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makeOrbitMackeyFunctor -- constructs the orbit Mackey functor of a C_p-module

Description

Given a module $M$ with $C_p$-action specified by an automorphism $\text{c}\colon M\to M$, there is a naturally associated orbit Mackey functor $\text{O}(M)$. The underlying module is the module $M$ with conjugation given by $\text{c}$. The fixed module is the quotient module $M_{C_p}$ of the action, represented as the cokernel of $1-\text{c}$. The restriction is the map $$\text{res}_e^{C_p} \colon M_{C_p} \to M$$ given by $1+\text{c}+\text{c}^2+\cdots+\text{c}^{p-1}$. The transfer is the map $$\text{tr}_e^{C_p} \colon M\to M_{C_p}$$ induced by the projection to the quotient.

i1 : makeOrbitMackeyFunctor(2,matrix{{0,1,0},{1,0,0},{0,0,1}})

o1 =                         Res : | 1 1 0 |
                                   | 1 1 0 |
                                   | 0 0 2 |
     cokernel | 1  -1 0 |  ------------------->   3  -┐ Conj : | 0 1 0 |
              | -1 1  0 | <-------------------  ZZ   <┘        | 1 0 0 |
              | 0  0  0 |    Tr : | 1 1 0 |                    | 0 0 1 |
                                  | 0 0 0 |
                                  | 0 0 1 |

o1 : CpMackeyFunctor

See also

Ways to use makeOrbitMackeyFunctor:

  • makeOrbitMackeyFunctor(ZZ,Matrix)

For the programmer

The object makeOrbitMackeyFunctor is a method function.


The source of this document is in CpMackeyFunctors/Documentation/ConstructorsDoc.m2:181:0.