Let $k$ be an algebraically closed field of positive characteristics. Let $X$ be the wonderful compactification of $PGL_3$ (see for example section 6 of "Frobenius Splitting Methods in Geometry and Representation Theory" by Brion and Kumar for the definition). Is there any explicit description of $X$ similar to the one for $PGL_2$ when we get $\mathbb{P}^3$? Can an irreducible representation that is taken in the definition of $X$ be described explicitly?
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3$\begingroup$ It is the same as the much older notion of the “space of complete collineations” in Type $A_n$, i.e., for $\textbf{PGL}_{n+1}$. So for $n$ equal to $2$, it is the blowing up of $\mathbb{P}^8$ along the image of the Segre embedding of $\mathbb{P}^2\times \mathbb{P}^2$. $\endgroup$– Jason StarrCommented Jun 17, 2022 at 10:20
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2$\begingroup$ You can take then irreducible representation to be the one whose highest weight is the sum of the fundamental weights, I.e., the highest weight irreducible representation that occurs in the decomposition of the tensor product of all (nonzero) exterior powers of the standard representation (the two one-dimensional representations do not change anything, so you are free to exclude them). $\endgroup$– Jason StarrCommented Jun 17, 2022 at 10:26
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