# Explicit Central Elements of ${U_{q}(\mathfrak{gl}(N+1))}$

Jeffrey Kuan111Columbia University, Department of Mathematics. Keke
Zhang222Columbia University, Undergraduate.

###### Abstract

Using Drinfeld’s central element construction and Jimbo’s fusion of
$R$–matrices, we construct central elements of the quantum group
${U_{q}(\mathfrak{gl}(N+1))}$. These elements are explicitly written in terms
of the generators.

##  1 Introduction

Several recent works (e.g. [CGRS, CGRS16, Kuab, Kuaa, Kua16]), motivated by
mathematical physics, have used explicit central elements of various quantum
groups. For the applications in those paper, the central elements need to be
explicitly written in terms of the generators of the quantum group.

Previous work of [ZGB91b] applies Drinfeld’s central element construction
[Dri90] to universal $R$–matrices in order to construct central elements of
quantum groups and to determine their eigenvalues on irreducible highest
weight modules. In particular, by using Jimbo’s [Jim86] explicit formula for
the $R$–matrix of ${U_{q}(\mathfrak{gl}(N+1))}$, further work by the same
authors [GZB91] explicitly writes a quantum Casimir element of
${U_{q}(\mathfrak{gl}(N+1))}$ with a formula for its eigenvalues. Using an
explicit formula for the universal $R$–matrix in [KR90], the authors [ZGB91a]
write an explicit (but somewhat complicated) expression for general Casimir
elements in quantum groups.

In this paper, we apply Drinfeld’s central element construction to the fused
$R$–matrices of ${U_{q}(\mathfrak{gl}(N+1))}$ in [Jim86], rather than the
universal $R$–matrices of [KR90]. The resulting central elements appear to be
slightly simpler than the previous expressions. The proof requires some new
ingredients, notably relations between the root vectors [Xi94] and some
elementary knowledge of coset representatives of symmetric groups.

We also note the work in [Li10], which shows that the Casimir elements (and a
trivial central element) generate the entire center of
${U_{q}(\mathfrak{gl}(N+1))}$. Additionally, the paper [LXZ18] explicitly
writes two algebraically independent central elements in
${U_{q}(\mathfrak{gl}(N+1))}$. In principle, it should be possible to match
those central elements to the ones here, but we do not pursue this direction
here.

Acknowledgements: Jeffrey Kuan was supported by the Minerva Foundation and NSF
grant DMS-1502665. Both authors were supported by the 2017 Columbia
Mathematics REU program, which was funded by Columbia University. We thank Yi
Sun for helpful discussions.

##  2 Notations and Backgrounds

###  2.1 Symmetric Groups

Define the usual action of the symmetric group $S_{m}$ on $\mathbb{N}^{m}$ by
$\sigma(x_{1},\cdots,x_{m})=(x_{\sigma(1)},\cdots,x_{\sigma(m)})$. For any
$A\in\mathbb{N}^{m}$, define $H_{A}\leq S_{m}$ to be the subgroup
$\\{\sigma\in S_{m}:\sigma(A)=A\\}$. Let $D_{A}\subset S_{m}$ be the set of
(left) coset representatives of $H_{A}$ with the fewest inversions: in other
words, $\sigma\in D_{A}$ if and only if
$\operatorname{inv}(\sigma)\leq\operatorname{inv}(\tau)$ for every
$\tau\in\sigma H_{A}$.

######  Example 1.

Take $A=(3,3,3,2,2,1).$ Then $H_{A}=S_{3}\times S_{2}\times S_{1}\leq S_{6}$.
We have $(34)\in D_{A}$ with $(34)\cdot A=(3,3,2,3,2,1)$, and $(35)\in D_{A}$
with $(35)\cdot A=(3,3,2,2,3,1)$. However, $(134)\cdot A=(34)\cdot A$ and
$\operatorname{inv}((134))>\operatorname{inv}((34))$, so $(134)\notin D_{A}$.

We recall (see e.g. [Car85]) that each coset of $H_{A}$ has a unique
representative $\sigma\in D_{A}$, and that
$\operatorname{inv}(\sigma\tau)=\operatorname{inv}(\sigma)+\operatorname{inv}(\tau)$
for every $\tau\in H_{A}$.

######  Lemma 1.

Suppose that $\tau,\tau^{\prime}\in D_{A}$. Then there exist a sequence of
elements $\tau_{0},\ldots,\tau_{l}\in D_{A}$ such that
$\tau_{0}=\tau^{\prime},\tau_{l}=\tau$ and every $\tau_{j+1}\tau_{j}^{-1}$ is
a transposition for $j=0,\ldots,l-1$.

###### Proof.

It suffices to prove this statement when either $\tau$ or $\tau^{\prime}$ is
the identity permutation $e$, because the two sequences can be concatenated.
So suppose that $\tau$ is an arbitrary element of $D_{A}$ and
$\tau^{\prime}=e$. Let $s_{k}\cdots s_{1}$ be a minimal word representation of
$\tau$ and set $\tau_{j}=s_{j}\cdots s_{1}$. Then
$\tau_{j+1}\tau_{j}^{-1}=s_{j+1}$, which is a transposition. So it remains to
show that $\tau_{j}\in D_{A}$. If it were not, then there would exist a
transposition $s\in H_{A}$ such that
$\operatorname{inv}(\tau_{j}s)=\operatorname{inv}(\tau_{j})-1$. But then
$\operatorname{inv}(\tau s)=\operatorname{inv}(\tau)-1$, contradicting the
assumption that $\tau\in D_{A}$.

Now suppose that $\tau=e$ and $\tau^{\prime}$ is an arbitrary element of
$D_{A}$. By the previous paragraph, there exist a sequence of elements
$\tilde{\tau}_{0}=e,\tilde{\tau}_{1},\ldots,\tilde{\tau}_{l}=\tau^{\prime}$ in
$D_{A}$ such that every $\tilde{\tau}_{j+1}\tilde{\tau}_{j}^{-1}$ is a
transposition. Setting $\tau_{j}=\tilde{\tau}_{l-j}$, we have that
$\tau_{0}=\tau^{\prime},\ldots,\tau_{l}=e$ is a sequence of elements in
$D_{A}$ and ${\tau}_{l-j-1}{\tau}_{l-j}^{-1}$ is a transposition for every
$j$. The latter equality is equivalent to the condition that for every $k$,
$\tau_{k}=s\tau_{k+1}$ for some transposition $s$. Since this is also
equivalent to the condition that $\tau_{k+1}\tau_{k}^{-1}$ is a transposition
for every $k$, this finishes the proof. ∎

Let $\mathcal{B}_{m}^{(N)}$ denote the set of sequences
$\mu=(\mu_{0},\cdots,\mu_{N})$ of non-negative integers such that
$\mu_{0}+\cdots+\mu_{N}=m$. For any $\mu\in\mathcal{B}_{m}^{(N)}$, let
$H^{\mu}\leq S_{m}$ denote the subgroup $S_{\mu_{0}}\times\cdots\times
S_{\mu_{N}}$, and likewise let $D^{\mu}$ denote the set of left coset
representatives with the fewest inversions. Define $\mathcal{B}_{m}$ to be the
union $\bigcup_{N\geq 1}\mathcal{B}_{m}^{(N)}$.

Let $\mathcal{W}_{m}\subset\mathbb{N}^{m}$ denote the subset of elements
$(i_{1},\ldots,i_{m})$ satisfying $i_{1}\leq\cdots\leq i_{m}$. For
${\mathbf{i}}\in\mathcal{W}_{m}$, and assuming that $N\geq i_{m}$, let
$\mu^{(N)}({\mathbf{i}})\in\mathcal{B}_{m}^{(N)}$ be defined by

$\mu^{(N)}({\mathbf{i}})_{k}=|\\{l\in\\{1,\ldots,m\\}:i_{l}=k\\}|.$

For ${\mathbf{i}}\in\mathcal{W}_{m}$ satisfying $i_{m}\leq N$, we have a
natural isomorphism between the subgroups $H_{{\mathbf{i}}}$ and
$H^{\mu^{(N)}({\mathbf{i}})}$. Thus there is also a natural bijection between
$D_{{\mathbf{i}}}$ and $D^{\mu^{(N)}({\mathbf{i}})}$.

Given ${\mathbf{i}}\in\mathcal{W}_{m}$, define the equivalence relation
$\sim_{{\mathbf{i}}}$ on $S_{m}$ by

$\tau\sim_{{\mathbf{i}}}\sigma\text{ if and only if }\sigma^{-1}\tau\in
H_{{\mathbf{i}}}.$

In words, $\tau\sim_{{\mathbf{i}}}\sigma$ if and only if they are in the same
left coset of $H_{{\mathbf{i}}}$. For any $\tau\in S_{m}$, there is a unique
$\sigma\in D_{{\mathbf{i}}}$ such that $\tau\sim_{{\mathbf{i}}}\sigma$. Define
$d_{{\mathbf{i}}}(\tau)=\operatorname{inv}(\sigma^{-1}\tau)$. In other words,
if $\tau$ is written uniquely as $\tau=\sigma\xi$ for $\sigma\in
D_{{\mathbf{i}}}$ and $\xi\in H_{{\mathbf{i}}}$, then
$d_{{\mathbf{i}}}(\tau)=\operatorname{inv}(\xi)$. We will also let
$\sigma_{{\mathbf{i}}}(\tau)$ and $\xi_{{\mathbf{i}}}(\tau)$ denote the two
permutations in the unique expression $\tau=\sigma\xi$.

Finally, given $\tau\in S_{m}$, let $\bar{\tau}$ denote the reversed
permutation $\bar{\tau}(k)=\tau(m+1-k)$.

We conclude this section by noting the following identity.

