why the group $GL(6,V)$ has an open orbit? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T22:57:26Z http://mathoverflow.net/feeds/question/118801 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118801/why-the-group-gl6-v-has-an-open-orbit why the group $GL(6,V)$ has an open orbit? Hassan Jolany 2013-01-13T12:01:24Z 2013-04-13T17:17:02Z <p>N.Hitchen in his paper about geometry of three forms wrote that "for a Real vector space $V$ of dimension six, the group $GL(6,V)$ has an open orbit and he referenced it to a thesis which was written in 1907 , how can we prove this fact, is there any open access reference? </p> http://mathoverflow.net/questions/118801/why-the-group-gl6-v-has-an-open-orbit/118824#118824 Answer by Robert Bryant for why the group $GL(6,V)$ has an open orbit? Robert Bryant 2013-01-13T16:26:31Z 2013-01-14T12:59:40Z <p>In any case, the proof is very simple. Consider the $3$-form $$\phi_0 = dx^1\wedge dx^2\wedge dx^3 + dx^4\wedge dx^5\wedge dx^6.$$ I claim that the subgroup $G\subset\mathrm{GL}(6,\mathbb{R})$ that stabilizes $\phi_0$ consists of the obvious subgroup $G_0=\mathrm{SL}(3,\mathbb{R})\times\mathrm{SL}(3,\mathbb{R})$ that fixes the two summands plus the discrete part that switches the two summands. Since this subgroup has dimension $8{+}8=16$ while $\mathrm{GL}(6,\mathbb{R})$ has dimension $36$, it follows that the $\mathrm{GL}(6,\mathbb{R})$-orbit of $\phi_0$ has dimension $20= 36-16$, which is the dimension of $\Lambda^3(\mathbb{R}^6)$. Thus, this orbit is open.</p> <p>To see the claim, just look at the vectors $v\in \mathbb{R}^6$ that satisfy $\bigl(\iota_v(\phi_0)\bigr)^2=0$. Clearly, such a vector must lie in either the 'first' $\mathbb{R}^3$ or the 'second' $\mathbb{R}^3$. Thus, $G$ must carry each of these subspaces into either itself or the other subspace. The subgroup of $G$ that fixes each subspace must be of index $2$ in $G$, and is clearly $G_0$.</p> <p>What is not so obvious is that there exist exactly $2$ open orbits. One is the one listed above, and the other is $$\phi_1 = \mathrm{Re}\bigl((dx^1+i\ dx^4)\wedge(dx^2+i\ dx^5)\wedge(dx^3+i\ dx^6)\bigr),$$ whose stabilizer has an index two subgroup that is isomorphic to $\mathrm{SL}(3,\mathbb{C})$.</p> <p><em>Remark:</em> By the way, the existence of the open orbit shouldn't be surprising. It's a general fact that $\mathrm{GL}(n,\mathbb{R})$ has an open orbit in $\Lambda^k(\mathbb{R}^n)$ for $k>0$ whenever $n^2$, the dimension of $\mathrm{GL}(n,\mathbb{R})$, is as large as $n\choose k$, the dimension of $\Lambda^k(\mathbb{R}^n)$. This happens for all $n$ when $k\in\lbrace1,2,n{-}2,n{-}1,n\rbrace$ and otherwise only for $$(n,k)\in\lbrace (6,3), (7,3), (7,4), (8,3), (8,5)\rbrace.$$</p>