geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
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Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
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
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
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.
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
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.
For an application in the study of characteristic classes see
An application in a counting problem:
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Last revised on June 5, 2024 at 12:00:39. See the history of this page for a list of all contributions to it.