text:
\begin{document}

\title[An Integral Basis for the Onsager Algebra]{An Integral Basis for the Universal Enveloping Algebra of the Onsager Algebra}

\author{Angelo Bianchi}
\email{acbianchi@unifesp.br}
\address{Universidade Federal de S\~ao Paulo }
\author{Samuel Chamberlin}
\email{samuel.chamberlin@park.edu}
\address{Park University, 8700 River Park Drive Box No. 30, Parkville, MO 64152}

\begin{abstract}
We construct an integral form for the universal enveloping algebra of the \textit{Onsager algebra} and an explicit integral basis for this integral form. We also formulate straightening identities among some products of basis elements.

\end{abstract}

\maketitle

MSC2020: Primary 17B65; Secondary 17B05

\tableofcontents

\section*{Introduction}

The focus of this paper is the Onsager algebra, which took on the name of the Nobel Prize winning Lars Onsager (1903-1976). This algebra appeared for the first time in Onsager's solution of the two-dimensional Ising model in a zero magnetic field,  \cite{O44}. In this work, two non-commuting matrices appeared related to the transfer matrix associated to this model. By analyzing the structure of the algebra generated by these matrices, Onsager derived a complex Lie algebra. There are a few different realizations of the Onsager algebra. We refer the reader to \cite{el} for a survey with details. The realization used here is as an equivariant map algebra. Equivariant map algebras have the form $\lie g_\mathcal A:=(\mathfrak{g}\otimes_{\mathbb C}\mathcal{A})^\Gamma$, where $\mathfrak{g}$ is a finite-dimensional complex simple Lie algebra, $\mathcal{A}$ is a commutative, associative, unital $\mathbb C$-algebra, and $\Gamma$ is a group acting on $\lie g$ and $\cal A$ (and hence diagonally on $\lie g_\mathcal A$).

We highlight the work of Roan \cite{R91} who first investigated the relationship between the Onsager algebra and the $\mathfrak{sl}_2$-loop algebra, folowed by Benkart and  Terwilliger \cite{BT07} and  Hartwig and Terwilliger \cite{HT07}. Since then, a few authors have written papers on this topic, see for instance \cite{DR00,E07}.

Suitable integral forms and bases for the universal enveloping algebras of the complex simple finite-dimensional Lie algebras were formulated by Kostant in 1966 (cf. \cite{K}), after Chevalley investigated integral forms for the classical Lie algebras in 1955. This led to the
construction of the classical Chevalley groups. In 1978, Garland formulated useful integral forms and bases for the universal enveloping algebras of the (untwisted) loop algebras, \cite{G}. Mitzman extended Garland's integral bases to the universal enveloping algebras of the twisted loop algebras in 1983, \cite{M}. The second author formulated suitable integral forms and bases for the universal enveloping algebras of the (untwisted) map algebras and (untwisted) map superalgebras, \cite{BC,C}.

Once one has access to suitable integral forms and bases for a particular Lie algebra one can study its representation theory in positive characteristic via its hyperalgebra. This was done for the (untwisted) hyper-loop algebras by Jakeli\'c and Moura using Garland's integral bases, \cite{JM1}. The first author and Moura extended this to the twisted hyper-loop algebras using Mitzman's integral bases, \cite{BM}. The authors extended these results to the (untwisted) hyper-multiloop, hyper-multicurrent, and hyper-map algebras this year using the integral bases formulated by the first author, \cite{BiC1,BiC2}.

When we view the Onsager algebra as an equivariant map algebra, the defining automorphism (the standard Chevalley involution) is not an isomorphism of the Dynkin diagram. No integral forms for any equivariant map algebras have been formulated in such a case in the literature.

Section \ref{pre} is dedicated to the algebraic preliminaries, including a review of the Onsager algebra. In Section \ref{main}, we construct an integral form and an integral basis for the Onsager algebra, the main result of the paper is stated as Theorem \ref{thm}. Section \ref{proof} contains all relevant proofs for the main result of the paper, which includes the necessary straightening identities.

\

\noindent\textbf{Note on the ArXiv version}  For the interested reader, the .tex file of the ArXiv version of this paper includes some details of straightforward proofs omitted in the .pdf file. These details can be displayed by activating the \texttt{details} toggle in the .tex file and recompiling it.

\section{Preliminaries}\label{pre}

Throughout this work we denote by $\mathbb C$, $\mathbb Z$, $\mathbb Z_+$ and $\mathbb N$ the sets of complex numbers, integers, nonnegative integers and positive integers, respectively.

\

If $\mathcal{A}$ is an  $\mathbb C$-algebra, an \textbf{integral form} $\mathcal{A}_{\mathbb Z}$ of $\mathcal{A}$ is a ${\mathbb Z}$-algebra such that $\mathcal{A}_{\mathbb Z}\otimes_{\mathbb Z}\mathbb C=\mathcal{A}$. An \textbf{integral basis} for $\mathcal{A}$ is a ${\mathbb Z}$-basis for $\mathcal{A}_{\mathbb Z}$.

\subsection{The \texorpdfstring{$\lie{sl}_2$}{}-algebra, its loop algebra, and a Chevalley involution} \label{sl2}

Let $\lie{sl}_2$ be the set of complex traceless order two matrices with Lie $\mathbb C$-algebra structure with bracket given by the commutator. Recall the standard basis of $\mathfrak{sl}_2$,
\begin{align*}
    \left\{ x^+ = \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right), \ h = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right), \ x^- = \left( \begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array} \right) \right\},
\end{align*}
satisfying $[x^+,x^-] = h$, $[h,x^+] = 2x^+$, and $[h,x^-] = -2x^-$.

In what follows an unadorned tensor means a tensor product over the field $\mathbb C$. Let $\mathbb C[t,t^{-1}] $ be the set of complex Laurent polynomials in one variable. Then the (untwisted) loop algebra of $\lie{sl}_2$ is the Lie algebra $\widetilde{\lie{sl}_2}:=\lie{sl}_2\otimes \mathbb C[t,t^{-1}] $ with Lie bracket given by bilinearly extending the bracket
$$[z\otimes f,z'\otimes g]=[z,z']\otimes fg,$$
where $z,z'\in\lie{sl}_2$ and $f,g\in\mathbb C [t,t^{-1}]$.

 The \emph{Chevalley involution} on $\mathfrak{sl}_2$ is defined by the map $$M \mapsto \overline{M} := \left( \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right) M \left(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}\right) $$ for all $M \in \mathfrak{sl}_2$. Therefore, under this involution we have
$$x^+ \mapsto   x^-, \quad x^- \mapsto x^+, \quad  h \mapsto -h.$$
Notice that the Chevalley involution is an involution defined as an automorphism of order two.

The Chevalley involution on $\mathfrak{sl}_2$ induces an involution on $\widetilde{\mathfrak{sl}_2}$, also called the \textit{Chevalley involution} on $\widetilde{\mathfrak{sl}_2}$, and denoted by $\omega$ where
$$ \omega(g\otimes f(t)) := \overline{g}\otimes f(t^{-1}) , \quad \text{for } f(t) \in \mathbb C[t,t^{-1}],\ g \in \mathfrak{sl}_2.$$

\subsection{Universal enveloping algebras}

For a Lie algebra $\lie a$, we denote by $U(\lie a)$ the corresponding universal enveloping algebra of the Lie algebra $\lie a$. Given $u\in U(\lie a)$ and $k\in{\mathbb Z}$ define the divided powers and binomials of $u$, respectively, as follows: $u^{(k)}=\binom{u}{k}=0$, if $k<0$, and
$$u^{(k)}:=\frac{u^k}{k!}\textnormal{ and }\binom{u}{k}:=\frac{u(u-1)\dots(u-k+1)}{k!},$$
if $k\geq0$.

Define $T^0(\lie a):=\mathbb C$, and for all $j\geq1$, define $T^j(\lie a):=\lie a^{\otimes j}$, $T(\lie a):=\bigoplus_{j=0}^\infty T^j(\lie a)$, and $T_j(\lie a):=\bigoplus_{k=0}^jT^k(\lie a)$. Then, set $U_j(\lie a)$ to be the image of $T_j(\lie a)$ under the canonical surjection $T(\lie a)\to U(\lie a)$, and for any $u\in U(\lie a)$ \emph{define the degree of $u$} by $$\deg u:=\min_{j}\{u\in U_j(\lie a)\}.$$

\subsection{The Onsager Algebra and one of its historical realizations}

We begin by with the original definition of the Onsager algebra and its relation to $\widetilde{\lie{sl_2}}$. We use the notation in \cite{el} instead of the original notation in \cite{O44}.

\

The Onsager algebra, denoted $\mathcal{O}$, is the (nonassociative) $\mathbb C$-algebra with basis $\{A_m, G_l ~ | ~ m \in \mathbb{Z}, l \in \mathbb{N}_+\}$ and antisymmetric product given by:
\begin{align*}
	[A_l,A_m] &= 2G_{l-m}, \text{ for } l > m,  \\
  [G_l,A_m] &= A_{m+l} - A_{m-l},   \\
  [G_l,G_m] &= 0.
\end{align*}

\

By considering the $\mathbb C$-linear map $\gamma:\mathcal{O} \rightarrow \widetilde{\lie{sl}_2}$ defined by
\begin{align}\label{onsager1}
A_m\longmapsto x^+\otimes t^m + x^-\otimes t^{-m}\qquad\qquad G_l\longmapsto\frac{1}{2}\left(h\otimes (t^l-t^{-l})\right)
\end{align}
for $m\in\mathbb{Z}$ and $l\in\mathbb{N}$, one can see that the Onsager algebra is isomorphic to the Lie subalgebra of $\widetilde{\lie{sl}_2}$ fixed by the Chevalley involution $\omega$ introduced in the Subsection \ref{sl2} (see details in \cite{R91}).

\subsection{The Onsager Algebra as an equivariant map algebra.}

Following the language of equivariant map algebras as in \cite{NSS2}, let $\Gamma:=\langle\sigma\rangle\cong{\mathbb Z}_2$ be a group of order two. We act on $\lie{sl}_2$ by the involution:
\begin{align*}
    \sigma\cdot(x^\pm)=-x^\mp\\
    \sigma\cdot(h)=-h,
\end{align*}
and on $\mathbb C[t, t^{-1}]$ by
$$\sigma\cdot(t) = t^{-1}.$$
This induces the diagonal action of $\Gamma$ on $\widetilde{\lie{sl}_2}$ given by
$$\sigma\cdot(g\otimes f)=\sigma\cdot(g)\otimes\sigma\cdot(f),$$ for all $g\in\lie{sl}_2$ and all $f\in \mathbb C[t,t^{-1}] $.
Therefore, the Onsager algebra $\mathcal O$ is also isomorphic to the Lie subalgebra of $\widetilde{\lie{sl}_2}$ which is fixed by this action of $\Gamma$, according to \cite[Remark 3.11 and Lemma 3.3]{NSS}.

\section{A better realization of the Onsager Algebra \texorpdfstring{$\mathcal O$}{} } \label{main}

The realization of the Onsager algebra in \eqref{onsager1} is not suitable to construct an integral form for $\mathcal O$ due to the intrinsic difficulty to deal with brackets between elements in \eqref{onsager1} and, hence, to produce useful identities and  $\mathbb Z$-subalgebras in $U(\mathcal O)$. The following realization of $\mathcal O$ allows straightening identities and the construction of  suitable subalgebras, which then gives rise to a ``triangular type decomposition'' and an integral form for $U(\mathcal O)$.
We start with the following basis for $\lie{sl}_2$ and then build the appropriate basis for $\mathcal O$ from this one.

We define the following elements in $\lie {sl}_2$: $$h^\Gamma:=-i\left(x^+-x^-\right), \quad x_+^\Gamma:=\frac{1}{2}\left(x^++x^--ih\right), \quad \text{ and } \quad x_-^\Gamma:=\frac{1}{2}\left(x^++x^-+ih\right).$$
Then, $\mathrm{span}\left\{h^\Gamma\right\}\cong\mathbb C$, $\lie{h}^*_\Gamma\cong\mathbb C$,  $\mathrm{span}\{x_-^\Gamma,h^\Gamma,x_+^\Gamma\}\cong\lie{sl}_2$, and $\sigma\left(x_\pm^\Gamma\right)=-x_\pm^\Gamma$.

Given $j,k,l\in{\mathbb Z}_+$, define
\begin{align*}
    h_k^\Gamma:=h^\Gamma\otimes\left(t^k+t^{-k}\right),\quad
    x_{j}^{\Gamma,+}:=x_+^\Gamma\otimes\left(t^j-t^{-j}\right), \quad \text{ and } \quad x_{l}^{\Gamma,-}:=x_-^\Gamma\otimes\left(t^l-t^{-l}\right),
\end{align*}
and note that $h_{-k}^\Gamma=h_k^\Gamma$, $x_{-j}^{\Gamma,\pm}=-x_{j}^{\Gamma,\pm}$, and $x_{0}^{\Gamma,\pm}=0$.

Therefore, by defining
$$u_j:=\left(x^+-x^-\right)\otimes\left(t^j+t^{-j}\right), \quad v_k:=\left(x^++x^-\right)\otimes\left(t^k-t^{-k}\right), \quad \text{ and } \quad w_l:=h\otimes\left(t^l-t^{-l}\right),$$
for all $j,k,l\in{\mathbb Z}$, then, using a similar argument to that in \cite[Proposition 3.2.2]{el}, we see that the set $\left\{u_j,v_k,w_l\ |\ j\in{\mathbb Z}_+,\ k,l\in\mathbb N\right\}$ is a $\mathbb C$-basis for $\cal O$ and, hence, the relations $$u_j=ih_j^\Gamma, \quad  v_k=x_k^{\Gamma,+}+x_k^{\Gamma,-},\text{ and}\quad w_l=-i\left(x_l^{\Gamma,-}-x_l^{\Gamma,+}\right)$$ demonstrate that the set $\mathcal B:=\left\{x_{l}^{\Gamma,-},h_k^\Gamma,x_{j}^{\Gamma,+}\ |\ k\in{\mathbb Z}_+,\ j,l\in\mathbb N\right\}$ spans $\cal O$. Furthermore, since $\cal B$ is clearly linearly independent it is a basis for $\cal O$.