######  Lemma 2.

For ${\mathbf{i}}\in\mathcal{W}_{m}$, set $\mu=\mu^{(N)}({\mathbf{i}})$. Then

$-N\mu_{0}-(N-2)\mu_{1}+\cdots+N\mu_{N}=-Nm+2(i_{1}+\ldots+i_{m}).$

###### Proof.

By definition,

$\mu_{k}=\left|l\in\\{1,\ldots,m\\}:i_{l}=k\right|.$

We thus re–write

$\displaystyle-N\mu_{0}-(N-2)\mu_{1}+\cdots+N\mu_{N}$
$\displaystyle=-N(\mu_{0}+\ldots+\mu_{N})+2\mu_{1}+4\mu_{2}+\ldots+2N\mu_{N}$
$\displaystyle=-Nm+2\mu_{1}+4\mu_{2}+\ldots+2N\mu_{N}$

and note that

$2(i_{1}+\ldots+i_{m})=2\sum_{k=1}^{N}k\cdot\left|l\in\\{1,\ldots,m\\}:i_{l}=k\right|=2\sum_{k=1}^{N}k\cdot\mu_{k},$

which shows the identity. ∎

###  2.2 Quantum Groups

We use the following notation modified from [Jim86]. Define
$U_{q}(\mathfrak{sl}(N+1))$ to be the associative algebra over $\mathbb{C}$
generated by the symbols $q^{\pm h_{i}/2},\hat{e}_{\pm,i},(1\leq i\leq N)$
under the following relations:

$\displaystyle\begin{split}&q^{h_{i}/2}\cdot q^{-h_{i}/2}=q^{-h_{i}/2}\cdot
q^{h_{i}/2}=1,q^{h_{i}/2}\cdot q^{h_{i^{\prime}}/2}=q^{h_{i^{\prime}}/2}\cdot
q^{h_{i}/2},\\\ &q^{h_{i}/2}\hat{e}_{\pm,i^{\prime}}q^{-h_{i}/2}=q^{\pm
a_{ii^{\prime}}/2}\hat{e}_{\pm,i^{\prime}},\\\
&[\hat{e}_{+,i},\hat{e}_{-,i^{\prime}}]=\delta_{i,i^{\prime}}\frac{q^{h_{i}}-q^{-h_{i}}}{q-q^{-1}}\\\
&\hat{e}_{\pm,i}\hat{e}_{\pm,i^{\prime}}=\hat{e}_{\pm,i^{\prime}}\hat{e}_{\pm,i},(|i-i^{\prime}|\geq
2),\\\ &\hat{e}_{\pm,i}^{2}\hat{e}_{\pm,i\pm
1}-(q+q^{-1})\hat{e}_{\pm,i}\hat{e}_{\pm,i\pm
1}\hat{e}_{\pm,i}+\hat{e}_{\pm,i\pm 1}\hat{e}_{\pm,i}^{2}=0(1\leq i,i\pm 1\leq
N)\end{split}$ (1)

Here, $(a_{i,i^{\prime}})_{1\leq i,i^{\prime}\leq N}$ denotes the Cartan
matrix of type $A_{N}$, i.e.,

$a_{ii^{\prime}}=\begin{cases}2\quad(i=i^{\prime});\\\ -1\quad(i=i^{\prime}\pm
1);\\\ 0\quad(\text{otherwise})\end{cases}$ (2)

Then define $U_{q}(\mathfrak{gl}(N+1))$ by adjoining to
$U_{q}(\mathfrak{sl}(N+1))$ the elements $q^{\pm\epsilon_{i}/2}(0\leq i\leq
N)$ so that $q^{h_{i}}=q^{\epsilon_{i-1}-\epsilon_{i}}$ and that
$q^{\epsilon_{0}+\cdots+\epsilon_{N}}$ belongs to the center.

The $m$–fold co–product is the algebra homomorphism

$\Delta^{(m)}:{U_{q}(\mathfrak{gl}(N+1))}\rightarrow\underbrace{{U_{q}(\mathfrak{gl}(N+1))}\otimes\cdots\otimes{U_{q}(\mathfrak{gl}(N+1))}}_{m}$

such that

$\displaystyle\begin{split}&\Delta^{(m)}(q^{\pm\epsilon_{i}/2})=q^{\pm\epsilon_{i}/2}\otimes\cdots\otimes
q^{\pm\epsilon_{i}/2}\\\
&\Delta^{(m)}(\hat{e}_{\pm,i})=\sum_{v=1}^{m}\hat{e}^{(v)}_{\pm,i},\end{split}$
(3)

where

$\hat{e}^{(v)}_{\pm,i}=\underbrace{q^{h_{i}/2}\otimes\cdots\otimes
q^{h_{i}/2}}_{v-1}\otimes\hat{e}_{\pm,i}\otimes\underbrace{q^{-h_{i}/2}\otimes\cdots\otimes
q^{-h_{i}/2}}_{m-v}.$

We also have the reversed co–product

$\displaystyle\begin{split}&\bar{\Delta}^{(m)}(q^{\pm\epsilon_{i}/2})=q^{\pm\epsilon_{i}/2}\otimes\cdots\otimes
q^{\pm\epsilon_{i}/2}\\\
&\bar{\Delta}^{(m)}(\hat{e}_{\pm,i})=\sum_{v=1}^{m}\underbrace{q^{-h_{i}/2}\otimes\cdots\otimes
q^{-h_{i}/2}}_{v-1}\otimes\hat{e}_{\pm,i}\otimes
q^{h_{i}/2}\otimes\cdots\otimes q^{h_{i}/2}\end{split}$ (4)

We write $\Delta,\bar{\Delta}$ for $\Delta^{(2)},\bar{\Delta}^{(2)}$. The map
$\Delta$ endows ${U_{q}(\mathfrak{gl}(N+1))}$ with a structure of a
bi–algebra. It is also a Hopf algebra, but we will not need the counit and
antipode.

Consider the following relations for $R=R(\lambda,\mu)$:

$\displaystyle R\Delta(u)=\bar{\Delta}(u)R\quad(\forall
u\in{U_{q}(\mathfrak{gl}(N+1))})$ $\displaystyle
R(\lambda,\mu)(\lambda\hat{e}_{0}\otimes
q^{-h_{0}/2}+q^{h_{0}/2}\otimes\mu\hat{e}_{0})=(q^{-h_{0}/2}\otimes\lambda\hat{e}_{0}+\mu\hat{e}_{0}\otimes
q^{h_{0}/2}R(\lambda,\mu))$ (5)

This admits a unique (up to a multiplicative constant) solution
$R\in\operatorname{End}(V\otimes V)$, where $V:=\mathbb{C}^{N+1}$ is the
defining representation of ${U_{q}(\mathfrak{gl}(N+1))}$. From (5), it is
clear that $R(\lambda,\mu)$ only depends on $\lambda$ and $\mu$ through their
ratio $\lambda/\mu$, so we write $R(\lambda/\mu)=R(\lambda,\mu)$.

Let us consider (5) in
${U_{q}(\mathfrak{gl}(N+1))}\otimes\operatorname{End}({\mathbb{C}}^{N+1})$.
Then the solution is given by (see [Jim86]) $R(x)=\sum_{0\leq i,j\leq
N}\hat{E}_{ij}^{\prime}(x)\otimes e_{ji}$, where

$\displaystyle\hat{E}^{\prime}_{ij}(x)=\begin{cases}(xq^{(\epsilon_{i}+\epsilon_{j}-1)/2})^{\mp
1}{E}^{\prime}_{ij}\qquad(i\lessgtr j)\\\
(xq^{\epsilon_{i}}-x^{-1}q^{-\epsilon_{i}})/(q-q^{-1})\qquad(i=j)\end{cases}$
(6)

Here, $E^{\prime}_{ij}$ are the root vectors defined recursively by

$E^{\prime}_{ij}=E_{ik}^{\prime}E_{kj}^{\prime}-q^{\pm
1}E_{kj}^{\prime}E_{ik}^{\prime},\quad(i\lessgtr k\lessgtr j),\quad
E^{\prime}_{i-1,i}=\hat{e}_{+,i},\quad E^{\prime}_{i,i-1}=\hat{e}_{-,i}$ (7)

and $e_{ji}$ is the usual matrix which acts on the canonical basis
$\\{I_{0},\ldots,I_{N}\\}$ of $\mathbb{C}^{N+1}$ by

$e_{ji}I_{k}=\delta_{ik}I_{j}.$

In [GZB91], the authors define

$\hat{E}_{ij}=\begin{cases}q^{(E_{ii}+E_{jj}-1)/2}E_{ij},\quad i\neq j,\\\
q^{E_{ii}}/(q-q^{-1}),\quad i=j,\end{cases}$

where the modified root vectors are

$E_{ij}=E_{ik}E_{kj}-q^{-1}E_{kj}E_{ik},\quad(i\lessgtr k\lessgtr j),\quad
E^{\prime}_{i-1,i}=\hat{e}_{+,i},\quad E^{\prime}_{i,i-1}=\hat{e}_{-,i}$ (8)

Using Jimbo’s results, they show that

$\displaystyle\begin{split}R&=\sum_{N\geq i\geq j\geq 0}\hat{E}_{ij}\otimes
e_{ji}\in{U_{q}(\mathfrak{gl}(N+1))}\otimes\operatorname{End}(V)\\\
R^{T}&=\sum_{N\geq i\geq j\geq 0}\hat{E}_{ji}\otimes
e_{ij}\in{U_{q}(\mathfrak{gl}(N+1))}\otimes\operatorname{End}(V)\end{split}$
(9)

satisfy

$\displaystyle R\Delta(u)$ $\displaystyle=\bar{\Delta}(u)R$ $\displaystyle
R^{T}\bar{\Delta}(u)$ $\displaystyle={\Delta}(u)R^{T}$

We can solve (5) even more generally. Consider the $m$-fold tensor of the
defining representation $V^{\otimes m}$, and let $P_{m}V^{\otimes m}$ be the
symmetric projection. Then the solution of (5) in
${U_{q}(\mathfrak{gl}(N+1))}\otimes\operatorname{End}(P_{m}V^{\otimes m})$ is
given by the fused $R$–matrix [Jim86]

$R_{0,\\{1,2,\cdots,m\\}}(x)=R_{0m}(x)R_{0m-1}(xq)\cdots
R_{01}(xq^{m-1})P_{m}\in{U_{q}(\mathfrak{gl}(N+1))}\otimes\operatorname{End}(P_{m}V^{\otimes
m}).$ (10)

Therefore, the fused $R$–matrices

$\displaystyle R_{0,\\{1,2,\cdots,m\\}}$ $\displaystyle=R_{0m}R_{0m-1}\cdots
R_{01}P_{m}\in
U_{q}(\mathfrak{gl}_{N+1})\otimes\operatorname{End}(P_{m}V^{\otimes m})$
$\displaystyle R^{T}_{0,\\{1,2,\cdots,m\\}}$
$\displaystyle=R^{T}_{0m}R^{T}_{0m-1}\cdots R^{T}_{01}P_{m}\in
U_{q}(\mathfrak{gl}_{N+1})\otimes\operatorname{End}(P_{m}V^{\otimes m})$

satisfy

$\displaystyle R_{0,\\{1,2,\cdots,m\\}}\Delta(u)$
$\displaystyle=\bar{\Delta}(u)R_{0,\\{1,2,\cdots,m\\}}$ (11) $\displaystyle
R^{T}_{0,\\{1,2,\cdots,m\\}}\bar{\Delta}(u)$
$\displaystyle={\Delta}(u)R^{T}_{0,\\{1,2,\cdots,m\\}}$ (12)

in ${U_{q}(\mathfrak{gl}(N+1))}\otimes\operatorname{End}(P_{m}V^{\otimes m})$
for all $u\in{U_{q}(\mathfrak{gl}(N+1))}$.

