11
$\begingroup$

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.

Improve this question
$\endgroup$
5
  • $\begingroup$ A good answer to this question might also make a good answer to mathoverflow.net/questions/42653/… . $\endgroup$
    – Qiaochu Yuan
    Apr 13, 2011 at 18:20
  • $\begingroup$ 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. $\endgroup$
    – ARupinski
    Apr 13, 2011 at 19:18
  • 1
    $\begingroup$ 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 birdtracks.eu . 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). $\endgroup$
    – Theo Johnson-Freyd
    Apr 14, 2011 at 1:20
  • $\begingroup$ @Theo: that book looks promising for at least a partial answer. Will definitely have a look through it. $\endgroup$
    – ARupinski
    Apr 14, 2011 at 2:17
  • 2
    $\begingroup$ If $V$ is self-dual, then I think the $S^{(2,1)}$ case corresponds to having an invariant Lie bracket. $\endgroup$
    – John Wiltshire-Gordon
    May 22, 2012 at 1:49

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.