 We can define an ordering, $\leq$, on $\cal B$ such that $x_{l}^{\Gamma,-}\leq h_{k}^{\Gamma}\leq x_{j}^{\Gamma,+}$ whenever $k\in\mathbb Z_+$ and $j,l\in\mathbb N$, $x_{l}^{\Gamma,\pm}\leq x_{j}^{\Gamma,\pm}$ whenever $l,j\in\mathbb N$ with $l\leq j$, and $h_{k}^{\Gamma}\leq h_{m}^{\Gamma}$ whenever $k,m\in\mathbb Z_+$ with $k\leq m$. Note that
\begin{align*}
    \left[x_{j}^{\Gamma,+},x_{l}^{\Gamma,-}\right]
    &=h^\Gamma_{j+l}-h^\Gamma_{j-l}=h^\Gamma_{j+l}-h^\Gamma_{|j-l|};\\
    \left[h_k^\Gamma,x_{j}^{\Gamma,+}\right]
    &=2\left(x^{\Gamma,+}_{j+k}+x^{\Gamma,+}_{j-k}\right)=2\left(x^{\Gamma,+}_{j+k}-x^{\Gamma,+}_{k-j}\right);\\
    \left[h_k^\Gamma,x_{l}^{\Gamma,-}\right]
    &=-2\left(x^{\Gamma,-}_{l+k}-x^{\Gamma,-}_{k-l}\right)=-2\left(x^{\Gamma,-}_{l+k}+x^{\Gamma,-}_{l-k}\right).
\end{align*}

\subsection{An integral form and integral basis for \texorpdfstring{$U(\mathcal O)$}{}.}

    Define $U_{\mathbb Z}(\mathcal O)$ to be the ${\mathbb Z}$-subalgebra of $U(\mathcal O)$ generated by
    $$\left\{\left(x_k^{\Gamma,\pm}\right)^{(s)}\ \bigg|\ k\in\mathbb N,\ s\in{\mathbb Z}_+\right\},$$
    $U_{\mathbb Z}^+(\mathcal O)$ and $U_{\mathbb Z}^-(\mathcal O)$ to be the ${\mathbb Z}$-subalgebras of $U_{\mathbb Z}(\mathcal O)$ generated respectively by
$$\left\{\left(x_k^{\Gamma,+}\right)^{(s)}\ \bigg|\ k\in\mathbb N,\ s\in{\mathbb Z}_+\right\}\textnormal{ and }\left\{\left(x_k^{\Gamma,-}\right)^{(s)}\ \bigg|\ k\in\mathbb N,\ s\in{\mathbb Z}_+\right\},$$
and $U_{\mathbb Z}^0(\mathcal O)=U\left(\left(\lie g^0\otimes\mathbb C [t,t^{-1}]\right)^\Gamma\right)\bigcap U_{\mathbb Z}(\mathcal O)$.

We need the following definitions in order to give our integral basis and to state our straightening identities.

Given $j,l\in\mathbb N$, define $\Lambda^\Gamma_{j,l,1}=-\left(h^\Gamma_{j+l}-h^\Gamma_{j-l}\right)$ and $D^{\Gamma,\pm}_{u,1}(j,l)\in U(\mathcal O)$, for $u\in{\mathbb Z}_+$, recursively as follows:
\begin{align*}
    D^{\Gamma,+}_{0,1}(j,l)
    &=\left(x_{j}^{\Gamma,+}\right);\\
    D^{\Gamma,-}_{0,1}(j,l)
    &=\left(x_{l}^{\Gamma,-}\right);\\
    D^{\Gamma,\pm}_{u,1}(j,l)
    &=\pm\frac{1}{2}\left[D^{\Gamma,\pm}_{u-1,1}(j,l),\Lambda_{j,l,1}^\Gamma\right].
\end{align*}

Furthermore, for $j,k,l\in\mathbb N$, set $p^{\Gamma}_k(j,l):=\left[x_{j}^{\Gamma,+},D^{\Gamma,-}_{k-1,1}(j,l)\right]$ and for $j,l\in\mathbb N$, and $k\in{\mathbb Z}$, define $\Lambda^\Gamma_{j,l,k}$ by equating coefficients in the following formal series:
    $$\Lambda_{j,l}^\Gamma(u):=\sum_{r=0}^\infty\Lambda_{j,l,r}^\Gamma u^r=\exp\left(-\sum_{s=1}^\infty\frac{p_s^\Gamma(j,l)}{s}u^s\right).$$
In particular, $\Lambda^\Gamma_{j,l,k}=0$ for $k<0$, and $\Lambda^\Gamma_{j,l,0}=1$.

\begin{prop}
    Given $j,k,l\in\mathbb N$,
    \begin{equation*}
        \Lambda^\Gamma_{j,l,k}=-\frac{1}{k}\sum_{i=1}^kp_i^\Gamma(j,l)\Lambda^\Gamma_{j,l,k-i}.
    \end{equation*}
\end{prop}
\begin{proof}
    We use an argument similar to that in \cite[Lemma 3.2]{CP}. Differentiating both sides of
    $$\log\Lambda^\Gamma_{j,l}(u)=-\sum_{s=1}^\infty\frac{p_s^\Gamma(j,l)}{s}u^s$$
    with respect to $u$ and then multiplying by $u$ gives
    $$u\frac{\left(\Lambda^\Gamma_{j,l}\right)'(u)}{\Lambda^\Gamma_{j,l}(u)}=-\sum_{s=1}^\infty p_s^\Gamma(j,l)u^s.$$
    Multiplying both sides of this equation by $\Lambda^\Gamma_{j,l}(u)$ gives
    $$\sum_{k=1}^\infty k\Lambda_{j,l,k}^\Gamma u^k=-\sum_{s=1}^\infty p_s^\Gamma(j,l)u^s\Lambda_{j,l}^\Gamma(u).$$
    Expanding $\Lambda_{j,l}^\Gamma(u)$ and the product on the right side and equating coefficients gives the result.
\end{proof}

Given a PBW monomial with respect to the order on $\mathcal B$, we construct an ordered monomial in the elements of the set
$$M:=\left\{\left(x^{\Gamma,+}_j\right)^{(r)},\Lambda^\Gamma_{j,l,k},\left(x^{\Gamma,-}_l\right)^{(s)}\ \bigg|\ j,l\in\mathbb N,\ k,r,s\in{\mathbb Z}_+\right\}$$
via the correspondence
$$\left(x^{\Gamma,\pm}_j\right)^k\leftrightarrow\left(x^{\Gamma,\pm}_j\right)^{(k)}\quad\text{ and }\quad\left(p_1^\Gamma(l,m)\right)^{r}\leftrightarrow\Lambda^\Gamma_{l,m,r}.$$

\

The main goal of this paper is to prove the following theorem, whose proof is in Section \ref{proof} (see \cite{BC,C,G,H,M} for analogs in different settings).

\begin{thm}\label{thm}
    The subalgebra $U_{\mathbb Z}(\mathcal O)$ is a free ${\mathbb Z}$-module and the set of ordered monomials constructed from $M$ is a ${\mathbb Z}$ basis of $U_{\mathbb Z}(\mathcal O)$.
\end{thm}

This theorem implies the following:
\begin{align*}
    \mathbb C\otimes_{\mathbb Z} U_{\mathbb Z}(\mathcal O)&\cong U(\mathcal O),\\
    \mathbb C\otimes_{\mathbb Z} U_{\mathbb Z}^\pm(\mathcal O)&\cong U\left(\mathrm{span}\left\{x_j^{\Gamma,\pm}\ |\ j\in\mathbb N\right\}\right),\\
    \mathbb C\otimes_{\mathbb Z} U_{\mathbb Z}^0(\mathcal O)&\cong U\left(\mathrm{span}\left\{h_k^{\Gamma}\ |\ k\in{\mathbb Z}_+\right\}\right).\\
\end{align*}
In particular, $U_{\mathbb Z}(\mathcal O)$ is an \emph{integral form} of $U(\mathcal O)$.

\subsection{Identities for \texorpdfstring{$D^{\Gamma,\pm}_{u,v}(j,l)$}{}.}
Define $D^{\Gamma,\pm}_{u,v}(j,l)\in U(\mathcal O)$ recursively as follows:
\begin{align*}
    D^{\Gamma,\pm}_{u,v}(j,l)
    &=0\textnormal{ if }v<0;\\
    D^{\Gamma,\pm}_{u,0}(j,l)
    &=\delta_{u,0};\\
    D^{\Gamma,\pm}_{u,v}(j,l)
    &=\frac{1}{v}\sum_{i=0}^uD^{\Gamma,\pm}_{i,1}(j,l)D^{\Gamma,\pm}_{u-i,v-1}(j,l).
\end{align*}

The following proposition gives the $D^{\Gamma,\pm}_{u,v}(j,l)$ as the coefficients of a power series in the indeterminate $w$. The proof of the first equation is done by induction on $v$ and we omit the details. The second equation follows by the Multinomial Theorem.

\begin{prop}\label{Duv}
    For all $j,l,u,v\in\mathbb N$
    \begin{align*}
        D^{\Gamma,\pm}_{u,v}(j,l)
        &=\sum_{\substack{k_0,\ldots,k_u\in{\mathbb Z}_+\\k_0+\dots k_u=v\\k_1+2k_2+\dots+uk_u=u}}\left(D^{\Gamma,\pm}_{0,1}(j,l)\right)^{(k_0)}\dots\left(D^{\Gamma,\pm}_{u,1}(j,l)\right)^{(k_u)}\\
        &=\left(\left(\sum_{m\geq0}D^{\Gamma,\pm}_{m,1}(j,l)w^{m+1}\right)^{(v)}\right)_{u+v}
    \end{align*}
    \hfill\qedsymbol
\end{prop}

It can be easily shown by induction that $D^{\Gamma,+}_{0,v}(j,l)=\left(x_{j}^{\Gamma,+}\right)^{(v)}$ and $D^{\Gamma,-}_{0,v}(j,l)=\left(x_{l}^{\Gamma,-}\right)^{(v)}$.

A straightforward calculation shows that
\begin{align}
    \left[\left(x_{k}^{\Gamma,+}\right),\Lambda_{j,l,1}^\Gamma\right]
    &=2\left(x_{k+j+l}^{\Gamma,+}\right)+2\left(x_{k-j-l}^{\Gamma,+}\right)-2\left(x_{k+j-l}^{\Gamma,+}\right)-2\left(x_{k-j+l}^{\Gamma,+}\right).\label{xk+Lambda_1}
\end{align}

The following proposition gives the $D^{\Gamma,\pm}_{u,1}(j,l)$ as ${\mathbb Z}$-linear combinations of the $\left(x^{\Gamma,\pm}_{n}\right)$.

\begin{prop}\label{Du1}
    For all $u\in{\mathbb Z}_+$ and $j,l\in\mathbb N$
    \begin{align}
        D^{\Gamma,+}_{u,1}(j,l)
        &=\sum_{k=0}^{\left\lfloor\frac{u-1}{2}\right\rfloor}\sum_{i=0}^{u+1}(-1)^{k+i}\binom{u}{k}\binom{u+1}{i}\left(x_{(u+1-2i)j+(u-2k)l}^{\Gamma,+}\right)\nonumber\\
        &+((u+1)\mod2)\sum_{i=0}^{\frac{u}{2}}(-1)^{\frac{u}{2}+i}\binom{u}{\frac{u}{2}}\binom{u+1}{i}\left(x_{(u+1-2i)j}^{\Gamma,+}\right)\label{D+u1};\\
        D^{\Gamma,-}_{u,1}(j,l)
        &=\sum_{k=0}^{\left\lfloor\frac{u-1}{2}\right\rfloor}\sum_{i=0}^{u+1}(-1)^{k+i}\binom{u}{k}\binom{u+1}{i}\left(x_{(u+1-2i)l+(u-2k)j}^{\Gamma,-}\right)\nonumber\\
        &+((u+1)\mod2)\sum_{i=0}^{\frac{u}{2}}(-1)^{\frac{u}{2}+i}\binom{u}{\frac{u}{2}}\binom{u+1}{i}\left(x_{(u+1-2i)l}^{\Gamma,-}\right)\label{D-u1}.
    \end{align}
\end{prop}

The proof of \eqref{D+u1} has been omitted to shorten the text due to its straightforwardness.  \eqref{D-u1} follows from \eqref{D+u1} by applying the automorphism $\tau:\cal O\to\cal O$ given by $\tau\left(x^{\Gamma,\pm}_j\right)=x^{\Gamma,\mp}_j$ and $\tau\left(h^{\Gamma}_k\right)=-h^\Gamma_k$ and switching $j$ and $l$.

\subsection{Straightening Identities}
In this section we state all necessary straightening identities.

Given $u\in{\mathbb Z}_+$ and $j,k,m\in\mathbb N$ define
\begin{equation*}
    D^{\Gamma,\pm}_u(j,k,m)=\sum_{n=0}^u\sum_{v=0}^u(-1)^{n+v}\binom{u}{n}\binom{u}{v}\left(x_{j+(u-2n)k+(u-2v)m}^{\Gamma,\pm}\right).
\end{equation*}

\begin{prop}\label{straightening}
    For all $j,l,k,m\in\mathbb N$ and $n,r,s\in{\mathbb Z}_+$
    \begin{align}
        \Lambda^\Gamma_{j,l,r}\Lambda^\Gamma_{k,m,n}
        &=\Lambda^\Gamma_{k,m,n}\Lambda^\Gamma_{j,l,r}\\
        \left(x_j^{\Gamma,\pm}\right)^{(r)}\left(x_j^{\Gamma,\pm}\right)^{(s)}
        &=\binom{r+s}{s}\left(x_j^{\Gamma,\pm}\right)^{(r+s)}\\
        \left(x_j^{\Gamma,+}\right)^{(r)}\left(x_l^{\Gamma,-}\right)^{(s)}
        &=\sum_{\substack{m,n,q\in{\mathbb Z}_+\\m+n+q\leq\min\{r,s\}}}(-1)^{m+n+q}D^{\Gamma,-}_{m,s-m-n-q}(j,l)\Lambda^\Gamma_{j,l,n}D^{\Gamma,+}_{q,r-m-n-q}(j,l)\\
        \left(x_j^{\Gamma,+}\right)^{(r)}\Lambda^\Gamma_{k,m,n}
        &=\sum_{i=0}^n\Lambda_{k,m,n-i}^\Gamma\prod_{\substack{v_0\ldots v_u\in{\mathbb Z}_+\\\sum v_u=r\\\sum uv_u=i}}\left((u+1)D^{\Gamma,+}_{u}(j,k,m)\right)^{(v_u)}\\
        \Lambda^\Gamma_{k,m,n}\left(x_l^{\Gamma,-}\right)^{(s)}
        &=\sum_{i=0}^n\prod_{\substack{v_0\ldots v_u\in{\mathbb Z}_+\\\sum v_u=s\\\sum uv_u=i}}\left((u+1)D^{\Gamma,-}_{u}(l,k,m)\right)^{(v_u)}\Lambda_{k,m,n-i}^\Gamma
    \end{align}
\end{prop}
(5) and (6) are clear. Before proving (7)--(9)  we will state and prove a necessary corollary and prove Theorem \ref{thm}.

\begin{cor}
    For all $j,l\in\mathbb N$ and $u,v\in{\mathbb Z}_+$
    \begin{align*}
        (i)_u&\ \ D^{\Gamma,\pm}_{u,v}(j,l)\in U_{\mathbb Z}(\mathcal O);\\
        (ii)_u&\ \ \Lambda^\Gamma_{j,l,u}\in U_{\mathbb Z}(\mathcal O).
    \end{align*}
\end{cor}
\begin{proof}
    We will prove both statements simultaneously by induction on $u$ according to the scheme
    $$(ii)^u\Rightarrow (i)^u\Rightarrow (ii)^{u+1}$$
    where $(i)^u$ is the statement that $(i)_m$ holds for all $m\leq u$ and similarly for $(ii)^u$. $(i)_0$, $(ii)_0$ and $(ii)_1$ are all clearly true. Assume that $(ii)^u$ and $(i)^{u-1}$ hold for some $u\in{\mathbb Z}_+$. Then, by Proposition \ref{straightening}(7) we have
    \begin{align*}
        \left(x_j^{\Gamma,+}\right)^{(u+v)}\left(x_l^{\Gamma,-}\right)^{(u)}
        &=\sum_{\substack{m,n,q\in{\mathbb Z}_+\\m+n+q\leq u}}(-1)^{m+n+q}D^{\Gamma,-}_{m,u-m-n-q}(j,l)\Lambda^\Gamma_{j,l,n}D^{\Gamma,+}_{q,u+v-m-n-q}(j,l)\\
        &=(-1)^{u}D^{\Gamma,+}_{u,v}(j,l)+\sum_{\substack{m,n,q\in{\mathbb Z}_+\\m+n+q\leq u\\m,q<u}}(-1)^{m+n+q}D^{\Gamma,-}_{m,u-m-n-q}(j,l)\Lambda^\Gamma_{j,l,n}D^{\Gamma,+}_{q,u+v-m-n-q}(j,l)
    \end{align*}
    The summation is in $U_{\mathbb Z}(\mathcal O)$ by the induction hypothesis and $(ii)^u$. Therefore, $(i)^u$ holds.

