Global Representation Ring complementary code

This repo contains complementary code for the paper Global Representation Ring and Knutson Index (authors: Dylan Johnston, Diego Martín Duro, Dmitriy Rumynin). arxiv: https://arxiv.org/abs/2403.18498

To run: Download and open the .ipynb files on a Jupyter notebook and run with a Sagemath kernel. We used Sagemath 9.8 but other versions should work also. The GAP package is required.

Note: The code was written in a Jupyter Notebook as this appeared to be the easiest way to use Sagemath on a Windows machine.

Contents

1. Overview
2. global_table.ipynb
3. SameCharSameBurnside.ipynb
4. SplittingLemma.ipynb
5. BrauerPairs.ipynb

1. Overview

Let $G$ be a finite group and $X$ a $G$-set. Then $X$ can be decomposed into a disjoint of orbits, $X_i$. By the Orbit Stabiliser Theorem the $X_i$ are isomorphic (as sets) to $G/\text{Stab}(x_i)$ for any $x_i \in X_i$. Note that choosing a different $x_i' \in X_i$ results in quotienting by a conjugate of $\text{Stab}(x_i)$ - but these two $G$-sets are of course isomorphic.

The upshot is that all finite $G$-sets are generated by $G$-sets of the form $G/H$, where $H$ is a subgroup of $G$. Moreover, we don't need to consider all subgroups $H$, but only one per equivalence class of subgroups up to conjugation.

Let $G$ be a finite group and $N$ a normal subgroup. The main idea behind the (reduced) global representation ring is to consider "$G$ equivariant vector bundles on finite $G/N$ sets." That is, $G$ equivariant vector bundles on $G$-sets of the form $G/H$ where $N < H < G$.

Note: The global representation ring (as in [Witherspoon, The Ring of Equivariant Vector Bundles on Finite Sets, 1995]) occurs when $N={e}$, this is why the prefix 'reduced' is included above.

It is well known that $G$-equivariant vector bundles on $G/H$ are in $1-1$ correspondence with $H$-representations. [Segal, Equivariant K-theory, 1968] Therefore our (reduced) global representation ring is generated by pairs $(G/H, V)$ where $V$ is an irreducible $H$-representation and $H$ is a subgroup with $N < H < G$.

There also exists a mark on such pairs. Namely, let $K < G$ with $N < K$, and let $k \in K$. Then we define the mark $(G/H,V)(K,k) = \sum_{g \in (G/H)^K} \chi_V(g^{-1}kg)$ where by abuse of notation $g \in (G/H)^K$ means take a representative in $G$ for each coset in $(G/H)^K$, and $\chi_V$ denotes the character of $V$.

This information can be encoded in a table - with rows labelled by (simple $G$-set, irreducible representation) pairs and columns labelled by (subgroup, element) pairs and with entries of the table given by the marks.

The final thing to note is that if any two elements $k,k' \in K$ are conjugate by an element in the normaliser $N_G(K)$ then all marks will be equal (that is, the columns corresponding to $k$ and $k'$ will be identical), therefore we only need to take one element for each orbit of $N_G(K)$ acting on $K$. Similarly, we only need one irreducible representation in each orbit of $N_G(H)$ acting on $\text{Irr}(H)$, as two irreducible representations in the same orbit will have identical rows in the table.

This repo contains code which investigates questions and properties of the reduced global tables.

2. global_table.ipynb

This is the main script in the repo. The main function is 'reduced_global_table(G,N,print_it = True)'. You pass a GAP group $G$ and a normal subgroup $N$ and the function returns the reduced global table as a numpy array if print_it = False, otherwise, the table is printed (with no return value).

Example of how to use.

To compute the global table of $C_5$ and store to variable 'my_table' do the following:

#global_table.ipynb script here

G = gap.SmallGroup(5,1) 
N = gap.Subgroups(G)[1] #gap indexing starts from 1. The identity subgroup is always at index 1. 
my_table = reduced_global_table(G,N,print_it = False)

#do something with the table

3. SameCharSameBurnside.ipynb

The character table and Burnside table of marks can be seen from a global table. Therefore, when looking for two groups with the same global table a good place to start is to find two groups with the same character table and Burnside table of marks. This is the purpose of this script.

4. SplittingLemma.ipynb

There is a notion of a Knutson subindex of a $G$-set $G/H$ in the Burnside ring (written $\mathcal{K}_r^S(G/H)$). See the arxiv paper for more details, specifically Definition 2.2 and the short discussion after Lemma 3.1. We had two implications, namely

• A sequence $1 \rightarrow N \rightarrow G \rightarrow \Gamma \rightarrow 1$ splits $\implies$ $1 \rightarrow N_p \rightarrow G_p \rightarrow \Gamma_p \rightarrow 1$ splits, where $(-)_p$ denotes taking Sylow p-subgroups (some care is needed here on which Sylow p-subgroups are taken, see the paper for details).
• A sequence $1 \rightarrow N \rightarrow G \rightarrow \Gamma \rightarrow 1$ splits $\implies$ $\mathcal{K}_r^S(G/N) = 1$.

The backward implications are false, e.g. gap.SmallGroup($48,33$) is a counterexample to both. The question is whether a group being a counterexample to one backwards implication implies it is a counterexample to the other. This is true! (cf. Theorem 3.9). This script was used to collect empirical evidence for this theorem (when it was merely a conjecture).

5. BrauerPairs.ipynb

Two groups are said to be a Brauer pair if they have the same character tables and power maps. This script aims to find Brauer pairs. This was used to investigate whether there is a link between groups having the same global table and being a Brauer pair. However, we later found that no such link exists.