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toH -- Complete symmetric (h-) basis representation

Synopsis

Description

Given a symmetric function f, the function toH yields a representation of f as a polynomial in the complete symmetric functions.

If f is an element of a Schur ring S then the output fh is an element of the Symmetric ring associated to S (see symmetricRing).

i1 : R = symmetricRing 7;
 
i2 : toH(h_3*e_3)

      3                2
o2 = h h  - 2h h h  + h
      1 3     1 2 3    3

o2 : R
 
i3 : S = schurRing(s,4)

o3 = S

o3 : SchurRing
 
i4 : toH S_{3,2,1}

        6     4       2 2     3                2    2
o4 = - h  + 4h h  - 3h h  - 3h h  + 3h h h  - h  + h h
        1     1 2     1 2     1 3     1 2 3    3    1 4

o4 : QQ[e ..e , p ..p , h ..h ]
         1   4   1   4   1   4

This also works over tensor products of Symmetric/Schur rings.

i5 : R = schurRing(r, 4, EHPVariables => (a,b,c));
 
i6 : S = schurRing(R, s, 2, EHPVariables => (x,y,z));
 
i7 : T = schurRing(S, t, 3);
 
i8 : A = symmetricRing T;
 
i9 : f = (r_1+s_1+t_1)^2

o9 = t  + t    + (2r  s  + 2r s  )t  + (s  + s    + 2r s  + (r  +
      2    1,1      () 1     1 ()  1     2    1,1     1 1     2  
     ------------------------------------------------------------------------
     r   )s  )t
      1,1  ()  ()

o9 : T
 
i10 : toH f

       2                    2            2
o10 = h  + (2y  + 2b )h  + y  + 2b y  + b
       1      1     1  1    1     1 1    1

o10 : A

See also

Ways to use toH :

For the programmer

The object toH is a method function.