    Again, by Proposition \ref{straightening}(7), we have
    \begin{align*}
        \left(x_j^{\Gamma,+}\right)^{(u+1)}\left(x_l^{\Gamma,-}\right)^{(u+1)}
        &=\sum_{\substack{m,n,q\in{\mathbb Z}_+\\m+n+q\leq u+1}}(-1)^{m+n+q}D^{\Gamma,-}_{m,u+1-m-n-q}(j,l)\Lambda^\Gamma_{j,l,n}D^{\Gamma,+}_{q,u+1-m-n-q}(j,l)\\
        &=(-1)^{u+1}\Lambda^\Gamma_{j,l,u+1}\\
        &+\sum_{\substack{m,n,q\in{\mathbb Z}_+\\m+n+q\leq u+1\\n<u+1}}(-1)^{m+n+q}D^{\Gamma,-}_{m,u+1-m-n-q}(j,l)\Lambda^\Gamma_{j,l,n}D^{\Gamma,+}_{q,u+1-m-n-q}(j,l)\\
    \end{align*}
    The summation is in $U_{\mathbb Z}(\mathcal O)$ by the induction hypothesis and $(i)^u$. Therefore, $(ii)^{u+1}$ holds.
\end{proof}

\section{Proof of the main result }\label{proof}

\subsection{Proof of Theorem \ref{thm}}

In this section we will prove Theorem \ref{thm}. The proof will proceed by induction on the degree of monomials in $U_{\mathbb Z}(\mathcal O)$ and the following lemmas and proposition.

\begin{lem}\label{brackets}
    For all $r,s,n\in{\mathbb Z}_+$ and $j,k,l,m\in\mathbb N$
    \begin{enumerate}
        \item $\left[\left(x_j^{\Gamma,+}\right)^{(r)},\left(x_l^{\Gamma,-}\right)^{(s)}\right]$ is in the $\mathbb Z$-span of $\mathcal B$ and has degree less that $r+s$.

        \item $\left[\left(x_j^{\Gamma,+}\right)^{(r)},\Lambda^\Gamma_{k,m,n}\right]$ is in the $\mathbb Z$-span of $\mathcal B$ and has degree less that $r+n$.

        \item $\left[\Lambda^\Gamma_{k,m,n},\left(x_l^{\Gamma,-}\right)^{(s)}\right]$ is in the $\mathbb Z$-span of $\mathcal B$ and has degree less that $s+n$.
    \end{enumerate}
\end{lem}
\begin{proof}
    For (1), by Proposition \ref{straightening}(7) we have
    \begin{equation*}
        \left[\left(x_j^{\Gamma,+}\right)^{(r)},\left(x_l^{\Gamma,-}\right)^{(s)}\right]
        =\sum_{\substack{m,n,q\in{\mathbb Z}_+\\0<m+n+q\leq\min\{r,s\}}}(-1)^{m+n+q}D^{\Gamma,-}_{m,s-m-n-q}(j,l)\Lambda^\Gamma_{j,l,n}D^{\Gamma,+}_{q,r-m-n-q}(j,l).
    \end{equation*}
    By Proposition \ref{Du1} and the definition of $D^{\Gamma,\pm}_{u,v}(j,l)$ we see that the terms of the summation are in the $\mathbb Z$-span of $\mathcal B$ and have degree $r+s-2m-n-2q<r+s$. For (2) and (3) we have, by Proposition \ref{straightening}(8) and (9),
    \begin{align*}
        \left[\left(x_j^{\Gamma,+}\right)^{(r)},\Lambda^\Gamma_{k,m,n}\right]
        &=\sum_{i=1}^n\Lambda_{k,m,n-i}^\Gamma\prod_{\substack{v_0\ldots v_u\in{\mathbb Z}_+\\\sum v_u=r\\\sum uv_u=i}}\left((u+1)D^{\Gamma,+}_{u}(j,k,m)\right)^{(v_u)},\\
        \left[\Lambda^\Gamma_{k,m,n},\left(x_l^{\Gamma,-}\right)^{(s)}\right]
        &=\sum_{i=1}^n\prod_{\substack{v_0\ldots v_u\in{\mathbb Z}_+\\\sum v_u=s\\\sum uv_u=i}}\left((u+1)D^{\Gamma,-}_{u}(l,k,m)\right)^{(v_u)}\Lambda_{k,m,n-i}^\Gamma.
    \end{align*}
    In either case the right-hand side is in the ${\mathbb Z}$-span of $\mathcal B$ and has degree $n-i+r<n+r$ or $n-i+s<n+s$ respectively by the definition of $D^{\Gamma,\pm}_{u}(j,k,m)$.
\end{proof}

The following proposition gives the $p^{\Gamma}_u(j,l)$ as ${\mathbb Z}$-linear combinations of the $h^{\Gamma}_{n}$.

\begin{prop}\label{p_u}
    Given $u\in\mathbb N$
    \begin{align*}
        p^{\Gamma}_u(j,l)
        &=\sum_{k=0}^{\left\lfloor\frac{u-1}{2}\right\rfloor}\sum_{i=0}^{u}(-1)^{k+i}\binom{u}{k}\binom{u}{i}\left(h_{(u-2i)j+(u-2k)l}^{\Gamma}\right)\nonumber\\
        &+((u+1)\mod2)\sum_{i=0}^{u}(-1)^{\frac{u}{2}+i}\binom{u-1}{\frac{u-2}{2}}\binom{u}{i}\left(h_{(u-2i)j}^{\Gamma}\right)
    \end{align*}\hfill\qedsymbol
\end{prop}

The proof is omitted due to straightforwardness.

\begin{lem}\label{LambdaLambda}
    For all $j,l\in\mathbb N$ and $k,m\in\mathbb N$
    \begin{equation*}
        \Lambda_{j,l,k}^\Gamma\Lambda_{j,l,m}^\Gamma=\binom{k+m}{k}\Lambda_{j,l,k+m}^\Gamma+u
    \end{equation*}
    where $u$ is in the ${\mathbb Z}$-span of the set $\left\{\Lambda_{i_1,n_1,q_1}^\Gamma\dots\Lambda_{i_s,n_s,q_s}^\Gamma\ \Big|\ i_1,\ldots,i_s,n_1\ldots,n_s\in\mathbb N,\ q_1,\ldots,q_s\in{\mathbb Z}_+\right\}$ and has degree less than $k+m$.
\end{lem}
\begin{proof}
    $\Lambda_{j,l,k}^\Gamma$ corresponds to $\Lambda_{k-1}$ from \cite{G} under the map $\tau:U\left((\lie g^0\otimes\mathbb C [t,t^{-1}])^\Gamma\right)\to\mathbb C[X_1,X_2,\ldots]$ given by $\tau(p_i^\Gamma(j,l))=-X_i$. Therefore, by Lemma 9.2 in \cite{G}, it suffices to show that $p_i^\Gamma(j,l)$ is an integer linear combination of $\Lambda^\Gamma_{i,n,q}$. To that end we have, for all $n\in{\mathbb Z}_+$
    \begin{align}
        p_{2n+1}^\Gamma(j,l)
        &=\sum_{i=0}^{n}\sum_{k=0}^{n}(-1)^{k+i+1}\binom{2n+1}{i}\binom{2n+1}{k}\Lambda^\Gamma_{(2n+1-2i)j,(2n+1-2k)l,1}.\label{p2n+1}\\
        p^\Gamma_{2n}(j,l)
        &=\sum_{i=0}^{n-1}\sum_{k=0}^{2n}(-1)^{k+i+1}\binom{2n-1}{i}\binom{2n}{k}\Lambda^\Gamma_{(2n-1-2i)j+(2n-2k)l,j,1}.\label{p2n}
    \end{align}
    The proofs of \eqref{p2n+1} and \eqref{p2n} are straightforward and have been omitted to shorten the text.
\end{proof}

We can now prove Theorem \ref{thm}.
\begin{proof}
    Let $B$ be the set of ordered monomials constructed from $M$. The PBW Theorem implies that $B$ is a $\mathbb C$-linearly independent set. Thus it is a ${\mathbb Z}$-linearly independent set.

    The proof that the ${\mathbb Z}$-span of $B$ is $U_{\mathbb Z}(\cal O)$ will proceed by induction on the degree of monomials in $U_{\mathbb Z}(\cal O)$. Since $\Lambda^\Gamma_{j,l,1}=-\left(h^\Gamma_{j+l}-h^\Gamma_{j,l}\right)$, any degree one monomial is in the ${\mathbb Z}$-span of $B$. Now take any monomial, $m$, in $U_{\mathbb Z}(\cal O)$. If $m\in B$, then we are done. If not then either the factors of $m$ are not in the correct order or $m$ has ``repeated'' factors with the following forms
    \begin{equation}\label{repeated}
        \left(x_j^{\Gamma,\pm}\right)^{(r)}\text{ and }\left(x_j^{\Gamma,\pm}\right)^{(s)}\text{ or }\Lambda^\Gamma_{j,l,r}\text{ and }\Lambda^\Gamma_{j,l,s},\ j,l,r,s\in\mathbb N.
    \end{equation}
    If the factors of $m$ are not in the correct order, then we can rearrange the factors of $m$ using the straightening identities in Proposition \ref{straightening}. Once this is done Lemma \ref{brackets} guarantees that each rearrangement will only produce ${\mathbb Z}$-linear combinations of monomials in the correct order with lower degree. These lower degree monomials are then in the ${\mathbb Z}$-span of $B$ by the induction hypothesis.

    If (possibly after rearranging factors as above) $m$ contains the products of the pairs of factors in \eqref{repeated} we apply Proposition \ref{straightening}(6) or Lemma \ref{LambdaLambda} respectively to consolidate these pairs of factors into single factors with integral coefficients.

    In the end we see that $m\in{\mathbb Z}$-span $B$. Thus the ${\mathbb Z}$-span of $B$ is $U_{\mathbb Z}(\cal O)$ and hence $B$ is an integral basis for $U_{\mathbb Z}(\cal O)$.
\end{proof}

All that remains is to prove Proposition \ref{straightening}(7)-(9).

\subsection{Proof of Proposition \ref{straightening} (8) and (9).} We first need the following propositions.

\begin{prop}\label{pnewD}
    For all $k,u\in{\mathbb Z}_+$
    \begin{equation*}
        p_{k+u+1}^\Gamma(j,l)=\left[D^{\Gamma,+}_{u,1}(j,l),D^{\Gamma,-}_{k,1}(j,l)\right]
    \end{equation*}
\end{prop}
\begin{proof}
    To prove the claim we proceed by induction on $u$. If $u=0$ the claim is true by definition. Now assume the claim for some $u\in{\mathbb Z}_+$. Then we have
    \begin{align*}
        \left[D^{\Gamma,+}_{u+1,1}(j,l),D^{\Gamma,-}_{k,1}(j,l)\right]
        &=\frac{1}{2}\left[\left[D^{\Gamma,+}_{u,1}(j,l),\Lambda_{j,l,1}^\Gamma\right],D^{\Gamma,-}_{k,1}(j,l)\right]\\
        &=-\frac{1}{2}\left[D^{\Gamma,-}_{k,1}(j,l),\left[D^{\Gamma,+}_{u,1}(j,l),\Lambda_{j,l,1}^\Gamma\right]\right]\\
        &=\frac{1}{2}\left[D^{\Gamma,+}_{u,1}(j,l),\left[\Lambda_{j,l,1}^\Gamma,D^{\Gamma,-}_{k,1}(j,l)\right]\right]+\frac{1}{2}\left[\Lambda_{j,l,1}^\Gamma,\left[D^{\Gamma,-}_{k,1}(j,l),D^{\Gamma,+}_{u,1}(j,l)\right]\right]\\
        &\textnormal{ by the Jacobi Identitiy}\\
        &=\left[D^{\Gamma,+}_{u,1}(j,l),D^{\Gamma,-}_{k+1,1}(j,l)\right]-\frac{1}{2}\left[\Lambda_{j,l,1}^\Gamma,p_{k+u+1}^\Gamma(j,l)\right]\textnormal{ by the induction hypothesis}\\
        &=p_{k+u+1}^\Gamma(j,l)\textnormal{ by the induction hypothesis again}
    \end{align*}
\end{proof}

We now give the proof of Proposition \ref{straightening} (8). The proof of $(9)$ is similar.

