Previously, I established that the set of conjugacy class digraphs of a finite group gives rise to a (commutative) association scheme. It consequently follows that the adjacency matrix of a conjugacy class digraph of a finite group is normal and hence is diagonalizable. I would now like to verify how to derive a general expression for the spectrum of a conjugacy class digraph of a finite group.

Suppose that \(G\) is a finite a group. Let \(\operatorname{Cl}(G)\) denote the set of conjugacy classes of \(G\), \(\mathbb{C}[G]\) denote the space of all total functions from \(G\) to \(\mathbb{C}\), \(X(G) \subset \mathbb{C}[G]\) denote the set of irreducible characters of \(G\) into \(\mathbb{C}\), \(\{ \delta_{g} : g \in G \}\) denote the standard basis for \(\mathbb{C}[G]\), \(\operatorname{Mat}_{G \times G}(\mathbb{C})\) denote the space of \(\lvert G \rvert \times \lvert G \rvert\) complex matrices whose rows and columns are indexed by the elements of \(G\) relative to a certain ordering of them, and \(\mathcal{A} := \{ A_{C} : C \in \operatorname{Cl}(G) \} \subset \operatorname{Mat}_{G \times G}(\mathbb{C})\) denote a set of adjacency matrices for the set of conjugacy class digraphs of \(G\). Define:

  • \[\left ( M_{f} \right ) \in \operatorname{Mat}_{G \times G}(\mathbb{C})\] \[\left ( M_{f} \right )_{x,y} := f(x^{-1} y)\]

    for all \(f \in \mathbb{C}[G]\).

  • \[\left ( \star \right ) : \mathbb{C}[G] \times \mathbb{C}[G] \to \mathbb{C}[G]\] \[\left ( f \star g \right )(x) := \sum\limits_{y \in G} f(y) g(y^{-1} x)\]

    turning \(\mathbb{C}[G]\) into the group ring of \(G\) over \(\mathbb{C}\).

Next, for all \(C \in \operatorname{Cl}(G)\) and all irreducible representations \(F \in \operatorname{Hom} \left (G, \operatorname{Aut}(\mathbb{C}[G] ) \right )\):

  • Interpret the group ring \(\mathbb{C}[G]\) as a module over itself, defining the scalar multiplication operation \((\ast) : \mathbb{C}[G] \times \mathbb{C}[G] \to \mathbb{C}[G]\) as \(\left ( \delta_{g} \ast f \right ) := F(g) \circ f\) for all \(g \in G\).

    Assume the facts that:

    1. \[\sum\limits_{c \in C} F(c) \in \operatorname{End}_{\ast}( \mathbb{C}[G])\]

      [ letting \(\operatorname{End}_{\ast}( \mathbb{C}[G])\) denote the space of module endomorphisms from \(\mathbb{C}[G]\) to itself ].

    2. \[\operatorname{End}_{\ast}( \mathbb{C}[G]) \subset \{ \lambda I : \lambda \in \mathbb{C} \}\]

      [ which is to say that Schur’s lemma is true in this context ].

It is then possible to deduce that for all \(C \in \operatorname{Cl}(G)\) and all irreducible characters \(\chi \in X(G)\)

there is a \(\lambda_{\chi}(C) \in \mathbb{C}\) such that for all \(x,y \in G\):

\[\left ( A_{C} \ M_{\chi} \right )_{x,y}\] \[= \sum\limits_{z \in G, \ xz^{-1} \in C} \chi(z^{-1}y)\] \[= \sum\limits_{c \in C} \chi(x^{-1} c y)\] \[= \sum\limits_{c \in C} \chi \left ( x (x^{-1} c y )x^{-1} \right )\] \[= \sum\limits_{c \in C} \chi \left ( c y x^{-1} \right )\] \[= \sum\limits_{c \in C} \operatorname{tr} \left ( F \left ( c y x^{-1} \right ) \right )\]

[ for some irreducible representation \(F \in \operatorname{Hom} \left (G, \operatorname{Aut}(\mathbb{C}[G] ) \right )\), assuming the fact that such an irreducible representation exists ]

\[= \sum\limits_{c \in C} \operatorname{tr} \left ( F \left ( c \right ) F \left ( y x^{-1} \right ) \right )\]

[ since \(F\) is a group homomorphism ]

\[= \sum\limits_{c \in C} \operatorname{tr} \left ( F \left ( c \right ) F \left ( y x^{-1} \right ) \right )\] \[= \operatorname{tr} \left ( \left ( \sum\limits_{c \in C} F \left ( c \right ) \right ) F \left ( y x^{-1} \right ) \right )\]

[ by linearity of trace ]

\[= \operatorname{tr} \left ( \left ( \overline { \lambda_{\chi}(C) } I \right ) \ F \left ( y x^{-1} \right ) \right )\]

[ by previous set of assumptions ]

\[= \overline { \lambda_{\chi}(C) } \ \operatorname{tr} \left ( F \left ( y x^{-1} \right ) \right )\]

[ again, by linearity of trace ]

\[= \overline { \lambda_{\chi}(C) } \ \chi \left ( y x^{-1} \right )\] \[= \overline { \lambda_{\chi}(C) \ \chi \left ( x^{-1} y \right ) }\]

[ assuming the fact that \(\chi \left ( g^{-1} \right ) = \overline { \chi \left ( g \right )}\) for all \(g \in G\) ]

\[= \overline{ \lambda_{\chi}(C) \ \left ( M_{\chi} \right)_{x,y} }\]

