GCExpression

Description

A general data type for representing elements of bracket rings and Grassmann-Cayley algebras. GC expressions can be assembled into matrices, and they support a number of the usual arithmetic operations: addition, multiplication, and scalar multiplication.

Multiplication on the Grassmann-Cayley algebra is the usual exterior product. This represents the span, or join, of linear subspaces in a given vector space.

The Grassmann-Cayley algebra is also endowed with a "shuffle product", representing the intersection or meet of linear subspaces. This is implemented in GCExpression ^ GCExpression.

i1 : B = bracketRing(4, 2)

o1 = B
      4,2

o1 : BracketRing

i2 : e = 3 * [1 2]_B + 7 * [3 4]_B  -- a GCExpression living in a BracketRing

o2 = 7*[34]+3*[12]

o2 : GCExpression

i3 : class e

o3 = GCExpression

o3 : Type

i4 : degree e

o4 = {1}

o4 : List

i5 : terms e

o5 = {7*[34], 3*[12]}

o5 : List

i6 : G = gc(a..d, 2)

o6 = Grassmann-Cayley Algebra generated by 1-extensors a..d

o6 : GCAlgebra

i7 : ge = (a * b)_G                 -- a GCExpression living in a GCAlgebra

o7 = a*b

o7 : GCExpression

i8 : class ge

o8 = GCExpression

o8 : Type

Types of GCExpression:

Bracket

Methods that use an object of class GCExpression:

bracketRing(GCExpression) -- see bracketRing -- Constructor for bracket rings
coefficients(GCExpression) (missing documentation)
commonRing(GCExpression,GCExpression) (missing documentation)
degree(GCExpression) (missing documentation)
factor(GCExpression) (missing documentation)
GCExpression * GCExpression (missing documentation)
GCExpression * GCMatrix (missing documentation)
GCExpression * Number (missing documentation)
GCExpression * RingElement (missing documentation)
GCExpression + GCExpression (missing documentation)
GCExpression - GCExpression (missing documentation)
GCExpression ^ GCExpression -- Shuffle product in the Grassmann-Cayley Algebra
GCExpression ^ RingElement (missing documentation)
GCExpression ^ ZZ (missing documentation)
GCExpression _ BracketRing -- Substituting top-degree Grassmann-Cayley elements into the bracket ring
GCExpression _ GCAlgebra (missing documentation)
GCMatrix * GCExpression (missing documentation)
net(GCExpression) (missing documentation)
normalForm(GCExpression) -- see normalForm -- Represent a bracket polynomial in a normal form with respect to a Groebner basis
Number * GCExpression (missing documentation)
ring(GCExpression) (missing documentation)
RingElement * GCExpression (missing documentation)
RingElement ^ GCExpression (missing documentation)
terms(GCExpression) (missing documentation)

For the programmer

The object GCExpression is a type, with ancestor classes HashTable < Thing.

The source of this document is in Brackets.m2:608:0.