\begin{proof}
We proceed by induction on $r$. The case $r=0$ is clear. In the case $r=1$ we need to show

    \begin{equation}\label{xj+Lambdan}
        \left(x_j^{\Gamma,+}\right)\Lambda^\Gamma_{k,m,n}=\sum_{i=0}^n\sum_{r=0}^{i}\sum_{s=0}^i(-1)^{r+s}(i+1)\binom{i}{r}\binom{i}{s}\Lambda^\Gamma_{k,m,n-i}\left(x_{j+(i-2r)k+(i-2s)m}^{\Gamma,+}\right).
    \end{equation}
    We will prove this by induction on $n$. The case $n=1$ is a straightforward calculation. Assume for some $n\in\mathbb N$ that \eqref{xj+Lambdan} holds for all $1\leq q\leq n$. Then we have
    \begin{align*}
        \left(x_j^{\Gamma,+}\right)\Lambda^\Gamma_{k,m,n}
        &=-\frac{1}{n}\sum_{i=1}^n\left(x_j^{\Gamma,+}\right)p^\Gamma_i(k,m)\Lambda^\Gamma_{k,m,n-i}\\
        &=-\frac{1}{n}\sum_{i=1}^np^\Gamma_i(k,m)\left(x_j^{\Gamma,+}\right)\Lambda^\Gamma_{k,m,n-i}\\
        &+\frac{2}{n}\sum_{i=1}^n\sum_{r=0}^i\sum_{s=0}^i(-1)^{r+s}\binom{i}{r}\binom{i}{s}\left(x_{j+(i-2r)k+(i-2s)m}^{\Gamma,+}\right)\Lambda^\Gamma_{k,m,n-i}\textnormal{ by Proposition }\ref{bracketxp}\\
        &=-\frac{1}{n}\sum_{i=1}^n\sum_{u=0}^{n-i}\sum_{r=0}^{u}\sum_{s=0}^u(-1)^{r+s}(u+1)\binom{u}{r}\binom{u}{s}p^\Gamma_i(k,m)\Lambda^\Gamma_{k,m,n-i-u}\left(x_{j+(u-2r)k+(u-2s)m}^{\Gamma,+}\right)\\
        &+\frac{2}{n}\sum_{i=1}^n\sum_{r=0}^i\sum_{s=0}^i\sum_{u=0}^{n-i}\sum_{v=0}^{u}\sum_{w=0}^u(u+1)(-1)^{r+s}(-1)^{v+w}\binom{i}{r}\binom{i}{s}\binom{u}{v}\binom{u}{w}\Lambda^\Gamma_{k,m,n-i-u}\\
        &\times\left(x_{j+(u+i-2(v+r))k+(u+i-2(w+s))m}^{\Gamma,+}\right)\textnormal{ by the induction hypothesis}\\
        &=-\frac{1}{n}\sum_{u=0}^{n-1}\sum_{r=0}^{u}\sum_{s=0}^u(-1)^{r+s}(u+1)\binom{u}{r}\binom{u}{s}\sum_{i=1}^{n-u}p^\Gamma_i(k,m)\Lambda^\Gamma_{k,m,n-i-u}\left(x_{j+(u-2r)k+(u-2s)m}^{\Gamma,+}\right)\\
        &+\frac{2}{n}\sum_{i=1}^n\sum_{r=0}^i\sum_{s=0}^i\sum_{u=i}^{n}\sum_{v=0}^{u-i}\sum_{w=0}^{u-i}(u-i+1)(-1)^{r+s}(-1)^{v+w}\binom{i}{r}\binom{i}{s}\binom{u-i}{v}\binom{u-i}{w}\Lambda^\Gamma_{k,m,n-u}\\
        &\times\left(x_{j+(u-2(v+r))k+(u-2(w+s))m}^{\Gamma,+}\right)\\
        &=\frac{1}{n}\sum_{u=0}^{n-1}\sum_{r=0}^{u}\sum_{s=0}^u(-1)^{r+s}(u+1)(n-u)\binom{u}{r}\binom{u}{s}\Lambda^\Gamma_{k,m,n-u}\left(x_{j+(u-2r)k+(u-2s)m}^{\Gamma,+}\right)\\
        &+\frac{2}{n}\sum_{i=1}^n\sum_{r=0}^i\sum_{s=0}^i\sum_{u=i}^{n}\sum_{v=r}^{u+r-i}\sum_{w=s}^{u+s-i}(u-i+1)(-1)^{r+s}(-1)^{v-r+w-s}\binom{i}{r}\binom{i}{s}\binom{u-i}{v-r}\binom{u-i}{w-s}\\
        &\times\Lambda^\Gamma_{k,m,n-u}\left(x_{j+(u-2v)k+(u-2w)m}^{\Gamma,+}\right)\\
        &=\frac{1}{n}\sum_{u=0}^{n-1}\sum_{r=0}^{u}\sum_{s=0}^u(-1)^{r+s}(u+1)(n-u)\binom{u}{r}\binom{u}{s}\Lambda^\Gamma_{k,m,n-u}\left(x_{j+(u-2r)k+(u-2s)m}^{\Gamma,+}\right)\\
        &+\frac{2}{n}\sum_{u=1}^{n}\sum_{v=0}^{u}\sum_{w=0}^{u}\sum_{i=1}^u\sum_{r=0}^{v}\sum_{s=0}^{w}(-1)^{v+w}(u-i+1)\binom{i}{r}\binom{i}{s}\binom{u-i}{v-r}\binom{u-i}{w-s}\Lambda^\Gamma_{k,m,n-u}\\
        &\times\left(x_{j+(u-2v)k+(u-2w)m}^{\Gamma,+}\right)\\
        &=\frac{1}{n}\sum_{u=0}^{n-1}\sum_{r=0}^{u}\sum_{s=0}^u(-1)^{r+s}(u+1)(n-u)\binom{u}{r}\binom{u}{s}\Lambda^\Gamma_{k,m,n-u}\left(x_{j+(u-2r)k+(u-2s)m}^{\Gamma,+}\right)\\
        &+\frac{2}{n}\sum_{u=1}^{n}\sum_{v=0}^{u}\sum_{w=0}^{u}(-1)^{v+w}\sum_{i=1}^u(u-i+1)\binom{u}{v}\binom{u}{w}\Lambda^\Gamma_{k,m,n-u}\left(x_{j+(u-2v)k+(u-2w)m}^{\Gamma,+}\right)\\
        &\textnormal{by the Chu–-Vandermonde identity}\\
        &=\Lambda^\Gamma_{k,m,n}\left(x_{j}^{\Gamma,+}\right)\\
        &+\frac{1}{n}\sum_{u=1}^{n}\sum_{r=0}^{u}\sum_{s=0}^u(-1)^{r+s}(u+1)(n-u)\binom{u}{r}\binom{u}{s}\Lambda^\Gamma_{k,m,n-u}\left(x_{j+(u-2r)k+(u-2s)m}^{\Gamma,+}\right)\\
        &+\frac{2}{n}\sum_{u=1}^{n}\sum_{v=0}^{u}\sum_{w=0}^{u}(-1)^{v+w}\left(u(u+1)-\frac{u(u+1)}{2}\right)\binom{u}{v}\binom{u}{w}\Lambda^\Gamma_{k,m,n-u}\left(x_{j+(u-2v)k+(u-2w)m}^{\Gamma,+}\right)\\
        &=\Lambda^\Gamma_{k,m,n}\left(x_{j}^{\Gamma,+}\right)\\
        &+\frac{1}{n}\sum_{u=1}^{n}\sum_{r=0}^{u}\sum_{s=0}^u(-1)^{r+s}(u+1)(n-u)\binom{u}{r}\binom{u}{s}\Lambda^\Gamma_{k,m,n-u}\left(x_{j+(u-2r)k+(u-2s)m}^{\Gamma,+}\right)\\
        &+\frac{1}{n}\sum_{u=1}^{n}\sum_{v=0}^{u}\sum_{w=0}^{u}(-1)^{v+w}u(u+1)\binom{u}{v}\binom{u}{w}\Lambda^\Gamma_{k,m,n-u}\left(x_{j+(u-2v)k+(u-2w)m}^{\Gamma,+}\right)\\
        &=\Lambda^\Gamma_{k,m,n}\left(x_{j}^{\Gamma,+}\right)+\sum_{u=1}^{n}\sum_{r=0}^{u}\sum_{s=0}^u(-1)^{r+s}(u+1)\binom{u}{r}\binom{u}{s}\Lambda^\Gamma_{k,m,n-u}\left(x_{j+(u-2r)k+(u-2s)m}^{\Gamma,+}\right)\\
        &=\sum_{u=0}^{n}\sum_{r=0}^{u}\sum_{s=0}^u(-1)^{r+s}(u+1)\binom{u}{r}\binom{u}{s}\Lambda^\Gamma_{k,m,n-u}\left(x_{j+(u-2r)k+(u-2s)m}^{\Gamma,+}\right)
    \end{align*}
    This concludes the proof of the $r=1$ case. Assume the proposition for some $r\in\mathbb N$. Then
    \begin{align*}
        (r+1)\left(x_{j}^{\Gamma,+}\right)^{(r+1)}\Lambda_{k,m,n}^\Gamma
        &=\left(x_{j}^{\Gamma,+}\right)\left(x_{j}^{\Gamma,+}\right)^{(r)}\Lambda_{k,m,n}^\Gamma\\
        &=\sum_{i=0}^n\left(x_{j}^{\Gamma,+}\right)\Lambda_{k,m,n-i}^\Gamma\prod_{\substack{v_0\ldots v_u\in{\mathbb Z}_+\\\sum v_u=r\\\sum uv_u=i}}\left((u+1)D^{\Gamma,+}_{u}(j,k,m)\right)^{(v_u)}\\
        &\textnormal{by the induction hypothesis}\\
        &=\sum_{i=0}^n\sum_{s=0}^{n-i}\Lambda^\Gamma_{k,m,n-i-s}(s+1)D_s^{\Gamma,+}(j,k,m)\prod_{\substack{v_0\ldots v_u\in{\mathbb Z}_+\\\sum v_u=r\\\sum uv_u=i}}\left((u+1)D^{\Gamma,+}_{u}(j,k,m)\right)^{(v_u)}\\
        &=\sum_{i=0}^n\sum_{s=0}^{n-i}\Lambda^\Gamma_{k,m,n-i-s}\prod_{\substack{v_0\ldots v_u\in{\mathbb Z}_+\\u\neq s\\\sum v_u=r-v_s\\\sum uv_u=i-sv_s}}\left((u+1)D^{\Gamma,+}_{u}(j,k,m)\right)^{(v_u)}\\
        &\times\left((s+1)D^{\Gamma,+}_{s}(j,k,m)\right)^{(v_s)}(s+1)D_s^{\Gamma,+}(j,k,m)\\
        &=\sum_{i=0}^n\sum_{s=0}^{n-i}\Lambda^\Gamma_{k,m,n-i-s}\prod_{\substack{v_0\ldots v_u\in{\mathbb Z}_+\\u\neq s\\\sum v_u=r-v_s\\\sum uv_u=i-sv_s}}\left((u+1)D^{\Gamma,+}_{u}(j,k,m)\right)^{(v_u)}\\
        &\times(v_s+1)\left((s+1)D^{\Gamma,+}_{s}(j,k,m)\right)^{(v_s+1)}\\
        &=\sum_{i=0}^n\sum_{s=i}^{n}\Lambda^\Gamma_{k,m,n-s}\prod_{\substack{v_0\ldots v_u\in{\mathbb Z}_+\\\sum v_u=r+1\\\sum uv_u=s}}(v_{s-i})\left((u+1)D^{\Gamma,+}_{u}(j,k,m)\right)^{(v_u)}\\
        &=\sum_{s=0}^{n}\Lambda^\Gamma_{k,m,n-s}\prod_{\substack{v_0\ldots v_u\in{\mathbb Z}_+\\\sum v_u=r+1\\\sum uv_u=s}}\sum_{i=0}^s(v_{s-i})\left((u+1)D^{\Gamma,+}_{u}(j,k,m)\right)^{(v_u)}\\
        &=\sum_{s=0}^{n}\Lambda^\Gamma_{k,m,n-s}\prod_{\substack{v_0\ldots v_u\in{\mathbb Z}_+\\\sum v_u=r+1\\\sum uv_u=s}}\sum_{i=0}^s(v_{s})\left((u+1)D^{\Gamma,+}_{u}(j,k,m)\right)^{(v_u)}\\
        &=(r+1)\sum_{s=0}^{n}\Lambda^\Gamma_{k,m,n-s}\prod_{\substack{v_0\ldots v_u\in{\mathbb Z}_+\\\sum v_u=r+1\\\sum uv_u=s}}\left((u+1)D^{\Gamma,+}_{u}(j,k,m)\right)^{(v_u)}\\
    \end{align*}
\end{proof}

\subsection{Proof of Proposition \ref{straightening} (7)}
We will proceed following the method of proof of Lemma 5.4 in \cite{C}, which was in turn modeled on the proof of Lemma 5.1 in \cite{CP}. We begin with some necessary propositions.

