plethysm -- Plethystic operations on representations

Usage:

pl = plethysm(f,g)

pl = f @ g
Inputs:
- f, a ring element, element of Symmetric ring or Schur ring
- g, a ring element, element of Symmetric ring or Schur ring
Outputs:
- pl, a ring element, element of the ring of g

Description

Given a symmetric function 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

Since the power-sum basis behaves multiplicatively under plethysm, one has the identity plethysm(p_m, p_n) = p_{mn}, and more generally any complete/power-sum pair commutes under plethysm, as shown below:

i10 : R = symmetricRing(QQ,8);

i11 : toS plethysm(p_2,p_3) == toS p_6

o11 = true

i12 : toS plethysm(h_3,p_2) == toS plethysm(p_2,h_3)

o12 = true

The symmetric and antisymmetric squares of an irreducible representation decompose via Sym^2 = plethysm(\{2\},-) and \Lambda^2 = plethysm(\{1,1\},-), and their sum recovers the ordinary tensor square:

i13 : S = schurRing(QQ,s,4);

i14 : sym2 = plethysm({2},s_{2,1})

o14 = s    + s      + s        + s
       4,2    3,2,1    3,1,1,1    2,2,2

o14 : S

i15 : wedge2 = plethysm({1,1},s_{2,1})

o15 = s      + s    + s      + s
       4,1,1    3,3    3,2,1    2,2,1,1

o15 : S

i16 : s_{2,1}*s_{2,1} - sym2 - wedge2

o16 = 0

o16 : S

Plethysm makes sense for representations of symmetric groups as well. In an Sn-flavored ring, plethysm(lambda, chi) applies the Schur functor S_lambda to the Sn-representation chi:

i17 : Sn = schurRing(QQ,c,4,GroupActing => "Sn");

i18 : plethysm({2,1},c_{2,1,1})

o18 = c    + c    + c
       3,1    2,2    2,1,1

o18 : Sn

Plethysm also works over tensor products of Schur rings, mixing a GL factor and an Sn factor:

i19 : G = schurRing(QQ,g,3);

i20 : N = schurRing(G,n,3,GroupActing => "Sn");

i21 : plethysm({2},g_1*n_{2,1})

o21 = g n  + g n    + g   n
       2 3    2 2,1    1,1 1,1,1

o21 : N

Ways to use plethysm:

plethysm(RingElement,RingElement)
plethysm(BasicList,RingElement) -- Plethystic operations on representations
plethysm(BasicList,ClassFunction) -- see plethysm(RingElement,ClassFunction) -- Plethystic operations on class functions
plethysm(RingElement,ClassFunction) -- Plethystic operations on class functions

For the programmer

The object plethysm is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:6262:0.