Therefore

$\Gamma_{m}:=R^{T}_{0,\\{1,2,\cdots,m\\}}R_{0,\\{1,2,\cdots,m\\}}\in{U_{q}(\mathfrak{gl}(N+1))}\otimes\operatorname{End}(P_{m}V^{\otimes
m})$ (13)

commutes with $\Delta(u)$ for all $u\in{U_{q}(\mathfrak{gl}(N+1))}$. By
Drinfeld’s central element construction [Dri90], the element ,

$C_{m}=\operatorname{id}\otimes\operatorname{tr}_{q}(\Gamma_{m})$ (14)

is central in ${U_{q}(\mathfrak{gl}(N+1))}$, where the quantum trace
$\operatorname{tr}_{q}$ of an operator $A$ is defined by

$\operatorname{tr}_{q}(A)=\operatorname{tr}(q^{-2h_{\rho}}A):=\operatorname{tr}(q^{-N\epsilon_{0}-(N-2)\epsilon_{1}-\cdots+N\epsilon_{N}}A)$
(15)

We also have the following relations between the root vectors:333 Similar
relations can be found in the paper [Xi94], but there appear to be some typos.
They can also be derived from (11) by applying $\operatorname{id}\otimes B$ to
both sides, for suitable linear maps $B$ on $\mathrm{End}(P_{2}V^{\otimes
2})$. One can also check that the relations hold in the explicit
representations in Remark 1 below. For $i<l$ and $j<k$,

${E}_{il}{E}_{jk}=\begin{cases}{E}_{jk}{E}_{il},&\quad i<j<k<l\text{ or
}i<l<j<k,\\\ q^{-1}{E}_{jk}{E}_{il},&\quad i=j<k<l\\\ q{E}_{jk}{E}_{il},&\quad
i<j<k=l\\\ {E}_{jk}{E}_{il}+(q-q^{-1}){E}_{jl}{E}_{ik},&\quad
i<j<l<k,\end{cases}$ (16)
${E}_{li}{E}_{kj}=\begin{cases}{E}_{kj}{E}_{li},&\quad i<j<k<l\text{ or
}i<l<j<k,\\\ q{E}_{kj}{E}_{li},&\quad i=j<k<l\\\ q^{-1}{E}_{kj}{E}_{li},&\quad
i<j<k=l\\\ {E}_{kj}{E}_{li}-(q-q^{-1}){E}_{lj}{E}_{ki},&\quad
i<j<l<k,\end{cases}$ (17)

By the first and fourth lines above,

$\displaystyle\hat{E}_{il}\hat{E}_{jk}+q^{-1}\hat{E}_{ik}\hat{E}_{jl}=\hat{E}_{jk}\hat{E}_{il}+q\hat{E}_{jl}\hat{E}_{ik},$
(18)
$\displaystyle\hat{E}_{li}\hat{E}_{kj}+q\hat{E}_{ki}\hat{E}_{lj}=\hat{E}_{kj}\hat{E}_{li}+q^{-1}\hat{E}_{lj}\hat{E}_{ki}.$
(19)

For any ${\mathbf{i}}$ and ${\mathbf{j}}$ in $\mathbb{N}^{m}$, define

$\hat{E}^{\pm}_{{\mathbf{i}}{\mathbf{j}}}:=\begin{cases}\hat{E}_{i_{1}j_{1}}\cdots\hat{E}_{i_{m}j_{m}},\quad\pm(i_{1}-j_{1}),\cdots,\pm(i_{m}-j_{m})\geq
0,\\\ 0,\quad\text{ else}.\end{cases}$

and let $e_{{\mathbf{j}}{\mathbf{i}}}:=e_{j_{1}i_{1}}\otimes\cdots\otimes
e_{j_{m}i_{m}}$. For ${\mathbf{i}},{\mathbf{j}}\in\mathcal{W}_{m}$, define

$\tilde{E}^{\pm}_{{\mathbf{i}}{\mathbf{j}}}=\sum_{\zeta\in
D_{{\mathbf{i}}}}q^{\pm(\operatorname{inv}(\tau)-\operatorname{inv}(\zeta))}\hat{E}^{\pm}_{\bar{\zeta}({\mathbf{i}})\bar{\tau}({\mathbf{j}})},$

where $\tau$ is an arbitrary element of $D_{{\mathbf{j}}}$. Note that the
notation does not depend on $\tau$, which is justified by the following lemma
and the relation
$\operatorname{inv}(\bar{\tau})=(m-1)m/2-\operatorname{inv}(\tau)$.

######  Lemma 3.

  1. 1.

For every $\tau,\tau^{\prime}\in D_{A}$, the following identity holds:

$\sum_{\sigma\in
D_{B}}q^{\mp(\operatorname{inv}(\sigma)-\operatorname{inv}(\tau))}\hat{E}^{\pm}_{\tau(A)\sigma(B)}=\sum_{\sigma\in
D_{B}}q^{\mp(\operatorname{inv}(\sigma)-\operatorname{inv}(\tau^{\prime}))}\hat{E}^{\pm}_{\tau^{\prime}(A)\sigma(B)}$
(20)

  2. 2.

Likewise, for every $\sigma,\sigma^{\prime}\in D_{B}$,

$\sum_{\tau\in
D_{A}}q^{\mp(\operatorname{inv}(\tau)-\operatorname{inv}(\sigma))}\hat{E}^{\pm}_{\tau(A)\sigma(B)}=\sum_{\tau\in
D_{A}}q^{\mp(\operatorname{inv}(\tau)-\operatorname{inv}(\sigma^{\prime}))}\hat{E}^{\pm}_{\tau(A)\sigma^{\prime}(B)}$
(21)

###### Proof.

For $m=2$, both cases are equivalent to the relations in (16) and (17).

Now suppose that $m>2$. We only prove part 1, as part 2 is similar. By Lemma
1, it suffices to consider the case when $\tau^{\prime}=s\tau$ where $s$ is a
transposition, and assume without loss of generality that
$\operatorname{inv}(\tau^{\prime})=\operatorname{inv}(\tau)+1$. Define the two
sets $D_{B}^{(1)}$ and $D_{B}^{(2)}$ by

$\displaystyle D_{B}^{(1)}$ $\displaystyle=\\{\sigma\in D_{B}:s\sigma(B)=B\\}$
$\displaystyle D_{B}^{(2)}$ $\displaystyle=\\{\sigma\in D_{B}:s\sigma(B)\neq
B\\}$

First, note that by the second and third lines of (16),

$\sum_{\sigma\in
D_{B}^{(1)}}q^{\operatorname{inv}(\sigma)-\operatorname{inv}(\tau)}\hat{E}^{-}_{\tau(A)\sigma(B)}=\sum_{\sigma\in
D_{B}^{(1)}}q^{\operatorname{inv}(\sigma)-\operatorname{inv}(s\tau)}\hat{E}^{-}_{s\tau(A)\sigma(B)}.$

Partition the set $D_{B}^{(2)}$ into two sets $D_{-}$ and $D_{+}$ of equal
cardinality, where $\sigma\in D_{-}$ if and only if
$\operatorname{inv}(\sigma)<\operatorname{inv}(s\sigma)$. Then

$\displaystyle\sum_{\sigma\in
D_{B}^{(2)}}q^{\operatorname{inv}(\sigma)-\operatorname{inv}(\tau)}\hat{E}^{-}_{\tau(A)\sigma(B)}$
$\displaystyle=\sum_{\sigma\in
D_{-}}\left(q^{\operatorname{inv}(\sigma)-\operatorname{inv}(\tau)}\hat{E}^{-}_{\tau(A)\sigma(B)}+q^{\operatorname{inv}(\sigma)+1-\operatorname{inv}(\tau)}\hat{E}^{-}_{\tau(A)s\sigma(B)}\right)$
$\displaystyle\stackrel{{\scriptstyle\eqref{comm2}}}{{=}}\sum_{\sigma\in
D_{-}}\left(q^{\operatorname{inv}(\sigma)+1-\operatorname{inv}(\tau^{\prime})}\hat{E}^{-}_{s\tau(A)s\sigma(B)}+q^{\operatorname{inv}(\sigma)-1-\operatorname{inv}(\tau)}\hat{E}^{-}_{s\tau(A)\sigma(B)}\right)$
$\displaystyle=\sum_{\sigma\in
D_{-}}\left(q^{\operatorname{inv}(\sigma)+1-\operatorname{inv}(\tau^{\prime})}\hat{E}^{-}_{\tau^{\prime}(A)s\sigma(B)}+q^{\operatorname{inv}(\sigma)-\operatorname{inv}(\tau^{\prime})}\hat{E}^{-}_{\tau^{\prime}(A)\sigma(B)}\right)$
$\displaystyle=\sum_{\sigma\in
D_{B}^{(2)}}q^{\operatorname{inv}(\sigma)-\operatorname{inv}(\tau^{\prime})}\hat{E}^{-}_{\tau^{\prime}(A)\sigma(B)},$

as needed. The proof for $E^{+}$ is similar, where one uses (19) instead of
(18). ∎

######  Example 2.

Consider $m=2$ and $N=3$. Set ${\mathbf{j}}=(0,1)$ and ${\mathbf{i}}=(2,3)$.
Then

$\tilde{E}_{{\mathbf{j}}{\mathbf{i}}}^{-}=\hat{E}_{13}\hat{E}_{02}+q\hat{E}_{03}\hat{E}_{12}=q^{-1}(\hat{E}_{12}\hat{E}_{03}+q\hat{E}_{02}\hat{E}_{13}),$

with the equality following from (18). Additionally,

$\tilde{E}_{{\mathbf{i}}{\mathbf{j}}}^{+}=\hat{E}_{31}\hat{E}_{20}+q^{-1}\hat{E}_{21}\hat{E}_{30}=q\left(\hat{E}_{30}\hat{E}_{21}+q^{-1}\hat{E}_{20}\hat{E}_{31}\right),$

with the equality following from (19).

##  3 Statements and proofs

The main theorem is the following expression for central elements of
${U_{q}(\mathfrak{gl}(N+1))}$.

######  Theorem 1.

The element given by

$C_{m}=\sum_{{\mathbf{i}},{\mathbf{j}}\in\mathcal{W}_{m}}q^{2i_{1}+\ldots+2i_{m}-Nm}\tilde{E}_{{\mathbf{j}}{\mathbf{i}}}^{-}\tilde{E}_{{\mathbf{i}}{\mathbf{j}}}^{+}$
(22)

is central in ${U_{q}(\mathfrak{gl}(N+1))}$.

######  Example 3.

Consider $m=1$. Then

$C_{1}=(q-q^{-1})^{-2}\sum_{i=0}^{N}q^{2i-N}q^{2\epsilon_{i}}+\sum_{0\leq
j<i\leq N}q^{2i-N-1}q^{2\epsilon_{i}+2\epsilon_{j}}E_{ji}E_{ij},$

which is (up to a constant) the central element $C$ from [GZB91].

