# Has there been any application of tensor species?

Joyal's combinatorial species, endofunctors in the category of finite sets with bijections $\mathbf B$ have found numerous applications. One generalisation is given by so-called "tensor species" (also "tensorial species", or, "linear species" - not to be confused with the species on totally ordered sets in the book by Bergeron, Labelle and Leroux) which are defined as functors from $\mathbf B$ into the category of finite dimensional vector spaces (say, over the complex numbers) with linear transformations $\mathbf{Vect}$.

I wonder whether there have been any "practical" applications of tensor species? I know of a very short list of articles dealing with them (eg. by Méndez) but hardly any spelled out examples. I wonder whether I overlooked something.

Note that for any combinatorial species $F$ we cann regard $F[\{1,2,\dots,n\}]$ as a finite set with an action of the symmetric group. Similarly, if $F$ is a tensor species, we can regard $F[{1,2,\dots,n}]$ as a linear representation of the symmetric group. Thus, I am mostly interested in examples that use the combinatorial operations for greater clarity of a construction.

• I suppose that if you are interested in representations of symmetric groups, then this forms an ample supply of such. As for actual applications, I can't say. – Spice the Bird May 4 '12 at 9:49
• Yes, I am aware of this. In fact, I like the spirit of the sample application that Méndez gives (he uses the product formula...), but I am hoping for something more substantial. – Martin Rubey May 4 '12 at 10:46

For example one can define an operad very concise as a monoid in species under a certain monoidal structure. Without this language, it takes quite a while to write down all the compatibilities with the various $S_n$ actions (Although I find it illuminating to write down an "elementary" definition never the less).