I would like to introduce the character theory of (commutative) coherent algebras, with the goal of arriving at what it means for a coherent algebra to be self-dual and how the theory of duality of in coherent algebras generalizes the Fourier theory of finite abelian groups in a way that could be useful towards e.g. the classification of distance-regular graphs.

Suppose that \(\mathcal{W}\) is a commutative coherent algebra (of order \(n\) and dimension \(d\)).

Let \(\Gamma(\mathcal{W})\) and \(\Lambda(\mathcal{W})\) denote the set of Schur-primitive and primitive matrices in \(W\) respectively:

Note that a non-zero \(01\)-matrix \(M \in \mathcal{W}\) is said to be Schur-primitive if \(M \circ N \in \operatorname{span} \{ M \}\) for all \(N \in \mathcal {W}\), and likewise a non-zero idempotent matrix \(M \in \mathcal{W}\) is said to be primitive if \(MN \in \operatorname{span} \{ M \}\) for all \(N \in \mathcal {W}\).

I would propose that \(\mathcal{W} := \operatorname{span} \left ( \Gamma(\mathcal{W}) \right ) = \operatorname{span} \left ( \Lambda(\mathcal{W}) \right )\) .

  1. Let \(\Gamma(\mathcal{W})^{\perp}\) denote the orthogonal complement of \(\Gamma(\mathcal{W})\) relative to the trace inner product \(\langle \cdot, \cdot \rangle\) on \(\operatorname{Mat}_{n \times n}(\mathbb{C})\).

    Observe that \(\langle M, N \rangle = \operatorname{sum} \left ( \overline{M} \circ N \right )\) and hence \(M \circ N = 0\) if and only if \(\langle M, N \rangle = 0\) for all \(M \in \mathcal{W}\) and \(N \in \operatorname{span} \left ( \Gamma(\mathcal{W}) \right )\).

    \[\implies \Gamma(\mathcal{W})^{\perp} = \{ M \in \mathcal{W} : M \circ N = 0 \text{ for all } N \in \operatorname{span} \left ( \Gamma(\mathcal{W}) \right )\}\]

    Now, observe then that \(\Gamma(\mathcal{W})\) is Schur-closed.

    It follows that for all \(M \in \Gamma(\mathcal{W})^{\perp}\) with a non-zero entry \(c \in \mathbb{C}\), the matrix \(N \circ N - N \in \Gamma(\mathcal{W})^{\perp}\) where \(N := \frac{1}{c} M\) has fewer non-zero entries than \(M\).

    This then means that the matrix in \(\Gamma(\mathcal{W})^{\perp}\) with the least number of non-zero entries must be a \(01\)-matrix, and moreover must either be Schur-primitive or the zero matrix.

    But, there are no Schur-primitive matrices in \(\Gamma(\mathcal{W})^{\perp}\).

    \[\implies \Gamma(\mathcal{W})^{\perp} = \{ 0 \}\]

    It is then ultimately apparent that \(\mathcal{W} = \operatorname{span} \left ( \Lambda(\mathcal{W}) \right )\).

  2. Let \(\sigma(A)\) denote the set of distinct eigenvalues of \(A\) for all \(A \in \Gamma(\mathcal{W})\).

    Observe that \(\Gamma(\mathcal{W})\) is a set of commuting normal matrices.

    \(\implies\) There is a unitary matrix \(U \in \operatorname{Mat}_{n \times n}(\mathbb{C})\) that simultaneously diagonalizes \(\Gamma(\mathcal{W})\).

    Let \(S := \{ P_{j} : 1 \leq j \leq n \}\) where \(P_{j} := \left ( U e_{j} \right ) \left ( U e_{j} \right )^{T} \in \operatorname{Mat}_{n \times n}(\mathbb{C})\) for all \(1 \leq j \leq n\).

    Now, observe too that the elements of \(S\) are orthogonal projections, and \(PQ = 0\) for all \(P, Q \in S\) such that \(P \neq Q\).

    Let \(\Omega(A) := \left \{ \sum\limits_{P \in T_{\lambda}} P : \lambda \in \sigma(A) \right \}\) where \(T_{\lambda} := \{ P \in S : AP = \lambda P \}\) for all \(\lambda \in \sigma(A)\) and all \(A \in \Gamma(\mathcal{W})\).

    It then follows that \(\Omega(A)\) is a set of orthogonal projections onto the eigenspaces of \(A\) for all \(A \in \Gamma(\mathcal{W})\).

    Let \(\Omega(\mathcal{W})\) denote the set of non-zero matrices in \(\operatorname{Mat}_{n \times n}(\mathbb{C})\) of the form

    \(\prod\limits_{A \in \Gamma(\mathcal{W})} M_{A}\) for some \(\{ M_{A} : A \in \Gamma(\mathcal{W}), M_{A} \in \Omega(A) \} \subseteq \operatorname{Mat}_{n \times n}(\mathbb{C})\).

    \[\implies \mathcal{W} = \operatorname{span} \left ( \bigcup_{A \in \Gamma(\mathcal{W})} \Omega(A) \right ) \subseteq \operatorname{span} \left ( \Omega(\mathcal{W}) \right )\]

    Lastly, recall that every normal matrix \(M \in \operatorname{Mat}_{n \times n}(\mathbb{C})\) has a unique set \(\Omega(M)\) consisting of orthogonal projections onto each of its eigenspaces, and that \(\Omega(M) \subseteq \{ f(M) : f(x) \in \mathbb{C}[x] \}\) as well (i.e. they are all polynomials in \(M\)).

    \(\implies \operatorname{span} \left ( \Omega(\mathcal{W}) \right ) \subseteq \mathcal{W}\).

    It can then ultimately be observed that \(\Omega(\mathcal{W}) = \Lambda(\mathcal{W})\), and hence \(\mathcal{W} = \operatorname{span} \left ( \Lambda(\mathcal{W}) \right )\).

So \(\mathcal{W}\) is indeed spanned by its set of Schur-primitive matrices, and dually, its set of primitive matrices as well.

Now, suppose that orderings \(\Gamma(\mathcal{W}) = \{ A_{j} : 1 \leq j \leq d \}\) and \(\Lambda(\mathcal{W}) = \{ E_{j} : 1 \leq j \leq d \}\) are given on the sets of Schur-primitive and primitive matrices in \(\mathcal{W}\) respectively.

Then, since \(\mathcal{W} := \operatorname{span} \left ( \Gamma(\mathcal{W}) \right ) = \operatorname{span} \left ( \Lambda(\mathcal{W}) \right )\), there is a matrix \(P \in \operatorname{Mat}_{d \times d}(\mathbb{C})\) such that \(A_{j} = \sum\limits_{k = 1}^{d} P_{j,k} \ E_k\) for all \(1 \leq j \leq d\), known as the character table of \(\mathcal{W}\) (relative to this ordering).

Dually, the character table \(P\) of \(W\) is invertible, and \(E_{j} = \sum\limits_{k = 1}^{d} P_{j,k}^{-1} \ A_{k}\) for all \(1 \leq j \leq d\).

\(\mathcal{W}\) is said to be self-dual if \(P \overline{P} = n I\).

A prototypical example of a family of self-dual coherent algebras are the coherent algebras arising from spans of regular representations of finite abelian groups as groups of permutation matrices. I will return in another post to elaborate on how the theory of self-dual coherent algebras generalizes the Fourier theory of finite abelian groups in a way that could be useful towards e.g. the classification of distance-regular graphs, as mentioned.