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This question is mostly about understanding the notation used in the following article: Alex Eskin, Andrei Okounkov, Pillowcases and quasimodular forms, in: Victor Ginzburg (ed.), Algebraic Geometry and Number Theory in Honor of Vladimir Drinfeld's 50th Birthday, Birkhäuser 2006.

page 8 about the 2-quotients of a partition. It claims that every partition ${\bf{\lambda}}:=(\lambda_1, \lambda_2,\ldots,)$ has two partitions $\bf{\alpha, \beta}$ such that

$$\Big\{\lambda_i -1 +1/2\Big\}=\Big\{\alpha_i -i+1/2 +\bar{p}_0(\lambda) \Big\}\sqcup \Big\{\beta_i -i+1/2+\bar{p}_0(\lambda) \Big\} $$

I mostly interested in balanced partition so I can forget about $\bar{p}_0(\lambda) $. I don't understand what is the definition of $\alpha_i, \beta_i$. Is it related to Frobenius coordinates?

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I suspect $\alpha_i$ and $\beta_i$ refer to heights in the Russian way to describe Young diagrams (rotate the English notation 135 degrees), this makes it into a piecewise linear function, and this interpretation has a few nice applications.

Information regarding quotients, (and its relation to character values in particular), can be found here. There is the article

Olivier Brunat and Rishi Nath. Cores and quotients of partitions through the Frobenius symbol. ArXiv e-prints, 2019.

which apparently does what the title suggest, in case the Frobenius description of partitions is relevant.

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