skewpoly.hpp File Reference

SkewPolynomialRing — polynomial ring with a designated set of anticommuting variables. More...

#include "poly.hpp"

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Classes
class	SkewPolynomialRing
	PolyRing subclass for skew-commutative (exterior-style) polynomial rings: the listed skewvars anticommute among themselves and square to zero. More...

Detailed Description

SkewPolynomialRing — polynomial ring with a designated set of anticommuting variables.

Note

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Declares SkewPolynomialRing, the PolyRing subclass for the super-commutative case: a set of variables passed to create(K, M, skewvars) anticommutes pairwise and squares to zero (x_i x_j = -x_j x_i, x_i^2 = 0), while every other variable commutes with everything. Construction routes through initialize_skew(skewvars) and delegates the bookkeeping to the inherited SkewMultiplication skew_ field (declared in skew.hpp); arithmetic enters the skew code path through the is_skew_ flag inherited from PolynomialRing. The overridden mult_by_term walks the second factor and produces the sign by counting the transpositions needed to move its skew variables past those of the first.

Reports is_skew_commutative_ring() true and is_commutative_ring() false; has_gcd() is overridden to false because the ring is not a domain (any skew generator x_i is a zero divisor with itself). The antipode(f) method implements the R <-> R^op isomorphism by twisting each monomial of skew degree d by (-1)^{d choose 2}. Exterior algebras are the familiar special case (every variable skew); the broader setup with a strict subset is what makes the class more general than a pure exterior algebra.

See also

skew.hpp

poly.hpp

polyring.hpp

Definition in file skewpoly.hpp.