  Let $H$ be a semisimple Hopf algebra over an algebraically closed field
$\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$ and $p\nmid
2\dim_{\mathbbm{k}}(H)$. In this paper, we consider the smash product
semisimple Hopf algebra $H\#\mathbbm{k}G$, where $G$ is a cyclic group of order
$n:=2\dim_{\mathbbm{k}}(H)$. Using irreducible representations of $H$ and those
of $\mathbbm{k}G$, we determine all non-isomorphic irreducible representations
of $H\#\mathbbm{k}G$. There is a close relationship between the Grothendieck
algebra $(G_0(H\#\mathbbm{k}G)\otimes_{\mathbb{Z}}\mathbbm{k},*)$ of
$H\#\mathbbm{k}G$ and the Grothendieck algebra
$(G_0(H)\otimes_{\mathbb{Z}}\mathbbm{k},*)$ of $H$. To establish this
connection, we endow with a new multiplication operator $\star$ on
$G_0(H)\otimes_{\mathbb{Z}}\mathbbm{k}$ and show that the Grothendieck algebra
$(G_0(H\#\mathbbm{k}G)\otimes_{\mathbb{Z}}\mathbbm{k},\ast)$ is isomorphic to
the direct sum of
$(G_0(H)\otimes_{\mathbb{Z}}\mathbbm{k},*)^{\oplus\frac{n}{2}}$ and
$(G_0(H)\otimes_{\mathbb{Z}}\mathbbm{k},\star)^{\oplus\frac{n}{2}}$.

Representation Theory;Category Theory