\begin{prop}\label{bracketxp}
    For all $i,j,k,l,m\in\mathbb N$
    \begin{enumerate}
        \item
        \begin{equation*}
            \left[\left(x_j^{\Gamma,+}\right),p_i^\Gamma(k,m)\right]=-2D_i^{\Gamma,+}(j,k,m)
        \end{equation*}
        \item
        \begin{equation*}
            \left[p_i^\Gamma(k,m),\left(x_l^{\Gamma,-}\right)\right]=-2D_i^{\Gamma,-}(l,k,m)
        \end{equation*}
    \end{enumerate}
\end{prop}
\begin{proof}
    We will only prove $(1)$ in detail, because the proof of $(2)$ is similar. By Proposition \ref{p_u} we have
    \begin{align*}
        \left[\left(x_j^{\Gamma,+}\right),p_i^\Gamma(k,m)\right]
        &=\sum_{r=0}^{\left\lfloor\frac{i-1}{2}\right\rfloor}\sum_{s=0}^i(-1)^{r+s}\binom{i}{r}\binom{i}{s}\left[\left(x_j^{\Gamma,+}\right),\left(h^\Gamma_{(i-2s)k+(i-2r)m}\right)\right]\\
        &+((i+1)\mod2)\sum_{s=0}^i(-1)^{\frac{i}{2}+s}\binom{i-1}{\frac{i-2}{2}}\binom{i}{s}\left[\left(x_j^{\Gamma,+}\right),\left(h^\Gamma_{(i-2s)k}\right)\right]\\
        &=-2\sum_{r=0}^{\left\lfloor\frac{i-1}{2}\right\rfloor}\sum_{s=0}^i(-1)^{r+s}\binom{i}{r}\binom{i}{s}\left(\left(x_{j+(i-2s)k+(i-2r)m}^{\Gamma,+}\right)+\left(x_{j-(i-2s)k-(i-2r)m}^{\Gamma,+}\right)\right)\\
        &-2((i+1)\mod2)\sum_{s=0}^i(-1)^{\frac{i}{2}+s}\binom{i-1}{\frac{i-2}{2}}\binom{i}{s}\left(\left(x_{j+(i-2s)k}^{\Gamma,+}\right)+\left(x_{j-(i-2s)k}^{\Gamma,+}\right)\right)\\
        &=-2\sum_{r=0}^{\left\lfloor\frac{i-1}{2}\right\rfloor}\sum_{s=0}^i(-1)^{r+s}\binom{i}{r}\binom{i}{s}\left(x_{j+(i-2s)k+(i-2r)m}^{\Gamma,+}\right)\\
        &-2\sum_{r=0}^{\left\lfloor\frac{i-1}{2}\right\rfloor}\sum_{s=0}^i(-1)^{r+s}\binom{i}{r}\binom{i}{s}\left(x_{j-(i-2s)k-(i-2r)m}^{\Gamma,+}\right)\\
        &-2((i+1)\mod2)\sum_{s=0}^i(-1)^{\frac{i}{2}+s}\binom{i-1}{\frac{i-2}{2}}\binom{i}{s}\left(x_{j+(i-2s)k}^{\Gamma,+}\right)\\
        &-2((i+1)\mod2)\sum_{s=0}^i(-1)^{\frac{i}{2}+s}\binom{i-1}{\frac{i-2}{2}}\binom{i}{s}\left(x_{j-(i-2s)k}^{\Gamma,+}\right)\\
        &=-2\sum_{r=0}^{\left\lfloor\frac{i-1}{2}\right\rfloor}\sum_{s=0}^i(-1)^{r+s}\binom{i}{r}\binom{i}{s}\left(x_{j+(i-2s)k+(i-2r)m}^{\Gamma,+}\right)\\
        &-2\sum_{r=\left\lfloor\frac{i+2}{2}\right\rfloor}^{i}\sum_{s=0}^i(-1)^{i-r+i-s}\binom{i}{r}\binom{i}{s}\left(x_{j+(i-2s)k+(i-2r)m}^{\Gamma,+}\right)\\
        &-2((i+1)\mod2)\sum_{s=0}^i(-1)^{\frac{i}{2}+s}\binom{i-1}{\frac{i-2}{2}}\binom{i}{s}\left(x_{j+(i-2s)k}^{\Gamma,+}\right)\\
        &-2((i+1)\mod2)\sum_{s=0}^i(-1)^{\frac{3i}{2}-s}\binom{i-1}{\frac{i-2}{2}}\binom{i}{s}\left(x_{j+(i-2s)k}^{\Gamma,+}\right)\\
        &=-2\sum_{r=0}^{\left\lfloor\frac{i-1}{2}\right\rfloor}\sum_{s=0}^i(-1)^{r+s}\binom{i}{r}\binom{i}{s}\left(x_{j+(i-2s)k+(i-2r)m}^{\Gamma,+}\right)\\
        &-2\sum_{r=\left\lfloor\frac{i+2}{2}\right\rfloor}^{i}\sum_{s=0}^i(-1)^{r+s}\binom{i}{r}\binom{i}{s}\left(x_{j+(i-2s)k+(i-2r)m}^{\Gamma,+}\right)\\
        &-2((i+1)\mod2)\sum_{s=0}^i(-1)^{\frac{i}{2}+s}\binom{i}{\frac{i}{2}}\binom{i}{s}\left(x_{j+(i-2s)k}^{\Gamma,+}\right)\\
        &=-2\sum_{r=0}^{i}\sum_{s=0}^i(-1)^{r+s}\binom{i}{r}\binom{i}{s}\left(x_{j+(i-2s)k+(i-2r)m}^{\Gamma,+}\right)
    \end{align*}
\end{proof}

\begin{prop}\label{bracketpD}
    For all $i,j,k,m\in\mathbb N$ and $u\in{\mathbb Z}_+$
    \begin{equation*}
        \left[p_m^\Gamma(j,l),D_{u,1}^{\Gamma,+}(j,l)\right]=2D_{m+u,1}^{\Gamma,+}(j,l)
    \end{equation*}
\end{prop}
\begin{proof}
    We will proceed by induction on $u$. The case $u=0$ is Proposition \ref{bracketxp}. Assume the claim for some $u\in\mathbb N$. Then we have
    \begin{align*}
        \left[p_m^\Gamma(j,l),D_{u+1,1}^{\Gamma,+}(j,l)\right]
        &=\frac{1}{2}\left[p_m^\Gamma(j,l),\left[D_{u,1}^{\Gamma,+}(j,l),\Lambda^\Gamma_{j,l,1}\right]\right]\\
        &=-\frac{1}{2}\left(\left[D_{u,1}^{\Gamma,+}(j,l),\left[\Lambda^\Gamma_{j,l,1},p_m^\Gamma(j,l)\right]\right]+\left[\Lambda^\Gamma_{j,l,1},\left[p_m^\Gamma(j,l),D_{u,1}^{\Gamma,+}(j,l)\right]\right]\right)\\
        &\textnormal{ by the Jacobi Identity}\\
        &=-\frac{1}{2}\left[\Lambda^\Gamma_{j,l,1},\left[p_m^\Gamma(j,l),D_{u,1}^{\Gamma,+}(j,l)\right]\right]\\
        &=-\left[\Lambda^\Gamma_{j,l,1},D_{m+u,1}^{\Gamma,+}(j,l)\right]\textnormal{ by the induction hypothesis}\\
        &=\left[D_{m+u,1}^{\Gamma,+}(j,l),\Lambda^\Gamma_{j,l,1}\right]\\
        &=2D_{m+u+1,1}^{\Gamma,+}(j,l)
    \end{align*}
\end{proof}

\begin{prop}\label{Du1Lambda}
    For all $u,n\in{\mathbb Z}_+$ and $j,l\in\mathbb N$
    \begin{equation*}
        D^{\Gamma,+}_{u,1}(j,l)\Lambda^\Gamma_{j,l,n}=\sum_{i=0}^n(i+1)\Lambda^\Gamma_{j,l,n-i}D^{\Gamma,+}_{i+u,1}(j,l)
    \end{equation*}
\end{prop}
\begin{proof}
    We will proceed by induction on $u$. The case $u=0$ is Proposition \ref{straightening}(8). Assume the claim for some $u\in{\mathbb Z}_+$. Then we have
    \begin{align*}
        \left[D^{\Gamma,+}_{u+1,1}(j,l),\Lambda^\Gamma_{j,l,n}\right]
        &=\frac{1}{2}\left[\left[D^{\Gamma,+}_{u,1}(j,l),\Lambda^\Gamma_{j,l,1}\right],\Lambda^\Gamma_{j,l,n}\right]\\
        &=\frac{1}{2}\left[D^{\Gamma,+}_{u,1}(j,l),\left[\Lambda^\Gamma_{j,l,1},\Lambda^\Gamma_{j,l,n}\right]\right]+\frac{1}{2}\left[\Lambda^\Gamma_{j,l,1},\left[\Lambda^\Gamma_{j,l,n},D^{\Gamma,+}_{u,1}(j,l)\right]\right]\\
        &\textnormal{ by the Jacobi Identity}\\
        &=-\frac{1}{2}\sum_{i=1}^n(i+1)\left[\Lambda^\Gamma_{j,l,1},\Lambda^\Gamma_{j,l,n-i}D^{\Gamma,+}_{i+u,1}(j,l)\right]\textnormal{ by the induction hypothesis}\\
        &=-\frac{1}{2}\sum_{i=1}^n(i+1)\Lambda^\Gamma_{j,l,1}\Lambda^\Gamma_{j,l,n-i}D^{\Gamma,+}_{i+u,1}(j,l)+\frac{1}{2}\sum_{i=1}^n(i+1)\Lambda^\Gamma_{j,l,n-i}D^{\Gamma,+}_{i+u,1}(j,l)\Lambda^\Gamma_{j,l,1}\\
        &=\frac{1}{2}\sum_{i=1}^n(i+1)\Lambda^\Gamma_{j,l,n-i}\left[D^{\Gamma,+}_{i+u,1}(j,l),\Lambda^\Gamma_{j,l,1}\right]\\
        &=\sum_{i=1}^n(i+1)\Lambda^\Gamma_{j,l,n-i}D^{\Gamma,+}_{i+u+1,1}(j,l)
    \end{align*}
\end{proof}

\begin{prop}\label{LambdaD+}
    For all $i,k\in{\mathbb Z}_+$ and $j,l\in\mathbb N$ we have
    \begin{equation*}
        \Lambda^\Gamma_{j,l,i}D_{k,1}^{\Gamma,+}(j,l)=D_{k,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i}-2D_{k+1,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i-1}+D_{k+2,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i-2}
    \end{equation*}
\end{prop}
\begin{proof}
    We will proceed by induction on $i$. The cases $i=0,1$ are straightforward. Assume the claim for some $i\in\mathbb N$. Then we have
    \begin{align*}
        (i+1)\Lambda^\Gamma_{j,l,i+1}D_{k,1}^{\Gamma,+}(j,l)
        &=-\sum_{m=1}^{i+1}p_m^\Gamma(j,l)\Lambda^\Gamma_{j,l,i+1-m}D_{k,1}^{\Gamma,+}(j,l)\\
        &=-\sum_{m=1}^{i+1}p_m^\Gamma(j,l)\left(D_{k,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i+1-m}-2D_{k+1,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i-m}+D_{k+2,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i-1-m}\right)\\
        &\textnormal{ by the induction hypothesis}\\
        &=-\sum_{m=1}^{i+1}p_m^\Gamma(j,l)D_{k,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i+1-m}+2\sum_{m=1}^{i+1}p_m^\Gamma(j,l)D_{k+1,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i-m}\\
        &-\sum_{m=1}^{i+1}p_m^\Gamma(j,l)D_{k+2,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i-1-m}\\
        &=-D_{k,1}^{\Gamma,+}(j,l)\sum_{m=1}^{i+1}p_m^\Gamma(j,l)\Lambda^\Gamma_{j,l,i+1-m}-2\sum_{m=1}^{i+1}D_{k+m,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i+1-m}\\
        &+2D_{k+1,1}^{\Gamma,+}(j,l)\sum_{m=1}^{i+1}p_m^\Gamma(j,l)\Lambda^\Gamma_{j,l,i-m}+4\sum_{m=1}^{i+1}D_{k+m+1,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i-m}\\
        &-D_{k+2,1}^{\Gamma,+}(j,l)\sum_{m=1}^{i+1}p_m^\Gamma(j,l)\Lambda^\Gamma_{j,l,i-1-m}-2\sum_{m=1}^{i+1}D_{k+m+2,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i-1-m}\\
        &\textnormal{ by Proposition }\ref{bracketpD}\\
        &=(i+1)D_{k,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i+1}-2D_{k+1,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i}-2D_{k+2,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i-1}\\
        &-2\sum_{m=3}^{i+1}D_{k+m,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i+1-m}-2iD_{k+1,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i}+4\sum_{m=2}^{i+2}D_{k+m,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i+1-m}\\
        &+(i-1)D_{k+2,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i-1}-2\sum_{m=3}^{i+3}D_{k+m,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i+1-m}\\
        &=(i+1)D_{k,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i+1}-2\sum_{m=3}^{i+1}D_{k+m,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i+1-m}-2(i+1)D_{k+1,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i}\\
        &+4D_{k+2,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i-1}+4\sum_{m=3}^{i+1}D_{k+m,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i+1-m}+(i-3)D_{k+2,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i-1}\\
        &-2\sum_{m=3}^{i+1}D_{k+m,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i+1-m}\\
        &=(i+1)D_{k,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i+1}-2(i+1)D_{k+1,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i}+(i+1)D_{k+2,1}^{\Gamma,+}(j,l)\Lambda^\Gamma_{j,l,i-1}
    \end{align*}
\end{proof}

\begin{prop}\label{uD}
    For all $j,l\in\mathbb N$ and $u,v\in{\mathbb Z}_+$
    \begin{enumerate}
        \item
        \begin{equation*}
            uD_{u,v}^{\Gamma,\pm}(j,l)=\sum_{i=0}^uiD_{i,1}^{\Gamma,\pm}(j,l)D_{u-i,v-1}^{\Gamma,\pm}(j,l)
        \end{equation*}
        \item
        \begin{equation*}
            (u+v)D_{u,v}^{\Gamma,\pm}(j,l)=\sum_{i=0}^u(i+1)D_{i,1}^{\Gamma,\pm}(j,l)D_{u-i,v-1}^{\Gamma,\pm}(j,l)
        \end{equation*}
    \end{enumerate}
\end{prop}
\begin{proof}
    We will proceed by induction on $v$. The cases $v=0,1$ are straightforward. Assume the formula for some $v\in{\mathbb Z}_+$. Then, for $(1)$, we have
    \begin{align*}
        uD_{u,v+1}^{\Gamma,\pm}(j,l)
        &=\frac{u}{v+1}\sum_{i=0}^uD_{i,1}^{\Gamma,\pm}(j,l)D_{u-i,v}^{\Gamma,\pm}(j,l)\\
        &=\frac{1}{v+1}\sum_{i=0}^uD_{i,1}^{\Gamma,\pm}(j,l)(u-i)D_{u-i,v}^{\Gamma,\pm}(j,l)+\frac{1}{v+1}\sum_{i=0}^uiD_{i,1}^{\Gamma,\pm}(j,l)D_{u-i,v}^{\Gamma,\pm}(j,l)\\
        &=\frac{1}{v+1}\sum_{i=0}^uD_{i,1}^{\Gamma,\pm}(j,l)\sum_{k=0}^{u-i}kD_{k,1}^{\Gamma,\pm}(j,l)D_{u-i-k,v-1}^{\Gamma,\pm}(j,l)+\frac{1}{v+1}\sum_{i=0}^uiD_{i,1}^{\Gamma,\pm}(j,l)D_{u-i,v}^{\Gamma,\pm}(j,l)\\
        &\textnormal{ by the induction hypothesis}\\
        &=\frac{1}{v+1}\sum_{k=0}^{u}kD_{k,1}^{\Gamma,\pm}(j,l)\sum_{i=0}^{u-k}D_{i,1}^{\Gamma,\pm}(j,l)D_{u-i-k,v-1}^{\Gamma,\pm}(j,l)+\frac{1}{v+1}\sum_{i=0}^uiD_{i,1}^{\Gamma,\pm}(j,l)D_{u-i,v}^{\Gamma,\pm}(j,l)\\
        &=\frac{v}{v+1}\sum_{k=0}^{u}kD_{k,1}^{\Gamma,\pm}(j,l)D_{u-k,v}^{\Gamma,\pm}(j,l)+\frac{1}{v+1}\sum_{i=0}^uiD_{i,1}^{\Gamma,\pm}(j,l)D_{u-i,v}^{\Gamma,\pm}(j,l)\\
        &=\sum_{i=0}^uiD_{i,1}^{\Gamma,\pm}(j,l)D_{u-i,v}^{\Gamma,\pm}(j,l).
    \end{align*}
    For $(2)$, we have
    \begin{align*}
        (u+v)D_{u,v}^{\Gamma,\pm}(j,l)
        &=uD_{u,v}^{\Gamma,\pm}(j,l)+vD_{u,v}^{\Gamma,\pm}(j,l)\\
        &=\sum_{i=0}^uiD_{i,1}^{\Gamma,\pm}(j,l)D_{u-i,v-1}^{\Gamma,\pm}(j,l)+\sum_{i=0}^uD_{i,1}^{\Gamma,\pm}(j,l)D_{u-i,v-1}^{\Gamma,\pm}(j,l)\textnormal{ by }(1)\\
        &=\sum_{i=0}^u(i+1)D_{i,1}^{\Gamma,\pm}(j,l)D_{u-i,v-1}^{\Gamma,\pm}(j,l)
    \end{align*}
\end{proof}

