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

Function: schurRing
Usage:

S = schurRing R
Inputs:
- R, 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:
- S, a Schur ring,

Description

Given a ring R, the function schurRing attempts to return a Schur ring S that is associated to R in a natural way. Namely, if the attribute R.Schur points to a Schur ring, then the function returns that ring. If R is already a Schur ring, then the ring R is returned. Otherwise, if the Schur level of R is at least one, then the function constructs a Schur ring over the base ring A of R, having the same relative dimension over A as R. If the Schur level of R is zero, then an error is returned.

i1 : R = schurRing(QQ,r,6);

i2 : schurRing R

o2 = R

o2 : SchurRing

i3 : Q = symmetricRing(QQ,3);

i4 : A = schurRing Q;

i5 : schurRing Q

o5 = A

o5 : SchurRing

Passing an existing Schur ring to schurRing simply returns it, which is convenient as a guard when a function wants to accept either a Symmetric ring or a Schur ring as input.

i6 : S = schurRing(QQ,s,4);

i7 : schurRing S === S

o7 = true

For a Symmetric ring with the "Sn" interpretation, the associated Schur ring inherits this flavor and is cached on the ring.

i8 : Rsn = symmetricRing(QQ,4,GroupActing => "Sn");

i9 : Ssn = schurRing Rsn;

i10 : numgens Ssn

o10 = 4

The construction also works over polynomial coefficient rings, producing a Schur ring with the same parameters as coefficients.

i11 : Rx = symmetricRing(QQ[x],3);

i12 : Sx = schurRing Rx;

i13 : coefficientRing Sx

o13 = QQ[x]

o13 : PolynomialRing

Ways to use this method:

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

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:5579:0.

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

Description

See also

Ways to use this method: