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NCRingElement -- exported from the NCAlgebra package, used to encode tensors

Functions and methods returning an object of class NCRingElement:

  • CAxisTensor(ZZ,NCPolynomialRing) -- see CAxisTensor -- the signature tensor of the canonical axis path at a given level
  • CMonTensor(ZZ,NCPolynomialRing) -- see CMonTensor -- the signature tensor of the canonical monomial path at a given level
  • matrixAction(Matrix,NCRing,NCRing) -- see matrixAction -- Diagonal matrix action on a tensor.
  • matrixAction(Matrix,NCRingElement,NCRing) -- see matrixAction -- Diagonal matrix action on a tensor.

Methods that use an object of class NCRingElement:

  • adjointWord(NCRingElement,NCPolynomialRing,List) -- see adjointWord -- image of a word through the shuffle algebra homomorphism induced by a polynomial map
  • antipode(NCRingElement) -- The antipode of a tensor
  • baseName(NCRingElement) -- Returns the base name of a generator of an NCRing
  • coordinates(NCRingElement) -- see coordinates -- Computes coordinates relative to a given basis
  • degree(NCRingElement) -- Returns the degree of an NCRingElement
  • halfshuffle(NCRingElement,NCRingElement) -- see halfshuffle -- compute the half-shuffle of an ordered pair of words
  • NCRingElement >> NCRingElement -- see halfshuffle -- compute the half-shuffle of an ordered pair of words
  • inner(NCRingElement,NCRingElement) -- see inner -- compute the inner product of two tensors.
  • NCRingElement @ NCRingElement -- see inner -- compute the inner product of two tensors.
  • isCentral(NCRingElement) -- see isCentral -- Determines if an element is central
  • isCentral(NCRingElement,NCGroebnerBasis) -- see isCentral -- Determines if an element is central
  • isConstant(NCRingElement) -- Returns whether the NCRingElement is constant
  • isHomogeneous(NCRingElement) -- see isHomogeneous(NCIdeal) -- Determines whether the input defines a homogeneous object
  • isLeftRegular(NCRingElement,ZZ) -- see isLeftRegular -- Determines if a given (homogeneous) element is regular in a given degree
  • isRightRegular(NCRingElement,ZZ) -- see isLeftRegular -- Determines if a given (homogeneous) element is regular in a given degree
  • isNormal(NCRingElement) -- Determines if a given NCRingElement is normal
  • leadCoefficient(NCRingElement) -- Returns the lead monomial of an NCRingElement
  • leadMonomial(NCRingElement) -- Returns the lead monomial of an NCRingElement
  • leadTerm(NCRingElement) -- Returns the lead term of an NCRingElement
  • leftMultiplicationMap(NCRingElement,List,List) -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
  • leftMultiplicationMap(NCRingElement,ZZ) -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
  • leftMultiplicationMap(NCRingElement,ZZ,ZZ) -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
  • rightMultiplicationMap(NCRingElement,List,List) -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
  • rightMultiplicationMap(NCRingElement,ZZ) -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
  • rightMultiplicationMap(NCRingElement,ZZ,ZZ) -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
  • List * NCRingElement -- Scales a list by an NCRingElement on the right
  • lyndonShuffle(NCRingElement) -- see lyndonShuffle -- compute the representation of a tensor as a shuffle polynomial in Lyndon words
  • Matrix * NCRingElement -- see matrixAction -- Diagonal matrix action on a tensor.
  • monomials(NCRingElement) -- Returns the monomials appearing in the NCRingElement
  • ncIdeal(NCRingElement) -- see ncIdeal -- Define a two-sided ideal in a noncommutative ring
  • ncLeftIdeal(NCRingElement) -- see ncLeftIdeal -- Define a left ideal in a noncommutative ring
  • NCMatrix * NCRingElement -- Product of NCMatrices
  • ncRightIdeal(NCRingElement) -- see ncRightIdeal -- Define a right ideal in a noncommutative ring
  • NCRingElement % NCGroebnerBasis -- Reduces a NCRingElement by a NCGroebnerBasis
  • NCRingElement * List -- Scales a list by an NCRingElement on the left
  • NCRingElement * NCMatrix -- Product of NCMatrices
  • NCRingMap NCRingElement -- Apply an NCRingMap to an element or matrix
  • normalAutomorphism(NCRingElement) -- see normalAutomorphism -- Computes the automorphism determined by a normal homogeneous element
  • normalFormBergman(NCRingElement,NCGroebnerBasis) -- see normalFormBergman -- Calls Bergman for a normal form calculation
  • oreExtension(NCRing,NCRingMap,NCRingElement) -- see oreExtension -- Creates an Ore extension of a noncommutative ring
  • oreExtension(NCRing,NCRingMap,NCRingMap,NCRingElement) -- see oreExtension -- Creates an Ore extension of a noncommutative ring
  • oreIdeal(NCRing,NCRingMap,NCRingElement) -- see oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring
  • oreIdeal(NCRing,NCRingMap,NCRingMap,NCRingElement) -- see oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring
  • ring(NCRingElement) -- Returns the NCRing of an NCRingElement
  • NCRingElement ** NCRingElement -- see shuffle -- shuffle product of two words
  • NCRingElement ⧢ NCRingElement -- see shuffle -- shuffle product of two words
  • shuffle(NCRingElement,NCRingElement) -- see shuffle -- shuffle product of two words
  • sig(Path,NCRingElement) -- see sig -- compute the signature of a piecewise polynomial path.
  • size(NCRingElement) -- Returns the number of terms of an NCRingElement
  • sparseCoeffs(NCRingElement) -- see sparseCoeffs -- Converts ring elements into vectors over the coefficient ring
  • support(NCRingElement) -- Returns the variables appearing in the NCRingElement
  • NCRingElement @ ZZ -- see tensorArray -- k-th level component of a tensor.
  • tensorArray(NCRingElement) -- see tensorArray -- k-th level component of a tensor.
  • tensorArray(NCRingElement,ZZ) -- see tensorArray -- k-th level component of a tensor.
  • tensorExp(NCRingElement,ZZ) -- see tensorExp -- Compute a component of the exponential of a tensor.
  • tensorLog(NCRingElement,ZZ) -- see tensorLog -- Compute a component of the logarithm of a tensor.
  • tensorParametrization(NCRingElement) -- see tensorParametrization -- Constructs the morphism that maps a word to its coefficient in the given tensor.
  • terms(NCRingElement) -- Returns the terms of an NCRingElement
  • toString(NCRingElement) -- Converts an NCRingElement to a string
  • wordFormat(NCRingElement) -- see wordFormat -- display a tensor in word notation
  • wordString(NCRingElement) -- see wordString -- a string representing a word in wordFormat

For the programmer

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

The source of this document is in PathSignatures/documentation.m2:159:0.