# plethysm -- Plethystic operations on representations

## Synopsis

• Usage:
pl = plethysm(f,g)
pl = f @ g
• Inputs:
• f, , element of Symmetric ring or Schur ring
• g, , element of Symmetric ring or Schur ring
• Outputs:
• pl, , element of the ring of g

## Description

Given a symmetric functions f and the character g of a virtual representation of a product of general linear and symmetric groups, the method computes the character of the plethystic composition of f and g. The result of this operation will be an element of the ring of g. We use the binary operator @ as a synonym for the plethysm function.

 i1 : R = symmetricRing(QQ,5); i2 : pl = plethysm(h_2,h_3) 6 4 2 2 3 3 2 2 o2 = e - 5e e + 7e e - 2e + 3e e - 6e e e + e - 2e e + 3e e + e e 1 1 2 1 2 2 1 3 1 2 3 3 1 4 2 4 1 5 o2 : R i3 : toS pl o3 = s + s 6 4,2 o3 : schurRing (QQ, s, 5) i4 : S = schurRing(QQ,q,3); i5 : h_2 @ q_{2,1} o5 = q + q + q 4,2 3,2,1 2,2,2 o5 : S i6 : plethysm(q_{2,1},q_{2,1}) o6 = q + q + 2q + q + q + 3q 6,2,1 5,4 5,3,1 5,2,2 4,4,1 4,3,2 o6 : S i7 : T = schurRing(S,t,2,GroupActing => "Sn"); i8 : plethysm(q_{1,1,1}-q_{2,1}+q_{3},q_{2,1}*t_2-t_{1,1}) o8 = (q - q - q + q + q - q + 2q )t - q t 6,3 6,2,1 5,4 5,2,2 4,4,1 4,3,2 3,3,3 2 () 1,1 o8 : T i9 : p_3 @ (q_{2,1}*t_2-t_{1,1}) o9 = (q - q - q + q + q - q + 2q )t - q t 6,3 6,2,1 5,4 5,2,2 4,4,1 4,3,2 3,3,3 2 () 1,1 o9 : T

## For the programmer

The object plethysm is .