\newcommand{\mb}[1]{\mathbb{#1}}
\newcommand{\mf}[1]{\mathbf{#1}}
\newcommand{\F}{\mb{F}}
\newcommand{\G}{\mb{G}}
\newcommand{\R}{\mb{R}}
\newcommand{\Z}{\mb{Z}}
\newcommand{\abs}[1]{\left| #1 \right|}
\newcommand{\norm}[1]{\left\| #1 \right\|}
\newcommand{\isomorphic}{\simeq}
\newcommand{\To}{\rightarrow}
\newcommand{\Emph}[1]{\emph{\textcolor{blue}{#1}}}
\newcommand{\Cay}[1]{\operatorname{Cay}\left(#1\right)}
\newcommand{\diag}[1]{\operatorname{diag}\left(#1\right)}
\newcommand{\dual}[1]{\widetilde{#1}}
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\newcommand{\weight}[1]{\operatorname{wt}\left(#1\right)}
\newcommand{\weightclass}[1]{\operatorname{wc}\left(#1\right)}
\newenvironment{proofof}[1]{\noindent\emph{Proof of #1.}}{\qed}