Proof of The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity

Each result cited is universally quantified over the data in its own statement.

Preliminaries. Let $\Pi_{N}$ be the nonempty finite index set of the definition of the logarithmic energy, let $M$ be its number of elements, and fix a bijection $\sigma:[M]\to\Pi_{N}$, which exists as recorded in Sum over a Finite Index Set; by that definition $\sum_{p\in\Pi_{N}}g(p)=\sum_{m=1}^{M}g(\sigma(m))$ for every map $g:\Pi_{N}\to\mathbb{R}$. Each $p\in\Pi_{N}$ is written $p=(p_{1},p_{2})$, so that $p_{1}<p_{2}$, and for $x\in W_{N}$ the number $x_{p_{1}}-x_{p_{2}}$ is positive, as recorded in that definition. We use the antisymmetry

Indeed, if $k=j$ both sides are $0$; if $k\ne j$, then $x_{j}-x_{k}=-(x_{k}-x_{j})$ and $\bigl(-(x_{k}-x_{j})\bigr)\bigl(-(x_{k}-x_{j})^{-1}\bigr)=1$ by claim 2 of Zero Products and Elementary Identities in a Field, so $(x_{j}-x_{k})^{-1}=-(x_{k}-x_{j})^{-1}$.

Claim 1. Let $x\in W_{N}$ and put $t_{m}=x_{\sigma(m)_{1}}-x_{\sigma(m)_{2}}$ for $m\in[M]$, a positive number. Let $\mu_{1}=t_{1}$ and, for $m<M$, let $\mu_{m+1}$ be an element with $\mu_{m+1}\le\mu_{m}$, $\mu_{m+1}\le t_{m+1}$ and $\mu_{m+1}\in\{\mu_{m},t_{m+1}\}$, given by claim 9 of Elementary Order Arithmetic in an Ordered Field. By induction on $m$, $\mu_{m}$ is one of $t_{1},\dots,t_{m}$, hence positive, and $\mu_{m}\le t_{m'}$ for every $m'\le m$, by transitivity of $\le$. Since $\sigma$ is surjective, $\mu=\mu_{M}$ satisfies $0<\mu$ and $\mu\le x_{i}-x_{j}$ for every $(i,j)\in\Pi_{N}$. Let $\delta=\mu\cdot2^{-1}$; by claim 8 of Elementary Order Arithmetic in an Ordered Field, $0<\delta$ and $\delta+\delta=\mu$.

Let $y$ lie in the open ball $B_{d_{E}}(x,\delta)$, so $\lVert x-y\rVert=d_{E}(x,y)<\delta$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$. The $k$th coordinate of $x-y$ is $x_{k}-y_{k}$ (Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$), so claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and mixed transitivity (claim 2 of Elementary Order Arithmetic in an Ordered Field) give $|x_{k}-y_{k}|<\delta$, that is $-\delta<x_{k}-y_{k}<\delta$ by claim 9 of Properties of the Absolute Value in an Ordered Field, for every $k\in[N]$. Let $(i,j)\in\Pi_{N}$. Adding suitable terms (claim 1 of Elementary Order Arithmetic in an Ordered Field) gives $x_{i}-\delta<y_{i}$ and $y_{j}<x_{j}+\delta$, hence $-(x_{j}+\delta)<-y_{j}$ by claim 4 there, and claim 3 there yields

As $\mu\le x_{i}-x_{j}$, the left side is nonnegative by claim 3 of Elementary Arithmetic in an Ordered Field, so $0<y_{i}-y_{j}$ by claim 2 of Elementary Order Arithmetic in an Ordered Field, and $y_{j}<y_{i}$ by claim 1 there. Thus $y\in W_{N}$ by the definition of $W_{N}$, so $B_{d_{E}}(x,\delta)\subseteq W_{N}$. Hence $W_{N}$ is open in $(\mathbb{R}^{N},d_{E})$, which is openness in the sense of the setting (by Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$).

