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# makeFixedPointMackeyFunctor -- constructs the fixed-point Mackey functor of a C_p-module

- Usage:
    `makeFixedPointMackeyFunctor(p,C)`
- Inputs:
  - p, an integer, a prime number $p$
  - C, a matrix, an order $p$ automorphism $c$
- Outputs:
  - a C_p Mackey Functor, the fixed-point Mackey functor of the $C_p$-module specified by $\text{c}$.

## Description

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

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

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

    o1 : CpMackeyFunctor

## See also

- constructing examples of Mackey functors -- info about how to construct explicit Mackey functors
- makeOrbitMackeyFunctor -- constructs the orbit Mackey functor of a C_p-module

## Ways to use makeFixedPointMackeyFunctor:

- `makeFixedPointMackeyFunctor(ZZ,Matrix)`

## For the programmer

The object makeFixedPointMackeyFunctor is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/CpMackeyFunctors/Documentation/ConstructorsDoc.m2:154:0.