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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:

Question: 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 $\rho$?

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.

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.

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.

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A good answer to this question might also make a good answer to… . – Qiaochu Yuan Apr 13 '11 at 18:20
I remember seeing that question awhile back, and admitting things like triality would certainly be along the lines of what I am wondering about. Since in the $k=2$ case self-dual representations split further into two types, I am interested not only in possible generalizations of duality/triality (perhaps beyond $n$-ality) but also how representations further split into different types within these actions and ultimately what these types might correspond to algebraically. – ARupinski Apr 13 '11 at 19:18
If you are in the compact-connected-simply-connected case, which is to say the semisimple-Lie-algebra-over-C case, then you will be interested in Cvitanović's book . He uses the Schur functors and multilinear forms to classify all finite-dimensional simple Lie algebras over C, and to work out part of their representation theories (the part that's most directly controlled by the defining representation). – Theo Johnson-Freyd Apr 14 '11 at 1:20
@Theo: that book looks promising for at least a partial answer. Will definitely have a look through it. – ARupinski Apr 14 '11 at 2:17
If $V$ is self-dual, then I think the $S^{(2,1)}$ case corresponds to having an invariant Lie bracket. – John Wiltshire-Gordon May 22 '12 at 1:49

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