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

Description

These methods describe the result of applying plethystic operations to a virtual character of a symmetric group. These operations are described either via a symmetric function f, or a partition lambda. Since cF corresponds to an S_n- representation, the option GroupActing is irrelevant in this case.

i1 : cF = new ClassFunction from {{2} => 1, {1,1} => -1};
 
i2 : pl1 = plethysm({1,1},cF)

o2 = ClassFunction{{1, 1} => 1}
                   {2} => 1

o2 : ClassFunction
 
i3 : R = symmetricRing 5;
 
i4 : pl2 = plethysm(e_1+e_2,cF)

o4 = ClassFunction{{1, 1} => 0}
                   {2} => 2

o4 : ClassFunction
 
i5 : S = schurRing R;
 
i6 : symmetricFunction(cF,S)

o6 = -s
       1,1

o6 : S
 
i7 : symmetricFunction(pl1,S)

o7 = s
      2

o7 : S
 
i8 : symmetricFunction(pl2,S)

o8 = s  - s
      2    1,1

o8 : S

Applying the partition \{2\} to the sign character of S_2 gives Sym^2 of the sign, which is the trivial representation of S_4:

i9 : sgn = new ClassFunction from {{2} => -1, {1,1} => 1};
 
i10 : pl = plethysm({2},sgn)

o10 = ClassFunction{{1, 1} => 1}
                    {2} => 1

o10 : ClassFunction
 
i11 : symmetricFunction(pl, schurRing(QQ, s, 4, GroupActing => "Sn"))

o11 = s
       2

o11 : schurRing (QQ, s, 4)

Plethysm by a power-sum class function is Adams-type: p_k sends a representation to its k-th Adams operation (on class functions).

i12 : cF2 = new ClassFunction from {{3} => 2, {2,1} => 0, {1,1,1} => -1};
 
i13 : R2 = symmetricRing 4;
 
i14 : plethysm(p_2, cF2)

o14 = ClassFunction{{1, 1, 1} => -1}
                    {2, 1} => -1
                    {3} => 2

o14 : ClassFunction

Ways to use this method:

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:6375:0.