mathbb{N}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, we consider the mixed Hodge-Deligne polynomials of quotients \n \n \n \n \n \n X<\/m:mi>\n <\/m:mrow>\n \n r<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n \n \/<\/m:mtext>\n \n F<\/m:mi>\n <\/m:math>\n {X}^{r}\\hspace{-0.15em}\\text{\/}\\hspace{-0.08em}F<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, where \n \n \n \n F<\/m:mi>\n <\/m:math>\n F<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> acts diagonally, and compute them for certain classes of varieties \n \n \n \n X<\/m:mi>\n <\/m:math>\n X<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with simple mixed Hodge structures (MHSs). A particularly interesting case is when \n \n \n \n X<\/m:mi>\n <\/m:math>\n X<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is the maximal torus of an affine reductive group \n \n \n \n G<\/m:mi>\n <\/m:math>\n G<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, and \n \n \n \n F<\/m:mi>\n <\/m:math>\n F<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is its Weyl group. As an application, we obtain explicit formulas for the Hodge-Deligne and \n \n \n \n E<\/m:mi>\n <\/m:math>\n E<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-polynomials of (the distinguished component of) \n \n \n \n G<\/m:mi>\n <\/m:math>\n G<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-character varieties of free abelian groups. In the cases \n \n \n \n G<\/m:mi>\n =<\/m:mo>\n G<\/m:mi>\n L<\/m:mi>\n \n (<\/m:mo>\n \n n<\/m:mi>\n ,<\/m:mo>\n C<\/m:mi>\n \n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n G=GL\\left(n,{\\mathbb{C}}\\hspace{-0.1em})<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n \n \n S<\/m:mi>\n L<\/m:mi>\n \n (<\/m:mo>\n \n n<\/m:mi>\n ,<\/m:mo>\n C<\/m:mi>\n \n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n SL\\left(n,{\\mathbb{C}}\\hspace{-0.1em})<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, we get even more concrete expressions for these polynomials, using the combinatorics of partitions.<\/jats:p>","DOI":"10.1515\/math-2021-0038","type":"journal-article","created":{"date-parts":[[2021,5,25]],"date-time":"2021-05-25T06:53:25Z","timestamp":1621925605000},"page":"338-362","source":"Crossref","is-referenced-by-count":10,"title":["Hodge-Deligne polynomials of character varieties of free abelian groups"],"prefix":"10.1515","volume":"19","author":[{"given":"Carlos","family":"Florentino","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica, Faculdade de Ci\u00eancias, Univ. de Lisboa, Edf. C6, Campo Grande , 1749-016 , Lisboa , Portugal"}]},{"given":"Jaime","family":"Silva","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, ISEL - Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Em\u00eddio Navarro, 1 , 1959-007 , Lisboa , Portugal"}]}],"member":"374","published-online":{"date-parts":[[2021,5,21]]},"reference":[{"key":"2022022201425494487_j_math-2021-0038_ref_001","doi-asserted-by":"crossref","unstructured":"N. J. Hitchin\n, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59\u2013126.","DOI":"10.1112\/plms\/s3-55.1.59"},{"key":"2022022201425494487_j_math-2021-0038_ref_002","doi-asserted-by":"crossref","unstructured":"C. T. Simpson\n, Moduli of representations of the fundamental group of a smooth projective variety II, Inst. Hautes \u00c9tudes Sci. Publ. 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