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# makeRealRepresentationMackeyFunctor -- constructs the real representation Mackey functor

- Usage:
    `makeRealRepresentationMackeyFunctor(p)`
- Inputs:
  - p, an integer, a prime number $p$
- Outputs:
  - a C_p Mackey Functor, the real representation Mackey functor for the group $C_p$

## Description

The real representation Mackey functor of a group $G$ is defined by sending $G/H$ to the Grothendieck group of real orthogonal $G$-representations. The transfer and restriction come from induction and restriction of $G$-representations. When $G$ is a cyclic group of prime order p, with p odd, this admits a nice form. The underlying module is given by $\ZZ$ with trivial conjugation action, while the fixed module is $\mathbb{Z}\{\lambda_{0},\lambda_1,\dots,\lambda_{(p-1)/2}\}$. The element $\lambda_i$ for $i>0$ represents the two dimensional real representation given by rotation by $(2\pi i)/p$ radians. The element $\lambda_0$ represents the trivial one dimensional representation. Restriction is defined as \[\text{res}_e^{C_p} \colon \mathbb{Z}\{\lambda_{0},\lambda_1,\dots,\lambda_{(p-1)/2}\} \to \mathbb{Z}\] by sending \[\lambda_i\mapsto \begin{cases} 2 & i>0 \\ 1 & i=0. \end{cases} \] Transfer is of the form \[\text{tr}_e^{C_p} \colon \mathbb{Z} \to \mathbb{Z}\{\lambda_{0},\lambda_1,\dots,\lambda_{(p-1)/2}\}\] by sending $x\mapsto x\cdot \left(\sum^{(p-1)/2}_{0} \lambda_i\right)$.

    i1 : makeRealRepresentationMackeyFunctor(5)

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

    o1 : CpMackeyFunctor

## See also

- constructing examples of Mackey functors -- info about how to construct explicit Mackey functors
- makeComplexRepresentationMackeyFunctor -- constructs the complex representation Mackey functor

## Ways to use makeRealRepresentationMackeyFunctor:

- `makeRealRepresentationMackeyFunctor(ZZ)`

## For the programmer

The object makeRealRepresentationMackeyFunctor 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.m2:264:0.