symmetricRing(Ring) -- The Symmetric ring corresponding to a given (Schur) ring.

Synopsis

Function: symmetricRing
Usage:

R = symmetricRing S
Inputs:
- S, a ring,
Optional inputs:
- EHPVariables => ..., default value (e,h,p), Specifies sequence of symbols representing e-, h-, and p-functions
- GroupActing => ..., default value "GL", Specifies the group that is acting
- SVariable => ..., default value s, Specifies symbol representing s-functions
Outputs:
- R, a ring,

Description

Given a (Schur) ring S, the function symmetricRing returns a (Symmetric) ring R that is associated to S in a natural way. Namely, if the attribute S.symmetricRing points to a ring, then the function returns that ring. If S is not a Schur ring, then the function returns S. Otherwise, if S is a Schur ring, then the function constructs a polynomial ring over the Symmetric ring R_A of the base ring A of R, having the same relative dimension over R_A as S over A.

i1 : A = schurRing(QQ,a,6);

i2 : B = schurRing(A,b,3);

i3 : symmetricRing B

o3 = QQ[e ..e , p ..p , h ..h ][e ..e , p ..p , h ..h ]
         1   6   1   6   1   6   1   3   1   3   1   3

o3 : PolynomialRing

i4 : symmetricRing ZZ

o4 = ZZ

o4 : Ring

Ways to use this method: