# Plethysm

Plethysm was introduced has an operation on symmetric polynomials by D.E. Littlewood in his paper

Polynomial Concomitants and Invariant Matrices,
J. London Math. Soc. (1936) s1-11(1): 49-55.
doi: 10.1112/jlms/s1-11.1.49,

with the notation $\lambda\otimes \mu$ for what we now write

$\large s_\mu\circ s_\lambda \qquad {\rm or}\qquad s_\mu[s_\lambda]$.

Here, $s_\mu=s_\mu(x_1,x_2,x_3,\ldots)$ stands for the Schur symmetric polynomials. Plethysm of symmetric polynomials is entirely characterized by the fact that it satisfies the following properties (with $p_k=p_k(x_1,x_2,x_3,\ldots)$ standing for the power sum symmetric polynomials):

• $(f+f’)\circ g = (f\circ g)+(f’\circ g)$,
• $(f\cdot f’)\circ g = (f\circ g)\cdot (f’\circ g)$,
• $p_k\circ (g +g’)= (p_k\circ g) + (p_k\circ g’)$,
• $p_k\circ (g \cdot g’)= (p_k\circ g) \cdot (p_k\circ g’)$,
• $p_k \circ p_j = p_{kj}$,

since any polynomial may be expanded as a linear combination of products of power sum polynomials. If one interprets the symmetric polynomials involved as characters of $GL(V)$-modules constructed through polynomial functors going from the category of finite dimensional vector spaces to itself, then plethysm corresponds to functor compositions. Plethysm may also be naturally understood from the point of view of $\lambda$-ring theory.

An important case for Representation Theory, Algebraic Geometry, and Geometric Complexity Theory,  is the plethysm $s_\mu\circ s_\lambda$ of Schur polynomials. For instance, using the above rule, and well-known change of basis formulas for symmetric polynomials, one may calculate that

$\large s_4\circ s_2= s_8+s_{62}+s_{44}+s_{422}+s_{2222},$

which corresponds exactly to the Schur functor decomposition:

$\large S^4\circ S^2=S^8+S^{62}+S^{44}+S^{422}+S^{2222}.$

Recall that, for $V$ a vector space admitting $\{x_1,x_2,\ldots,x_d\}$ as a basis, then the vector space $S^d(V)$ affords as basis the set of degree $d$ “monomials” in the “variables” $x_i$.