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My question is rather "simple" to ask : what is the dimension of the quotient variety $GL_3/U$, where $U$ is the (closed) group of upper triangular unipotent matrices (= upper triangular matrices with 1's on the diagonal).

Before working on that quotient variety, I've worked on $SL_2/U$ and I could determine the dimension thanks to the isomorphism $SL_2/U \simeq \mathbb{A}^2 \setminus \{(0,0)\}$ (I did use the fact that an open set of an affine variety has the same dimension, is that true?).

Thank you for your help !

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    Here $\overline{P}$ is just the natural "complement" of $U$, the subgroup of lower-triangular matrices. This statement goes by the name of "LU decomposition" in undergraduate linear algebra textbooks.2012-01-12

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