Abstract
We compute the zeta functions enumerating graded ideals in the graded Lie rings associated with the free d-generator Lie rings \(\mathfrak {f}_{c,d}\) of nilpotency class c for all \(c\leqslant 2\) and for \((c,d)\in \{(3,3),(3,2),(4,2)\}\). We apply our computations to obtain information about \(\mathfrak {p}\)-adic, reduced, and topological zeta functions, in particular pertaining to their degrees and some special values.
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References
Bruns, W., Herzog, J.: Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)
Butler, L.M.: A unimodality result in the enumeration of subgroups of a finite abelian group. Proc. Am. Math. Soc. 101(4), 771–775 (1987)
du Sautoy, M.P.F.: Counting \(p\)-groups and nilpotent groups. Publ. Math. I.H.E.S 92, 63–112 (2000)
du Sautoy, M.P.F., Grunewald, F.J.: Analytic properties of zeta functions and subgroup growth. Ann. Math. (2) 152, 793–833 (2000)
du Sautoy, M.P.F., Woodward, L.: Zeta Functions of Groups and Rings. Lecture Notes in Mathematics, vol. 1925. Springer, Berlin (2008)
Evseev, A.: Reduced zeta functions of Lie algebras. J. Reine Angew. Math. 633, 197–211 (2009)
Grunewald, F.J., Segal, D., Smith, G.C.: Subgroups of finite index in nilpotent groups. Invent. Math. 93, 185–223 (1988)
Hall Jr., M.: A basis for free Lie rings and higher commutators in free groups. Proc. Am. Math. Soc. 1, 575–581 (1950)
Rossmann, T.: Computing topological zeta functions of groups, algebras, and modules, I. Proc. Lond. Math. Soc. (3) 110(5), 1099–1134 (2015)
Rossmann, T.: Stability results for local zeta functions of groups and related structures. Math. Proc. Camb. Philos. Soc. (2017). https://doi.org/10.1017/S0305004117000585
Rossmann, T.: Computing local zeta functions of groups, algebras, and modules. Trans. Am. Math. Soc. 370, 4841–4879 (2018). https://doi.org/10.1090/tran/7361
Schein, M.M., Voll, C.: Normal zeta functions of the Heisenberg groups over number rings I—the unramified case. J. Lond. Math. Soc. (2) 91(1), 19–46 (2015)
Stanley, R.P.: Hilbert functions of graded algebras. Adv. Math. 28, 57–83 (1978)
Voll, C.: Zeta functions of groups and enumeration in Bruhat–Tits buildings. Am. J. Math. 126, 1005–1032 (2004)
Voll, C.: Normal subgroup growth in free class-\(2\)-nilpotent groups. Math. Ann. 332, 67–79 (2005)
Voll, C.: Functional equations for zeta functions of groups and rings. Ann. Math. (2) 172(2), 1181–1218 (2010)
Voll, C.: Local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms. Int. Math. Res. Not. IMRN (2017). https://doi.org/10.1093/imrn/rnx186
Witt, E.: Treue Darstellung Liescher Ringe. J. Reine Angew. Math. 177, 152–160 (1937)
Acknowledgements
Our work was supported by DFG Sonderforschungsbereich 701 “Spectral Structures and Topological Methods in Mathematics” at Bielefeld University. Seungjai Lee was also supported by A23200000 fund from the National Institute for Mathematical Sciences (NIMS). We acknowledge numerous helpful conversations with Tobias Rossmann.
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Lee, S., Voll, C. Enumerating graded ideals in graded rings associated to free nilpotent Lie rings. Math. Z. 290, 1249–1276 (2018). https://doi.org/10.1007/s00209-018-2062-9
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DOI: https://doi.org/10.1007/s00209-018-2062-9