Shifted character observables $p_{\rho }^{\mathrm{\#}}$ and profile moments ${\tilde{p}}_k$

(a) Shifted character observables. For a partition $\rho ⊢r$ (Partitions, English diagrams, and conjugation) and $\lambda ⊢n$ set $p_{\rho }^{\mathrm{\#}}(\lambda ):=\{\begin{array}{ll}{n}^{\downarrow r}{\displaystyle \frac{{\chi }_{\rho \cup 1^{n-r}}^{\lambda }}{{\dim }_{\mathbb{C}}{S}^{\lambda }}},& n\ge r,\\ 0,& n<r,\end{array}$ where $n^{\downarrow r}=n(n-1)\cdots (n-r+1)$ is the falling factorial of The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$, $\rho \cup 1^{n-r}=(\rho ,1,\dots ,1)⊢n$ is the padded partition, ${\chi }^{\lambda }$ is the complex irreducible character of $S_n$ indexed by $\lambda$, and ${\dim }_{\mathbb{C}}{S}^{\lambda }={\chi }_{(1^n)}^{\lambda }=f^{\lambda }>0$ by Standard polytabloids form a basis of a complex Specht module. Character values are the power-sum coefficients, ${\chi }^{\lambda }(\mu )=⟨s_{\lambda },p_{\mu }⟩$ for $\mu ⊢n$ (Irreducible symmetric-group character values are power-sum coefficients). For $n\ge r$ the quotient is a finite real number: a permutation and its inverse have the same cycle type and are conjugate (reverse the order within each cycle), while a complex character satisfies $\chi (g^{-1})=\overline{\chi (g)}$ and is constant on conjugacy classes (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). Therefore $\chi (g)=\overline{\chi (g)}$. The definition for $n<r$ is consistent with $n^{\downarrow r}=0$: in particular $p_1^{\mathrm{\#}}(\lambda )=n^{\downarrow 1}\frac{{\chi }_{(1^n)}^{\lambda }}{{\dim }_{\mathbb{C}}{S}^{\lambda }}=n\frac{f^{\lambda }}{f^{\lambda }}=n(\lambda ⊢n,n\ge 1)\mathrm{.}$

(b) Profile moments. For $\omega \in D_0$ with profile ${\sigma }_{\omega }=\frac{1}{2}(\omega -\mathrm{\mid }x\mathrm{\mid })$ (Continual diagrams, Russian profiles, and the $\sqrt{n}$-scaling of a Young diagram) and $k\ge 2$ define the profile moment ${\tilde{p}}_k[\omega ]:=k(k-1){\int }_{\mathbb{R}}{x}^{k-2}{\sigma }_{\omega }(x)dx,{\tilde{p}}_1[\omega ]:=0.$ The integrand is continuous, since ${\sigma }_{\omega }$ is Lipschitz, and compactly supported. Its integral is the Riemann integral over any compact interval containing its support, so it exists and is finite by A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion. If in addition $\omega$ is piecewise linear with finitely many corners, ${\sigma }_{\omega }$ is continuous, compactly supported and piecewise $C^1$, and applying integration by parts on each linear piece and summing (the boundary terms cancel, since ${\sigma }_{\omega }$ vanishes at the ends and its values at the interior corners enter twice with opposite signs) gives ${\tilde{p}}_k[\omega ]=-k{\int }_{\mathbb{R}}{x}^{k-1}{\sigma }_{\omega }^{\mathrm{\prime }}(x)dx(k\ge 2),$ in agreement with the source's formula (2.2), where ${\sigma }_{\omega }^{\mathrm{\prime }}$ exists except at the finitely many corners; this is the form used for the Young-diagram profiles of this page. The convention ${\tilde{p}}_1=0$ is the source's.

(c) Scaling. For every $s>0$ and $k\ge 2$, the substitution $y=sx$ (Monotone change of variable for Riemann-integrable functions, applied on a compact interval containing the support) in the definition of the $s$-scaling ${\omega }_s(x)=s^{-1}\omega (sx)$ gives ${\tilde{p}}_k[{\omega }_s]=k(k-1){\int }_{\mathbb{R}}{x}^{k-2}\frac{1}{2}(s^{-1}\omega (sx)-\mathrm{\mid }x\mathrm{\mid })dx=k(k-1)s^{-k}{\int }_{\mathbb{R}}{y}^{k-2}{\sigma }_{\omega }(y)dy=s^{-k}{\tilde{p}}_k[\omega ],$ because $\frac{1}{2}(s^{-1}\mathrm{\mid }sx\mathrm{\mid }-\mathrm{\mid }x\mathrm{\mid })=0$; hence for $\lambda ⊢n$, $n\ge 1$, the $\sqrt{n}$-scaled profile $\bar{\lambda }$ of Continual diagrams, Russian profiles, and the $\sqrt{n}$-scaling of a Young diagram satisfies ${\tilde{p}}_k[\bar{\lambda }]=n^{-k\mathrm{/}2}{\tilde{p}}_k[\lambda (\cdot )]\mathrm{.}$ No choice principle is used.