Plethystic substitution

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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.

Definition[edit]

The formal definition of plethystic substitution relies on the fact that the ring of symmetric functions is generated as an R-algebra by the power sum symmetric functions

For any symmetric function and any formal sum of monomials , the plethystic substitution f[A] is the formal series obtained by making the substitutions

in the decomposition of as a polynomial in the pk's.

Examples[edit]

If denotes the formal sum , then .

One can write to denote the formal sum , and so the plethystic substitution is simply the result of setting for each i. That is,

.

Plethystic substitution can also be used to change the number of variables: if , then is the corresponding symmetric function in the ring of symmetric functions in n variables.

Several other common substitutions are listed below. In all of the following examples, and are formal sums.

  • If is a homogeneous symmetric function of degree , then

  • If is a homogeneous symmetric function of degree , then

, where is the well-known involution on symmetric functions that sends a Schur function to the conjugate Schur function .

  • The substitution is the antipode for the Hopf algebra structure on the Ring of symmetric functions.
  • The map is the coproduct for the Hopf algebra structure on the ring of symmetric functions.
  • is the alternating Frobenius series for the exterior algebra of the defining representation of the symmetric group, where denotes the complete homogeneous symmetric function of degree .
  • is the Frobenius series for the symmetric algebra of the defining representation of the symmetric group.

External links[edit]

References[edit]

  • M. Haiman, Combinatorics, Symmetric Functions, and Hilbert Schemes, Current Developments in Mathematics 2002, no. 1 (2002), pp. 39–111.