Let $q\in\mathbb{R}^{N}$ have $k$th coordinate $N-k+1$. If $1\le i<j\le N$, then $i<j$ as real numbers, so $-j<-i$ and $N+1-j<N+1-i$ by claims 4 and 1 of Elementary Order Arithmetic in an Ordered Field; that is $q_{j}<q_{i}$, and $q\in W_{N}$.

Claim 2. Let $J=(0,\infty)$, an interval as recorded in The Natural Logarithm. Every $t\in J$ is an interior point of $J$, since $t\cdot2^{-1}$ and $t+1$ lie in $J$ and $t\cdot2^{-1}<t<t+1$, by claims 8, 6, 1 and 3 of Elementary Order Arithmetic in an Ordered Field. For $p\in\Pi_{N}$ and $k\in[N]$ put $c_{k}(p)=1$ if $k=p_{1}$, $c_{k}(p)=-1$ if $k=p_{2}$, and $c_{k}(p)=0$ otherwise (well defined as $p_{1}\ne p_{2}$), and define functions on $W_{N}$, which is open by Claim 1, by

(a) Partial derivatives along slices. Let $f:J\to\mathbb{R}$ be differentiable at every point of $J$, with derivative $f'$. We show that for $x\in W_{N}$ and $k\in[N]$ the partial derivative of $f\circ L_{p}$ with respect to the $k$th variable exists at $x$ and

By claim 1 of Slice Function and the Partial Derivative there are $r_{0}>0$ and the interval $I=\{s:x_{k}-r_{0}<s<x_{k}+r_{0}\}$, with $x_{k}$ an interior point, such that $x[s]\in W_{N}$ for $s\in I$, where $x[s]$ is $x$ with $k$th coordinate replaced by $s$. Put $\gamma(s)=L_{p}(x[s])$ for $s\in I$; then $\gamma(s)\in J$, and checking the cases $k=p_{1}$, $k=p_{2}$ and $k\notin\{p_{1},p_{2}\}$ gives $\gamma(s)=L_{p}(x)+c_{k}(p)(s-x_{k})$. For $h\ne0$ with $x_{k}+h\in I$ the difference quotient $(\gamma(x_{k}+h)-\gamma(x_{k}))/h$ equals $c_{k}(p)$, so $\gamma$ is differentiable at $x_{k}$ with $\gamma'(x_{k})=c_{k}(p)$ by Derivative at an Interior Point. Since $\gamma(x_{k})=L_{p}(x)$ is an interior point of $J$, Chain Rule for One-Dimensional Derivatives shows that $f\circ\gamma$ is differentiable at $x_{k}$ with derivative $f'(L_{p}(x))\,c_{k}(p)$. As $f\circ\gamma$ is the slice function of $f\circ L_{p}$ at $x$ in the $k$th variable, claim 2 of Slice Function and the Partial Derivative gives (2.1).

Applying (2.1) to $f=\log$, whose derivative is $\log'(t)=t^{-1}$ by The Natural Logarithm, gives $\partial_{k}\varphi_{p}(x)=c_{k}(p)\rho_{p}(x)$. Applying it to $f=r$, $r(t)=t^{-1}$, which is differentiable at every $t\in J$ with $r'(t)=-(t^{-1})^{2}$ by claim 2 of Reciprocal Rule for One-Dimensional Derivatives (as $0\notin J$), and noting $\rho_{p}=r\circ L_{p}$, gives $\partial_{l}\rho_{p}(x)=-c_{l}(p)\rho_{p}(x)^{2}$ for $l\in[N]$. Since $\partial_{k}\varphi_{p}=c_{k}(p)\rho_{p}$ as functions on $W_{N}$, claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set gives