\begin{lem}\label{LambdaD+x-}
    For all $n,v\in{\mathbb Z}_+$ and $j,l\in\mathbb N$
    \begin{align*}
        \sum_{i=0}^n\Lambda^\Gamma_{j,l,i}D^{\Gamma,+}_{n-i,v}(j,l)\left(x^{\Gamma,-}_l\right)
        &=-(n+1)\sum_{u=0}^{n+1}\Lambda^\Gamma_{j,l,n+1-u}D^{\Gamma,+}_{u,v-1}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{k=0}^{n-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-m-k}D^{\Gamma,+}_{k,v}(j,l)
    \end{align*}
\end{lem}

\begin{proof}
    We will proceed by induction on $v$. If $v=0$ we have
    \begin{equation*}
        \Lambda^\Gamma_{j,l,n}\left(x^{\Gamma,-}_l\right)=\sum_{m=0}^n(m+1)D^{\Gamma,-}_{m}(l,j,l)\Lambda^\Gamma_{j,l,n-m}.
    \end{equation*}
    Assume the result for some $v\in{\mathbb Z}_+$. Then we have
    \begin{align*}
        &(v+1)\sum_{i=0}^n\Lambda^\Gamma_{j,l,i}D^{\Gamma,+}_{n-i,v+1}(j,l)\left(x^{\Gamma,-}_l\right)
       {\Gamma,+}_{n-i-k,v}(j,l)\left(x^{\Gamma,-}_l\right)\\
        &=\sum_{i=0}^n\sum_{k=0}^{n-i}\Lambda^\Gamma_{j,l,i}D^{\Gamma,+}_{k,1}(j,l)D^{\Gamma,+}_{n-i-k,v}(j,l)\left(x^{\Gamma,-}_l\right)\\
        &=\sum_{k=0}^n\sum_{i=0}^{n-k}\Lambda^\Gamma_{j,l,i}D^{\Gamma,+}_{k,1}(j,l)D^{\Gamma,+}_{n-i-k,v}(j,l)\left(x^{\Gamma,-}_l\right)\\
        &=\sum_{k=0}^n\sum_{i=0}^{n-k}\Big(D^{\Gamma,+}_{k,1}(j,l)\Lambda^\Gamma_{j,l,i}-2D^{\Gamma,+}_{k+1,1}(j,l)\Lambda^\Gamma_{j,l,i-1}\\
        &+D^{\Gamma,+}_{k+2,1}(j,l)\Lambda^\Gamma_{j,l,i-2}\Big)D^{\Gamma,+}_{n-i-k,v}(j,l)\left(x^{\Gamma,-}_l\right)\\
        &\textnormal{ by Proposition }\ref{LambdaD+}\\
        &=\sum_{k=0}^{n}D^{\Gamma,+}_{k,1}(j,l)\sum_{i=0}^{n-k}\Lambda^\Gamma_{j,l,i}D^{\Gamma,+}_{n-i-k,v}(j,l)\left(x^{\Gamma,-}_l\right)\\
        &-2\sum_{k=0}^nD^{\Gamma,+}_{k+1,1}(j,l)\sum_{i=0}^{n-k}\Lambda^\Gamma_{j,l,i-1}D^{\Gamma,+}_{n-i-k,v}(j,l)\left(x^{\Gamma,-}_l\right)\\
        &+\sum_{k=0}^nD^{\Gamma,+}_{k+2,1}(j,l)\sum_{i=0}^{n-k}\Lambda^\Gamma_{j,l,i-2}D^{\Gamma,+}_{n-i-k,v}(j,l)\left(x^{\Gamma,-}_l\right)\\
        &=\sum_{k=0}^{n}D^{\Gamma,+}_{k,1}(j,l)\sum_{i=0}^{n-k}\Lambda^\Gamma_{j,l,i}D^{\Gamma,+}_{n-i-k,v}(j,l)\left(x^{\Gamma,-}_l\right)\\
        &-2\sum_{k=0}^nD^{\Gamma,+}_{k+1,1}(j,l)\sum_{i=0}^{n-k-1}\Lambda^\Gamma_{j,l,i}D^{\Gamma,+}_{n-i-k-1,v}(j,l)\left(x^{\Gamma,-}_l\right)\\
        &+\sum_{k=0}^nD^{\Gamma,+}_{k+2,1}(j,l)\sum_{i=0}^{n-k-2}\Lambda^\Gamma_{j,l,i}D^{\Gamma,+}_{n-i-k-2,v}(j,l)\left(x^{\Gamma,-}_l\right)\\
        &=\sum_{k=0}^{n}D^{\Gamma,+}_{k,1}(j,l)\Bigg(-(n-k+1)\sum_{u=0}^{n-k+1}\Lambda^\Gamma_{j,l,n-k+1-u}D^{\Gamma,+}_{u,v-1}(j,l)\\
        &+\sum_{m=0}^{n-k}\sum_{r=0}^{n-k-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-k-m-r}D^{\Gamma,+}_{r,v}(j,l)\Bigg)\\
        &-2\sum_{k=0}^nD^{\Gamma,+}_{k+1,1}(j,l)\Bigg(-(n-k)\sum_{u=0}^{n-k}\Lambda^\Gamma_{j,l,n-k-u}D^{\Gamma,+}_{u,v-1}(j,l)\\
        &+\sum_{m=0}^{n-k-1}\sum_{r=0}^{n-k-1-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-k-1-m-r}D^{\Gamma,+}_{r,v}(j,l)\Bigg)\\
        &+\sum_{k=0}^nD^{\Gamma,+}_{k+2,1}(j,l)\Bigg(-(n-k-1)\sum_{u=0}^{n-k-1}\Lambda^\Gamma_{j,l,n-k-1-u}D^{\Gamma,+}_{u,v-1}(j,l)\\
        &+\sum_{m=0}^{n-k-2}\sum_{r=0}^{n-k-2-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-k-2-m-r}D^{\Gamma,+}_{r,v}(j,l)\Bigg)\\
        &\textnormal{ by the induction hypothesis}\\
        &=-\sum_{k=0}^{n}D^{\Gamma,+}_{k,1}(j,l)(n-k+1)\sum_{u=0}^{n-k+1}\Lambda^\Gamma_{j,l,n-k+1-u}D^{\Gamma,+}_{u,v-1}(j,l)\\
        &+\sum_{k=0}^{n}D^{\Gamma,+}_{k,1}(j,l)\sum_{m=0}^{n-k}\sum_{r=0}^{n-k-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-k-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &+2\sum_{k=0}^nD^{\Gamma,+}_{k+1,1}(j,l)(n-k)\sum_{u=0}^{n-k}\Lambda^\Gamma_{j,l,n-k-u}D^{\Gamma,+}_{u,v-1}(j,l)\\
        &-2\sum_{k=0}^nD^{\Gamma,+}_{k+1,1}(j,l)\sum_{m=0}^{n-k-1}\sum_{r=0}^{n-k-1-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-k-1-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &-\sum_{k=0}^nD^{\Gamma,+}_{k+2,1}(j,l)(n-k-1)\sum_{u=0}^{n-k-1}\Lambda^\Gamma_{j,l,n-k-1-u}D^{\Gamma,+}_{u,v-1}(j,l)\\
        &+\sum_{k=0}^nD^{\Gamma,+}_{k+2,1}(j,l)\sum_{m=0}^{n-k-2}\sum_{r=0}^{n-k-2-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-k-2-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &=-\sum_{k=0}^{n}D^{\Gamma,+}_{k,1}(j,l)(n-k+1)\sum_{u=0}^{n-k+1}\Lambda^\Gamma_{j,l,n-k+1-u}D^{\Gamma,+}_{u,v-1}(j,l)\\
        &+\sum_{k=0}^{n}D^{\Gamma,+}_{k,1}(j,l)\sum_{m=0}^{n-k}\sum_{r=0}^{n-k-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-k-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &+2\sum_{k=1}^{n+1}D^{\Gamma,+}_{k,1}(j,l)(n-k+1)\sum_{u=0}^{n-k+1}\Lambda^\Gamma_{j,l,n-k-u+1}D^{\Gamma,+}_{u,v-1}(j,l)\\
        &-2\sum_{k=1}^{n+1}D^{\Gamma,+}_{k,1}(j,l)\sum_{m=0}^{n-k}\sum_{r=0}^{n-k-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-k-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &-\sum_{k=2}^{n+2}D^{\Gamma,+}_{k,1}(j,l)(n-k+1)\sum_{u=0}^{n-k+1}\Lambda^\Gamma_{j,l,n-k+1-u}D^{\Gamma,+}_{u,v-1}(j,l)\\
        &+\sum_{k=2}^{n+2}D^{\Gamma,+}_{k,1}(j,l)\sum_{m=0}^{n-k}\sum_{r=0}^{n-k-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-k-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &=-D^{\Gamma,+}_{0,1}(j,l)(n+1)\sum_{u=0}^{n+1}\Lambda^\Gamma_{j,l,n+1-u}D^{\Gamma,+}_{u,v-1}(j,l)-D^{\Gamma,+}_{1,1}(j,l)n\sum_{u=0}^{n}\Lambda^\Gamma_{j,l,n-u}D^{\Gamma,+}_{u,v-1}(j,l)\\
        &-\sum_{k=2}^{n}D^{\Gamma,+}_{k,1}(j,l)(n-k+1)\sum_{u=0}^{n-k+1}\Lambda^\Gamma_{j,l,n-k+1-u}D^{\Gamma,+}_{u,v-1}(j,l)\\
        &+D^{\Gamma,+}_{0,1}(j,l)\sum_{m=0}^{n}\sum_{r=0}^{n-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &+D^{\Gamma,+}_{1,1}(j,l)\sum_{m=0}^{n-1}\sum_{r=0}^{n-1-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-1-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &+\sum_{k=2}^{n}D^{\Gamma,+}_{k,1}(j,l)\sum_{m=0}^{n-k}\sum_{r=0}^{n-k-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-k-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &+2D^{\Gamma,+}_{1,1}(j,l)n\sum_{u=0}^{n}\Lambda^\Gamma_{j,l,n-u}D^{\Gamma,+}_{u,v-1}(j,l)\\
        &+2\sum_{k=2}^{n}D^{\Gamma,+}_{k,1}(j,l)(n-k+1)\sum_{u=0}^{n-k+1}\Lambda^\Gamma_{j,l,n-k-u+1}D^{\Gamma,+}_{u,v-1}(j,l)\\
        &-2D^{\Gamma,+}_{1,1}(j,l)\sum_{m=0}^{n-1}\sum_{r=0}^{n-1-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-1-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &-2\sum_{k=2}^{n}D^{\Gamma,+}_{k,1}(j,l)\sum_{m=0}^{n-k}\sum_{r=0}^{n-k-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-k-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &-\sum_{k=2}^{n}D^{\Gamma,+}_{k,1}(j,l)(n-k+1)\sum_{u=0}^{n-k+1}\Lambda^\Gamma_{j,l,n-k+1-u}D^{\Gamma,+}_{u,v-1}(j,l)\\
        &+\sum_{k=2}^{n}D^{\Gamma,+}_{k,1}(j,l)\sum_{m=0}^{n-k}\sum_{r=0}^{n-k-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-k-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &=-(n+1)D^{\Gamma,+}_{0,1}(j,l)\sum_{u=0}^{n+1}\Lambda^\Gamma_{j,l,n+1-u}D^{\Gamma,+}_{u,v-1}(j,l)+nD^{\Gamma,+}_{1,1}(j,l)\sum_{u=0}^{n}\Lambda^\Gamma_{j,l,n-u}D^{\Gamma,+}_{u,v-1}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{r=0}^{n-m}(m+1)D^{\Gamma,+}_{0,1}(j,l)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &-\sum_{m=0}^{n-1}\sum_{r=0}^{n-1-m}(m+1)D^{\Gamma,+}_{1,1}(j,l)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-1-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &=-(n+1)\sum_{u=0}^{n+1}\sum_{i=0}^{n+1-u}(i+1)\Lambda^\Gamma_{j,l,n+1-u-i}D^{\Gamma,+}_{i,1}(j,l)D^{\Gamma,+}_{u,v-1}(j,l)\\
        &+n\sum_{u=0}^{n}\sum_{i=0}^{n-u}(i+1)\Lambda^\Gamma_{j,l,n-u-i}D^{\Gamma,+}_{i+1,1}(j,l)D^{\Gamma,+}_{u,v-1}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{r=0}^{n-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{0,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{r=0}^{n-m}(m+1)p^\Gamma_{m+1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &-\sum_{m=0}^{n-1}\sum_{r=0}^{n-m-1}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{1,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r-1}D^{\Gamma,+}_{r,v}(j,l)\\
        &-\sum_{m=0}^{n-1}\sum_{r=0}^{n-1-m}(m+1)p^\Gamma_{m+2}(j,l)\Lambda^\Gamma_{j,l,n-1-m-r}D^{\Gamma,+}_{r,v}(j,l)\textnormal{ by Propositions }\ref{Du1Lambda}\textnormal{ and }\ref{pnewD}\\
        &=-(n+1)\sum_{u=0}^{n+1}\sum_{i=0}^{n+1-u}(i+1)\Lambda^\Gamma_{j,l,n+1-u-i}D^{\Gamma,+}_{i,1}(j,l)D^{\Gamma,+}_{u,v-1}(j,l)\\
        &+n\sum_{u=0}^{n+1}\sum_{i=0}^{n-u+1}i\Lambda^\Gamma_{j,l,n-u-i+1}D^{\Gamma,+}_{i,1}(j,l)D^{\Gamma,+}_{u,v-1}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{r=0}^{n-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{0,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{r=0}^{n-m}(m+1)p^\Gamma_{m+1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &-\sum_{m=0}^{n-1}\sum_{r=0}^{n-m-1}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{1,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r-1}D^{\Gamma,+}_{r,v}(j,l)\\
        &-\sum_{m=0}^{n}\sum_{r=0}^{n-m}mp^\Gamma_{m+1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &=-\sum_{u=0}^{n+1}\sum_{i=0}^{n+1-u}(n+i+1)\Lambda^\Gamma_{j,l,n+1-u-i}D^{\Gamma,+}_{i,1}(j,l)D^{\Gamma,+}_{u,v-1}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{r=0}^{n-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{0,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r,v}(j,l)+\sum_{m=0}^{n}\sum_{r=0}^{n-m}p^\Gamma_{m+1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &-\sum_{m=0}^{n-1}\sum_{r=0}^{n-m-1}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{1,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r-1}D^{\Gamma,+}_{r,v}(j,l)\\
        &=-\sum_{i=0}^{n+1}\sum_{u=0}^{n+1-i}(n+i+1)\Lambda^\Gamma_{j,l,n+1-u-i}D^{\Gamma,+}_{i,1}(j,l)D^{\Gamma,+}_{u,v-1}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{r=0}^{n-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{0,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r,v}(j,l)+\sum_{r=0}^{n}\sum_{m=0}^{n-r}p^\Gamma_{m+1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &-\sum_{m=0}^{n-1}\sum_{r=0}^{n-m-1}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{1,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r-1}D^{\Gamma,+}_{r,v}(j,l)\\
        &=-\sum_{i=0}^{n+1}\sum_{u=i}^{n+1}(n+i+1)\Lambda^\Gamma_{j,l,n+1-u}D^{\Gamma,+}_{i,1}(j,l)D^{\Gamma,+}_{u-i,v-1}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{r=0}^{n-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{0,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r,v}(j,l)+\sum_{r=0}^{n}\sum_{m=1}^{n-r+1}p^\Gamma_{m}(j,l)\Lambda^\Gamma_{j,l,n-m-r+1}D^{\Gamma,+}_{r,v}(j,l)\\
        &-\sum_{m=0}^{n-1}\sum_{r=0}^{n-m-1}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{1,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r-1}D^{\Gamma,+}_{r,v}(j,l)\\
        &=-nv\sum_{u=0}^{n+1}\Lambda^\Gamma_{j,l,n+1-u}D^{\Gamma,+}_{u,v}(j,l)-\sum_{u=0}^{n+1}(u+v)\Lambda^\Gamma_{j,l,n+1-u}D^{\Gamma,+}_{u,v}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{r=0}^{n-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{0,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r,v}(j,l)-\sum_{r=0}^{n+1}(n-r+1)\Lambda^\Gamma_{j,l,n-r+1}D^{\Gamma,+}_{r,v}(j,l)\\
        &-\sum_{m=0}^{n-1}\sum_{r=0}^{n-m-1}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{1,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r-1}D^{\Gamma,+}_{r,v}(j,l)\\
        &=-(v+1)(n+1)\sum_{u=0}^{n+1}\Lambda^\Gamma_{j,l,n+1-u}D^{\Gamma,+}_{u,v}(j,l)+\sum_{m=0}^{n}\sum_{r=0}^{n-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{0,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &-\sum_{m=0}^{n-1}\sum_{r=0}^{n-m-1}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{1,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r-1}D^{\Gamma,+}_{r,v}(j,l)\\
        &=-(v+1)(n+1)\sum_{u=0}^{n+1}\Lambda^\Gamma_{j,l,n+1-u}D^{\Gamma,+}_{u,v}(j,l)+\sum_{m=0}^{n}\sum_{r=0}^{n-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{0,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &-\sum_{m=0}^{n}\sum_{r=1}^{n-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{1,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r-1,v}(j,l)\\
        &=-(v+1)(n+1)\sum_{u=0}^{n+1}\Lambda^\Gamma_{j,l,n+1-u}D^{\Gamma,+}_{u,v}(j,l)+\sum_{m=0}^{n}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{0,1}(j,l)\Lambda^\Gamma_{j,l,n-m}D^{\Gamma,+}_{0,v}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{k=0}^{1}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{k,1}(j,l)\Lambda^\Gamma_{j,l,n-m-1}D^{\Gamma,+}_{1-k,v}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{r=2}^{n-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{0,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r,v}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{r=2}^{n-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{1,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r-1,v}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{r=2}^{n-m}\sum_{k=2}^{r}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{k,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r-k,v}(j,l)\\
        &-2\sum_{m=0}^{n}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{1,1}(j,l)\Lambda^\Gamma_{j,l,n-m-1}D^{\Gamma,+}_{0,v}(j,l)\\
        &-2\sum_{m=0}^{n}\sum_{r=2}^{n-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{1,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r-1,v}(j,l)\\
        &-2\sum_{m=0}^{n}\sum_{r=2}^{n-m}\sum_{k=2}^{r}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{k,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r-k,v}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{r=2}^{n-m}\sum_{k=2}^{r}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{k,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r-k,v}(j,l)\\
        &=-(v+1)(n+1)\sum_{u=0}^{n+1}\Lambda^\Gamma_{j,l,n+1-u}D^{\Gamma,+}_{u,v}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{r=0}^{n-m}\sum_{k=0}^{r}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{k,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r-k,v}(j,l)\\
        &-2\sum_{m=0}^{n}\sum_{r=1}^{n-m}\sum_{k=1}^{r}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{k,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r-k,v}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{r=2}^{n-m}\sum_{k=2}^{r}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{k,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r-k,v}(j,l)\\
        &=-(v+1)(n+1)\sum_{u=0}^{n+1}\Lambda^\Gamma_{j,l,n+1-u}D^{\Gamma,+}_{u,v}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{r=0}^{n-m}\sum_{k=0}^{r}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{k,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r-k,v}(j,l)\\
        &-2\sum_{m=0}^{n}\sum_{r=0}^{n-m}\sum_{k=0}^{r}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{k+1,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r-1}D^{\Gamma,+}_{r-k,v}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{r=0}^{n-m}\sum_{k=0}^{r}(m+1)D^{\Gamma,-}_{m,1}(j,l)D^{\Gamma,+}_{k+2,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r-2}D^{\Gamma,+}_{r-k,v}(j,l)\\
        &=-(v+1)(n+1)\sum_{u=0}^{n+1}\Lambda^\Gamma_{j,l,n+1-u}D^{\Gamma,+}_{u,v}(j,l)\\
        &+\sum_{m=0}^{n}\sum_{r=0}^{n-m}\sum_{k=0}^{r}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Big(D^{\Gamma,+}_{k,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}\\
        &-2D^{\Gamma,+}_{k+1,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r-1}+D^{\Gamma,+}_{k+2,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r-2}\Big)D^{\Gamma,+}_{r-k,v}(j,l)\\
        &=-(v+1)(n+1)\sum_{u=0}^{n+1}\Lambda^\Gamma_{j,l,n+1-u}D^{\Gamma,+}_{u,v}(j,l)+\sum_{m=0}^{n}\sum_{r=0}^{n-m}\sum_{k=0}^{r}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{k,1}(j,l)D^{\Gamma,+}_{r-k,v}(j,l)\\
        &\textnormal{ by Proposition }\ref{LambdaD+}\\
        &=-(v+1)(n+1)\sum_{u=0}^{n+1}\Lambda^\Gamma_{j,l,n+1-u}D^{\Gamma,+}_{u,v}(j,l)+(v+1)\sum_{m=0}^{n}\sum_{r=0}^{n-m}(m+1)D^{\Gamma,-}_{m,1}(j,l)\Lambda^\Gamma_{j,l,n-m-r}D^{\Gamma,+}_{r,v+1}(j,l)
    \end{align*}
\end{proof}

