SYMF::SfTheta --
applies the Theta-automorphism
Call(s)
SYMF::SfTheta(sf,q <,t>)
Parameterssf | - | any symmetric function |
q, t | - | any names or expressions |
IntroductionThe SYMF::SfTheta function realizes a certain multiplicative
automorphism of the ring of symmetric functions. It is defined
on power-sum functions as follows:
SYMF::SfTheta(sf, q) gives the image of sf under the
transformation:
p[i] -->> q x p[i].
SYMF::SfTheta(sf, q, t) is the image of sf under the transformation:
p[i] -->> (1-q^i)/(1-t^i) x p[i].
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 FunctionsMuPAD Combinat, an open source algebraic combinatorics package