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internalProduct -- Internal product of symmetric functions/class functions

Description

This method computes the internal (Kronecker) product of two homogeneous symmetric functions of the same degree. If we think of these functions as virtual characters of some symmetric group, then their internal product is just the character of the tensor product of the corresponding virtual representations. We use the binary operator ** as a shorthand for internalProduct.

The complete symmetric function of degree n corresponds to the trivial S_n-representation and is therefore the unit of the representation ring of S_n:

i1 : R = symmetricRing(QQ,5);
 
i2 : S = schurRing(QQ,s,3);
 
i3 : internalProduct(h_3,s_{2,1})

o3 = s
      2,1

o3 : S
 
i4 : toE(h_3 ** e_3)

o4 = e
      3

o4 : QQ[e ..e , p ..p , h ..h ]
         1   3   1   3   1   3

The square of the sign representation is the trivial representation:

i5 : toH internalProduct(e_3,e_3)

o5 = h
      3

o5 : QQ[e ..e , p ..p , h ..h ]
         1   3   1   3   1   3

Working in a Schur ring directly, Kronecker products of Schur functions give the decomposition of tensor products of irreducible S_n-representations. The Kronecker square of s_{2,1} (the standard representation of S_3) decomposes as:

i6 : T = schurRing(QQ,t,3);
 
i7 : internalProduct(t_{2,1}, t_{2,1})

o7 = t  + t    + t
      3    2,1    1,1,1

o7 : T

The exterior square of the sign representation of S_3 is the trivial representation, which on the symmetric-function side is the identity s_{1,1,1} \otimes s_{1,1,1} = s_3:

i8 : internalProduct(t_{1,1,1}, t_{1,1,1})

o8 = t
      3

o8 : T

See also

Ways to use internalProduct:

For the programmer

The object internalProduct is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:7502:0.