SYMF::SfTheta
--
applies the Theta-automorphism
SYMF::SfTheta(sf,q <,t>)
sf | - | any symmetric function |
q, t | - | any names or expressions |
The 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.
>> 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 /
MuPAD Combinat, an open source algebraic combinatorics package