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

Function: symmetricRing
Usage:

R = symmetricRing S
Inputs:
- S, a ring,
Optional inputs:
- Basis => ..., default value "Schur", Specifies the basis to use for a Schur ring
- 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
- OddOrEven => ..., default value null, Select type $B_n$ or type $D_n$ for an orthogonal Schur ring
- 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

For a plain Schur ring, the associated Symmetric ring is a polynomial ring in the elementary, complete, and power-sum generators.

i5 : S = schurRing(QQ,s,4);

i6 : R = symmetricRing S;

i7 : (R.eVariable 2)^2

      2
o7 = e
      2

o7 : R

i8 : toS ((R.eVariable 2)^2)

o8 = s    + s      + s
      2,2    2,1,1    1,1,1,1

o8 : S

The construction plays well with coefficient rings of different flavors, and with the symmetric-group interpretation via GroupActing => "Sn".

i9 : Ssn = schurRing(QQ,c,4,GroupActing => "Sn");

i10 : Rsn = symmetricRing Ssn;

i11 : numgens Rsn

o11 = 12

On a tower of Schur rings, symmetricRing produces a tower of Symmetric rings mirroring the coefficient-ring structure.

i12 : A = schurRing(QQ,a,3);

i13 : B = schurRing(A,b,2);

i14 : RB = symmetricRing B;

i15 : coefficientRing RB

o15 = QQ[e ..e , p ..p , h ..h ]
          1   3   1   3   1   3

o15 : PolynomialRing

Ways to use this method:

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

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:5644:0.

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

Description

See also

Ways to use this method: