CpMackeyFunctors -- for Cp-Mackey functor computations

Description

This Macaulay2 package implements basic homological algebra for Mackey functors of a cyclic group $C_p$ of prime order. Some highlights of our functionality include:

free resolutions of $C_p$-Mackey functors
Ext and Tor computations for (cohomological) Mackey functors
the box product and internal hom
constructions of various common examples of Mackey functors
the ability to generate random Mackey functors over $C_p$
explicit examples from the equivariant algebra literature

Authors

Version

This documentation describes version 1.0 of CpMackeyFunctors, released July 4, 2025.

Citation

If you have used this package in your research, please cite it as follows:

@misc{CpMackeyFunctorsSource,
  title = {{CpMackeyFunctors: A \emph{Macaulay2} package. Version~1.0}},
  author = {Thomas Brazelton and David Chan and Benjamin Mudrak and Ben Spitz and Chase Vogeli and Chenglu Wang and Michael R. Zeng and Sasha Zotine},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
}

Exports

Types
- CpMackeyFunctor -- The type of Cp Mackey functors
- MackeyFunctorHomomorphism -- the class of Mackey functor homomorphisms
Functions and commands
- boxProduct -- box product of Cp-Mackey functors
- complexLinearizationMap -- returns the complex linearization map for the prime p
- drawVerticalCpMackeyFunctor -- reorients the net of a Mackey functor vertically
- ExtCoh -- computes Ext of two cohomological Cp-Mackey fuctors
- internalHom -- returns the internal hom Mackey functor between two Mackey functors.
- isCohomological -- checks if a CpMackeyFunctor is cohomological
- isTrivialMackeyFunctor -- checks if a Mackey functor is trivial or not
- makeBurnsideMackeyFunctor -- constructs the Burnside Mackey functor
- makeComplexRepresentationMackeyFunctor -- constructs the complex representation Mackey functor
- makeCpMackeyFunctor -- make a Cp Mackey functor
- makeFixedPointMackeyFunctor -- constructs the fixed-point Mackey functor of a C_p-module
- makeFixedSignMackeyFunctor -- constructs the fixed-point Mackey functor for the sign action of $C_2$ on $\mathbb{Z}$
- makeFixedTrivMackeyFunctor -- constructs the fixed-point Mackey functor for the trivial action of $C_p$ on $\mathbb{Z}/p$
- makeKGroupMackeyFunctor -- constructs the $K$-group Mackey functor
- makeOrbitMackeyFunctor -- constructs the orbit Mackey functor of a C_p-module
- makeRandomCpMackeyFunctor -- generate a random Cp-Mackey functor
- makeRandomMackeyFunctorHomomorphism -- generate a random Cp-Mackey functor homomorphism
- makeRealRepresentationMackeyFunctor -- constructs the real representation Mackey functor
- makeUnderlyingFreeMackeyFunctor -- constructs the free Mackey functor on an underlying generator
- makeZeroMackeyFunctor -- constructs the zero Mackey functor for the group
- makeZeroOnUnderlyingMackeyFunctor -- constructs a Mackey functor which is zero on underlying
- realLinearizationMap -- returns the real linearization map for the prime p
- resolutionCohomological -- constructs a free resolution in cohomological Mackey functors of specified length
- TorCoh -- computes Tor of two cohomological Cp-Mackey fuctors
Methods
- boxProduct(CpMackeyFunctor,CpMackeyFunctor) -- see boxProduct -- box product of Cp-Mackey functors
- CpMackeyFunctor ** CpMackeyFunctor -- see boxProduct -- box product of Cp-Mackey functors
- CpMackeyFunctor ⊠ CpMackeyFunctor -- see boxProduct -- box product of Cp-Mackey functors
- boxProduct(CpMackeyFunctor,MackeyFunctorHomomorphism) -- see boxProduct(MackeyFunctorHomomorphism,CpMackeyFunctor) -- induced maps on box products
- boxProduct(MackeyFunctorHomomorphism,CpMackeyFunctor) -- induced maps on box products
- CpMackeyFunctor ** MackeyFunctorHomomorphism -- see boxProduct(MackeyFunctorHomomorphism,CpMackeyFunctor) -- induced maps on box products
- CpMackeyFunctor ⊠ MackeyFunctorHomomorphism -- see boxProduct(MackeyFunctorHomomorphism,CpMackeyFunctor) -- induced maps on box products
- MackeyFunctorHomomorphism ** CpMackeyFunctor -- see boxProduct(MackeyFunctorHomomorphism,CpMackeyFunctor) -- induced maps on box products
- MackeyFunctorHomomorphism ⊠ CpMackeyFunctor -- see boxProduct(MackeyFunctorHomomorphism,CpMackeyFunctor) -- induced maps on box products
- cokernel(MackeyFunctorHomomorphism) -- cokernel of a Mackey functor homomorphism
- complexLinearizationMap(ZZ) -- see complexLinearizationMap -- returns the complex linearization map for the prime p
- CpMackeyFunctor == CpMackeyFunctor -- see CpMackeyFunctor -- The type of Cp Mackey functors
- directSum(CpMackeyFunctor) -- see CpMackeyFunctor -- The type of Cp Mackey functors
- net(CpMackeyFunctor) -- see CpMackeyFunctor -- The type of Cp Mackey functors
- CpMackeyFunctor ++ CpMackeyFunctor -- direct sum of Cp-Mackey functors
- degree(MackeyFunctorHomomorphism) (missing documentation)
- drawVerticalCpMackeyFunctor(CpMackeyFunctor) -- see drawVerticalCpMackeyFunctor -- reorients the net of a Mackey functor vertically
- Ext^ZZ(CpMackeyFunctor,CpMackeyFunctor) -- see Ext -- computes Ext of two Cp-Mackey fuctors
- ExtCoh(ZZ,CpMackeyFunctor,CpMackeyFunctor) -- see ExtCoh -- computes Ext of two cohomological Cp-Mackey fuctors
- freeResolution(CpMackeyFunctor,ZZ) -- constructs a free Mackey functor resolution of specified length
- Hom(CpMackeyFunctor,CpMackeyFunctor) -- computes the Hom group between two Cp Mackey functors
- Hom(CpMackeyFunctor,MackeyFunctorHomomorphism) -- computes the induced map on Hom groups
- Hom(MackeyFunctorHomomorphism,CpMackeyFunctor) -- see Hom(CpMackeyFunctor,MackeyFunctorHomomorphism) -- computes the induced map on Hom groups
- inducedMap(CpMackeyFunctor,CpMackeyFunctor) -- tries to induce a natural map between two Mackey functors
- inducedMap(CpMackeyFunctor,CpMackeyFunctor,MackeyFunctorHomomorphism) -- see inducedMap(CpMackeyFunctor,CpMackeyFunctor) -- tries to induce a natural map between two Mackey functors
- inducedMap(CpMackeyFunctor,CpMackeyFunctor,Matrix,Matrix) -- see inducedMap(CpMackeyFunctor,CpMackeyFunctor) -- tries to induce a natural map between two Mackey functors
- internalHom(CpMackeyFunctor,CpMackeyFunctor) -- see internalHom -- returns the internal hom Mackey functor between two Mackey functors.
- internalHom(CpMackeyFunctor,MackeyFunctorHomomorphism) -- returns the induced map on an internal hom.
- internalHom(MackeyFunctorHomomorphism,CpMackeyFunctor) -- see internalHom(CpMackeyFunctor,MackeyFunctorHomomorphism) -- returns the induced map on an internal hom.
- isCohomological(CpMackeyFunctor) -- see isCohomological -- checks if a CpMackeyFunctor is cohomological
- isIsomorphism(MackeyFunctorHomomorphism) -- checks if a Mackey functor homomorphism is an isomorphism
- isTrivialMackeyFunctor(CpMackeyFunctor) -- see isTrivialMackeyFunctor -- checks if a Mackey functor is trivial or not
- kernel(MackeyFunctorHomomorphism) -- kernel of a Mackey functor homomorphism
- - MackeyFunctorHomomorphism -- see MackeyFunctorHomomorphism -- the class of Mackey functor homomorphisms
- directSum(MackeyFunctorHomomorphism) -- see MackeyFunctorHomomorphism -- the class of Mackey functor homomorphisms
- inverse(MackeyFunctorHomomorphism) -- see MackeyFunctorHomomorphism -- the class of Mackey functor homomorphisms
- MackeyFunctorHomomorphism + MackeyFunctorHomomorphism -- see MackeyFunctorHomomorphism -- the class of Mackey functor homomorphisms
- MackeyFunctorHomomorphism ++ MackeyFunctorHomomorphism -- see MackeyFunctorHomomorphism -- the class of Mackey functor homomorphisms
- MackeyFunctorHomomorphism - MackeyFunctorHomomorphism -- see MackeyFunctorHomomorphism -- the class of Mackey functor homomorphisms
- MackeyFunctorHomomorphism == MackeyFunctorHomomorphism -- see MackeyFunctorHomomorphism -- the class of Mackey functor homomorphisms
- MackeyFunctorHomomorphism ^ ZZ -- see MackeyFunctorHomomorphism -- the class of Mackey functor homomorphisms
- net(MackeyFunctorHomomorphism) -- see MackeyFunctorHomomorphism -- the class of Mackey functor homomorphisms
- prune(MackeyFunctorHomomorphism) -- see MackeyFunctorHomomorphism -- the class of Mackey functor homomorphisms
- ZZ * MackeyFunctorHomomorphism -- see MackeyFunctorHomomorphism -- the class of Mackey functor homomorphisms
- MackeyFunctorHomomorphism * MackeyFunctorHomomorphism -- composition of Mackey functor homomorphisms
