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

Even simple plethysms of Schur functions are not obvious a priori. For example, Sym^2 of the antisymmetric square $\Lambda^2 V = S_{1,1}V$ breaks up as:

i7 : S = schurRing(QQ,s,4);

i8 : plethysm({2},s_{1,1})

o8 = s    + s
      2,2    1,1,1,1

o8 : S

Applying a partition directly lets one extract isotypic summands, e.g.\ the two pieces of V^{\otimes 2} for an Sn-representation:

i9 : Sn = schurRing(QQ,c,4,GroupActing => "Sn");

i10 : sym2chi = plethysm({2}, c_{3,1})

o10 = c  + c    + c
       4    3,1    2,2

o10 : Sn

i11 : wed2chi = plethysm({1,1}, c_{3,1})

o11 = c
       2,1,1

o11 : Sn

i12 : c_{3,1}*c_{3,1} - sym2chi - wed2chi

o12 = 0

o12 : Sn

For representations of products of groups, the plethysm is applied diagonally; here on a GL x GL tensor product:

i13 : A = schurRing(QQ,a,3);

i14 : B = schurRing(A,b,2);

i15 : plethysm({2},a_1*b_1)

o15 = a b  + a   b
       2 2    1,1 1,1

o15 : B

Ways to use this method:

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

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:6325:0.