Plethystic substitution

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

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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

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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.
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References

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