Plethystic substitution is a shorthand notation for a common kind of substitution in the algebra of symmetric functions and that of symmetric polynomials. It is essentially basic substitution of variables, but allows for a change in the number of variables used.
The formal definition of plethystic substitution relies on the fact that the ring of symmetric functions Λ R ( x 1 , x 2 , … ) is generated as an R-algebra by the power sum symmetric functions
p k = x 1 k + x 2 k + x 3 k + ⋯ .
For any symmetric function f and any formal sum of monomials A = a 1 + a 2 + ⋯ , the plethystic substitution f[A] is the formal series obtained by making the substitutions
p k ⟶ a 1 k + a 2 k + a 3 k + ⋯
in the decomposition of f as a polynomial in the pk's.
If X denotes the formal sum X = x 1 + x 2 + ⋯ , then f [ X ] = f ( x 1 , x 2 , … ) .
One can write 1 / ( 1 − t ) to denote the formal sum 1 + t + t 2 + t 3 + ⋯ , and so the plethystic substitution f [ 1 / ( 1 − t ) ] is simply the result of setting x i = t i − 1 for each i. That is,
f [ 1 1 − t ] = f ( 1 , t , t 2 , t 3 , … ) .
Plethystic substitution can also be used to change the number of variables: if X = x 1 + x 2 + ⋯ , x n , then f [ X ] = f ( x 1 , … , x n ) is the corresponding symmetric function in the ring Λ R ( x 1 , … , x n ) of symmetric functions in n variables.
Several other common substitutions are listed below. In all of the following examples, X = x 1 + x 2 + ⋯ and Y = y 1 + y 2 + ⋯ are formal sums.
If f is a homogeneous symmetric function of degree d , then f [ t X ] = t d f ( x 1 , x 2 , … )
If f is a homogeneous symmetric function of degree d , then f [ − X ] = ( − 1 ) d ω f ( x 1 , x 2 , … ) , where ω is the well-known involution on symmetric functions that sends a Schur function s λ to the conjugate Schur function s λ ∗ .
The substitution S : f ↦ f [ − X ] is the antipode for the Hopf algebra structure on the Ring of symmetric functions. p n [ X + Y ] = p n [ X ] + p n [ Y ] The map Δ : f ↦ f [ X + Y ] is the coproduct for the Hopf algebra structure on the ring of symmetric functions. h n [ X ( 1 − t ) ] is the alternating Frobenius series for the exterior algebra of the defining representation of the symmetric group, where h n denotes the complete homogeneous symmetric function of degree n . h n [ X / ( 1 − t ) ] is the Frobenius series for the symmetric algebra of the defining representation of the symmetric group.