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coefficientRing(SchurRing) -- Coefficient ring of a Schur ring

Description

Given a Schur ring S, the function returns its coefficient ring. The coefficient ring may be any commutative ring that Macaulay2 supports, including other Schur rings obtained by iterating schurRing.

i1 : S = schurRing(ZZ[x],s,4);
 
i2 : coefficientRing S

o2 = ZZ[x]

o2 : PolynomialRing
 
i3 : A = schurRing(QQ,a,3);
 
i4 : B = schurRing(A,b,2);
 
i5 : coefficientRing B

o5 = A

o5 : SchurRing

For a tower of Schur rings, coefficientRing peels off one layer at a time, allowing the user to navigate the entire construction.

i6 : T = schurRing(B,t,2);
 
i7 : coefficientRing T

o7 = B

o7 : SchurRing
 
i8 : coefficientRing coefficientRing T

o8 = A

o8 : SchurRing

One can build Schur rings over finite-field coefficients as well.

i9 : P = schurRing(ZZ/5,p,4);
 
i10 : coefficientRing P

      ZZ
o10 = --
       5

o10 : QuotientRing
 
i11 : (p_{2,1} + p_{1})^2

o11 = p    + p      + p    + 2p      + p        + 2p    + p      + p        +
       4,2    4,1,1    3,3     3,2,1    3,1,1,1     3,1    2,2,2    2,2,1,1  
      -----------------------------------------------------------------------
      2p    + 2p      + p  + p
        2,2     2,1,1    2    1,1

o11 : P

See also

Ways to use this method:

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:5203:0.