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numgens(SchurRing) -- Number of generators of Schur ring.

Description

Given a Schur ring S, the function numgens outputs the number of generators of S. This is equal to the relative dimension of S over its base ring, and also to the maximal number of parts of a partition allowed as an index for the elements of S.

i1 : R = schurRing(QQ,r,6);
 
i2 : numgens R

o2 = 6
 
i3 : S = schurRing(s);
 
i4 : numgens S

o4 = infinity

o4 : InfiniteNumber

When a Schur ring is built on top of another Schur ring as its coefficient ring, numgens measures only the outermost (relative) layer. Nested Schur rings thus model tensor products of representation rings, each layer tracked by its own numgens.

i5 : A = schurRing(QQ,a,3);
 
i6 : B = schurRing(A,b,2);
 
i7 : numgens B

o7 = 2
 
i8 : numgens coefficientRing B

o8 = 3

The numgens value for a stable Schur ring (built with n => infinity or without an explicit rank) is infinity, reflecting the fact that no partition length is excluded.

i9 : Sinf = schurRing(QQ,u);
 
i10 : numgens Sinf

o10 = infinity

o10 : InfiniteNumber
 
i11 : u_{4,3,2,1}

o11 = u
       4,3,2,1

o11 : Sinf

The same rule applies to the symplectic and orthogonal flavors.

i12 : Sp = schurRing(QQ,sp,4,GroupActing => "Sp");
 
i13 : numgens Sp

o13 = 4
 
i14 : O = schurRing(QQ,o,5,GroupActing => "O",OddOrEven => "Odd");
 
i15 : numgens O

o15 = 5

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.m2:5514:0.