(b) Continuity. $L_{p}=\pi_{p_{1}}+(-1)\pi_{p_{2}}$ is smooth on $W_{N}$ by claims 2 and 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, hence continuous on $W_{N}$ relative to $W_{N}$ as a map into $(\mathbb{R},d_{\mathbb{R}})$ by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. The function $\log$ is continuous on $J$ relative to $J$ by Differentiability at an Interior Point Implies Continuity There, applied at every point of $J$, so $\varphi_{p}=\log\circ L_{p}$ is continuous on $W_{N}$ by claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map. The function $\rho_{p}$ is continuous on $W_{N}$ by claim 2 of Continuity of the Reciprocal of a Nonvanishing Real-Valued Function on a Metric Space, hence so are $c_{k}(p)\rho_{p}$ and $-c_{k}(p)c_{l}(p)\rho_{p}\rho_{p}$ by claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space. By claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions, all these functions are continuous in the Euclidean sense at every point of $W_{N}$.

(c) Regularity of $\varphi_{p}$. By (a), (b) and clause 1 of C^k Maps on a Euclidean Open Set, $\varphi_{p}$ is of class $C^{1}$ on $W_{N}$ (it is continuous and $\partial_{k}\varphi_{p}=c_{k}(p)\rho_{p}$ exists and is continuous), and so is each $\partial_{k}\varphi_{p}=c_{k}(p)\rho_{p}$ (it is continuous, and its partial derivatives (2.2) exist and are continuous). By clauses 2 and 3 there, $\varphi_{p}$ is of class $C^{2}$ on $W_{N}$.

(d) Regularity of $H$. For $m\in[M]$ let $\Psi_{m}(x)=\sum_{m'=1}^{m}\varphi_{\sigma(m')}(x)$ on $W_{N}$. By the recursion of claim 1 of Properties of Finite Sums ($\Psi_{1}=\varphi_{\sigma(1)}$ and $\Psi_{m+1}=\Psi_{m}+\varphi_{\sigma(m+1)}$) and claims 1 and 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, applied to $\Psi_{m}$ and $\varphi_{\sigma(m+1)}$ and then to $\partial_{k}\Psi_{m}$ and $\partial_{k}\varphi_{\sigma(m+1)}$, induction on $m$ shows that $\Psi_{m}$ is of class $C^{2}$ on $W_{N}$ with $\partial_{k}\Psi_{m}(x)=\sum_{m'=1}^{m}\partial_{k}\varphi_{\sigma(m')}(x)$ and $\partial_{k}\partial_{k}\Psi_{m}(x)=\sum_{m'=1}^{m}\partial_{k}\partial_{k}\varphi_{\sigma(m')}(x)$. By the definition of $H$ and the choice of $\sigma$, $H=(-\beta)\Psi_{M}$, so claims 1 and 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set show that $H$ is of class $C^{2}$ on $W_{N}$, in the sense of the setting (see clause 2 of the calculus setting), and, reading the sums along $\sigma$ back as sums over $\Pi_{N}$ and using (a) and (2.2) with $l=k$,

where the second form uses claim 4 of Properties of a Sum over a Finite Index Set with $\lambda=-1$ and claim 2 of Zero Products and Elementary Identities in a Field.

(e) Reindexing. Fix $x\in W_{N}$, $k\in[N]$ and $F:\mathbb{R}\to\mathbb{R}$ with $F(0)=0$. We show

Let $E=\{p\in\Pi_{N}:p_{1}=k\text{ or }p_{2}=k\}$ and $G=[N]\setminus\{k\}$; $G$ is nonempty since $1\ne2$ both lie in $[N]$ as $N\ge2$. Define $\theta:G\to E$ by $\theta(j)=(k,j)$ if $k<j$ and $\theta(j)=(j,k)$ if $j<k$. The coordinates of $\theta(j)$ are $k$ and $j$, so $\theta$ is injective; and if $p\in E$ with $p_{1}=k$ then $p=\theta(p_{2})$, while if $p_{2}=k$ then $p=\theta(p_{1})$, so $\theta$ is a bijection and $E$ is nonempty. For $j\in G$ we have $c_{k}(\theta(j))\rho_{\theta(j)}(x)=a_{kj}(x)$: if $k<j$ this is $1\cdot(x_{k}-x_{j})^{-1}$, and if $j<k$ it is $-(x_{j}-x_{k})^{-1}=a_{kj}(x)$ by (A). For $p\in\Pi_{N}\setminus E$, $c_{k}(p)=0$, so the term is $F(0)=0$ (claim 1 of Zero Products and Elementary Identities in a Field); likewise $F(a_{kk}(x))=F(0)=0$. Hence, by claim 4 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set, claim 2 of Properties of a Sum over a Finite Index Set along $\theta$, claim 4 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set again, and claim 1 of Properties of a Sum over a Finite Index Set,

