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classFunction(BasicList) -- Character of irreducible representation of symmetric group

Description

Given a partition l of N, the method computes the character of the irreducible S_N-representation corresponding to the partition l.

i1 : R = symmetricRing(QQ,7);
 
i2 : cF = classFunction({3,2,1})

o2 = ClassFunction{{1, 1, 1, 1, 1, 1} => 16}
                   {3, 1, 1, 1} => -2
                   {3, 3} => -2
                   {5, 1} => 1

o2 : ClassFunction
 
i3 : toS(symmetricFunction(cF,R))

o3 = s
      3,2,1

o3 : schurRing (QQ, s, 7)

Enumerating the irreducible characters of S_4 and pairing them with scalarProduct recovers the orthonormality relations from representation theory -- the diagonal entries are 1 and the off-diagonal entries are 0:

i4 : chars = for lam in partitions 4 list classFunction toList lam;
 
i5 : matrix for a in chars list for b in chars list scalarProduct(a,b)

o5 = | 1 0 0 0 0 |
     | 0 1 0 0 0 |
     | 0 0 1 0 0 |
     | 0 0 0 1 0 |
     | 0 0 0 0 1 |

              5       5
o5 : Matrix QQ  <-- QQ

Irreducible characters of different shapes are orthogonal. Here we verify orthogonality for three partitions of 6:

i6 : c1 = classFunction({3,2,1});
 
i7 : c2 = classFunction({4,1,1});
 
i8 : c3 = classFunction({2,2,2});
 
i9 : scalarProduct(c1,c2)

o9 = 0

o9 : QQ
 
i10 : scalarProduct(c2,c3)

o10 = 0

o10 : QQ

Self-pairings return 1 for each irreducible, independent of the shape, and the trivial (row shape) and sign (column shape) characters are the 1-dimensional irreducibles:

i11 : scalarProduct(c1,c1)

o11 = 1

o11 : QQ
 
i12 : classFunction({5})

o12 = 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

o12 : ClassFunction
 
i13 : classFunction({1,1,1,1,1})

o13 = 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

o13 : ClassFunction

See also

Ways to use this method:

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:7279:0.