Schur functors and categorified plethysm (Q6617075)

It is known that the Grothendieck ring of the category of Schur functors is the ring of symmetric functions, which has a rich structure being largely encapsulated in the fact that it is a plethory. This paper shows that similarly the category of Schur functors is a 2-plethory, which descends to give the plethory structure on symmetric functions.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] gives an overview of the classical theory of Schur functors, introducing the category \textsf{Poly} of polynomial species, which are a special kind of linear species in the sense of \textit{A. Joyal} [Adv. Math. 42, 1--82 (1981; Zbl 0491.05007)]. The authors then give the abstract definition of Schur functors as endomorphisms of the forgetful 2-functor\N\[\NU:\text{2-}\boldsymbol{Rig}\rightarrow\boldsymbol{Cat}\N\]\Nshowing that any polynomial species gives such a Schur functor.\N\N\item[\S 3] is dedicated to establishing the first main result (Theorem 9), which says that the category \textsf{Schur} of abstract Schur functors is equivalent to the category \textsf{Poly}. En route, the authors prove (Theorems 10 and 11) that \textsf{Poly} is the underlying category of the free 2-rig on one generator, which is called \(\overline{k\mathsf{S}}\), and that this 2-rig represents the 2-functor \(U:\text{2-}\boldsymbol{Rig} \rightarrow\boldsymbol{Cat}\).\N\N\item[\S 4] defines 2-birigs, a categorlfication of the notion of biring, showing (Theorem 22) that \textsf{Schurex} has a 2-birig structure from its equivalence with the free 2-rig on one generator.\N\N\item[\S 5] exposes an alternative perspective on birigs, showing that birigs and birings are examples of the more general notion of \(M\)-bialgebras with a monad \(M\) on \textsf{Set}. The category of \ admits a substitution (non-symmetric) monoidal structure, which allows of defining \(M\)\textit{-plethories} as monoids with respect to this monoidal structure [arXiv:1102.3549]. The authors then use this perspective to categorify the notion of rig-plethory, obtaining the concept of 2-plethory. It is shown (Theorem 39) that \textsf{Schur} is equipped with the structure of a 2-plethory.\N\N\item[\S 6] begins the decategorification process by studying the rig of isomorphisms classes of objects in \textsf{Schur}, which is denoted by \(\Lambda_{+}\) and whose elements are called positive symmetric functors. The authors equip \(\Lambda_{+}\) with a birig structure using the 2-birig structure on \textsf{Schur} (Theorem 45), and they equip \(\Lambda_{+}\) with a rig-plethory structure using the 2-plethory structure on \textsf{Schur} (Theorem 50).\N\N\item[\S 7] studies the group completion of \(\Lambda_{+}\), which is \(\Lambda\). The authors make \(\Lambda\) into a biring (Theorem 62), making it into a ring-plethory (Theorem 63).\N\end{itemize}