Semidirect -- action of semidirect product with torus

Consider a polynomial ring $R$ and an $R$-module $M$ with a $\mathbb{Z}^r$-grading corresponding to the action of a torus $T$. Let $G$ be a finite group acting on $R$ and $M$ in a way that is compatible with multiplication. Then the semidirect product $T\rtimes G$ acts on $R$ and $M$. For a given degree $d \in \mathbb{Z}^r$, the graded components $R_d$ and $M_d$ need not be representations of $G$. However, if $\mathcal{O}$ is the orbit of $d$ under the action of $G$ on the character group $\mathbb{Z}^r$ of $T$, then $\bigoplus_{d\in\mathcal{O}} R_d$ and $\bigoplus_{d\in\mathcal{O}} M_d$ are representations of $T\rtimes G$. Starting with version 2.3, the BettiCharacters package allows one to compute the characters of $G$ on these representations using the Semidirect option of the action method, and specifying a single degree $d$ in the orbit $\mathcal{O}$.

The value of the Semidirect option is a list of two functions. The first function takes as input a degree $d$ and returns its orbit $\mathcal{O}$ as output. This function is stored in the action under the key degreeOrbit. The second function takes as input a degree $d$ and returns a user-chosen representative $d'$ from the orbit $\mathcal{O}$ of $d$. This function is stored in the action under the key degreeRepresentative. When computing the actors or the characters of $G$ on $\bigoplus_{d\in\mathcal{O}} R_d$ and $\bigoplus_{d\in\mathcal{O}} M_d$, the values are stored only for the chosen representative $d'$, and computing the actors or characters of $G$ for another degree in the same orbit produces the same result as for $d'$. By default, both functions are set to the identity, which corresponds to the action of the direct product $T\times G$.

A typical use case is that of the symmetric group $\mathfrak{S}_n$ acting on a fine graded polynomial ring $\Bbbk [x_1,\dots,x_n]$ by permuting the variables. The symmetric group also acts by permuting the entries of the degrees $d \in \mathbb{Z}^n$. In this case, the orbit of $d$ consists of all its permutations, which can be obtained with the function uniquePermutations. As a representative of this orbit we choose the unique degree $d$ whose entries are sorted in nonincreasing order from left to right; this can be obtained with the function rsort.

We illustrate this use case. First, consider the action on the polynomial ring.

As expected, the character is the same if we compute it for a different degree in the same orbit.

Next, consider the quotient by an ideal stable under the group action.

Similarly, the Semidirect option can be used for actions on complexes and for computing Betti characters of a module.

Functions with optional argument named Semidirect:

• *(CharacterDecomposition,CharacterTable,Semidirect=>...)
• action(...,Semidirect=>...)
• character(...,Semidirect=>...)