I am reading a paper by McKay in 1971, the name of the paper is Irreducible Representations of Odd Degree. There is a theorem said: ${m}_{2}({S}_{n})=2^r$, where $n=\sum 2^{k_i}$, ${k}_{1}>{k}_{2}>\cdots>{k}_{a}$, and $r=\sum{k}_{i}$. The ${m}_{2}(G)$ here means the number of irreducible representations of odd degree. I have two questions about this proof. Firstly, what is the $k_i$ in this theorem? (not mentioned in the paper) Secondly, he chose an element $x$ with cycle type ${2}^{k_1},\dots,2^{k_a}$ and he said the character of $x$ is obtained from the diagram by successively removing hooks of length ${2}^{k_1},\dots,2^{k_a}$, what does this mean? Is there a simple example of this?
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1$\begingroup$ The $k_i$ are surely just the places where one has a $1$ when we write $n$ in base $2$? $\endgroup$– ancient mathematicianCommented Apr 29 at 6:45
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$\begingroup$ I think $k_i$ should be natual numbers, but I am not very sure about that. So I decided to ask this first :) $\endgroup$– AndyCommented Apr 29 at 6:49
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$\begingroup$ @ancientmathematician I think so :) $\endgroup$– AndyCommented Apr 29 at 8:21
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