Taking $F(t)=t$ and $F(t)=t^{2}$ in (2.4), (2.3) becomes $\partial_{k}H(x)=-\beta\sum_{j=1}^{N}a_{kj}(x)$ and $\partial_{k}\partial_{k}H(x)=\beta\sum_{j=1}^{N}a_{kj}(x)^{2}$. By Trace of a Real Square Matrix and Hessian Matrix of a C^2 Function, and claim 3 of Properties of Finite Sums,

Claim 3. Fix $x\in W_{N}$ and $z\in\mathbb{R}^{N}$, write $a_{kj}=a_{kj}(x)$, and let $Q=\sum_{k=1}^{N}\sum_{j=1}^{N}a_{kj}z_{k}$. By Interchange of a Finite Double Sum and renaming the bound indices, then (A) and claim 3 of Properties of Finite Sums with $\lambda=-1$,

Adding the two expressions for $Q$ and using claims 2 and 3 of Properties of Finite Sums, $Q+Q=\sum_{k=1}^{N}\sum_{j=1}^{N}a_{kj}(z_{k}-z_{j})$; since $Q+Q=2Q$ and $2^{-1}$ exists (claim 8 of Elementary Order Arithmetic in an Ordered Field), $Q=2^{-1}\sum_{k}\sum_{j}a_{kj}(z_{k}-z_{j})$. By Gradient of a Real-Valued Function on a Euclidean Open Set, Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$, Claim 2 and claim 3 of Properties of Finite Sums,

Claim 4. Let $x,y\in W_{N}$ and $z=x-y$, so $z_{k}=x_{k}-y_{k}$. For $k,j\in[N]$ put $d_{kj}=x_{k}-x_{j}$, $e_{kj}=y_{k}-y_{j}$ and $w_{kj}=z_{k}-z_{j}=d_{kj}-e_{kj}$. By claim 3 of Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$, Claim 3 at $x$ and at $y$, and claims 2 and 3 of Properties of Finite Sums,

Each summand is nonnegative. If $k=j$ it is $0\cdot w_{kk}=0$ (claim 1 of Zero Products and Elementary Identities in a Field). Let $k\ne j$ and write $d=d_{kj}$, $e=e_{kj}$. If $k<j$, then $(k,j)\in\Pi_{N}$ and $d,e$ are positive, so $0<de$ by claim 5 of Elementary Order Arithmetic in an Ordered Field; if $j<k$, then $-d,-e$ are positive and $de=(-d)(-e)$ is positive by claim 2 of Zero Products and Elementary Identities in a Field and the same claim 5. In both cases $(de)^{-1}$ is positive (claim 7 there), and field arithmetic gives $e^{-1}-d^{-1}=(d-e)(de)^{-1}$, so

Since $0\le(d-e)^{2}$ by claim 2 of Nonnegativity of Squares in an Ordered Field, claim 5 of Elementary Arithmetic in an Ordered Field (with $x=0$) and claim 1 of Zero Products and Elementary Identities in a Field give $0\le(d-e)^{2}(de)^{-1}$. By claim 5 of Properties of Finite Sums, applied to the inner and then the outer sum, the double sum is nonnegative; as $\tfrac{\beta}{2}$ is positive (claim 8 of Elementary Order Arithmetic in an Ordered Field), claim 5 of Elementary Arithmetic in an Ordered Field gives $0\le\bigl(DH(x)-DH(y)\bigr)\cdot(x-y)$.