Now, we prove Proposition \ref{straightening} (7):
\begin{proof}
    We proceed by induction on $s$. The case $s=0$ holds, because $D_{0,r}^{\Gamma,+}(j,l)=\left(x_j^{\Gamma,+}\right)^{(r)}$. Assume the formula is true for some $s\in{\mathbb Z}_+$. Then we have
    $$(s+1)\left(x_j^{\Gamma,+}\right)^{(r)}\left(x_l^{\Gamma,-}\right)^{(s+1)}$$
    \begin{align*}
        &=\left(x_j^{\Gamma,+}\right)^{(r)}\left(x_l^{\Gamma,-}\right)^{(s)}\left(x_l^{\Gamma,-}\right)\\
        &=\sum_{\substack{m,n,q\in{\mathbb Z}_+\\m+n+q\leq\min\{r,s\}}}(-1)^{m+n+q}D^{\Gamma,-}_{m,s-m-n-q}(j,l)\Lambda^\Gamma_{j,l,n}D^{\Gamma,+}_{q,r-m-n-q}(j,l)\left(x_l^{\Gamma,-}\right)\\
        &\textnormal{by the induction hypothesis}\\
        &=\sum_{\substack{m,i,q\in{\mathbb Z}_+\\m+i+q\leq\min\{r,s\}}}(-1)^{m+i+q}D^{\Gamma,-}_{m,s-m-i-q}(j,l)\Lambda^\Gamma_{j,l,i}D^{\Gamma,+}_{q,r-m-i-q}(j,l)\left(x_l^{\Gamma,-}\right)\\
        &=\sum_{\substack{m,i,n\in{\mathbb Z}_+\\m+n\leq\min\{r,s\}}}(-1)^{m+n}D^{\Gamma,-}_{m,s-m-n}(j,l)\Lambda^\Gamma_{j,l,i}D^{\Gamma,+}_{n-i,r-m-n}(j,l)\left(x_l^{\Gamma,-}\right)\\
        &=\sum_{\substack{m,n\in{\mathbb Z}_+\\m+n\leq\min\{r,s\}}}(-1)^{m+n}D^{\Gamma,-}_{m,s-m-n}(j,l)\sum_{i=0}^n\Lambda^\Gamma_{j,l,i}D^{\Gamma,+}_{n-i,r-m-n}(j,l)\left(x_l^{\Gamma,-}\right)\\
        &=\sum_{\substack{m,n\in{\mathbb Z}_+\\m+n\leq\min\{r,s\}}}(-1)^{m+n}D^{\Gamma,-}_{m,s-m-n}(j,l)\bigg(-(n+1)\sum_{u=0}^{n+1}\Lambda^\Gamma_{j,l,n+1-u}D^{\Gamma,+}_{u,r-m-n-1}(j,l)\\
        &+\sum_{i=0}^{n}\sum_{k=0}^{n-i}(i+1)D^{\Gamma,-}_{i,1}(j,l)\Lambda^\Gamma_{j,l,n-i-k}D^{\Gamma,+}_{k,r-m-n}(j,l)\bigg)\textnormal{ by Lemma }\ref{LambdaD+x-}\\
        &=\sum_{\substack{m,n\in{\mathbb Z}_+\\m+n\leq\min\{r,s\}}}(-1)^{m+n+1}D^{\Gamma,-}_{m,s-m-n}(j,l)(n+1)\sum_{u=0}^{n+1}\Lambda^\Gamma_{j,l,n+1-u}D^{\Gamma,+}_{u,r-m-n-1}(j,l)\\
        &+\sum_{\substack{m,n\in{\mathbb Z}_+\\m+n\leq\min\{r,s\}}}(-1)^{m+n}\sum_{i=0}^{n}(i+1)D^{\Gamma,-}_{i,1}(j,l)D^{\Gamma,-}_{m,s-m-n}(j,l)\sum_{k=0}^{n-i}\Lambda^\Gamma_{j,l,n-i-k}D^{\Gamma,+}_{k,r-m-n}(j,l)\\
        &=\sum_{\substack{m,n\in{\mathbb Z}_+\\m+n\leq\min\{r,s\}}}(-1)^{m+n+1}D^{\Gamma,-}_{m,s-m-n}(j,l)(n+1)\sum_{u=0}^{n+1}\Lambda^\Gamma_{j,l,u}D^{\Gamma,+}_{n+1-u,r-m-n-1}(j,l)\\
        &+\sum_{\substack{m,n\in{\mathbb Z}_+\\m+n\leq\min\{r,s\}}}(-1)^{m+n}\sum_{i=0}^{n}(i+1)D^{\Gamma,-}_{i,1}(j,l)D^{\Gamma,-}_{m,s-m-n}(j,l)\sum_{k=0}^{n-i}\Lambda^\Gamma_{j,l,k}D^{\Gamma,+}_{n-i-k,r-m-n}(j,l)\\
        &=\sum_{\substack{m,n,u\in{\mathbb Z}_+\\m+n\leq\min\{r,s\}}}(-1)^{m+n+1}(n+1)D^{\Gamma,-}_{m,s-m-n}(j,l)\Lambda^\Gamma_{j,l,u}D^{\Gamma,+}_{n+1-u,r-m-n-1}(j,l)\\
        &+\sum_{\substack{m,n\in{\mathbb Z}_+\\m+n\leq\min\{r,s\}}}(-1)^{m+n}\sum_{k=0}^{n}\sum_{i=0}^{n-k}(i+1)D^{\Gamma,-}_{i,1}(j,l)D^{\Gamma,-}_{m,s-m-n}(j,l)\Lambda^\Gamma_{j,l,k}D^{\Gamma,+}_{n-i-k,r-m-n}(j,l)\\
        &=\sum_{\substack{m,u,q\in{\mathbb Z}_+\\m+q+u\leq\min\{r,s\}}}(-1)^{m+q+u+1}(q+u+1)D^{\Gamma,-}_{m,s-m-q-u}(j,l)\Lambda^\Gamma_{j,l,u}D^{\Gamma,+}_{q+1,r-m-q-u-1}(j,l)\\
        &+\sum_{\substack{m,q,k\in{\mathbb Z}_+\\m+q+k\leq\min\{r,s\}}}(-1)^{m+q+k}\sum_{i=0}^{q}(i+1)D^{\Gamma,-}_{i,1}(j,l)D^{\Gamma,-}_{m,s-m-q-k}(j,l)\Lambda^\Gamma_{j,l,k}D^{\Gamma,+}_{q-i,r-m-q-k}(j,l)\\
        &=\sum_{\substack{m,u,q\in{\mathbb Z}_+\\1\leq m+q+u\leq\min\{r,s\}+1}}(-1)^{m+q+u}(q+u)D^{\Gamma,-}_{m,s+1-m-q-u}(j,l)\Lambda^\Gamma_{j,l,u}D^{\Gamma,+}_{q,r-m-q-u}(j,l)\\
        &+\sum_{i=0}^{\min\{r,s\}}\sum_{\substack{m,q,k\in{\mathbb Z}_+\\m+q+k\leq\min\{r,s\}}}(-1)^{m+q+k}(i+1)D^{\Gamma,-}_{i,1}(j,l)D^{\Gamma,-}_{m,s-m-q-k}(j,l)\Lambda^\Gamma_{j,l,k}D^{\Gamma,+}_{q-i,r-m-q-k}(j,l)\\
        &=\sum_{\substack{m,u,q\in{\mathbb Z}_+\\1\leq m+q+u\leq\min\{r,s+1\}}}(-1)^{m+q+u}(q+u)D^{\Gamma,-}_{m,s+1-m-q-u}(j,l)\Lambda^\Gamma_{j,l,u}D^{\Gamma,+}_{q,r-m-q-u}(j,l)\\
        &+\sum_{i=0}^{\min\{r,s\}}\sum_{\substack{m,q,k\in{\mathbb Z}_+\\m+q+k\leq\min\{r,s\}}}(-1)^{m+q+k}(i+1)D^{\Gamma,-}_{i,1}(j,l)D^{\Gamma,-}_{m-i,s-m-q-k}(j,l)\Lambda^\Gamma_{j,l,k}D^{\Gamma,+}_{q,r-m-q-k}(j,l)\\
        &=\sum_{\substack{m,u,q\in{\mathbb Z}_+\\1\leq m+q+u\leq\min\{r,s+1\}}}(-1)^{m+q+u}(q+u)D^{\Gamma,-}_{m,s+1-m-q-u}(j,l)\Lambda^\Gamma_{j,l,u}D^{\Gamma,+}_{q,r-m-q-u}(j,l)\\
        &+\sum_{\substack{m,q,k\in{\mathbb Z}_+\\m+q+k\leq\min\{r,s\}}}(-1)^{m+q+k}\sum_{i=0}^{m}(i+1)D^{\Gamma,-}_{i,1}(j,l)D^{\Gamma,-}_{m-i,s-m-q-k}(j,l)\Lambda^\Gamma_{j,l,k}D^{\Gamma,+}_{q,r-m-q-k}(j,l)\\
        &=\sum_{\substack{m,u,q\in{\mathbb Z}_+\\1\leq m+q+u\leq\min\{r,s+1\}}}(-1)^{m+q+u}(q+u)D^{\Gamma,-}_{m,s+1-m-q-u}(j,l)\Lambda^\Gamma_{j,l,u}D^{\Gamma,+}_{q,r-m-q-u}(j,l)\\
        &+\sum_{\substack{m,q,k\in{\mathbb Z}_+\\m+q+k\leq\min\{r,s\}}}(-1)^{m+q+k}(s+1-q-k)D^{\Gamma,-}_{m,s+1-m-q-k}(j,l)\Lambda^\Gamma_{j,l,k}D^{\Gamma,+}_{q,r-m-q-k}(j,l)\\
        &\textnormal{ by Proposition }\ref{uD}(ii)\\
        &=\sum_{\substack{m,n,q\in{\mathbb Z}_+\\m+n+q\leq\min\{r,s+1\}}}(-1)^{m+n+q}(q+n)D^{\Gamma,-}_{m,s+1-m-n-q}(j,l)\Lambda^\Gamma_{j,l,n}D^{\Gamma,+}_{q,r-m-n-q}(j,l)\\
        &+\sum_{\substack{m,n,q\in{\mathbb Z}_+\\m+n+q\leq\min\{r,s+1\}}}(-1)^{m+n+q}(s+1-q-n)D^{\Gamma,-}_{m,s+1-m-n-q}(j,l)\Lambda^\Gamma_{j,l,n}D^{\Gamma,+}_{q,r-m-n-q}(j,l)\\
        &=(s+1)\sum_{\substack{m,n,q\in{\mathbb Z}_+\\m+n+q\leq\min\{r,s+1\}}}(-1)^{m+n+q}D^{\Gamma,-}_{m,s+1-m-n-q}(j,l)\Lambda^\Gamma_{j,l,n}D^{\Gamma,+}_{q,r-m-n-q}(j,l)
    \end{align*}
\end{proof}


