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scalarProduct(ClassFunction,ClassFunction) -- Standard scalar product of class functions

Description

Given virtual characters ch1 and ch2, the method computes the standard pairing between ch1 and ch2.

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

     17
o3 = --
     10

o3 : QQ

The irreducible characters of S_n indexed by distinct partitions are orthonormal. For partitions of 5, we can verify the full orthogonality relations:

i4 : S = schurRing(QQ,s,5);
 
i5 : cF32 = classFunction s_{3,2};
 
i6 : cF41 = classFunction s_{4,1};
 
i7 : cF221 = classFunction s_{2,2,1};
 
i8 : scalarProduct(cF32, cF32)

o8 = 1

o8 : QQ
 
i9 : scalarProduct(cF32, cF41)

o9 = 0

o9 : QQ
 
i10 : scalarProduct(cF41, cF221)

o10 = 0

o10 : QQ

Decomposing an arbitrary virtual character into irreducibles by pairing with each Schur class function recovers the multiplicities:

i11 : psi = cF32 + 2*cF41 - cF221;
 
i12 : {scalarProduct(psi, cF32), scalarProduct(psi, cF41), scalarProduct(psi, cF221)}

o12 = {1, 2, -1}

o12 : List

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.m2:7447:0.