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tags: Assignments
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# Assignment 4

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Due Friday September 30
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1. In this question, we assume that our group $G$ has inverse-closed conjugacy classes.

a. Let $C_i$, $C_j$, $C_k$ be three conjugacy classes of a finite group $G$. Show that there is a constant $p_{i,j}^k$ such that for all $x,y\in G$ with $xy^{-1}\in C_k$, there are $p_{i,j}^k$ elements $z\in G$ such that $xz^{-1}\in C_i$ and $zy^{-1}\in C_j$. __(3 marks)__

> In general, the conjugacy class association scheme arises from gluing inverse-pairs of conjugacy classes of a finite group $G$. So if $C_i=c^G\cup (c^{-1})^G$, then $(x,y)\in R_i$ if and only if $xy^{-1} \in C_i$.

b. The group algebra (over $\mathbb{C}$) of a finite group $G$ is the set of all formal combinations $\sum_{g\in G}\alpha_g g$ where $\alpha_g\in\mathbb{C}$, equipped with natural pointwise addition, multiplication, and scalar multiplication. Show that the Bose-Mesner Algebra of the conjugacy class association scheme of a finite group $G$ is isomorphic to the centre of the group algebra of $G$. (Hint: There is a natural isomorphism from the adjacency matrices to the group algebra). __(3 marks)__

> Indeed, the minimal idempotents correspond to irreducible characters of $G$, and if $T$ is the character table of $G$, then $$diag(m_0,\ldots,m_d)\,P=T\, diag(k_0,\ldots,k_d)$$ where $P$ is the matrix of eigenvalues of the conjugacy class association scheme.

c. Show that a finite group $G$ is simple if and only if the graph of every nontrivial relation $R_i$ (i.e., not the equality relation) of its conjugacy class association scheme is connected. __(4 marks)__

2. Let $\mathcal{A}$ be the conjugacy class association scheme of the group $G:=PSL(2,9)$ Use the `AssociationSchemes` package to compute:
- The matrix of eigenvalues $P$ of $\mathcal{A}$;
- The matrix of dual eigenvalues $Q$ of $\mathcal{A}$.

Provide your code and its output. (You will need to use `for` loops to construct the relation matrix. You might find the `First` and `Display` operations handy). Also, you will want to realise `E(5)` as a 5-th root of unity $e^{2\pi i/5}$ and compute some values outside of GAP. __(5 marks)__