plethysm

The composite of Schur functors is again a Schur functor. The process of composing Schur functors is known as **plethysm**, especially in situations where we want an explicit “formula” for how the Schur functors compose.

For example, the $n^{th}$ exterior power functor $\Lambda^n$ is a Schur functor, as is the $n^{th}$ symmetric power functor $S^n$. So, we might want to describe the composite

$V \mapsto \Lambda^2 S^2(V)$

as a direct sum of Schur functors coming from Young diagrams. This is the sort of problem people study when they talk about “plethysm”.

Let $S$ be the skeletal category of finite sets and bijections and $C$ a symmetric additive monoidal category with monoidal product $\otimes$ and unit object $\mathbf{1}$. Objects in the category of contravariant functors $C^{S^{op}}$ can be descibed as collections $M = \{M(n), n\geq 0\}$ of objects $M(n)$ in $C$ with action of a symmetric group $\Sigma_n$ on $n$ letters. The category $C^{S^{op}}$ acts on $C$ by the polynomial functors

$M : V \mapsto \oplus_{n\geq 0} M(n)\otimes_{\Sigma_n} V^{\otimes n}$

The composition of such functors defines a monoidal product on $C^{S^{op}}$ called the **plethysm product**. This way we obtain a symmetric monoidal category. The monoids in that category are the (symmetric) $C$-operads.

In Richard P. Stanley’s book *Enumerative Combinatorics*, he discusses the origin of the term ‘plethysm’ in Volume 2, Appendix 2. He says that the term was introduced in

- D. E. Littlewood,
*Invariant theory, tensors and group characters*, Philos. Trans. Roy. Soc. London. Ser. A.**239**, (1944), 305–365.

The term ‘plethysm’ was suggested to Littlewood by M. L. Clark after the Greek word *plethysmos*, or πληθυσμός, which means ‘multiplication’ in modern Greek (though apparently the meaning goes back to ancient Greek). The related term *plethys* in Greek means ‘a big number’ or ‘a throng’, and this in turn comes from the Greek verb *plethein*, which means ‘to be full’, ‘to increase’, ‘to fill’, etc.

Some aspects of plethysm appear (partly through exercises) in the textbook W. Fulton, J. Harris, *Representation theory*.

- A. M. Garsia, G. Tesler,
*Plethystic formulas for Macdonald $q, t$-Kostka coefficients*, Advances in Math.**123**(1996) 144–222, MR1420484; A. M. Garsia, J. Remmel,*Plethystic formulas and positivity for $q,t$-Kostka coefficients*, Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996), 245–262, Progr. Math.**161**, Birkhäuser 1998, MR99j:05189d

For an application in the study of characteristic classes see

- Dragutin Svrtan,
*New plethysm operation, Chern characters of exterior and symmetric powers with applications to Stiefel-Whitney classes of Grassmannians*, Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991). Theoret. Comput. Sci.**117**(1993), no. 1-2, 289–301, http://dx.doi.org/10.1016/0304-3975(93)90320-S

An application in a counting problem:

- Thomas Kahle, Mateusz Michalek,
*Plethysm and lattice point counting*arxiv/1408.5708

Revised on August 26, 2014 07:22:45
by Zoran Škoda
(178.17.116.68)