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Plethystic substitution

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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

The formal definition of plethystic substitution relies on the fact that the ring of symmetric functions Λ R ( x 1 , x 2 , … ) {\displaystyle \Lambda _{R}(x_{1},x_{2},\ldots )} is generated as an R-algebra by the power sum symmetric functions

p k = x 1 k + x 2 k + x 3 k + ⋯ . {\displaystyle p_{k}=x_{1}^{k}+x_{2}^{k}+x_{3}^{k}+\cdots .}

For any symmetric function f {\displaystyle f} and any formal sum of monomials A = a 1 + a 2 + ⋯ {\displaystyle A=a_{1}+a_{2}+\cdots } , the plethystic substitution f is the formal series obtained by making the substitutions

p k ⟶ a 1 k + a 2 k + a 3 k + ⋯ {\displaystyle p_{k}\longrightarrow a_{1}^{k}+a_{2}^{k}+a_{3}^{k}+\cdots }

in the decomposition of f {\displaystyle f} as a polynomial in the pk's.

Examples

If X {\displaystyle X} denotes the formal sum X = x 1 + x 2 + ⋯ {\displaystyle X=x_{1}+x_{2}+\cdots } , then f [ X ] = f ( x 1 , x 2 , … ) {\displaystyle f=f(x_{1},x_{2},\ldots )} .

One can write 1 / ( 1 − t ) {\displaystyle 1/(1-t)} to denote the formal sum 1 + t + t 2 + t 3 + ⋯ {\displaystyle 1+t+t^{2}+t^{3}+\cdots } , and so the plethystic substitution f [ 1 / ( 1 − t ) ] {\displaystyle f} is simply the result of setting x i = t i − 1 {\displaystyle x_{i}=t^{i-1}} for each i. That is,

f [ 1 1 − t ] = f ( 1 , t , t 2 , t 3 , … ) {\displaystyle f\left=f(1,t,t^{2},t^{3},\ldots )} .

Plethystic substitution can also be used to change the number of variables: if X = x 1 + x 2 + ⋯ , x n {\displaystyle X=x_{1}+x_{2}+\cdots ,x_{n}} , then f [ X ] = f ( x 1 , … , x n ) {\displaystyle f=f(x_{1},\ldots ,x_{n})} is the corresponding symmetric function in the ring Λ R ( x 1 , … , x n ) {\displaystyle \Lambda _{R}(x_{1},\ldots ,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 + ⋯ {\displaystyle X=x_{1}+x_{2}+\cdots } and Y = y 1 + y 2 + ⋯ {\displaystyle Y=y_{1}+y_{2}+\cdots } are formal sums.

  • If f {\displaystyle f} is a homogeneous symmetric function of degree d {\displaystyle d} , then
    f [ t X ] = t d f ( x 1 , x 2 , … ) {\displaystyle f=t^{d}f(x_{1},x_{2},\ldots )}
  • If f {\displaystyle f} is a homogeneous symmetric function of degree d {\displaystyle d} , then
    f [ − X ] = ( − 1 ) d ω f ( x 1 , x 2 , … ) {\displaystyle f=(-1)^{d}\omega f(x_{1},x_{2},\ldots )} ,
where ω {\displaystyle \omega } is the well-known involution on symmetric functions that sends a Schur function s λ {\displaystyle s_{\lambda }} to the conjugate Schur function s λ ∗ {\displaystyle s_{\lambda ^{\ast }}} .
  • The substitution S : f ↦ f [ − X ] {\displaystyle S:f\mapsto f} is the antipode for the Hopf algebra structure on the Ring of symmetric functions.
  • p n [ X + Y ] = p n [ X ] + p n [ Y ] {\displaystyle p_{n}=p_{n}+p_{n}}
  • The map Δ : f ↦ f [ X + Y ] {\displaystyle \Delta :f\mapsto f} is the coproduct for the Hopf algebra structure on the ring of symmetric functions.
  • h n [ X ( 1 − t ) ] {\displaystyle h_{n}\left} is the alternating Frobenius series for the exterior algebra of the defining representation of the symmetric group, where h n {\displaystyle h_{n}} denotes the complete homogeneous symmetric function of degree n {\displaystyle n} .
  • h n [ X / ( 1 − t ) ] {\displaystyle h_{n}\left} is the Frobenius series for the symmetric algebra of the defining representation of the symmetric group.

External links

References

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