Are my PCA groups significantly different? I am studying the feeding behaviour of deep sea fishes, and have produced a dataset containing percentages for 20 different fatty acids (totalling 100%) for 32 individual fish. I have performed PCA (see below) and the data seems to group into 3 distinct groups, which correlates to the species of fish. Now I want to test the hypothosis that there is a significant difference between the 3 groups. I tried performing a MANOVA, but discovered in the process that my data is not normally distributed:
Shapiro-Wilk normality test
data:  norm_test
W = 0.62718, p-value < 2.2e-16
So, now I hope some of you can give your recommendations as to which test/tests to perform to determine whether the groups are significantly different or not. I would also appreciate suggestions to how to present my results, and additional work I can pursue to maximize the information contained in the data.
Here is an example of what my dataset looks like:


Thank you for your time!
/Kristian 
 A: That's a pretty obvious difference between the 3 groups, isn't it? Your figure seems convincing enough for me. I can't say that attaching a p-value to this would change my opinion either way. Basically, I don't think this is a place where p-values (or null hypothesis statistical testing more generally) are very useful. 
There are still plenty of ways to learn from your data without this framework. I can't really offer the best suggestions since I don't know too much about fish. But here are some things to consider, just based on your plot:
You could proceed by investigating exactly what it means that blue is separated strongly from the other two on PC1, while red & green are differentiated from each other on PC2. And also consider whether the apparently higher intraspecific variance in PC2 for red corroborates your expectations or can be explained. Looking at the loadings on the PCs would help you understand what these patterns mean biologically, as well as whether they are obvious or merit deeper investigation.
