SchurRingElement -- A type describing elements of a SchurRing

Description

Elements of any SchurRing -- whether a GL-ring, a symmetric-group ring, or an orthogonal/symplectic ring -- have type SchurRingElement. Products, sums and (in characteristic zero) rational scalings of Schur ring elements are again of this type.

i1 : S = schurRing(s,5)

o1 = S

o1 : SchurRing

i2 : a = s_{3,2,1}

o2 = s
      3,2,1

o2 : S

i3 : instance(a,SchurRingElement)

o3 = true

i4 : T = schurRing(S,t,3,GroupActing => "Sn")

o4 = T

o4 : SchurRing

i5 : b = t_{2,1}+t_3

o5 = t  + t
      3    2,1

o5 : T

i6 : instance(a*b,SchurRingElement)

o6 = true

The same type is used for symplectic characters. Multiplying two Sp-characters stays inside the ring:

i7 : Sp = schurRing(QQ,sp,3,GroupActing => "Sp");

i8 : u = sp_{2,1};

i9 : instance(u,SchurRingElement)

o9 = true

i10 : instance(u*u, SchurRingElement)

o10 = true

For an Sn-flavored Schur ring the ordinary ring multiplication is the tensor product of characters, while internalProduct gives the pointwise (Kronecker) product:

i11 : Sn = schurRing(QQ,c,4,GroupActing => "Sn");

i12 : internalProduct(c_{3,1}, c_{2,1,1})

o12 = c    + c    + c      + c
       3,1    2,2    2,1,1    1,1,1,1

o12 : Sn

i13 : instance(c_{3,1} * c_{2,1,1}, SchurRingElement)

o13 = true

In a two-level ring the SchurRingElement type is closed under multiplication across levels:

i14 : G = schurRing(QQ,g,3);

i15 : H = schurRing(G,h,2);

i16 : z = g_{1,1} * h_{2}

o16 = g   h
       1,1 2

o16 : H

i17 : instance(z, SchurRingElement)

o17 = true

Methods that use an object of class SchurRingElement:

dim(List,SchurRingElement) -- dimension of representation
dim(SchurRingElement) -- see dim(List,SchurRingElement) -- dimension of representation
dim(Thing,SchurRingElement) -- see dim(List,SchurRingElement) -- dimension of representation

For the programmer

The object SchurRingElement is a type, with ancestor classes RingElement < BasicList < Thing.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:6859:0.