TITLE: A question on the injectivity of a canonical map between galois cohomology groups
QUESTION [5 upvotes]: I'm currently reading the book "Galois theory of $p$-extensions" by Helmut Koch.
There, we calculate the cohomological dimension of the galois group $G(K/k)$ where $K$ is the maximal (normal) $p$-extension of $k$.
(Here $p$ is a prime and $k$ is a local field or global field of finite type, i.e finite extension of $\mathbb{Q}$ of $\mathbb{Q}_p$.)
As $G(K/k)$ is a pro-p group, we study $H^2(G(K/k), \mathbb{F}_p)$ where $\mathbb{F}_p$ is the finite field with $p$ elements with trivial group action.
Let $k'$ be the field generated by $k$ and the $p$'th roots of unity, and let $K'$ be the maximal $p$-extension of $k'$. Then there is a canonical group homomorphism from $G(K'/k')$ to $G(K/k)$ (restriction map).
This induces a homomorphism from $H^2(G(K/k),\mathbb{F}_p)$ to $H^2(G(K'/k'),\mathbb{F}_p)$.
The question is, is this map injective?

REPLY [2 votes]: I believe this map is always injective. Here is a quick argument: first note that $K'$ is normal over $k$ (because it is invariant under any automorphism of the algebraic closure of $k$ which preserves $k'$, and $k'$ is normal over $k$). This means that we can view the map $G(K'/k') \to G(K/k)$ as a composition of two maps
$$ G(K'/k') \stackrel{f}{\to} G(K'/k) \stackrel{g}{\to} G(K/k) .$$
It will hence suffice to show that both $f^*:H^2(G(K'/k),\mathbb{F}_p) \to H^2(G(K'/k'),\mathbb{F}_p)$ and $g^*:H^2(G(K/k),\mathbb{F}_p) \to H^2(G(K'/k),\mathbb{F}_p)$ are injective. Now $f: G(K'/k') \to G(K'/k)$ is an inclusion of a subgroup of finite index $[k':k]$, and hence there exists a corestriction map $f_!:H^2(G(K'/k'),\mathbb{F}_p) \to H^2(G(K'/k),\mathbb{F}_p)$ such that $f_! \circ f^*$ is multiplication by $[k':k]$. Since $[k':k]$ is coprime to $p$ and $H^2(G(K'/k),\mathbb{F}_p)$ is a $p$-torsion group it follows that $f^*$ is injective. To show that $g^*$ is injective, we note that $g: G(K'/k) \to G(K/k)$ is a surjective group homomorphism with kernel $G(K'/K)$, and so the Hochschild-Serre spectral sequence gives an exact sequence
$$ H^1(G(K'/K),\mathbb{F}_p)^{G(K/k)} \to H^2(G(K/k),\mathbb{F}_p) \to H^2(G(K'/k),\mathbb{F}_p) .$$
To show that $g^*$ is injective it will hence suffice to show that $H^1(G(K'/K),\mathbb{F}_p)^{G(K/k)} = 0$, i.e., that $K$ does not admit any non-trivial cyclic $p$-extensions inside $K'$ which are normal over $k$. But this is just because $K$ does not admit any non-trivial cyclic $p$-extensions which are normal over $k$: indeed, it is the maximal $p$-extension of $k$.


TITLE: Decomposition of Algebraic Numbers into Sets of Real and Rational Numbers
QUESTION [3 upvotes]: Show that a real number $\alpha$ is algebraic of degree at most $n$ if and only if there are real numbers $\{a_1, a_2, ..., a_n\}$ such that for every $k \in \mathbb{N}$ there exist rational numbers $\{b_{1,\space k}, ..., b_{n, \space k}\}$ such that $\alpha^k = a_1b_{1,\space k} + ... + a_nb_{n,\space k}$.
There are two hints: For the forward assertion use the Euclidean Algorithm and for the reverse use linear algebra.
I have been staring at this problem for hours and all of my attempts have led me nowhere. I would appreciate any suggestions on how to proceed or any other references.
One failed attempt: I tried to argue that $\alpha^k$ = $(\alpha^{n})^{k/n}$ and from here I tried to exponentiate $p(\alpha) - \alpha^n$ and proceed inductively. I ended up with a mess and did not think it would get me anywhere

REPLY [2 votes]: For the first direction, let me give you an example. Suppose $\alpha = 1 + \sqrt[3] 2$. Then $(\alpha - 1)^3 - 2 = 0$. So we have
$$ \alpha^3 - 3 \alpha^2 + 3 \alpha - 3 = 0. $$
Thus
$$ \alpha^3 = 3 \cdot \alpha^0 - 3 \cdot \alpha^1 + 3 \cdot \alpha^2. \tag{1} $$
Now I multiply both sides by $\alpha$ to obtain
$$ \alpha^4 = 3 \cdot \alpha^1 - 3 \cdot \alpha^2 + 3 \cdot \alpha^3. $$
Using $(1)$ we write this as
$$ \alpha^4 = 9 \cdot \alpha^0 - 6 \alpha^1 + 6 \alpha^2. $$
Likewise, if we multiply this by $\alpha$ and apply $(1)$ we obtain
$$ \alpha^5 = 18 \cdot \alpha^0 - 9 \cdot \alpha^1 + 12 \cdot \alpha^2. $$
Repeating this, we observe that we can always write
$$ \alpha^k = b_{0,k} \cdot \alpha^0 + b_{1,k} \cdot \alpha^1 + b_{2,k} \cdot \alpha^2 $$
for some rational numbers $b_{0,k}, b_{1,k}, b_{2,k}$.
Try using the polynomial division (the Euclidean Algorithm) to make this precise.

For the second direction, show that $1, \alpha, \alpha^2, \dots, \alpha^n$ are linearly dependent over $\mathbf{Q}$ by showing that they belong to a $\mathbf{Q}$-vector space of dimension at most $n$. What does linear dependence tell you?


TITLE: Find the general solution of $4y′′+ 3y′−y=e^{−x}+x$
QUESTION [4 upvotes]: Given $$4y′′+ 3y′−y = e^{−x} + x$$
Find the general solution

I identified this as a non-homogeneous linear equation
so I assumed that my solution,
$$ y = y_h  + y_p$$
$$ 4\lambda^2 + 3\lambda - 1 = 0$$
$$ \lambda = -1 \quad\text{and}\quad \lambda =1/4$$
$$ y_h  = Ae^{x/4} + Be^{-x}$$
However, I am not too sure what should I do to find $y_p$
I do know that my $G(x) = e^{-x} + x$ but am not too sure how to find $y_p$.

REPLY [0 votes]: HINT
\begin{align*}
4y^{\prime\prime} + 3y^{\prime} - y & = e^{-x} + x \Longleftrightarrow 4(y^{\prime\prime} + y^{\prime}) - (y^{\prime} + y) = e^{-x} + x \Longleftrightarrow\\\\
4(y^{\prime} + y)^{\prime} - (y^{\prime} + y) & = e^{-x} + x \Longleftrightarrow 4w^{\prime} - w = e^{-x} + x\quad\text{where}\quad w = y^{\prime} + y
\end{align*}


TITLE: Setting up a local-coordinate system in space-time using only a single clock and light beams
QUESTION [8 upvotes]: I have a question to ask about the operationalist view of space-time. I am a mathematician who happens to be interested in physics, so if anyone thinks that my question is a silly or vague one, please feel free to close it.
Many physics textbooks that discuss Special Relativity mention that one way of setting up a coordinate system in flat Minkowski space is to imagine that there is an infinite grid of meter rules that pervades the universe, with synchronized clocks located at each node of the grid. In order to obtain more precise measurements of space-time events, one simply divides the grid into more nodes. I feel that this presents problems, because what does it mean to say that ‘all the clocks are synchronized’? Any act of moving a clock from a starting point to its designated node is sure to involve some acceleration, and this is going to upset any synchronization attempt. Also, such a description is not satisfactory because it is impossible to deal with an infinite number of meter rules and clocks.
Things are made worse in curved space-time because of the further effect of gravitational time-dilation. Hence, I believe that the only logically consistent way to perform measurements of space-time events is to use a single atomic clock carried by a single observer and to use the world-lines of photons emitted by a light source carried by the observer to ‘carve out’ a local-coordinate system. The exact description of how this can be done is beyond me, so I would like to gather some useful information from anyone in the community. Thank you very much!

REPLY [5 votes]: The radar method is a general approach that works for non-inertial observers and curved spacetime.
Two co-ordinates of an event are given by your clock time at which the event intersects your future and past light cone, called retarded time and advanced time, ($\tau^+,\tau^-$, resp.). Or use a diagonal combination thereof: $\tau^\star = \frac{\tau^+ + \tau^-}{2}$, called radar time, and $\rho = c\frac{\tau^+ - \tau^-}{2}$, called radar distance.
This diagonal combination has the property that, in the case of an unaccelerated observer in flat spacetime, $\tau^\star$ and $\rho$ are equal to the usual measures (the "infinite grid of rulers and clocks" business).
Two other co-ordinates can be given by the incoming angles ($\Omega^+$) of the null geodesic from the event to you. This is the reception or retarded co-ordinate system. The dual system, the trasmission or advanced co-ordinate system, would use the outgoing angles ($\Omega^-$) of the null geodesic from you to the event.
For a non-rotating unaccelerated observer in flat spacetime, the two pairs of angles are equal to each other and to the usual polar co-ordinate angles.
In flat spacetime this will assign a unique co-ordinate to every reachable event, that is, every event in the observer's causal diamond. In curved spacetime it will assign at least one co-ordinate to every reachable event, however there may be duplicates. One can restrict to the boundary of the causal past and future, as described in answer I linked to above. Then, under certain causality assumptions, every reachable event gets a unique $\tau^\star$ and $\rho$. The surfaces of constant $\tau^\star$ and $\rho$ are then 2-D globally spacelike surfaces, but not always topologically $\mathcal{S}^2$, rather, they will be some subquotient of $\mathcal{S}^2$. That is, for a given $\tau^\star$ and $\rho$ some angle pairs $\Omega^+$ will not be valid (corresponding to parts of the light cone that have "fallen behind"), and some events on the boundary of validity will have more than one angle pair.


TITLE: function application order
QUESTION [0 upvotes]: In traditional mathematics, when we post-compose $x$ by $f$ we write $fx$ or $f(x)$, that is we prefix writing things right to left.
I realize some might be used to it, and it is absolutely trivial, but coming from a culture of left to right writing, this confuses me to no end and it badly hurts my senses.
Is there a standard way of writing things down in the correct order ?

REPLY [3 votes]: The notation ';' is sometimes used for this: if $f : A \to B$ and $g : B \to C$, we write $f;g$ to denote $g \circ f : A \to C$.
In the book "Categories, Allegories" by Freyd and Scedrov, the authors write $fg$ to denote what is usually written $g \circ f$.


TITLE: A lecture by Rudin
QUESTION [7 upvotes]: Is it available a written version of this lecture by Rudin on the relation between Fourier analysis and the birth of set theory?
https://youtu.be/hBcWRZMP6xs
If not Rudin himself, maybe someone else has collected these ideas in an article?

REPLY [8 votes]: I extracted the written text from Rudin's lecture, it starts out like this (see below). If there is an interest, with some effort I can provide the rest of the text, but it requires some editing and I'm unsure this is what the OP wants... Perhaps just listening to the talk is good enough?

It is a real honor to be invited to give this lecture and I'm sorry
professor Marden couldn't be here to listen to it.
I'm a bit out of my depth here today because I intend to give a talk
which is more historical than mathematical. I was told that there
would be many non mathematicians here in the audience and that I
should not write too many formulas.
A warning: the history of mathematics is a difficult field and it's
one that has been neglected and I think there are good reasons for
that. Mathematicians don't pay much attention to history. We name
theorems after someone and sometimes that someone has something to do
with the field and sometimes he didn't. Sometimes somebody later
proved what the first guy should have proved if you see what he was
really doing, and we don't ever go to the first original sources. I
think we read what someone said about it, so mathematicians are not
really qualified to do the history of mathematics. Historians are
maybe even less qualified because you can't write about it unless you
understand it and once you start studying mathematics for its own sake
you become interested in it and you don't want to take time to go to
the history.