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classFunction -- Converts symmetric function to class function

Description

Given a symmetric function f, homogeneous of degree N, the method computes the corresponding virtual character of the symmetric group S_N.

The character of the standard representation of S_5 is

i1 : R = symmetricRing(QQ,5);
 
i2 : classFunction(jacobiTrudi({4,1},R))

o2 = ClassFunction{{1, 1, 1, 1, 1} => 4}
                   {2, 1, 1, 1} => 2
                   {3, 1, 1} => 1
                   {3, 2} => -1
                   {5} => -1

o2 : ClassFunction

The character of the second exterior power of the standard representation of S_5 is

i3 : R = symmetricRing(QQ,5);
 
i4 : classFunction(jacobiTrudi({3,1,1},R))

o4 = ClassFunction{{1, 1, 1, 1, 1} => 6}
                   {2, 2, 1} => -2
                   {5} => 1

o4 : ClassFunction

The sign representation of S_n corresponds to the Schur polynomial of shape (1^n). Its class function takes the value sgn(\sigma) on a permutation of cycle type \rho

i5 : Ssign = schurRing(QQ,s,5);
 
i6 : classFunction(s_{1,1,1,1,1})

o6 = ClassFunction{{1, 1, 1, 1, 1} => 1}
                   {2, 1, 1, 1} => -1
                   {2, 2, 1} => 1
                   {3, 1, 1} => 1
                   {3, 2} => -1
                   {4, 1} => -1
                   {5} => 1

o6 : ClassFunction

Small-n character tables are assembled row-by-row from classFunction. For S_3 there are three conjugacy classes (1^3), (2,1), (3) and three irreducibles, and the values are collected below:

i7 : R3 = symmetricRing(QQ,3);
 
i8 : for lam in {{3},{2,1},{1,1,1}} list classFunction(jacobiTrudi(lam,R3))

o8 = {ClassFunction{{1, 1, 1} => 1}, ClassFunction{{1, 1, 1} => 2},
                    {2, 1} => 1                    {3} => -1       
                    {3} => 1                                       
     ------------------------------------------------------------------------
     ClassFunction{{1, 1, 1} => 1}}
                   {2, 1} => -1
                   {3} => 1

o8 : List

Tensor products of S_n-representations correspond to internalProduct of class functions. For S_4 the tensor square of the standard representation decomposes into irreducibles by pairing with each irreducible character via scalarProduct:

i9 : R4 = symmetricRing(QQ,4);
 
i10 : std = classFunction(jacobiTrudi({3,1},R4));
 
i11 : sq = internalProduct(std, std);
 
i12 : for lam in {{4},{3,1},{2,2},{2,1,1},{1,1,1,1}} list
        scalarProduct(sq, classFunction(jacobiTrudi(lam,R4)))

o12 = {1, 1, 1, 1, 0}

o12 : List

See also

Ways to use classFunction:

For the programmer

The object classFunction is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:7221:0.