constructing examples of Mackey functors -- info about how to construct explicit Mackey functors

The general constructor: New Mackey functors can be built using makeCpMackeyFunctor, which constructs a Mackey functor out of a restriction, transfer, and conjugation matrix, in that order. It takes as input a prime, and three matrices. For example, we can construct $\mathbb{F}_4$ as a $C_2$-Galois Mackey functor as follows:

The output of the makeCpMackeyFunctor method is a CpMackeyFunctor, which is a new type implemented in this package. Under the hood it is a hash table, encoding all the data of the Mackey functor. The underlying and fixed modules can be recovered with the Underlying and Fixed keys, and the homomorphisms in the data can be recovered from Res, Trans, and Conj keys.

Pruning: Any time we might need a nicer, more readable form of a Mackey functor, we can use the prune method to simplify the presentation of the underlying/fixed modules.

Algebraic constructors: We provide specific methods for constructing common examples of $C_p$-Mackey functors coming from algebra and representation theory. The Burnside Mackey functor is a prototypical example -- its underlying module is $\ZZ$ with trivial conjugation action, and its fixed module is the Burnside ring $A(C_p)$, which is the group completion of the monoid of isomorphism classes of finite $C_p$-sets.

Another example from algebra are the real and complex representation Mackey functors. Similarly to the Burnside Mackey functor, their underlying module is $\ZZ$, however their fixed module is the representation ring of $C_p$. Any finite $C_p$-set has an associated permutation representation, which induces what are called the real and complex linearization maps from the Burnside Mackey functor to the representation Mackey functor.

Fixed point and orbit Mackey functors: If $M$ is a $\ZZ[C_p]$-module, the fixed point Mackey functor of $M$ has $M$ as its underlying module and the fixed points $M^{C_p}$ as its fixed module. The conjugation action on $M$ is the $C_p$-action, and the transfer sums along the orbits of an element.

Similarly, given a $\ZZ[C_p]$-module $M$, we can form the orbit Mackey functor, whose underlying module is again $M$ with $C_p$-action yielding conjugation. The fixed point module is now the quotient module $M/C_p$, transfer is the quotient map, and now restriction is defined by summing along the orbits.

Free constructors: Analogous to how a free module can be constructed on a generator or set of generators, we can construct free Mackey functors. For $C_p$-Mackey functors, there are two modules, which lead to two different ideas of a "free" Mackey functor, namely a free $C_p$-Mackey functor on a generator in the underlying module, and a free $C_p$-Mackey functor on a generator in the fixed module. The free Mackey functor on a single underlying generator can be accessed as follows:

We often call this $\underline{B}$. Trying to construct a free Mackey functor on a single generator in the fixed module, we can verify that it recovers a similar construction, namely the Burnside Mackey functor, which we often denote by $\underline{A}$.

These two Mackey functors $\underline{A}$ and $\underline{B}$ are very special - they are the projective generators of the category $\text{Mack}_G$, and they play an important role in constructing resolutions.

Random constructor: Furthermore we have a random constructor which allow us to build a random Mackey functor over the group $C_p$. To implement this, we recall that we have projective generators $\underline{A}$ and $\underline{B}$, hence any Mackey functor $M$ can be written as a cokernel of some map of the form

\[ \underline{A}^k \oplus \underline{B}^\ell \to \underline{A}^n \oplus \underline{B}^m \to M \to 0.\]

We can specify $k$, $\ell$, $n$, and $m$ if we want, as the following example over $C_7$ shows:

If we don't want to specify this data, $n$ and $m$ will be chosen randomly between 1 and 10. To randomly choose between 1 and a higher number, we can use the GenBound option.

Remark: What makes this possible is the fact that it is computationally very easy to cook up a random map $\underline{A}^k \oplus \underline{B}^\ell \to \underline{A}^n \oplus \underline{B}^m$ using the universal properties of the functors $\underline{A}$ and $\underline{B}$. Since $\underline{A}$ corepresents the fixed module functor and $\underline{B}$ corepresents the underlying module functor, we can determine random maps of the form $\underline{A}^k \oplus \underline{B}^\ell \to M$ by picking a random list of elements from the fixed and underlying modules of any $M$.