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SFA::SfAExpand -- expands a SFA expression

Call(s)


SFA::SfAExpand(sfa)

Parameters

sfa- any valid expression in vlist

Introduction

The SFA::SfAExpand function expands, whenever possible, any expression in SFA. It is a direct implementation of the Lambda-ring structure, i.e. symmetric functions are considered as operators over the ring of polynomials.

Expansion is inductively propagated, thus enabling chains of plethysms.

Alphabets are either formal alphabets (A1, A2, ...), or real constants (any float object or any formal variable <> Ai, or variables (when explicitely so declared), or symmetric functions of SFA, or any linear combination of alphabets. For instance, 3/2*A1 + k*(A2 - s[3](-A1)) is a valid alphabet.

One may declare variables by setting e.g. SfAVars({{x}, {y}, z1, z2}). Here, z1, z2, all xi's, and all yi's will be held as variables within alphabets.

Rules used for e.g. power sums symmetric functions are:


where i and j are positive integers. Equivalent rules for the other bases are also implemented.

Example 1

>> muEC::SFA::SfAExpand( p[4,2](7+A1) );
                7 p[2](A1) + 7 p[4](A1) + p[4, 2](A1) + 49
>> muEC::SFA::SfAExpand( s[2](A1 - 3/2*A2 + k) );
      k              3 s[2](A2)   15 s[1, 1](A2)
      - + s[2](A1) + ---------- + -------------- + k s[1](A1) -
      2                   8              8
      
                                               2
         3 k s[1](A2)   3 s[1](A1) s[1](A2)   k
         ------------ - ------------------- + --
               2                 2             2
>> muEC::SFA::SfAVars( { {x}, {y}, z1 } );
                              {{x}, {y}, z1}
>> muEC::SFA::SfAExpand( p[3]( q*m[2,1]( x1-k+A2 ) - z1^2+C ) );
                                        2             6
      C - q p[9](A2) + q p[6, 3](A2) + k  q + k q - z1  -
      
                                             3         6
         k q p[3](A2) - k q p[6](A2) - k q x1  - k q x1  +
      
             3                6
         q x1  p[6](A2) + q x1  p[3](A2)

Related Functions

SfAVars, SYMF::SfEval, SYMF::SfPlethysm, SfACollect

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MuPAD Combinat, an open source algebraic combinatorics package