######  Example 4.

Consider $m=2$ and $N=3$. Then

$\displaystyle C_{2}$ $\displaystyle=\sum_{0\leq j\leq i\leq
3}q^{-6}q^{4i}\hat{E}_{ji}^{2}\hat{E}_{ij}^{2}$ $\displaystyle+\sum_{0\leq
j\leq i_{1}<i_{2}\leq
3}q^{-6+2i_{1}+2i_{2}}\hat{E}_{ji_{1}}\hat{E}_{ji_{2}}(\hat{E}_{i_{2}j}\hat{E}_{i_{1}j}+q^{-1}\hat{E}_{i_{1}j}\hat{E}_{i_{2}j})$
$\displaystyle+\sum_{0\leq j_{1}<j_{2}\leq i\leq
3}q^{-6+4i}(\hat{E}_{j_{1}i}\hat{E}_{j_{2}i}+q^{-1}\hat{E}_{j_{2}i}\hat{E}_{j_{1}i})\hat{E}_{ij_{2}}\hat{E}_{ij_{1}}$
$\displaystyle+\sum_{0\leq j_{1}\leq i_{1}<j_{2}\leq i_{2}\leq
3}q^{-6+2i_{1}+2i_{2}}\hat{E}_{j_{1}i_{1}}\hat{E}_{j_{2}i_{2}}\hat{E}_{i_{2}j_{2}}\hat{E}_{i_{1}j_{1}}$
$\displaystyle+\sum_{0\leq j_{1}<j_{2}\leq i_{1}<i_{2}\leq
3}q^{-6+2i_{1}+2i_{2}}(\hat{E}_{j_{1}i_{1}}\hat{E}_{j_{2}i_{2}}+q^{-1}\hat{E}_{j_{i}i_{2}}\hat{E}_{j_{2}i_{1}})(\hat{E}_{i_{2}j_{2}}\hat{E}_{i_{1}j_{1}}+q^{-1}\hat{E}_{i_{2}j_{1}}\hat{E}_{i_{1}j_{2}})$

The central element $C_{2}$ has $50$ terms, consisting of $10$ terms from
$0\leq j\leq i\leq 3$, $20$ terms from $0\leq j_{1}<j_{2}\leq i\leq 3$ and
$0\leq j\leq i_{1}<i_{2}\leq 3$, $15$ terms coming from $0\leq j_{1}\leq
i_{1}<j_{2}\leq i_{2}\leq 3$ and $5$ terms coming from $0\leq j_{1}<j_{2}\leq
i_{1}<i_{2}\leq 3$. One can also verify that $50$ is the correct number of
terms, from the fact that $|\mathcal{W}_{2}|=10$ and the set
$\\{\\{{\mathbf{i}},{\mathbf{j}}\\}:{\mathbf{i}},{\mathbf{j}}\in\mathcal{W}_{2}\\}$
has $55=10(10+1)/2$ elements, but the term
$\tilde{E}^{\pm}_{{\mathbf{j}}{\mathbf{i}}}$ is zero when
$\\{{\mathbf{i}},{\mathbf{j}}\\}$ is one of the $5$ sets
$\\{(0,2),(1,1)\\},\\{(0,3),(2,2)\\},\\{(0,3),(1,1)\\},\\{(1,3),(2,2)\\},\\{(1,2),(0,3)\\}$.

The representation $P_{2}\mathbb{C}^{4}$ of ${U_{q}(\mathfrak{gl}(N+1))}$ is
$10$–dimensional, which can be explicitly written using Example 2 below. By
multiplying $10\times 10$ matrices, one can check that $C_{2}$ acts as
$\text{const}\cdot\mathrm{Id}_{10}$, where the constant is
$(q+q^{-1})^{-4}q^{-6}(1+q^{2}+2q^{4}+q^{6}+q^{8}+q^{10}+q^{12}+q^{14}+q^{20})$.

######  Remark 1.

In [ZGB91a], the central element $C^{\Lambda_{0}}$ is defined, where
$\Lambda_{0}$ is the highest weight of a finite–dimensional irreducible module
$V(\Lambda_{0})$. The construction there is similar to the one here, with the
major difference being the use of explicit universal $R$–matrices in place of
fused $R$–matrices. Although it is not necessarily simple to check directly
that $C_{m}$ equals (up to a constant) $C^{\Lambda_{0}}$ for
$V(\Lambda_{0})=P_{m}V^{\otimes m}$, it is straightforward to check that their
eigenvalues are the same.

If $V(\Lambda_{0})$ has distinct weights $\lambda_{1},\ldots,\lambda_{r}$ with
multiplicities $d_{1},\ldots,d_{r}$, then the eigenvalue of $C^{\Lambda_{0}}$
on an irreducible module with highest weight $\Lambda$ is given by

$\sum_{k=1}^{r}d_{k}q^{2(\Lambda+\rho,\lambda_{k})}.$

Here, $\rho$ is half the sum of the positive roots, and $(\cdot,\cdot)$ is the
usual invariant bilinear form on $\mathfrak{h}^{*}$. Now take $\Lambda_{0}$ to
be the highest weight of $P_{m}V^{\otimes m}$; then the distinct weights are
elements of $\mathcal{B}^{(N)}_{m}$ with multiplicity $1$. Therefore the
eigenvalue is

$\sum_{\mu\in\mathcal{B}^{(N)}_{m}}q^{(2\rho,\mu)}q^{2(\Lambda,\mu)}$

If $C_{m}$ acts on the same module $V(\Lambda)$, then its eigenvalue can be
found by evaluating on the lowest weight vector, because then only the
diagonal terms $({\mathbf{i}}={\mathbf{j}})$ have a nonzero contribution. So
the eigenvalue is

$(q-q^{-1})^{-2m}q^{-Nm}\sum_{{\mathbf{i}}\in\mathcal{W}_{m}}q^{2i_{1}+\ldots+2i_{m}}q^{(2\mu^{(N)}({\mathbf{i}}),\Lambda)}.$

By Lemma 2, this is $(q-q^{-1})^{-2m}q^{-Nm}$ times the eigenvalue of
$C^{\Lambda_{0}}$.

###  3.1  Basis for $P_{m}V^{\otimes m}$

Before proving Theorem 1, we will write a basis for the symmetric projection
$P_{m}V^{\otimes m}$.

Let $I_{0},\cdots,I_{N}$ be the canonical basis of $V$, and define the action
on $V$ by

$\displaystyle\hat{e}_{+,i}I_{j}=\delta_{ij}I_{j-1}$ (23)
$\displaystyle\hat{e}_{-,i}I_{j-1}=\delta_{ij}I_{j}$ (24) $\displaystyle
q^{\pm\epsilon_{i}/2}I_{j}=q^{\pm\delta_{i}j/2}I_{j}.$ (25)

For $\mu\in\mathcal{B}_{m}^{(N)}$, define the vector

$v{(\mu)}=I_{0}^{\otimes\mu_{0}}\otimes\cdots\otimes I_{N}^{\otimes\mu_{N}}$
(26)

Define

$M({\mu})=\sum_{\sigma\in
D^{\mu}}q^{-\operatorname{inv}(\sigma)}\sigma(v(\mu)).$ (27)

Here, as before, $S_{m}$ acts on $V^{\otimes m}$ by permuting the order. By an
abuse of notation, for ${\mathbf{i}}\in\mathcal{W}_{m}$ we define

$v({\mathbf{i}})=v(\mu^{(N)}({\mathbf{i}})),\quad
M({\mathbf{i}})=M(\mu^{(N)}({\mathbf{i}})),$

where $N\geq i_{m}$. Note that these definitions do not depend on the value of
$N$. We briefly note the identity

$e^{\sigma(a)}_{\pm,i}\left(\sigma(v(\mu))\right)=\sigma\left(e^{(a)}_{\pm,i}\left(v(\mu)\right)\right)\text{
for all }\sigma\in S_{m}.$ (28)

We now show that the set $\\{M(\mu)\\}_{\mu\in\mathcal{B}^{(N)}_{m}}$ gives a
basis for $P_{m}V^{\otimes m}$. A more general statement appeared in [Kuab]
with a more complicated proof, but the expression here is more convenient for
calculations.

######  Theorem 2.

The set $\\{M(\mu)\\}_{\mu\in\mathcal{B}^{(N)}_{m}}$ gives a basis for
$P_{m}V^{\otimes m}$.

###### Proof.

Note that $|\mathcal{B}^{(N)}_{m}|=\operatorname{dim}P_{m}V^{\otimes m}$ and
$\\{M(\mu)\\}_{\mu\in\mathcal{B}^{(N)}_{m}}$ is a linearly independent set, so
it suffices to show that $M(\mu)\in P_{m}V^{\otimes m}$ for all
$\mu\in\mathcal{B}^{(N)}_{m}$. To show that every $M(\mu)$ is in
$P_{m}V^{\otimes m}$, it suffices to show that

$\Delta^{(m)}(\hat{e}_{-,i})(M(\mu))=q^{\frac{1}{2}(\mu_{i-1}+\mu_{i}-1)}(1+q^{-2}+q^{-4}+\cdots+q^{-2\mu_{i}})\cdot
M(\mu-\hat{i})$ (29)

where $\hat{i}=(0,\cdots,0,1,-1,0,\cdots,0)$ with the $-1$ occuring in the
$i$th position. By (3) and (24), $\Delta^{(m)}(\hat{e}_{-,i})(M(\mu))$ is in
the span of $\\{\tau(v(\mu-\hat{i}))\\}_{\tau\in S_{m}}$. Let $A(\tau)$ be the
coefficients in the expansion

$\Delta^{(m)}(\hat{e}_{-,i})(M(\mu))=\sum_{\tau\in
D^{\mu-\hat{i}}}A(\tau)\tau(v(\mu-\hat{i})).$

It suffices to show that
$A(\tau)=q^{\frac{1}{2}(\mu_{i-1}+\mu_{i}-1)}(1+q^{-2}+q^{-4}+\cdots+q^{-2\mu_{i}})q^{-\operatorname{inv}(\tau)}$
for all $\tau\in D^{\mu-\hat{i}}$. In fact, we show something stronger: for
all $\tau\in D^{\mu-\hat{i}}$, there exist elements
$\sigma^{(0)},\ldots,\sigma^{(\mu_{i})}\in D^{\mu}$ such that for $0\leq
j\leq\mu_{i}$,

$\hat{e}_{-,i}^{(a_{j})}\left(q^{-\operatorname{inv}(\sigma^{(j)})}\sigma^{(j)}(v(\mu))\right)=q^{\frac{1}{2}(\mu_{i-1}+\mu_{i}-1)}q^{-2j}q^{-\operatorname{inv}(\tau)}\tau(v(\mu-\hat{i})).$

We will proceed by induction on the value of $\operatorname{inv}(\tau)$, using
Lemma 1. The base case is when $\tau$ is the identity permutation. Then it is
straightforward to check that

$\sigma^{(j)}=(\mu_{[0,i-1]}\ \ \ \ \mu_{[0,i-1]}+1\ \ \ \ \cdots\ \ \ \
\mu_{[0,i-1]}+j),$

where $\mu_{[0,i-1]}=\mu_{0}+\ldots+\mu_{i-1}$, satisfies the necessary
conditions.

