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scalarProduct -- Standard pairing on symmetric functions/class functions

Description

This method computes the standard Hall scalar product on the ring \Lambda of symmetric functions. One way to define this product is by imposing that the collection of Schur functions s_{\lambda} form an orthonormal basis.

Alternatively, by the correspondence between symmetric functions and virtual characters of symmetric groups, this scalar product coincides with the standard scalar product on class functions.

The number of standard tableaux of shape \{4,3,2,1\} is:

i1 : R = symmetricRing(QQ,10);
 
i2 : S = schurRing(QQ,s,10);
 
i3 : scalarProduct(h_1^10,s_{4,3,2,1})

o3 = 768

o3 : QQ

The Schur basis is orthonormal: <s_\lambda, s_\mu> equals 1 if \lambda = \mu and 0 otherwise.

i4 : T = schurRing(QQ,t,5);
 
i5 : scalarProduct(t_{3,1,1}, t_{3,1,1})

o5 = 1

o5 : QQ
 
i6 : scalarProduct(t_{3,1,1}, t_{2,2,1})

o6 = 0

o6 : QQ
 
i7 : scalarProduct(t_{4,1}, t_{3,2})

o7 = 0

o7 : QQ

The power-sum basis is orthogonal with norms given by the centralizer sizes: <p_\rho, p_\rho> = z_\rho. Here we verify this for the three partitions of 3.

i8 : U = symmetricRing(QQ,4);
 
i9 : scalarProduct(p_3, p_3) == centralizerSize{0,0,1}

o9 = true
 
i10 : scalarProduct(p_2*p_1, p_2*p_1) == centralizerSize{1,1}

o10 = true
 
i11 : scalarProduct(p_1^3, p_1^3) == centralizerSize{3}

o11 = true

By the Cauchy identity, <p_1^n, p_1^n> counts the elements of S_n, i.e.\ it equals n!:

i12 : scalarProduct(p_1^4, p_1^4)

o12 = 24

o12 : QQ

See also

Ways to use scalarProduct:

For the programmer

The object scalarProduct is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:7338:0.