nLab plethysm

Overview

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”.

Plethysm product and symmetric operads

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 monoidal category. The monoids in that category are the (symmetric) $C$-operads.

Related concepts

• plethory

History and references

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, doi

An application in a counting problem:

• Thomas Kahle, Mateusz Michalek, Plethysm and lattice point counting, Found. Comput. Math. 16 (2016) 1241–1261 arxiv/1408.5708 doi

Categorified:

• John Baez, Joe Moeller, Todd Trimble, Schur functors and categorified plethysm, arXiv:2106.00190