This then means that for all for all \(C \in \operatorname{Cl}(G)\) and all irreducible characters \(\chi \in X(G)\):

\[\overline{ \lambda_{\chi}(C) \ \left ( M_{\chi} \right)_{e,e} } = \left ( \sum\limits_{c \in C} \chi(c) \right ) \left ( \frac { \left ( M_{\chi} \right)_{e,e} }{ \left ( M_{\chi} \right)_{e,e} } \right )\]

[ letting \(e\) denote the identity element of \(G\) ]

\[\implies \lambda_{\chi}(C) = \frac{1}{\chi(e)} \ \overline { \left ( \sum\limits_{c \in C} \chi(c) \right ) }\]

[ assuming the fact that \(\left ( M_{\chi} \right)_{e,e} = \chi(e) \in \mathbb{R}\) ].

Now, observe that for all \(f,g \in \mathbb{C}[G]\):

\[\left ( M_{f} \ M_{g} \right )_{x,y}\] \[= \sum\limits_{z \in G} f(x^{-1} z) \ g(z^{-1} y)\] \[= \sum\limits_{z \in G} f(x^{-1} z) \ g \left ( z^{-1} \left ( x \ x^{-1} \right ) y \right )\] \[= \sum\limits_{z \in G} f(x^{-1} z) \ g \left ( \left ( x^{-1} z \right )^{-1} \left ( x^{-1} y \right ) \right )\] \[= \sum\limits_{z \in G} f(z) \ g \left ( z^{-1} \left ( x^{-1} y \right ) \right )\] \[= \left ( f \star g \right ) \left ( x^{-1} y \right )\] \[= \left ( M_{f \star g} \right )_{x,y}\]

[ by definition ].

It follows from this that for all \(\chi \in X(G)\):

  • \[M_{\chi}^{2} = M_{\chi \star \chi} = \frac{ \lvert G \rvert }{ \chi(e) } \ M_{\chi}\]

    [ assuming that \(\left ( \chi \star \chi \right ) = \frac{ \lvert G \rvert }{ \chi(e) } \chi\) ].

Next, for all \(f \in \mathbb{C}[G]\): define \(E_{f} := \frac{ f(e) }{ \lvert G \rvert } M_{f} \in \operatorname{Mat}_{G \times G}(\mathbb{C})\).

Assume the facts that for all \(\chi \in X(G)\):

  • \(\chi \left ( g^{-1} \right ) = \overline{\chi(g)}\).
  • \(\sum\limits_{g \in G} \chi(g) \ \psi(g) = 0\) for all \(\psi \in X(G)\) such that \(\chi \neq \psi\).

It then follows that for all \(\chi, \psi \in X(G)\) such that \(\chi \neq \psi\):

  • \(E_{\chi} E_{\psi} = E_{\chi}^{*} E_{\psi} = 0\) .
\[\implies \left ( \sum\limits_{\chi \in X(G)} E_{\chi} \right )^{2}\] \[= \sum\limits_{\chi \in X(G)} E_{\chi}^{2}\] \[= \sum\limits_{\chi \in X(G)} E_{\chi}\]

and moreover

\[\operatorname{rank} \left ( \sum\limits_{\chi \in X(G)} E_{\chi} \right )\] \[= \operatorname{tr} \left ( \sum\limits_{\chi \in X(G)} E_{\chi} \right )\] \[= \sum\limits_{\chi \in X(G)} \operatorname{tr} \left ( E_{\chi} \right )\] \[= \sum\limits_{\chi \in X(G)} \left ( \chi(e) \right )^{2}\] \[= \lvert G \rvert\]

[ assuming that \(\sum\limits_{\chi \in X(G)} \left ( \chi(e) \right )^{2} = \lvert G \rvert\) ]

\[\implies \sum\limits_{\chi \in X(G)} E_{\chi} = I\]

Lastly, observe now that for all \(\chi \in X(G)\) and \(C \in \operatorname{Cl}(G)\):

  • \[\sum\limits_{\psi \in S} E_{\psi}\]

    is a projection onto \(\operatorname{null} \left ( A_{C} - \lambda_{\chi}(C) I \right )\)

    [ letting \(S := \{ \psi : \psi \in X(G) \operatorname{ and } \lambda_{\psi}(C) = \lambda_{\chi}(C) \}\) ].

    \[\implies \operatorname{dim} \left( \operatorname{null} \left ( A_{C} - \lambda_{\chi}(C) I \right ) \right )\] \[= \operatorname{tr} \left ( \sum\limits_{\psi \in S} E_{\psi} \right )\] \[= \sum\limits_{\psi \in S} \operatorname{tr} \left( E_{\psi} \right )\] \[= \sum\limits_{\psi \in S} \left (\psi (e) \right )^{2}\]

    .

Hence it is now possible to conclude that \(\phi_{C}(x) := \prod\limits_{\chi \in X(G)} \left ( x - \frac{1}{\chi(e)} \ \overline { \left ( \sum\limits_{c \in C} \chi(c) \right ) } \right )^{\left (\chi (e) \right )^{2}} \in \mathbb{C}[x]\) is the characteristic polynomial of \(A_{C}\) for all \(C \in \operatorname{Cl}(G)\).

A general expression for the spectrum of a conjugacy class digraph of a finite group is now immediate from the results verified in this post.