Now fix $\tau\in D^{\mu-\hat{i}}$, and suppose that the induction hypothesis
holds for $\tau$. Suppose that $\tilde{\tau}\in D^{\mu-\hat{i}}$ satisfies
$\tilde{\tau}=s\tau$ for some transposition $s$, and assume without loss of
generality that $\operatorname{inv}(\tilde{\tau})=\operatorname{inv}(\tau)+1$.
Define

$\tilde{\sigma}^{(j)}=\begin{cases}s\sigma^{(j)},&\text{ if }s\sigma^{(j)}\in
D^{\mu},\\\ \sigma^{(j)},&\text{ else. }\end{cases}$

We now aim to prove that

$\hat{e}^{(s(a_{j}))}_{-,i}\left(q^{-\operatorname{inv}(\tilde{\sigma}^{(j)})}\tilde{\sigma}^{(j)}(v(\mu))\right)=q^{\frac{1}{2}(\mu_{i-1}+\mu_{i}-1)}q^{-2j}q^{-\operatorname{inv}(\tilde{\tau})}\tilde{\tau}(v(\mu-\hat{i})).$
(30)

If $\tilde{\sigma}^{(j)}=s\sigma^{(j)}$, then (30) follows from (28) and the
induction hypothesis. Now assume that $\tilde{\sigma}^{(j)}=\sigma^{(j)}$.
Then $s=(a_{j}-1\ \ \ a_{j})$ and

$\tilde{\sigma}^{(j)}\left(v(\mu)\right)=\underbrace{I_{*}\otimes\cdots\otimes
I_{*}}_{a_{j}-2}\otimes I_{i-1}\otimes
I_{i-1}\otimes\underbrace{I_{*}\otimes\cdots\otimes I_{*}}_{m-a_{j}}.$

Using that

$P\left((\hat{e}_{i,-}\otimes q^{-h_{i}/2})(I_{i-1}\otimes
I_{i-1})\right)=q^{-1}(q^{h_{i}/2}\otimes\hat{e}_{i,-})(I_{i-1}\otimes
I_{i-1}),$

where $P(x\otimes y)=y\otimes x$ is the permutation operator, we have that

$\hat{e}^{(s(a_{j}))}_{-,i}\left(q^{-\operatorname{inv}(\tilde{\sigma}^{(j)})}\tilde{\sigma}^{(j)}(v(\mu))\right)=q^{-1}\hat{e}^{(a_{j})}_{-,i}\left(q^{-\operatorname{inv}({\sigma}^{(j)})}{\sigma}^{(j)}(v(\mu))\right).$

And now (30) follows from the induction hypothesis.

∎

######  Remark 2.

For all $\mu\in\mathcal{B}^{(N)}_{m}$, let $\tilde{M}(\mu)=c(\mu)M(\mu)$ where
$c(\mu)$ is defined inductively by

$\begin{split}c(m,0,\cdots,0)=1\\\
\frac{c(\mu-i)}{c(\mu)}=\frac{q^{\frac{\mu_{i-1}+\mu_{i}+1}{2}}(1-q^{-2\mu_{i}})}{q^{\mu_{i}+1}-q^{-\mu_{i}-1}},\end{split}$
(31)

then equation (29) is equivalent to

$\Delta^{(m)}(\hat{e}_{-,i})\tilde{M}(\mu)=\frac{q^{\mu_{i}+1}-q^{-\mu_{i}-1}}{q-q^{-1}}\tilde{M}(\mu-\hat{i}).$
(32)

In fact, one can show that

$\displaystyle\Delta^{(m)}(\hat{e}_{+,i})\tilde{M}(\mu)=\frac{q^{\mu_{i-1}+1}-q^{-\mu_{i-1}-1}}{q-q^{-1}}\tilde{M}(\mu+\hat{i})$
(33)
$\displaystyle\Delta^{(m)}(\hat{e}_{-,i})\tilde{M}(\mu)=\frac{q^{\mu_{i}+1}-q^{-\mu_{i}-1}}{q-q^{-1}}\tilde{M}(\mu-\hat{i})$
(34) $\displaystyle\Delta^{(m)}(q^{\pm
h_{i}/2})\tilde{M}(\mu)=q^{\pm\frac{1}{2}(\mu_{i-1}-\mu_{i})}\tilde{M}(\mu)$
(35)

defines a representation on $P_{m}V^{\otimes m}$. This is equivalent to the
representation in equation (3) of [KMMO16] and Lemma 3.1 of [Kuab].

######  Corollary 1.

Let ${\mathbf{i}},{\mathbf{j}}\in\mathcal{W}_{m}$. For any $\tau,\sigma\in
D_{{\mathbf{i}}}$ and any $\zeta\in D_{{\mathbf{j}}}$,

$\left(q^{\operatorname{inv}(\tau)}e_{\zeta({\mathbf{j}})\tau({\mathbf{i}})}-q^{\operatorname{inv}(\sigma)}e_{\zeta({\mathbf{j}})\sigma({\mathbf{i}})}\right)\Big{|}_{P_{m}^{+}V^{\otimes
m}}=0.$

Furthermore,

$\left(\sum_{\tau\in
D_{{\mathbf{j}}}}q^{-d_{{\mathbf{i}}}(\tau)}e_{\tau({\mathbf{j}})\tau({\mathbf{i}})}\right)M({\mathbf{i}})=M({\mathbf{j}}).$

###### Proof.

For any $\mu\in\mathcal{B}^{(N)}_{m}$ not equal to $\mu({\mathbf{i}})$, it is
straightforward that

$\left(q^{\operatorname{inv}(\tau)}e_{\zeta({\mathbf{j}})\tau({\mathbf{i}})}-q^{\operatorname{inv}(\sigma)}e_{\zeta({\mathbf{j}})\sigma({\mathbf{i}})}\right)M(\mu)=0-0=0.$

On the other hand,

$\displaystyle\left(q^{\operatorname{inv}(\tau)}e_{\zeta({\mathbf{j}})\tau({\mathbf{i}})}-q^{\operatorname{inv}(\sigma)}e_{\zeta({\mathbf{j}})\sigma({\mathbf{i}})}\right)M(\mu({\mathbf{i}}))$
$\displaystyle=q^{\operatorname{inv}(\tau)}e_{\zeta({\mathbf{j}})\tau({\mathbf{i}})}(q^{-\operatorname{inv}(\tau)}\tau(v(\mu({\mathbf{i}}))))-q^{\operatorname{inv}(\sigma)}e_{\zeta({\mathbf{j}})\sigma({\mathbf{i}})}(q^{-\operatorname{inv}(\sigma)}\sigma(v(\sigma({\mathbf{i}}))))$
$\displaystyle=\zeta(v(\mu({\mathbf{j}})))-\zeta(v(\mu({\mathbf{j}})))=0.$

For the second statement, we use that

$e_{\tau({\mathbf{j}})\tau({\mathbf{i}})}\sigma(v({\mathbf{i}}))=\begin{cases}\tau(v({\mathbf{j}})),&\tau\sim_{{\mathbf{i}}}\sigma,\\\
0,&\text{ else.}\end{cases}$

in order to show

$\displaystyle\left(\sum_{\tau\in
D_{{\mathbf{j}}}}q^{-d_{{\mathbf{i}}}(\tau)}e_{\tau({\mathbf{j}})\tau({\mathbf{i}})}\right)\sum_{\sigma\in
D_{{\mathbf{i}}}}q^{-\operatorname{inv}(\sigma)}\sigma(v({\mathbf{i}}))$
$\displaystyle=\sum_{\begin{subarray}{c}\tau\in D_{{\mathbf{j}}},\sigma\in
D_{{\mathbf{i}}}\\\
\tau\sim_{{\mathbf{i}}}\sigma\end{subarray}}q^{-d_{{\mathbf{i}}}(\tau)+\operatorname{inv}(\sigma)}e_{\tau({\mathbf{j}})\tau({\mathbf{i}})}\sigma(v({\mathbf{i}}))$
$\displaystyle=\sum_{\tau\in
D_{{\mathbf{j}}}}q^{-\operatorname{inv}(\tau)}\tau(v({\mathbf{j}})).$

∎

######  Example 5.

Some examples are

$\displaystyle(e_{21}\otimes e_{31}+q^{-1}e_{31}\otimes
e_{21})M(\mathbf{1,1})$ $\displaystyle=M(\mathbf{2,3}),$
$\displaystyle(e_{31}\otimes e_{32})M(\mathbf{1,2})$
$\displaystyle=M(\mathbf{3,3}),$ $\displaystyle(e_{31}\otimes
e_{42}+e_{42}\otimes e_{31})M(\mathbf{1,2})$ $\displaystyle=M(\mathbf{3,4}).$

With Corollary 1 as motivation, define for any
${\mathbf{i}},{\mathbf{j}}\in\mathcal{W}_{m}$,

$\tilde{e}_{{\mathbf{j}}{\mathbf{i}}}=\sum_{\tau\in
D_{{\mathbf{j}}}}q^{-d_{{\mathbf{i}}}(\tau)}e_{\tau({\mathbf{j}})\tau({\mathbf{i}})}.$

###  3.2 Proof of Theorem 1

We will now prove Theorem 1. Begin by re-writing $R_{0,\\{1,\ldots,m\\}}$. By
definition,

$\displaystyle R_{0,\\{1,\ldots,m\\}}$ $\displaystyle=R_{0m}R_{0m-1}\cdots
R_{01}$ $\displaystyle=\left(\sum_{i_{m}\geq
j_{m}}\hat{E}_{i_{m}j_{m}}\otimes\operatorname{id}^{\otimes m-1}\otimes
e_{j_{m}i_{m}}\right)\cdots\left(\sum_{i_{1}\geq
j_{1}}\hat{E}_{i_{1}j_{1}}\otimes
e_{j_{1}i_{1}}\otimes\operatorname{id}^{\otimes m-1}\right)$
$\displaystyle=\sum_{i_{m}\geq j_{m}}\cdots\sum_{i_{1}\geq
j_{1}}\hat{E}_{i_{m}j_{m}}\cdots\hat{E}_{i_{1}j_{1}}\otimes
e_{j_{1}i_{1}}\otimes\cdots\otimes e_{j_{m}i_{m}}$
$\displaystyle=\sum_{{\mathbf{i}},{\mathbf{j}}\in\mathcal{W}_{m}}\sum_{\zeta\in
D_{{\mathbf{i}}},\tau\in
D_{{\mathbf{j}}}}\hat{E}_{\bar{\zeta}({\mathbf{i}})\bar{\tau}({\mathbf{j}})}^{+}\otimes
e_{\tau({\mathbf{j}})\zeta({\mathbf{i}})}.$

From here, the goal is to write this expression as an element of
${U_{q}(\mathfrak{gl}(N+1))}\otimes\operatorname{End}(P_{m}(V^{\otimes m}))$.
By Corollary 1,

