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Let $\Gamma$ be a cocompact lattice in $\mathbb{R}^n$, eg. $\Gamma = A \mathbb{Z}^n$ for some $A \in SL_n \mathbb{R}$. Then any $k$-dimensional subspace $P$ which is rational in $\Gamma$ has a volume: …

This particular question (``consistently orienting the reductions") and all those arising from your above link are interesting. However, the answer to this question seems `easy' -- and i'm wondering …

The collection of marked unimodular lattices in the euclidean $n$-space corresponds to the symmetric space $S_n:=SO_n(\mathbb{R}) \backslash SL_n(\mathbb{R})$. … The collection of (marked) nondegenerate lattices in $n$-space corresponds to the collection $P_n$ of positive definite $n\times n$ symmetric real matrices. …

Finally, i understand the answer! It is Hermann's Convexity Theorem -- and Harish-Chandra's canonical embedding, which proves that EVERY symmetric space (of noncompact type) is conformally equivalent …

The following problem arises in computations with symplectic lattices: Suppose $W$ is a rational lagrangian subspace in the symplectic lattice $\Lambda$, and suppose we have a basis $w_1, w_2, \ldots$ …

(%Edited after abx comment%) I seek explicit linear integral representations $\rho: Aut(X,\omega) \to Sp_{2g}(\mathbb{Z})$, when $(X,\omega)$ is complex $g$-dimensional PPAV. I prefer explicit matri …

After reading the comments, I think the underlying algebraic question is: " if $G=U$ is a unipotent linear algebraic group defined over $\mathbb{Q}$, then is the arithmetic unipotent subgroup $U(\math …