E_7(Z)

E_7(Z)
Name	E_7(Z)
Type	Arithmetic subgroup
Field	Z (integers)
Related	E7, E_7(C), E_7(R), E_7(Q), E_7(F_p), Weyl group of E7

E_7(Z) is the integral form of the split exceptional Lie group of type E7, realized as an arithmetic subgroup of the algebraic group E7 defined over the integers. It sits among the exceptional groups studied alongside E8, E6, F4, and G2 and interacts with objects from Conway lattices, Weil representations, and Galois representations arising in arithmetic geometry. As an integral Chevalley group, it provides concrete incarnations used in connections with Atiyah–Bott, Langlands program, and string dualities investigated by Witten and Green.

Definition and basic properties

E_7(Z) is defined as the subgroup of the simply connected split algebraic group of type E7 fixed under integral structure, analogous to SL_n(Z) for type A. It inherits the root system described by Killing and Cartan and the Weyl group of type E7, which relates to Coxeter groups studied by Coxeter and Conway. The group is discrete in the real Lie group E_7(R) studied by Jacobson and has finite covolume analogous to arithmetic lattices in the sense of Borel and Prasad; it participates in reduction theory pioneered by Harish-Chandra and Borel with connections to Selberg-type spectral results.

Construction and integral form

One constructs E_7(Z) via Chevalley basis methods due to Chevalley and integral models developed by Springer and Steinberg. The integral Chevalley group is defined over Z using a root datum tied to the E7 Dynkin diagram classified by Dynkin and tables from Cartan. Alternate constructions employ lattice realizations related to the E7 lattice studied by Conway and Sloane or via Jordan algebras connected to work of Freudenthal and Tits. The group reduces modulo primes giving groups E_7(F_p) investigated by Taylor and Deligne in the context of finite groups of Lie type catalogued by Griess and Lusztig.

Arithmetic and group-theoretic structure

As an arithmetic group it satisfies properties established by Borel and Prasad: finite generation, congruence subgroup properties explored by Rapinchuk and Liebeck, and strong approximation related to Rapoport contexts. The group's parabolic subgroups correspond to subsystems connected to nodes in the E7 Dynkin diagram studied by Kac and Steinberg. Maximal subgroups have been classified in part by methods from Aschbacher-style analysis and the atlas of finite groups due to Conway and collaborators. Congruence kernels, profinite completions, and Serre’s conjectures involve comparisons to results of Serre and Prasad.

Representation theory and modules over Z

Integral representations of E_7(Z) stem from highest-weight theory of Cartan and Weyl, giving lattice modules such as the 56-dimensional fundamental module linked to Freudenthal constructions and the minimal representation studied by Vogan and Laumon. Modular representation theory over finite fields uses work of Lusztig and Jantzen, while integral lattices connect to Weil and Frenkel methods. Cohomological techniques by Borel and Lannes give integral cohomology computations for modules and variants relevant to Atiyah–Hirzebruch spectral sequences.

Applications in number theory and physics

E_7(Z) appears in automorphic constructions within the Langlands program and in conjectural correspondences explored by Langlands and Deligne. It features in explicit theta-lift phenomena tied to Weil representations and periods studied by Gelbart and Flicker. In theoretical physics, E7 integral structures arise in U-duality groups analyzed by Green, Witten, Hull, and in compactification schemes related to Calabi–Yau and K3 studied by Yau and Candelas. Black hole charge lattices and entropy formulas employ E7 arithmetic discussed by Strominger and Sen.

Cohomology, automorphic forms, and lattice connections

Cohomology of arithmetic groups for E7(Z) is treated through the frameworks of Borel, Serre, and Taussky-Todd, with automorphic forms on E7(R) linked to Eisenstein series studied by Harish-Chandra and residues analyzed by Arthur. Theta correspondences tie to Weil, while lattice connections leverage the E7 lattice and embeddings into the Leech lattice explored by Conway and Leech. Connections to sporadic groups enter via the moonshine program initiated by McKay and developed by Borcherds.

Computational aspects and known results for E7(Z)

Computations rely on algorithms from computational group theory by Frame, Sims, and implementations in systems like GAP and Magma used by researchers such as Conrad and Gordon. Known results include reduction mod p descriptions from Carter and character tables by GAP resources, generation by elementary unipotents following Chevalley and Steinberg, and partial computations of cohomology and automorphic spectra by Arthur and Illusie. Ongoing computational projects involve explicit congruence subgroup enumeration related to works by Bass and Serre and modularity investigations inspired by Wiles.

Category:Arithmetic groups