$R_{0,\\{1,\ldots,m\\}}=\sum_{{\mathbf{i}},{\mathbf{j}}\in\mathcal{W}_{m}}\sum_{\zeta\in
D_{{\mathbf{i}}},\tau\in
D_{{\mathbf{j}}}}q^{\operatorname{inv}(\sigma_{{\mathbf{i}}}(\tau))-\operatorname{inv}(\zeta)}\hat{E}^{+}_{\bar{\zeta}({\mathbf{i}})\bar{\tau}({\mathbf{j}})}\otimes
e_{\tau({\mathbf{j}})\sigma_{{\mathbf{i}}}(\tau)({\mathbf{i}})}.$

By definition, $\tau=\sigma_{{\mathbf{i}}}(\tau)\xi_{{\mathbf{i}}}(\tau)$ and
$d_{{\mathbf{i}}}(\tau)=\operatorname{inv}(\xi_{{\mathbf{i}}}(\tau))$, so
therefore

$R_{0,\\{1,\ldots,m\\}}=\sum_{{\mathbf{i}},{\mathbf{j}}\in\mathcal{W}_{m}}\sum_{\zeta\in
D_{{\mathbf{i}}},\tau\in
D_{{\mathbf{j}}}}q^{\operatorname{inv}(\tau)-d_{{\mathbf{i}}}(\tau)-\operatorname{inv}(\zeta)}\hat{E}^{+}_{\bar{\zeta}({\mathbf{i}})\bar{\tau}({\mathbf{j}})}\otimes
e_{\tau({\mathbf{j}})\tau({\mathbf{i}})}.$

By Lemma 3 and the identity
$\operatorname{inv}(\bar{\tau})=m(m-1)/2-\operatorname{inv}(\tau)$,

$\sum_{\zeta\in
D_{{\mathbf{i}}}}q^{\operatorname{inv}(\tau)-\operatorname{inv}(\zeta)}\hat{E}_{\bar{\zeta}({\mathbf{i}})\bar{\tau}({\mathbf{j}})}$

does not depend on $\tau$. Therefore, the $R$–matrix factors as

$\displaystyle R_{0,\\{1,\ldots,m\\}}$
$\displaystyle=\sum_{{\mathbf{i}},{\mathbf{j}}\in\mathcal{W}_{m}}\left(\sum_{\zeta\in
D_{{\mathbf{i}}}}q^{\operatorname{inv}(\tau)-\operatorname{inv}(\zeta)}\hat{E}^{+}_{\bar{\zeta}({\mathbf{i}})\bar{\tau}({\mathbf{j}})}\right)\otimes\left(\sum_{\tau\in
D_{{\mathbf{j}}}}q^{-d_{{\mathbf{i}}}(\tau)}e_{\tau({\mathbf{j}})\tau({\mathbf{i}})}\right)$
$\displaystyle=\sum_{{\mathbf{i}},{\mathbf{j}}\in\mathcal{W}_{m}}\tilde{E}_{{\mathbf{i}}{\mathbf{j}}}^{+}\otimes
e_{{\mathbf{j}}{\mathbf{i}}}$

A similar argument shows that

$R^{T}_{0,\\{1,\ldots,m\\}}=\sum_{{\mathbf{i}},{\mathbf{j}}\in\mathcal{W}_{m}}\tilde{E}^{-}_{{\mathbf{j}}{\mathbf{i}}}\otimes
e_{{\mathbf{i}}{\mathbf{j}}}$

Therefore,

$\Gamma_{m}=R^{T}_{0,\\{1,\ldots,m\\}}R_{0,\\{1,\ldots,m\\}}=\sum_{{\mathbf{i}},{\mathbf{j}}\in\mathcal{W}_{m}}\sum_{{\mathbf{i}}^{\prime},{\mathbf{j}}^{\prime}\in\mathcal{W}_{m}}\tilde{E}_{{\mathbf{j}}{\mathbf{i}}}^{-}\tilde{E}_{{\mathbf{i}}^{\prime}{\mathbf{j}}^{\prime}}^{+}\otimes
e_{{\mathbf{i}}{\mathbf{j}}}e_{{\mathbf{j}}^{\prime}{\mathbf{i}}^{\prime}}.$

By Corollary 1,

$C_{m}=(\operatorname{id}\otimes\operatorname{tr}_{q})(\Gamma_{m})=\sum_{{\mathbf{i}},{\mathbf{j}}\in\mathcal{W}_{m}}\tilde{E}_{{\mathbf{j}}{\mathbf{i}}}^{-}\tilde{E}_{{\mathbf{i}}{\mathbf{j}}}^{+}\otimes\operatorname{tr}_{q}(e_{{\mathbf{i}}{\mathbf{j}}}e_{{\mathbf{j}}{\mathbf{i}}})$

It just remains to calculate the quantum trace. It is given by

$\operatorname{tr}_{q}(e_{{\mathbf{i}}{\mathbf{j}}}e_{{\mathbf{j}}{\mathbf{i}}})=\operatorname{tr}_{q}(e_{{\mathbf{i}}{\mathbf{i}}})=\operatorname{tr}\left(q^{-NE_{00}-(N-2)E_{11}+\cdots+NE_{NN}}e_{{\mathbf{i}}{\mathbf{i}}}\right)=q^{-N\mu_{0}-(N-2)\mu_{1}+\cdots+N\mu_{N}},$

where $\mu=\mu^{(N)}({\mathbf{i}})$. Applying Lemma 2 finishes the proof.

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# Tilting Planets during Planet Scattering

Gongjie Li 11affiliationmark:  1 Center for Relativistic Astrophysics, School
of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
<EMAIL_ADDRESS>
###### Abstract

Observational constraints on planet spin-axis has recently become possible,
and revealed a system that favors a large spin-axis misalignment, a low
stellar spin-orbit misalignment and a high eccentricity. To explain the origin
of such systems, we propose a mechanism that could tilt the planet spin-axis
during planet-planet scattering, which are natural outcomes of in-situ
formation and disk migration. Specifically, we show that spin-orbit resonances
could occur for a short time period during the scattering processes, and
excite the misalignment of the planet spin-axis. This typically leads to
planets with large spin-misalignment and a wide range of eccentricities and
inclinations.

##  1\. Introduction

Tilt of the planetary spin-axis provides important constraints on the
formation and the subsequent dynamical evolution of the planetary systems
(Lissauer, 1993). For the Solar System planets, the tilt of the planets have
been well explored, and a large number of factors could influence the
obliquity (spin-orbit misalignment) of the planets. For instance, the
terrestrial planets all encountered chaotic obliquity variations in the past
due to secular resonances between planet spin-axis and orbital perturbation
(Ward, 1973; Laskar & Robutel, 1993; Touma & Wisdom, 1993; Li & Batygin,
2014), while the spin-axis of Mercury and Venus have been stabilized by tidal
dissipation and that of the Earth have been stabilized by the capture of the
Moon. For the gas giants, Jupiter’s obliquity remains low, while Saturn has a
larger obliquity of $26.7^{\circ}$. Both of them could be affected by
resonances with either Uranus or Neptune during planet migration (Ward &
Hamilton, 2004; Vokrouhlický & Nesvorný, 2015). Uranus has a large obliquity
above $90^{\circ}$, and this is likely due to giant impacts as well as spin-
orbit resonances in the presence of the circumplanetary disk (e.g., Slattery
et al., 1992; Morbidelli et al., 2012; Rogoszinski & Hamilton, 2020).

Little is known for extrasolar planets, comparing with those residing in the
solar system. Theoretical estimates on the obliquity variations of planets
have been done for those residing in the habitable region, motivated since
obliquity determines the latitudinal distribution of stellar radiation and is
important for snowball transition of planets (e.g., Kane & Torres, 2017; Shan
& Li, 2018; Saillenfest et al., 2019; Quarles et al., 2019, 2020). For
instance, it is shown that Kepler 62f and Kepler 186f do not require a massive
moon to stabilize their obliquity, different from Earth. In addition,
different mechanisms have been proposed to tilt the spin-axes of exoplanets
during planet formation, such as via planet-disk interactions (Millholland &
Batygin, 2019; Martin et al., 2020; Su & Lai, 2020), planet-planet
interactions (Ward & Hamilton, 2004; Vokrouhlický & Nesvorný, 2015; Quillen et
al., 2018; Millholland & Laughlin, 2019), planet collisions (Li & Lai, 2020)
and satellite migration (Saillenfest et al., 2021).

Many observational techniques have been proposed to constrain the obliquities
of planets (Barnes & Fortney, 2003; Gaidos & Williams, 2004; Carter & Winn,
2010; Schwartz et al., 2016). In particular, the first constraint on the
orientation of the spin-axis of a planet mass companion has been made for the
directly imaged system 2MASS J01225093–2439505 (2M0122), using the projected
rotation rates for the companion (Bryan et al., 2020). It was found that while
the stellar obliquity prefers alignment, the companion obliquity favors
misalignment. In addition, the planet mass companion is most likely in a high
eccentricity orbit. It was suggested that while secular spin-orbit resonances
(due to the low planet spin precession rate), collisions (due to high escape
velocity of the massive planet) and Kozai-Lidov oscillations (due to low
stellar obliquity) are all unlikely causing the tilt of the planet spin-axis,
gravito-turbulent disks provide a promising scenario to tilt the planetary
spin-axis.

To further explore mechanisms that lead to planets with large obliquities, low
stellar obliquities and high eccentricities, which are currently out of spin-
orbit resonances, we investigated spin-axis variations during planet-planet
scattering. Planet-planet scattering naturally leads to orbits with a wide
range of eccentricities and inclinaitons. In addition, planet-planet
scattering is a common outcome of planet formation, as planets form in a
nearly maximally packed configuration in the protoplanetary disk (Kokubo &
Ida, 2002; Goldreich et al., 2004; Ida & Lin, 2004). Once the disk dissipates,
mutual planetary perturbations will lead to eccentricity and inclination
growth, as well as close encounters between the planets (e.g., Rasio & Ford,
1996; Chambers et al., 1996; Lin & Ida, 1997; Adams & Laughlin, 2003; Ford &
Rasio, 2008; Chatterjee et al., 2008; Petrovich et al., 2014). It has been
found that planetary scattering could lead to high eccentricity planets and
planets with large stellar spin-orbit misalignment. More recently, it is
responsible for the formation of many of the warm jupiters (Mustill et al.,
2017; Frelikh et al., 2019; Anderson et al., 2020).

During planet-planet scattering, planets may collide with each other, and the
spin-axis of the merger product could be misaligned due to the conservation of
angular momentum and linear momentum (Li & Lai, 2020). However, the changes in
eccentricities and inclinations typically remain low during collisions. Thus,
in our paper, we focused on scattering of planets with no collisions. The
fraction of systems that avoid collisions increases with increasing planetary
semi-major axis and planetary masses.

