Wedge product matrices and orbits of principal congruence subgroups

Following Bump and Hoffstein in [2], the orbits in Γ∞(3)﹨Γ(3) are in bijection with sets of invariants satisfying certain relations. We explain how wedge product matrices give an alternative definition of the invariants of matrix orbits. This new method provides the possibility of performing similar...

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Bibliographic Details
Published in:Linear algebra and its applications 2024-09, Vol.696, p.1-28
Main Author: Chan, Yao Ming
Format: Article
Language:English
Subjects:
Bruhat decomposition
Principal congruence subgroup
Wedge product matrices
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Summary:Following Bump and Hoffstein in [2], the orbits in Γ∞(3)﹨Γ(3) are in bijection with sets of invariants satisfying certain relations. We explain how wedge product matrices give an alternative definition of the invariants of matrix orbits. This new method provides the possibility of performing similar computations with other congruence subgroups and arbitrary n×n matrices. Using Steinberg's refined version of the Bruhat decomposition, we construct an explicit choice of coset representative for each orbit in the orbit space Γ∞(3)﹨Γ(3) of 3×3 matrices over the PID of Eisenstein integers.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2024.05.008