- MackeyFunctorHomomorphism | MackeyFunctorHomomorphism -- Horizontal concatenation of Mackey functor homomorphisms
- MackeyFunctorHomomorphism || MackeyFunctorHomomorphism -- Vertical concatenation of Mackey functor homomorphisms
- makeBurnsideMackeyFunctor(ZZ) -- see makeBurnsideMackeyFunctor -- constructs the Burnside Mackey functor
- makeComplexRepresentationMackeyFunctor(ZZ) -- see makeComplexRepresentationMackeyFunctor -- constructs the complex representation Mackey functor
- makeCpMackeyFunctor(ZZ,Matrix,Matrix,Matrix) -- see makeCpMackeyFunctor -- make a Cp Mackey functor
- makeFixedPointMackeyFunctor(ZZ,Matrix) -- see makeFixedPointMackeyFunctor -- constructs the fixed-point Mackey functor of a C_p-module
- makeFixedSignMackeyFunctor(Sequence) -- see makeFixedSignMackeyFunctor -- constructs the fixed-point Mackey functor for the sign action of $C_2$ on $\mathbb{Z}$
- makeFixedTrivMackeyFunctor(ZZ) -- see makeFixedTrivMackeyFunctor -- constructs the fixed-point Mackey functor for the trivial action of $C_p$ on $\mathbb{Z}/p$
- makeKGroupMackeyFunctor(ZZ,ZZ,ZZ) -- see makeKGroupMackeyFunctor -- constructs the $K$-group Mackey functor
- makeOrbitMackeyFunctor(ZZ,Matrix) -- see makeOrbitMackeyFunctor -- constructs the orbit Mackey functor of a C_p-module
- makeRandomCpMackeyFunctor(ZZ) -- see makeRandomCpMackeyFunctor -- generate a random Cp-Mackey functor
- makeRandomCpMackeyFunctor(ZZ,List) -- see makeRandomCpMackeyFunctor -- generate a random Cp-Mackey functor
- makeRandomMackeyFunctorHomomorphism(CpMackeyFunctor,CpMackeyFunctor) -- see makeRandomMackeyFunctorHomomorphism -- generate a random Cp-Mackey functor homomorphism
- makeRealRepresentationMackeyFunctor(ZZ) -- see makeRealRepresentationMackeyFunctor -- constructs the real representation Mackey functor
- makeUnderlyingFreeMackeyFunctor(ZZ) -- see makeUnderlyingFreeMackeyFunctor -- constructs the free Mackey functor on an underlying generator
- makeZeroMackeyFunctor(ZZ) -- see makeZeroMackeyFunctor -- constructs the zero Mackey functor for the group
- makeZeroOnUnderlyingMackeyFunctor(ZZ,Module) -- see makeZeroOnUnderlyingMackeyFunctor -- constructs a Mackey functor which is zero on underlying
- map(CpMackeyFunctor,CpMackeyFunctor,Matrix,Matrix) -- constructs a map between two Mackey functors
- map(CpMackeyFunctor,CpMackeyFunctor,ZZ) -- constructs the zero map between two Mackey functors, or a multiple of the identity map
- prune(CpMackeyFunctor) -- prune a CpMackeyFunctor
- realLinearizationMap(ZZ) -- see realLinearizationMap -- returns the real linearization map for the prime p
- resolutionCohomological(CpMackeyFunctor,ZZ) -- see resolutionCohomological -- constructs a free resolution in cohomological Mackey functors of specified length
- source(MackeyFunctorHomomorphism) -- returns the source of a Mackey functor homomorphism
- target(MackeyFunctorHomomorphism) -- returns the target of a Mackey functor homomorphism
- Tor_ZZ(CpMackeyFunctor,CpMackeyFunctor) -- see Tor -- computes Tor of two Cp-Mackey fuctors
- TorCoh(ZZ,CpMackeyFunctor,CpMackeyFunctor) -- see TorCoh -- computes Tor of two cohomological Cp-Mackey fuctors
Symbols
- Conj -- the conjugation homomorphism in a Mackey functor
- Fixed -- the fixed module of a Mackey functor
- FixedMap -- the fixed-point level of a Mackey functor homomorphism
- GenBound -- optional input for generating random Mackey functors
- PrimeOrder -- the prime order of the group over which a Mackey functor is defined
- Res -- the restriction homomorphism in a Mackey functor
- Trans -- the transfer homomorphism in a Mackey functor
- Underlying -- the underlying module of a Mackey functor
- UnderlyingMap -- the underlying level of a Mackey functor homomorphism
Mutable symbols
- assertLevel -- controls whether constructors check for well-definedness
Other things
- assertLevel -- controls whether constructors check for well-definedness

For the programmer

The object CpMackeyFunctors is a package, defined in CpMackeyFunctors.m2, with auxiliary files in CpMackeyFunctors/.

The source of this document is in CpMackeyFunctors/Documentation/MainPageDoc.m2:27:0.