It is often assumed that the spin-axis have little changes during scattering,
since the sizes of the planets are small, and the total torque on the planets
during the scattering encounter are negligible to tilt the planet (Lee et al.,
2007). Nevertheless, here we show that spin-orbit resonances could take place
during the scattering events and tilt the planet spin-axis. This could lead to
misaligned planets with a wide range of eccentricities and inclinations, as
well as lower spin-rate comparing with that due to collisions. This effect is
similar to the tilt of the solar system giant planets during planet migration.

The paper is organized as the following: we describe our scattering
experiments in section 2, where section 2.1 presents the set up of the
scattering experiments, section 2.2 focuses on one illustrate example, and the
results of the Monte Carlo simulations of the scattering experiments are
summarized in section 2.3. Then, we explore a larger parameter space and
describe the dependence of planetary spin-axes inclination on the properties
of the planet system in section 3. In the end, we summarize the main findings
in section 4.

##  2\. Scattering Experiments

###  2.1. Experiment Setup

We run N-rigid-body simulations to study the spin-axis evolution during
planet-planet scattering, using the GRIT simulation package (Chen et al.,
2021). The GRIT simulation package contains a symplectic Lie-group integrator
that we developed to simulate systems with gravitationally interacting rigid
bodies (Chen et al., 2021). Different from the integrator developed by Touma &
Wisdom (1994), we do not assume the orbits to be near Keplerian, and thus,
this integrator is suitable to study the effects during planet-planet
scattering. Tidal dissipation is included in the GRIT simulation package.
However, we note that tides make negligible effects in the final distribution
of planetary obliquity, and thus we do not include tide for our scattering
experiments. General relativistic effects (first-order post-Newtonian) are
included in the scattering experiments since planets can be scattered very
close to the host star.

In our default set of simulations (results summarized in section 2.3), we
include three planets in each of the planet system. We set the initial
eccentricities of the planets to range between $0$ and $0.1$ distributed
uniformly and the initial inclination to range between $0^{\circ}$ and
$2^{\circ}$ uniformly. Higher initial eccentricities allow instabilities to
occur faster. We vary the initial orbital parameters (e.g., changing the
initial eccentricities to range between $0$ and $0.05$ in section 3). We set
the planets to be separated by $3.5R_{H,mutual}$ from each other, where the
mutual Hill radius is expressed as the following:

$\displaystyle
R_{H,mutual}=\Big{(}\frac{m_{1}+m_{2}}{3M_{star}}\Big{)}^{1/3}\frac{a_{1}+a_{2}}{2},$
(1)

where $m_{1}$ & $m_{1}$ and $a_{1}$ & $a_{2}$ are the masses and semi-major
axes of the two planets separately, and we placed the innermost planet at
$1$AU. The argument of pericenter, longitude of ascending node and mean
anomaly of the planets are uniformly distributed between $0$ and $2\pi$.

For each of the planets, we set their masses to range between $1m_{jup}$ and
$2m_{jup}$ distributed uniformly. The mass range is set arbitrarily, and we
discuss the effects with different planet masses in section 3. We set the
radius of the planets to be one jupiter radius, and the spin rate of the
planets to be $30\%$ of the break up spin rate. The oblateness of the planets
are obtained using Darwin-Radau relation (Murray & Dermott, 2000):

$\displaystyle\mathbb{C}\equiv\frac{C}{m_{p}R_{p}^{2}}=\frac{2}{3}\Big{[}1-\frac{2}{5}\Big{(}\frac{5}{2}\frac{q}{f}-1\Big{)}^{1/2}\Big{]},$
(2)

where $C$ is the planet’s moment of inertia around the rotational axis,
$m_{p}$ is the mass of the planet, $R_{p}$ is the radius of the planet, $q$
represents the relative importance of the centripetal acceleration and $f$
represents oblateness. We assume the moment of inertia coeffient
$\mathbb{C}=0.26$ for gas giants and obtain $f=0.067$. This is similar to the
oblatess of Jupiter ($f_{jup}=0.06487$), and smaller than that of Saturn
($f_{sat}=0.09796$). We assume the love number of the gas giants to be
$k_{2}=0.5$, following the $n=1$ polytrope density distribution. We assume the
stellar $J_{2}$ moment to be $2\times 10^{-7}$, similar to the Sun, though we
note that the stellar $J_{2}$ moment has little effects on the final
distribution of the planetary spin-axes.

We run the simulations for 1Myr ($10^{6}$ times the innermost planet’s
period). This is much longer than the instability time of the planets with a
separation of $3.5R_{H,mutual}$ ($\sim 300$yr) (Chatterjee et al., 2008). The
time step is set to be $10^{-4}$yr, less than $5\%$ of the spin period of the
planets.

###  2.2. Illustrative Example

To illustrate how the planetary spin-axes are tilted during planet scattering,
we include here an illustrative example using one of the Monte Carlo draws.
The mass of the planets are $1.1643m_{jup}$, $1.0090m_{jup}$ and
$1.6349m_{jup}$ separately, and the initial eccentricities and inclinations
are $0.052255$, $0.098151$ and $0.091897$, as well as $1.4622^{\circ}$,
$0.9179^{\circ}$ and $0.3547^{\circ}$ separately. Only one of the planet
($m_{2}$) survived after the close encounters. The orbital evolution of the
three planets are shown in Figure 1.

Figure 1.— Illustrative example of a scattering experiment. $\theta$ and
$\epsilon$ represent the tilt of the spin-axis relative to the reference plane
and the spin-orbit misalignment (obliquity) separately. Both planet $m_{1}$
and $m_{2}$’s spin axes are largely tilted during the scattering process, and
planet $m_{2}$’s inclination remains low $\sim 20^{\circ}$ after the
encounter.

As shown in Figure 1, eccentricities of all the planets are excited to large
values during the scattering, while inclinations of $m_{2}$ and $m_{3}$ remain
low. Both the spin-axes of $m_{1}$ and $m_{2}$ are tilted. In particular
$m_{1}$ has large and fast spin inclination ($\theta$) variations before it is
ejected (phase 2, yellow region). This is because inclination of $m_{1}$ is
excited above $50^{\circ}$, and spin-axis precession around the orbit drives
large amplitude spin-axis variations. In addition, the semi-major axis of
$m_{1}$ becomes small before the ejection, and this leads to fast spin-
precession rate and thus fast oscillations in $\theta$.

On the other hand, $m_{2}$’s obliquity ($\epsilon$) is slightly increased
during phase 1 (green region) due to orbital inclination increase. Then, its
spin-inclination ($\theta$) is largely tilted during the second close
encounter of the planets (phase 3, red region around $\sim 0.1$Myr to $\sim
0.2$Myr), while the orbital inclination remains low. After the ejection of
$m_{3}$, the obliquity of $m_{2}$ remains constant, and the spin-axis
inclination variations are caused by spin-axis precession around the orbital
normal direction due to spin-orbit coupling.

Figure 2.— Obliquity and spin evolution (top two panels) compared with the
spin and orbital precession rate (bottom panel). In the top two panels, the
blue lines represent the results of the N-rigid-body simulation, and the red
lines represent the secular results. The secular results largely agrees with
the N-rigid-body simulations, and disagreement is expected since secular
approximation is not valid during planetary close-encounter when the orbital
elements have rapid changes. The bottom panels shows the precession rates. It
illustrates that when the spin-precession rate coincides with that of the
orbital precession, the spin-axis of the planet is tilted.

How is the spin-axis of $m_{2}$ tilted? To investigate the mechanism that tilt
the spin-axis of the surviving planet $m_{2}$, we plot the obliquity and spin
evolution using both the N-rigid-body simulation and the secular approach. We
note that it is not appropriate to use secular method to estimate the
evolution of the system during planet-planet scattering, where the semi-major
axes change drastically. However, we use it as a probe to illustrate the
effects of the non-adiabatic resonance between the spin and orbital
precessions that enhanced spin-inclinations. This is analogous to the secular
spin-orbit resonance encounters during planet migration (e.g., Vokrouhlický &
Nesvorný, 2015; Millholland & Laughlin, 2019), while the semi-major axes
changes can be more drastic during scattering. We plot the precession rate of
the spin-axis and that of the orbital longitude of ascending node in Figure 2.
The secular results are obtained following the Hamiltonian listed below (e.g.,
Laskar & Robutel, 1993):

$\displaystyle
H(\xi,\psi,t)=\frac{1}{\alpha}\xi^{2}+\sqrt{1-\xi^{2}}(A(t)\sin{\psi}+B(t)\cos{\psi}),$
(3)

where $\xi=\cos{\epsilon}$ and $\psi$ is the spin-longitude projected in the
orbital plane, and $\alpha$ is the spin-precession coefficient (e.g., Ward &
Hamilton, 2004):

$\displaystyle\alpha$
$\displaystyle=\frac{k_{2}}{2\mathbb{C}}n\Big{[}\frac{\omega
R_{p}^{3}}{\omega_{b}a^{3}(1-e^{2})^{3}}\Big{]}^{1/2},$ (4)
$\displaystyle=5.9\times 10^{-4}{\rm yr^{-1}}$
$\displaystyle~{}~{}\times\Big{(}\frac{k_{2}/(2\mathbb{C})}{0.5/0.52}\Big{)}\Big{(}\frac{n}{2\pi
yr^{-1}}\Big{)}\Big{[}\frac{\omega/\omega_{b}}{0.3}\frac{(R_{p}/R_{jup})^{3}}{(a/1AU)^{3}(1-e^{2})^{3}}\Big{]}^{1/2}$

where $n$ is the orbital frequency $\omega$ and $\omega_{b}$ are the spin
velocity and the breakup velocity of the planet. In addition, $A(t)$ and
$B(t)$ reflect the orbital variations and are expressed as the following:

$\displaystyle A(t)=2(\dot{q}+p(q\dot{p}-p\dot{q}))/\sqrt{1-p^{2}-q^{2}}$ (5)
$\displaystyle B(t)=2(\dot{p}-q(q\dot{p}-p\dot{q}))/\sqrt{1-p^{2}-q^{2}},$ (6)

where $p=\sin{i}/2\sin{\Omega}$ and $q=\sin{i}/2\cos{\Omega}$. We obtain the
orbital evolution using the results of the N-rigid-body simulations and
integrate the spin-evolution using the secular approach before the ejection of
$m_{3}$.

For the precession rates, the orbital precession time is directly obtained
taking the time derivative of the longitude of ascending node ($\Omega$), and
we use the spin-axis precession coefficient $\alpha$ as a proxy for the spin-
axis precession rate.

