# 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.

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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

In the theory of algebraic operads, the language of "tensor species" is often used, see Chapter 5 of "Algebraic Operads, Jean-Louis Loday & Bruno Vallette, Grundlehren der mathematischen Wissenschaften, Volume 346, Springer-Verlag (2012).

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).

There are in fact many definitions in the theory of operads, which are a bit cumbersome to write down without talking about "tensor species".

And of course things like generating series for operads are of interest and operadic phenomena/constructions like Koszul duality give constraints/relations for their generating series.

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The theory of tensor species is equivalent to the theory of polynomial functors; so to this extent there is no call for a theory of tensor species as the theory of polynomial functors is well-developed. However this is, I suspect, missing the point of your question. My understanding is that the focus in combinatorial species is on species which satisfy polynomial equations of which there are many interesting examples. It then follows that the cycle index series will satisfy the same polynomial equation. This makes it natural to ask a more specific question of whether there are interesting tensor species/polynomial functors which satisfy polynomial equations (other than those arising from combinatorial species)? I would be interested to hear of any examples.

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Yes, your more specific question is precisely what I'm after. However, the equations involved need not be polynomial, plethysm (= composition) may also be involved. A prime example from combinatorial species is that set partitions are sets of non-empty sets. –  Martin Rubey May 4 '12 at 16:16