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SYMF::SfTheta -- applies the Theta-automorphism

Call(s)


SYMF::SfTheta(sf,q <,t>)

Parameters

sf- any symmetric function
q, t- any names or expressions

Introduction

The SYMF::SfTheta function realizes a certain multiplicative automorphism of the ring of symmetric functions. It is defined on power-sum functions as follows:

The result is given in the p-basis.

Example 1

>> muEC::SYMF::SfTheta( s[4,1], q );
        5                            4
      (q  p[1, 1, 1, 1, 1]) 1/30 + (q  p[2, 1, 1, 1]) 1/6 +
      
           3                     2
         (q  p[3, 1, 1]) 1/6 - (q  p[3, 2]) 1/6 - (q p[5]) 1/5
>> muEC::SYMF::SfTheta( p[3], q, t );
                                 3
                               (q  - 1) p[3]
                               -------------
                                    3
                                   t  - 1
>> muEC::SYMF::SfTheta( s[2,1], q, t );
            /        3            \       /   3           \
            | (q - 1)  p[1, 1, 1] |       | (q  - 1) p[3] |
            | ------------------- | 1/3 - | ------------- | 1/3
            |              3      |       |      3        |
            \       (t - 1)       /       \     t  - 1    /

Related Functions

SfOmega

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MuPAD Combinat, an open source algebraic combinatorics package