# Decomposition of certain representation of cyclic group by irreducible.

Let $C_{n}$ denotes cyclic group of order $n$. Let the set of real irreducible representations of $C_{n}$ can be listed as

$$\begin{cases} \{1,\xi,\xi^2 , \cdots \xi^{(n-1)/2}\}, \text{when n is odd} \\ \{1,\sigma,\xi,\xi^2 , \cdots \xi^{(n-2)/2}\}, \text{when n is even} \end{cases}$$

Let $S(t,n)= \{ t \times n \; \text{matrixes with row sum is zero and column sum is zero} \}$.Then $C_n$ act on $S(t,n)$ by permuting columns. Therefore $S(t,n)$ is a representation of $C_{n}.$

Question: How can one write $S(t,n)$ in terms of irreducible representation of $C_{n}$ for both cases ?

Any help will be appreciated.

Thank you.

• Presumably $S(t,n)$ consists of real matrices. Isn't this just $(t-1)\chi$, where $\chi$ is the sum of all irreducible representations other than $1$? – Derek Holt Aug 31 '15 at 12:01
• Yes , $S(t,n)$ consists of real matrices. – Surojit Aug 31 '15 at 12:03
• @DerekHolt : Can you give a proof of it ? What is the case when only row sum is zero ? Is it $t \chi$? – Surojit Aug 31 '15 at 12:11
• Yes. I think both statements follow from the fact that $S(1,n)$ (with just row sum zero) is the regular representation but without $1$. – Derek Holt Aug 31 '15 at 12:27