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On the eigenvalues of Cayley graphs


  • Dr. Modjtaba GHORBAN

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In authors proposed a formula for computing the spectrum of Cayley graph $\Gamma=Cay(G,S)$ with respect to the character table of $G$ where $S$ is a symmetric normal subset of $G$.

Let $q$ be a power of prime number $p$. A representation of degree $n$ of group $G$ is a homomorphism $\alpha: G \to GL(n,q)$, where $\alpha(g)=[g]_{\beta}$ for some basis $\beta$. A character table is a matrix whose rows and columns are correspond to the irreducible characters and the conjugacy classes of $G$, respectively.

Let $G$ be a group, for every element $g\in G$, we denote the conjugacy class of $g$ by $g^G$. Assume that $N$ be a normal subgroup of $G$ and $\widetilde{\chi}$ is a character of $G/N$, then the character $\chi$ of $G$ which is given by $$\chi(g)=\widetilde{\chi}(Ng)~~\forall g\in G$$ is called the lift of $\widetilde{\chi}$ to $G$.

Let $G$ and $H$ be two finite groups, then the direct product group $G\times H$ is a group whose elements are the Cartesian product of sets $G,H$ and for $(g_1,h_1),(g_2,h_2)\in G\times H$ the related binary operation is defined as $(g_1,h_1)(g_2,h_2)=(g_1g_2,h_1h_2).$

Theorem. Let $G$ and $H$ be two finite groups with irreducible characters $\varphi_1,\varphi_2, \cdots, \varphi_r$ and $\eta_1,\eta_2,\cdots, \eta_s$, respectively. Let $M(G)$ and $M(H)$ be character tables of $G$ and $H$, respectively. Then the direct product group $G\times H$ has exactly $rs$ irreducible characters $\varphi_i\eta_j$, where $1\leq i\leq r$ and $1\leq j\leq s$. In particular, the character table of group $G\times H$ is $$M(G\times H)=M(G)\otimes M(H),$$ where $\otimes$ denotes the Kronecker product.

[1] Diaconis, P., Shahshahani, M., (1981), Generating a random permutation with random transpositions, Zeit. fur Wahrscheinlichkeitstheorie verw. Gebiete, 57, pp. 159--179.