Invariants and orbits of $n$-tensors - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T06:07:46Z http://mathoverflow.net/feeds/question/67897 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67897/invariants-and-orbits-of-n-tensors Invariants and orbits of $n$-tensors Bugs Bunny 2011-06-15T21:22:39Z 2011-06-16T03:46:46Z <p>My question may be absolutely elementary and is probably answered in 19th century. A reference or a short clear argument would be highly appreciated.</p> <p>Let $V_1, \ldots V_n$ be finite dimensional vector spaces over the same field (may assume complex numbers). What are $GL(V_1)\times \ldots \times GL(V_n)$-orbits on $V_1 \otimes \ldots \otimes V_n$?</p> <p>The only invariant of an orbit I can see is "a multirank" $(k_1, \ldots k_n)$ where $k_i$ is the dimension of support of an element in $V_i$. The multirank satisfies inequalities $k_i \leq \prod_{j\neq i} k_j$. Would it be too naive to suggest that orbits are in 1-1 correspondence with legal multiranks?</p> http://mathoverflow.net/questions/67897/invariants-and-orbits-of-n-tensors/67904#67904 Answer by Gjergji Zaimi for Invariants and orbits of $n$-tensors Gjergji Zaimi 2011-06-16T00:57:48Z 2011-06-16T00:57:48Z <p>Here is a start, suppose that $V_i$ is $\mathbb C^{k_i}$ (and restricting to $k_1,k_2,\dots,k_n, n\geq 2$). The tuples $(k_1,k_2,\dots,k_n)$ for which the action of $GL_{k_1}\times\cdots\times GL_{k_n}$ on $\mathbb{C}^{k_1}\otimes \cdots\otimes \mathbb{C}^{k_n}$ has only finitely many orbits are $(k,l),(2,2,k),(2,3,k)$, for positive integers $k,l$. This was proven in </p> <blockquote> <p>V. G. Kac, "Some remarks on nilpotent orbits", J. Algebra 64 (1980), 190–213.</p> </blockquote> <p>These orbits are classified in "Orbits and their closures in the spaces $\mathbb{C}^{k_1}\otimes \cdots\otimes \mathbb{C}^{k_r}$" by P.G. Parfenov (<a href="http://www.ams.org/mathscinet-getitem?mr=1830474" rel="nofollow">MR</a>). This paper doesn't seem to be online, but I believe you can find a summary in section 5 <a href="http://arxiv.org/abs/0909.4262" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/67897/invariants-and-orbits-of-n-tensors/67906#67906 Answer by Tom Goodwillie for Invariants and orbits of $n$-tensors Tom Goodwillie 2011-06-16T01:21:38Z 2011-06-16T03:46:46Z <p>I don't think that's right, Bugs. Say that $V_1$ and $V_2$ are $d$-dimensional and $V_3=\mathbb C^2$. Then an element of $V_1\otimes V_2$ is a linear map $V_1^\star\to V_2$, generically an isomorphism; an element of $V_1\otimes V_2\otimes V_3$ is an ordered pair $(A,B)$ of these; the unordered $d$-tuple of eigenvalues of $B\circ A^{-1}$ is an invariant of the $GL(V_1)\times GL(V_2)$-action; and an element of $GL(V_3)$ will just perform some fractional linear transformation on all of these numbers, so that if $d\ge 4$ then there is a complex invariant here.</p>