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\title{Homework IX}
\author{Algebraic Combinatorics (Math 68)}
\date{\vspace{-.5em}Due November 13, 2019, at the \textbf{beginning of the class}}

\begin{document}
    \maketitle
    Collaboration among students to find key to the solution is encouraged, but each person must write the homework in his/her own words. You must write the name of the students with whom you work for each problem, as well as any written resource (web, book, etc.) that has been extensively used.
    
    You must write the appropriate justification as part of the solutions.\\
    
    \begin{enumerate}   
    \item Without looking in an external source (but you can talk to each other), find the table of characters of $S_4$. Give all the appropriate justifications. You have enough information not to use the Murnaghan-Nakayama rule.\\
    
    \item Without looking in an external source (but you can talk to each other), find the table of characters of $S_5$. Give all the appropriate justifications. Try to avoid as often as possible using the Murnaghan-Nakayama.\\
    
    \item Find the character for the Specht	module $\Yboxdim{5pt} S^{\yng(4,2,1)}$ (here, you should use the Murnaghan-Nakayama rule).\\
    
    \item Check that the character you found above is irreducible, using the orthogonality relations.\\
	\end{enumerate}

    \begin{center}
    	\textbf{Good luck!}
    \end{center}
\end{document}