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SchurRings :: ClassFunction

ClassFunction -- The class of all Class functions

Description

A class function (or virtual character of a symmetric group Sn) is a function that is constant on the conjugacy classes of Sn. Class functions for Sn are in one to one correspondence with symmetric functions of degree n. The class functions corresponding to actual representations of Sn are called characters.

The character of the standard representation of S3 is

i1 : S = schurRing(QQ,s,3);
i2 : classFunction(s_{2,1})

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

o2 : ClassFunction

The character of the sign representation of S5 is

i3 : S = schurRing(QQ,s,5);
i4 : classFunction(s_{1,1,1,1,1})

o4 = 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

o4 : ClassFunction

We can go back and forth between class functions and symmetric functions.

i5 : R = symmetricRing(QQ,3);
i6 : cF = new ClassFunction from {{1,1,1} => 2, {3} => -1};
i7 : sF = symmetricFunction(cF,R)

     1 3   1
o7 = -p  - -p
     3 1   3 3

o7 : R
i8 : toS sF

o8 = s
      2,1

o8 : schurRing (QQ, s, 3)
i9 : classFunction sF

o9 = ClassFunction{{1, 1, 1} => 2}
                   {3} => -1

o9 : ClassFunction

We can add, subtract, multiply, scale class functions:

i10 : S = schurRing(QQ,s,4);
i11 : c1 = classFunction(S_{2,1,1}-S_{4});
i12 : c2 = classFunction(S_{3,1});
i13 : c1 + c2

o13 = ClassFunction{{1, 1, 1, 1} => 5}
                    {2, 1, 1} => -1
                    {2, 2} => -3
                    {3, 1} => -1
                    {4} => -1

o13 : ClassFunction
i14 : c1 * c2

o14 = ClassFunction{{1, 1, 1, 1} => 6}
                    {2, 1, 1} => -2
                    {2, 2} => 2

o14 : ClassFunction
i15 : 3*c1 - c2*2

o15 = ClassFunction{{2, 1, 1} => -8}
                    {2, 2} => -4
                    {3, 1} => -3
                    {4} => 2

o15 : ClassFunction

Methods that use a Class function :

For the programmer

The object ClassFunction is a type, with ancestor classes HashTable < Thing.