As shown in Figure 2, obliquity of the planet is increased during the first
close encounter between the planets around $\sim 1$kyr. The inclination of
$m_{2}$ is excited, while the spin-axis inclination ($\theta$) is not affected
significantly during the close encounter. Thus, obliquity is increased due to
changes in the orbital inclination. Next, at $\sim 0.1$Myr, after the close-
encounter between $m_{2}$ and $m_{3}$, the semi-major axis of $m_{2}$ is
reduced, which leads to faster spin-precession rate. Meanwhile, the orbital
precession rate decreases as the planet companion $m_{3}$ scatters with
$m_{2}$ and migrates outward. This causes sweeping of spin-orbit resonance,
which drives obliquity excitation and the tilt of the spin-axis while the
inclination of the orbit remains low ($\sim 20^{\circ}$).

###  2.3. Scattering Results

We run $500$ scattering experiments to study the distribution in the tilt of
the planetary spin-axes during planet-planet scattering. We sample the orbital
parameters and masses of the planets as described in section 2.1. $\sim 60\%$
of the scatterings avoid planet-planet collisions. We focus on systems that
have no collisions, and the results are summarized in this section.

Figure 3.— Results of scattering experiments. Tilt of planet spin-axis as a
function of planet semi-major axis (top panel), eccentricity (middle panel)
and inclination (bottom panel). circles mark the runs that did not encounter
spin-orbit resonance crossing, and blue circles mark those encountered the
resonance crossing. Eccentricity and inclination can be excited to large
values during planet scattering, and there is a wide spread in eccentricity
and inclination for large tilt of planetary spin-axis. It is more easily to
tilt planet spin-axis with larger planet inclination, due to precession of the
spin-axis around the orbital normal.

Figure 3 shows the tilt of the spin-axis with respect to semi-major axes,
eccentricities and inclinations of the planets. The dearth of planets around
$\sim 1$AU is due to planet-planet scattering as the innermost planets start
at $1$AU. There is a wide spread in semi-major axes, eccentricities and
inclinations for planets with large spin-axis inclination ($\theta$). The
dependence of of the spin-axis inclination on the final eccentricity of the
planets is weak, but it is more likely to tilt the spin-axis of the planets
with larger final inclinations. This is because spin-axis precession around
the orbit leads to larger spin-axis inclination when the orbit is more
inclined relative to the reference plane. In addition, the obliquity
oscillation amplitude is larger with higher inclination under spin-orbit
resonances (e.g., Shan & Li, 2018).

To investigate the role of spin-orbit resonances, we mark the runs that
encountered spin-orbit resonance crossing (with matching spin-precession
frequency and orbital precession frequency) in blue. Spin-orbit resonances are
prevalent during scattering. Specifically, $80.2\%$ of the systems encountered
spin-orbit resonances, and all systems with spin-inclination excited above
$10^{\circ}$ encountered spin-orbit resonance crossing. Systems with lower
semi-major axes have a larger probability to encounter spin-orbit resonances
($97.8\%$ for $a<1$AU) comparing with their farther companions ($66.0\%$ for
$a>1$AU). Some systems have spin-inclination enhanced due to both orbit
precession and spin-orbit resonances. Overall, it is more likely to tilt the
planetary spin-axis with low semi-major axis due to faster spin-precession
rates that are both more likely to commensurate with that of orbital
precession, and to tilt the spin-axis around orbital normal more efficiently.

Precession timescales are longer for farther companions (e.g., $\sim 10$Myr at
$10$AU for Jupiter-like planet), and this will enhance planetary spin-
inclination over long timescales when planetary spin-axes precess around
inclined orbits. We note that we only focus on the outcomes shortly ($1$Myr)
after planet-planet scattering, and the detailed long term dynamics will be
discussed in a follow up study.

##  3\. Parameter Studies

We consider different scattering experiments varying the distribution of the
initial eccentricity, semi-major axes, as well as the mass of the planets in
this section. For each set of simulations, we run 200 experiments. The
distributions of the spin-axes inclinations as a function of planet final
semi-major axes are shown in Figure 4.

Figure 4.— Tilt of planet spin-axis as a function of planet semi-major axis
for different sets of scattering experiments. The set up of the simulations
are the same as that described in section 2.1, except that in the upper left
panel: innermost planet starts at 0.5 AU; upper right panel: innermost planet
starts at 3AU; lower left panel: initial eccentricities range between 0 and
0.05; and lower right panel: planet masses range between $10-20m_{jup}$.
Similar to Figure 3, red circles mark the runs that did not encounter spin-
orbit resonance crossing, and blue circles mark those encountered the
resonance crossing. It is more easily to tilt planet spin-axes for close-in
planets for all the scattering experiments, and the planetary spin-axes can be
tilted to larger values for planets with higher $v_{orb}$ to $v_{esc}$ ratio.

The upper panels of Figure 4 include different initial semi-major axes of the
innermost planets comparing with the default set of simulations in section
2.3. As the planets reside farther from the host star, the orbital velocity
decays, and planets can avoid collisions more easily (with lower ratio of the
orbital velocity to the escape velocity, see e.g., Petrovich et al. 2014).
However, the spin-axis inclination of the planets can be more easily excited,
when the planets start closer to the host stars due to faster spin precession
rates. Thus, less planets have their spin-axes tilted during scattering when
starting farther from the host star, even though more of them avoid
collisions.

The lower left panel shows the case when the planets start less eccentric ($e$
ranging between $0$ and $0.05$). It takes more time for the eccentricity to
grow larger due to planet interactions, and orbital instability occurs at a
later time. Nevertheless, spin-axis inclination distribution and its
dependence on semi-major axis are similar comparing to the default set of
simulations in section 3. Note that the distribution of the spin-axis
inclination is shown more clearly in Figure 5.

When the masses of the planets get larger, the escape velocities increase, and
planets experience faster ejection. Thus, as shown in the lower right panel of
Figure 4 with planet masses ranging between $10m_{jup}$ and $20m_{jup}$, less
number of planets have their spin-axes tilted to large values.

Figure 5.— Histogram of the spin-axes inclination for planets with final semi-
major axes interior (upper panel) and exterior (lower panel) to the initial
semi-major axes of the innermost planets. Planets with higher
$v_{orb}/v_{esc}$ ratio residing closer to the host star can more easily tilt
their spin-axes.

Similar to Figure 3, we mark the runs that encountered spin-orbit resonance
crossing in blue. It is more likely to encounter spin-orbit resonances when
the planet starts closer to the host star due to the faster precession rates
($89.0\%$ starting at $0.3$AU versus $59.6\%$ starting at $5$AU). Larger
planetary masses ($10m_{jup}$ and $20m_{jup}$) are less likely to encounter
spin-orbit resonances due to faster ejection processes ($37.9\%$), and lower
initial eccentricity ($e=0-0.05$) do not change the probability to encounter
the spin-orbit resonances significantly ($79.2\%$). As expected, spin-
inclinations are more likely to be enhanced for systems that encounter spin-
orbit resonance crossing.

For all the sets of simulations, planets with lower final semi-major axes can
have their spin-axes tilted more easily immediately (1 Myr) after the
scattering. This is both due to the higher probability to cross spin-orbit
resonances closer to the host star and due to the stronger non-adiabatic limit
farther from the host star that cannot efficiently excite spin-inclination.
Thus, in Figure 5 we divide the sample to two groups: those with final semi-
major axes lower than the initial semi-major axes of the innermost planets
(upper panel), and those that reside farther (middle panel).

For close-in planets, most of them could have large spin-axis inclination,
except those with large masses (or low $v_{orb}/v_{esc}$ ratio). Those that
initially reside closer to the host star can have spin-axes excited to larger
values. The differences in the initial eccentricities do not make significant
influence in the distribution of the final spin-axis inclinations.

For planets at large separation from their host star, a majority of them still
have low spin-axis inclination even though some planets could have their spin-
axes inclination excited. Comparing between different initial conditions of
the scatter experiments, the difference in the initial eccentricity also makes
little effects in the final distribution of the spin-axes inclination. When
the planetary masses are higher or start farther from the host star (lower
$v_{orb}/v_{esc}$ ratio), it is more difficult to excite the planetary spin-
axes due to faster planet ejection processes.

Combining the two groups together (bottom panel), $\sim 50\%$ of the
scattering experiments produce high spin-axis inclinations over $50^{\circ}$
when the planets start at $0.5$ AU, and $\sim 20-30\%$ produces high spin-axis
inclinations starting at $1-3$AU. Only around $\sim 10\%$ reach high spin-axis
inclination for massive planets ($10-20m_{jup}$).

After the completion of this paper, we became aware of a complementary study
that arrived at the same results independently and simultaneously based on
two-planet scattering (Hong et al., 2021). Our results about the dependence of
planet spin-axis inclination on semi-major axes agree with each other. In
addition, our results show that three-planet scatterings could lead to a
larger number of planets with higher obliquity due to longer instability time
and higher inclination excitation.

##  4\. Conclusion and Discussions

In this letter, we discuss planetary spin-axes variations during planet
scattering. We find that temporary spin-orbit resonances could lead to large
tilt of planetary spin-axes for planets with a wide range of eccentricities
and inclinations. It is more likely to tilt planetary spin-axes for planets
with lower masses and initially reside closer to the host star before
scattering occurs (higher $v_{orb}/v_{esc}$ ratio), which could allow slower
ejection processes and longer time to tilt the planets. In addition, it is
more likely to tilt the spin-axes of the planets that reside closer to the
host stars after planet-planet scattering.

We note that the spin-axis inclination of the massive planet in system 2M0122
is unlikely due to spin-orbit resonances during planet scattering. Orbiting
around a $0.4$M⊙ star, the escape velocity of the $12-27m_{jup}$ planet
greatly exceeds the orbital velocity. The ejection process is fast and could
quench obliquity excitation due to secular resonances. Thus, it is challenging
to produce planets with large obliquities in particularly at distances around
$\sim 50$ AU. It is likely that the tilt of the planet is due to turbulent
accretion or the tilt of circumplanetary disk (Bryan et al., 2020; Martin et
al., 2020). In addition, as a caveat, recent SPH simulations show that
collision rate could be further increased, and this will lower the probability
to tilt planetary spin-axes during planet-planet scattering (Li et al., 2021).

Many mechanisms have been proposed to explain the origin of tilted planetary
spin-axes. For instance, planet-disk interactions, turbulent accretion to
planets as well as collisions during planet-planet scattering can also lead to
misaligned planetary spin-axes. To distinguish from the other scenarios,
planet-planet scattering (with no collisions) could lead to planets with a
wide range of eccentricities and inclinations. In addition, planetary spin-
rate is nearly unchanged, different from those produced by planet collisions.
Obliquity presents an exciting and unique window into formation history, and
more systems with measured obliquities will enable statistical studies to
disentangle different planet formation mechanisms in the future.

## Acknowledgments

The author thanks Fred Adams, Smadar Naoz, Konstantin Batygin, Zeeve
Rogoszinski and Dong Lai for reading the manuscript and providing helpful
comments. The author also thanks the referee for helpful comments, which
greatly improved the quality of the letter. GL is grateful for the partial
support by NASA 80NSSC20K0641 and 80NSSC20K0522.

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