Claim 5. Fix $x\in W_{N}$, write $a_{kj}=a_{kj}(x)$ and $B_{k}=\sum_{j=1}^{N}a_{kj}$. By claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, Gradient of a Real-Valued Function on a Euclidean Open Set, Claim 2, claim 2 of Zero Products and Elementary Identities in a Field and claim 3 of Properties of Finite Sums,

By claim 3 of Properties of Finite Sums, used twice, $B_{k}^{2}=\sum_{j=1}^{N}a_{kj}B_{k}=\sum_{j=1}^{N}\sum_{l=1}^{N}a_{kj}a_{kl}$. For $k,j,l\in[N]$ let $b_{kjl}=a_{kj}a_{kl}$ if $j\ne l$ and $b_{kjl}=0$ if $j=l$. For fixed $k,j$, the numbers $a_{kj}a_{kl}-b_{kjl}$ ($l\in[N]$) vanish for $l\ne j$ and equal $a_{kj}^{2}$ for $l=j$, so by claims 7 and 2 of Properties of Finite Sums, $\sum_{l}a_{kj}a_{kl}=a_{kj}^{2}+\sum_{l}b_{kjl}$. Summing over $j$ and $k$ with claim 2 there,

We show $b_{kjl}+b_{jkl}+b_{lkj}=0$ for all $k,j,l\in[N]$. Each of the three terms has the form $b_{uvw}$ with $(u,v,w)$ a rearrangement of $(k,j,l)$. If two of $k,j,l$ coincide, then in each term either $v=w$, so $b_{uvw}=0$ by definition, or $v\ne w$ and $u\in\{v,w\}$, so $b_{uvw}=a_{uv}a_{uw}$ has a factor $a_{uu}=0$ and vanishes by claim 1 of Zero Products and Elementary Identities in a Field. If $k,j,l$ are distinct, put $A=x_{k}$, $B=x_{j}$, $C=x_{l}$, which are pairwise distinct; using (A) and field arithmetic over the common nonzero denominator $D=(A-B)(A-C)(B-C)$ (nonzero by claim 3 of Zero Products and Elementary Identities in a Field),

By Interchange of a Finite Double Sum, applied to the outer two sums, and renaming the bound indices, $\sum_{k}\sum_{j}\sum_{l}b_{jkl}=\sum_{j}\sum_{k}\sum_{l}b_{jkl}=T$. Applying the same lemma first to the inner two sums and then to the outer two, and renaming, $\sum_{k}\sum_{j}\sum_{l}b_{lkj}=\sum_{k}\sum_{l}\sum_{j}b_{lkj}=\sum_{l}\sum_{k}\sum_{j}b_{lkj}=T$. Summing the three-term identity over $k,j,l$ with claim 2 of Properties of Finite Sums (a sum of zeros being $0$ by claim 3 there with $\lambda=0$ and claim 1 of Zero Products and Elementary Identities in a Field) gives $T+T+T=0$, that is $(1+1+1)T=0$. Since $0<1+1+1$ by claims 6, 8 and 3 of Elementary Order Arithmetic in an Ordered Field, claim 3 of Zero Products and Elementary Identities in a Field gives $T=0$. Hence $\lVert DH(x)\rVert^{2}=\beta^{2}S(x)$, and by Claim 2, $\beta\operatorname{tr}\bigl(D^{2}H(x)\bigr)=\beta\cdot\beta S(x)=\beta^{2}S(x)=\lVert DH(x)\rVert^{2}$.

Claim 6. Let $x\in W_{N}$. By Claim 3 with $z=x$,

For $k\ne j$, $a_{kj}(x)(x_{k}-x_{j})=(x_{k}-x_{j})^{-1}(x_{k}-x_{j})=1=\epsilon_{kj}$; for $k=j$, $a_{kk}(x)(x_{k}-x_{k})=0=\epsilon_{kk}$ by claim 1 of Zero Products and Elementary Identities in a Field. So the double sum is $d_{N}$, termwise, and $DH(x)\cdot x=-\tfrac{\beta}{2}d_{N}$.