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SYMF::SfPlethysm -- plethysm of symmetric functions

Call(s)


SYMF::SfPlethysm(sf1,sf2 <,b1> <,b2>)

Parameters

sf1, sf2- any symmetric functions
b1, b2- any known bases

Introduction

The SYMF::SfPlethysm function computes the plethysm sf1[sf2].

One may specify that sf1 and sf2 are expressed on the bases b1 and b2 by using either SYMF::SfPlethysm(sf1, sf2, b1), SYMF::SfPlethysm(sf1, sf2, b1, b2).

The plethysm operation is defined as follows: let p[i] be the i-th power-sum symmetric function and sf1, sf2, sf3 be any symmetric functions, then,

        
        

The default is to compute plethysms on the p-basis and return the result expressed on the p-basis.

Special algorithms have been included to compute plethysms. In particular, the result is expressed on the basis of Schur functions when:

Example 1

>> muEC::SYMF::SfPlethysm( p[1]*s[2], h[1] );
                           p[2, 1]   p[1, 1, 1]
                           ------- + ----------
                              2           2
>> muEC::SYMF::SfPlethysm( e[3], s[2,1], e, s );
      s[2, 2, 2, 1, 1, 1] + s[3, 2, 2, 1, 1] + s[3, 3, 1, 1, 1] +
      
         s[4, 2, 1, 1, 1] + s[3, 2, 2, 2] + s[3, 3, 2, 1] +
      
         s[4, 2, 2, 1] + 2 s[4, 3, 1, 1] + s[5, 2, 1, 1] +
      
         s[6, 1, 1, 1] + s[3, 3, 3] + s[4, 3, 2] + s[5, 2, 2] +
      
         s[4, 4, 1] + s[5, 3, 1]

Related Functions

Char2Sf

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MuPAD Combinat, an open source algebraic combinatorics package