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internalProduct(ClassFunction,ClassFunction) -- Tensor product of virtual representations

Description

Given virtual characters ch1 and ch2, the method computes the character of the tensor product of corresponding virtual representations of the symmetric group.

i1 : ch1 = new ClassFunction from {{4,4} => 2, {8} => -1, {5,2,1} => 2, {3,2,2,1} => 1};
 
i2 : ch2 = new ClassFunction from {{2,2,2,2} => -4, {5,2,1} => 1, {3,2,2,1} => 3};
 
i3 : internalProduct(ch1,ch2)

o3 = ClassFunction{{3, 2, 2, 1} => 3}
                   {5, 2, 1} => 2

o3 : ClassFunction
 
i4 : ch1 * ch2

o4 = ClassFunction{{3, 2, 2, 1} => 3}
                   {5, 2, 1} => 2

o4 : ClassFunction

A classical example: the tensor square of the standard representation of S_4 has character values obtained by squaring the standard character. Pairing with itself recovers the multiplicities of the irreducibles in the tensor square:

i5 : S = schurRing(QQ,s,4);
 
i6 : std = classFunction s_{3,1};
 
i7 : sq = internalProduct(std, std)

o7 = ClassFunction{{1, 1, 1, 1} => 9}
                   {2, 1, 1} => 1
                   {2, 2} => 1
                   {4} => 1

o7 : ClassFunction
 
i8 : scalarProduct(sq, classFunction s_{4})

o8 = 1

o8 : QQ
 
i9 : scalarProduct(sq, classFunction s_{3,1})

o9 = 1

o9 : QQ
 
i10 : scalarProduct(sq, classFunction s_{2,2})

o10 = 1

o10 : QQ
 
i11 : scalarProduct(sq, classFunction s_{2,1,1})

o11 = 1

o11 : QQ

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.m2:7610:0.