Normalized shifted character observables ${\eta }_{\rho }$

Fix $n\ge 1$. For a partition $\rho$ of Shifted character observables $p_{\rho }^{\mathrm{\#}}$ and profile moments ${\tilde{p}}_k$ write $\mathrm{\mid }\rho {\mathrm{\mid }}_1:=\mathrm{\mid }\rho \mathrm{\mid }+m_1(\rho )$, where $m_1(\rho )$ is the multiplicity of the part $1$. For $\lambda ⊢n$ with $n\ge \mathrm{\mid }\rho \mathrm{\mid }$ define the normalized observable ${\eta }_{\rho }^{(n)}(\lambda ):=\frac{p_{\rho }^{\mathrm{\#}}(\lambda )}{n^{\mathrm{\mid }\rho {\mathrm{\mid }}_1\mathrm{/}2}{\prod }_{k\ge 2}{k}^{m_k(\rho )\mathrm{/}2}}\mathrm{.}$ For $n<\mathrm{\mid }\rho \mathrm{\mid }$ declare ${\eta }_{\rho }^{(n)}(\lambda ):=0$, consistently with $p_{\rho }^{\mathrm{\#}}(\lambda )=0$ there. Since $p_1^{\mathrm{\#}}(\lambda )=n$, the definition can be written in the equivalent localized form ${\eta }_{\rho }^{(n)}=\frac{p_{\rho }^{\mathrm{\#}}}{\left(p_1^{\mathrm{\#}}{)}^{m_1(\rho )}{\prod }_{k\ge 2}\right(k(p_1^{\mathrm{\#}}{)}^k{)}^{m_k(\rho )\mathrm{/}2}}\text{on }{Y}_n,$ which is the concrete evaluation of the source's localization $A_{\mathrm{e}\mathrm{x}\mathrm{t}}=A[(p_1^{\mathrm{\#}}{)}^{1\mathrm{/}2},(p_1^{\mathrm{\#}}{)}^{-1\mathrm{/}2}]$ on each $Y_n$; the equality uses $p_1^{\mathrm{\#}}(\lambda )=n$, a positive number for $n\ge 1$, so the square roots are ordinary positive real roots. In particular, for a single part $\rho =(k)$, $k\ge 2$, one has ${\eta }_{(k)}^{(n)}=p_k^{\mathrm{\#}}\mathrm{/}(\sqrt{k}{n}^{k\mathrm{/}2})={\eta }_k^{(n)}$, the observable of Joint convergence in distribution and the normalized cycle-character observables.

Each ${\eta }_{\rho }^{(n)}$ is a real function on the finite set $Y_n$, hence a random variable on $(Y_n,P_n)$ (The Plancherel measure on the partitions of $n$), and for every $n\ge \mathrm{\mid }\rho \mathrm{\mid }$ the expectation of Plancherel expectations of the shifted character observables gives ${\mathbb{E}}_{P_n}\left[{\eta }_{\rho }^{(n)}\right]=\{\begin{array}{ll}{\displaystyle \frac{n^{\downarrow \mathrm{\mid }\rho \mathrm{\mid }}}{n^{\mathrm{\mid }\rho {\mathrm{\mid }}_1\mathrm{/}2}\prod _{k\ge 2}{k}^{m_k(\rho )\mathrm{/}2}}},& \rho =(1^{\mathrm{\mid }\rho \mathrm{\mid }}),\\ 0,& \rho \mathrm{\ne }(1^{\mathrm{\mid }\rho \mathrm{\mid }}),\end{array}$ so the expectation vanishes whenever $m_1(\rho )=0$ and $\rho \mathrm{\ne }\mathrm{\varnothing }$, and equals ${\prod }_{j=0}^{\mathrm{\mid }\rho \mathrm{\mid }-1}(1-j\mathrm{/}n)\le 1$ for $\rho =(1^{\mathrm{\mid }\rho \mathrm{\mid }})$; in every case it is $O(1)$ uniformly in $n$. Moreover $\mid {\eta }_{\rho }^{(n)}(\lambda )\mid \le n^{(\mathrm{\mid }\rho \mathrm{\mid }-m_1(\rho ))\mathrm{/}2}{\prod }_{k\ge 2}{k}^{-m_k(\rho )\mathrm{/}2}$ for every $\lambda ⊢n$, $n\ge \mathrm{\mid }\rho \mathrm{\mid }$: by For a complex character, $\chi (1)=\dim V$, $\chi$ is a class function, and $\mathrm{\mid }\chi (g)\mathrm{\mid }\le \chi (1)$ with equality exactly at scalars every irreducible character satisfies $\mathrm{\mid }{\chi }_{\mu }^{\lambda }\mathrm{\mid }\le {\chi }_{(1^n)}^{\lambda }={\dim }_{\mathbb{C}}{S}^{\lambda }$, so $\mathrm{\mid }{p}_{\rho }^{\mathrm{\#}}(\lambda )\mathrm{\mid }\le n^{\downarrow \mathrm{\mid }\rho \mathrm{\mid }}\le n^{\mathrm{\mid }\rho \mathrm{\mid }}$, and dividing by $n^{\mathrm{\mid }\rho {\mathrm{\mid }}_1\mathrm{/}2}{\prod }_{k\ge 2}{k}^{m_k(\rho )\mathrm{/}2}=n^{(\mathrm{\mid }\rho \mathrm{\mid }+m_1(\rho ))\mathrm{/}2}{\prod }_{k\ge 2}{k}^{m_k(\rho )\mathrm{/}2}$ gives the displayed bound. This normalization is the one used in Hermite leading terms for normalized shifted characters and Kerov's central limit theorem for normalized cycle characters. No choice principle is used.