What do Multilinear Forms tell us about Representations? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T01:55:25Z http://mathoverflow.net/feeds/question/61568 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61568/what-do-multilinear-forms-tell-us-about-representations What do Multilinear Forms tell us about Representations? ARupinski 2011-04-13T17:11:00Z 2011-04-13T17:11:00Z <p>The last few days I have been calculating whether certain group representations are real, complex, or quaternionic. It is well-known that the type of the representation corresponds to what type of bilinear form the representation fixes (symmetric, antisymmetric, or neither). The existence of a symmetric bilinear form fixed by $\rho$ is equivalent to the existence of a copy of the trivial representation in $S^{(2)}\rho$ and similarly the existence of a antisymmetric bilinear form fixed by $\rho$ is equivalent to the existence of a copy of the trivial representation in $S^{(1,1)}\rho$ where $S^{\lambda}$ is the Schur functor of type $\lambda$. This got me to wondering:</p> <blockquote> <blockquote> <p><b>Question:</b> Given an irreducible representation $(\rho,V)$ of some group $G$, let $\phi:V^k\rightarrow\mathbb{C}$ be a $k$-linear form fixed by $\rho$ for some $k>2$. If $\phi$ is of type $\lambda$ (i.e. a copy of the trivial representation appears in $S^{(\lambda)}\rho$ with $\lambda$ a partition of $k$), what, if anything, does this tell us about $\rho$?</p> </blockquote> </blockquote> <p>To clarify a bit, in the $k=2$ case, symmetry or antisymmetry of the form corresponds to $\rho$ being realizable as a real or quaternionic matrix respectively; any sort of generalization of this realizability or other interesting things that can be said about the representation based on the type of the form is what I am curious about.</p> <p>One observation: the trivial representation appears in the decomposition of $V^{\otimes k}$ for some large enough $k$, so for any $\rho$ there is always a partition $\lambda$ of size $k$ such that $\rho$ fixes a $k$-linear form of type $\lambda$. Hence existence of some $k$-linear form as considered in the question is always guaranteed.</p> <p>Motivation is mainly curiosity, and if there is a more standard terminology than "$k$-linear form of type $\lambda$" for the forms I am considering, please let me know for future reference.</p>