1
$\begingroup$

The Wikipedia article on association schemes claims regarding Bose-Mesner algebras:

There is another algebra of $\left(n+1\right)\times \left(n+1\right)$ matrices which is isomorphic to ${\mathcal {A}}$, and is often easier to work with.

What is this algebra called? In which paper(s) is it defined and used? Unhelpfully, the statement is not cited.

$\endgroup$
1
$\begingroup$

The algebra is called intersection algebra. Here's why.

Let $\mathcal{A}$ be the Bose-Mesner algebra of an association scheme $(X,\mathcal{R})$ with $n$ classes and with intersection numbers $p_{ij}^k$. Let $D_0,D_1,\dots,D_n$ be the adjacency matrices that form a basis for the Bose-Mesner algebra. For $i=0,1,\dots,n$, let $L_i$ be an $(n+1)\times(n+1)$ matrix with $L_i=(p_{ij}^k)_{0\leq j,k\leq n}$. The $\mathbb{C}$-vector space $\mathcal{L}:=\langle L_0,L_1,\dots,L_n\rangle$ is a matrix subalgebra of $M_{n+1}(\mathbb{C})$. One way to prove this is by using the associativity of matrix multiplication: $D_i(D_jD_k)=(D_iD_j)D_k$. Moreover, we get an algebra isomorphism by sending $D_i\mapsto L_i$. This can be shown by using some properties of the intersection numbers. Hence, the algebra $\mathcal{L}$ is isomorphic to the Bose-Mesner algebra. In particular, the matrix $L_i$ has the same eigenvalues as the adjacency matrix $D_i$. This can be helpful if $|X|$ is large compared to $n$ since $D_i$ is an $|X|\times |X|$ matrix and $L_i$ is an $(n+1)\times(n+1)$ matrix.

If you look at this from a representation theoretical point of view, then the matrices $L_i$ correspond to the regular representation of $\mathcal{A}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.