FUNCTION: NcsfInternal - internal product

CALLING SEQUENCE:

NcsfInternal(ncsf_1, ncsf_2)

NCSF[NcsfInternal](ncsf_1, ncsf_2)

PARAMETERS:

ncsf_1, ncsf_2 = any noncommutative symmetric functions

SYNOPSIS:

• The NcsfInternal function calculates the internal product between the two noncommutative symmetric functions ncsf_1 and ncsf_2.

• We define such a product in the S-basis. Let I=[i_1, i_2, ..., i_p] and J=[j_1, j_2, ..., j_q] be two compositions: then

• NcsfInternal(S[I],S[J])=0 if i_1+i_2+...+i_p<>j_1+j_2+...+j_q, otherwise,

• NcsfInternal(S[I],S[J])=sum S[M] where the sum is over all matrices M=(m_ij) of nonnegative integers of order pxq with row sums vector I and column sums vector J, and S[M]=S[m_11] S[m_12] ... S[m_qp].

• The result is expressed in the S-basis.

• Whenever there is a conflict between the function name NcsfInternal and another name used in the same session, use the long form NCSF['NcsfInternal'].

EXAMPLES:

> with(NCSF):
> NcsfInternal(L[2,1,1]+Ph[2,2], R[2,1,1]);
 
                    3 S[2, 1, 1] - 4 S[2, 2] + 2 S[1, 1, 2]