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Hodge-Deligne polynomials of character varieties of free abelian groups
Open Mathematics ( IF 1.0 ) Pub Date : 2021-01-01 , DOI: 10.1515/math-2021-0038
Carlos Florentino 1 , Jaime Silva 2
Affiliation  

Let F F be a finite group and X X be a complex quasi-projective F F -variety. For r ∈ N r\in {\mathbb{N}} , we consider the mixed Hodge-Deligne polynomials of quotients X r / F {X}^{r}\hspace{-0.15em}\text{/}\hspace{-0.08em}F , where F F acts diagonally, and compute them for certain classes of varieties X X with simple mixed Hodge structures (MHSs). A particularly interesting case is when X X is the maximal torus of an affine reductive group G G , and F F is its Weyl group. As an application, we obtain explicit formulas for the Hodge-Deligne and E E -polynomials of (the distinguished component of) G G -character varieties of free abelian groups. In the cases G = G L ( n , C ) G=GL\left(n,{\mathbb{C}}\hspace{-0.1em}) and S L ( n , C ) SL\left(n,{\mathbb{C}}\hspace{-0.1em}) , we get even more concrete expressions for these polynomials, using the combinatorics of partitions.

中文翻译:

自由阿贝尔群的字符变体的Hodge-Deligne多项式

令FF为有限群,XX为复拟射影FF-变种。对于{\ mathbb {N}}中的r∈N r \,我们考虑商X r / F {X} ^ {r} \ hspace {-0.15em} \ text {/} \ hspace的混合Hodge-Deligne多项式{-0.08em} F,其中FF对角起作用,并针对具有简单混合Hodge结构(MHS)的某些类别的品种XX计算它们。一个特别有趣的情况是,当XX是仿射还原基团GG的最大圆环,而FF是它的Weyl基团时。作为一种应用,我们获得了自由阿贝尔族的GG-字符变体(的独特组成部分)的Hodge-Deligne和EE-多项式的显式公式。在G = GL(n,C)的情况下G = GL \ left(n,{\ mathbb {C}} \ hspace {-0.1em})和SL(n,C)SL \ left(n,{\ mathbb {C}} \ hspace {-0.1em})中,我们得到了这些多项式的更具体的表达式,
更新日期:2021-01-01
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