plethysm(BasicList,RingElement) -- Plethystic operations on representations

Function: plethysm
Usage:

pl = plethysm(lambda,g)
Inputs:
- lambda, a basic list, nonincreasing sequence of positive integers, or partition
- g, a ring element, element of Symmetric ring or Schur ring
Outputs:
- pl, a ring element, element of the ring of g

Description

The method computes the character of the representation obtained by applying the Schur functor S_{\lambda} to the representation with character g, where \lambda is a partition.

i1 : R = symmetricRing(QQ,3);

i2 : S = schurRing(QQ,q,3);

i3 : toE plethysm({2,1},e_1*e_2-e_3)

        4    4         2 2      3      3 2         2     3
o3 = e e  + e e e  - 4e e e  - e e  - e e  + 7e e e  - 3e
      1 2    1 2 3     1 2 3    2 3    1 3     1 2 3     3

o3 : R

i4 : plethysm({2,1,1},q_{1,1})

o4 = q
      3,3,2

o4 : S

i5 : T = schurRing(S,t,4,GroupActing => "Sn");

i6 : plethysm({1,1},q_1*t_{3,1})

o6 = q   t  + q   t    + q   t    + q t
      1,1 4    1,1 3,1    1,1 2,2    2 2,1,1

o6 : T

Ways to use this method:

plethysm(BasicList,RingElement) -- Plethystic operations on representations

The source of this document is in SchurRings.m2:2251:0.