# Theoretical study on $\Lambda_{c}^{+}\to\Lambda K^{+}\bar{K}^{0}$ decay and
$\Xi^{*}(1690)$ resonance

Si-Wei Liu<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy
of Sciences, Lanzhou 730000, China  School of Nuclear Sciences and Technology,
University of Chinese Academy of Sciences, Beijing 101408, China  Qing-Hua
Shen<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  School of Nuclear Sciences and Technology,
University of Chinese Academy of Sciences, Beijing 101408, China  Ju-Jun Xie
<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  School of Nuclear Sciences and Technology,
University of Chinese Academy of Sciences, Beijing 101408, China  Southern
Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese
Academy of Sciences, Huizhou 516000, Guangdong Province, China

###### Abstract

We present a theoretical study of $\Xi^{*}(1690)$ resonance in the
$\Lambda_{c}^{+}\to\Lambda K^{+}\bar{K}^{0}$ decay, where the weak interaction
part proceeds through the Cabibbo-favored process $c\to s+u\bar{d}$. Next, the
intermediate two mesons and one baryon state can be constructed with a pair of
$q\bar{q}$ with the vacuum quantum numbers. Finally, the $\Xi^{*}(1690)$ is
mainly produced from the final state interactions of $\bar{K}\Lambda$ in
coupled channels, and it is shown in the $\bar{K}\Lambda$ invariant mass
distribution. Besides, the scalar meson $a_{0}(980)$ and nucleon excited state
$N^{*}(1535)$ are also taken into account in the decaying channels
$K^{+}\bar{K}^{0}$ and $K^{+}\Lambda$, respectively. Within model parameters,
the $K^{+}\bar{K}^{0}$, $\bar{K}^{0}\Lambda$ and $K^{+}\Lambda$ invariant mass
distributions are calculated, and it is found that our theoretical results can
reproduce well the experimental measurements, especially for the clear peak
around $1690$ MeV in the $\bar{K}\Lambda$ spectrum. The proposed weak decay
process $\Lambda_{c}^{+}\to\Lambda K^{+}\bar{K}^{0}$ and the interaction
mechanism can provide valuable information on the nature of the
$\Xi^{*}(1690)$ resonance.

##  I Introduction

The study of the properties of the $\Xi^{*}$ states concentrates much
attention in hadron physics [1, 2, 3, 4, 5]. However, our knowledge about the
$\Xi$ spectrum is quite scarce [6]. Except for the ground $\Xi(1321)$ state
with spin-parity $J^{P}=1/2^{+}$ and $\Xi^{*}(1530)$ with $J^{P}=3/2^{+}$
being well established with four-star ratings, the situation of other
$\Xi^{*}$ excited states is still rather uncertain with less than three-star
ratings [6]. For some of them, their existence has not been confirmed. Hence,
further studies of the $\Xi^{*}$ resonances both on the experimental and
theoretical sides are necessary [7, 8, 9, 10].

For the study of $\Xi^{*}$ resonances, the $\bar{K}N$ scattering is available
[11]. Indeed, the $\Xi^{*}(1690)$ resonance was first observed in the
$\bar{K}\Sigma$ invariant mass spectrum [12] in the $K^{-}p$ reactions at
$4.2$ GeV. In the measurements of Ref. [12], the mass and total decay width of
$\Xi^{*}(1690)$ are established as $M=1694\pm 6$ MeV and $\Gamma=26\pm 6$ MeV
in the charged channel and $M=1684\pm 5$ MeV and $\Gamma=20\pm 4$ MeV in the
neutral channel. In Ref. [13], the $\Xi^{*}(1690)$ resonance was confirmed by
the WA89 collaboration in the neutral $\pi^{+}\Xi^{-}$ channel with measured
mass $M=1686\pm 4$ MeV and width $\Gamma=10\pm 6$ MeV. In Ref. [14], in
addition to the investigations of the $\Xi^{*}(1530)$ properties in the
$\Lambda^{+}_{c}\to K^{+}\pi^{+}\Xi^{-}$ decay process, evidence of the
existence of the $\Xi^{*}(1690)$ resonance with $J^{P}=1/2^{-}$ was also found
by $\it BABAR$ collaboration. In Ref. [15], the Belle collaboration presented
the first evidence for the process $\Lambda^{+}_{c}\to K^{+}\Xi^{*0}(1690)\to
K^{+}K^{-}\Sigma^{+}$, and the fitted mass and width of $\Xi^{*}(1690)$ are
$M=1688\pm 2$ MeV and $\Gamma=11\pm 4$ MeV. Furthermore, contribution of the
$\Xi^{*0}(1690)$ to the $\Lambda^{+}_{c}\to K^{+}\Xi^{*0}(1690)\to
K^{+}\bar{K}^{0}\Lambda$ reaction was also found [15]. In Ref. [16], the
$\Xi(1690)$ resonance was also observed in the $\bar{K}^{0}\Lambda$ channel in
the decay $\Lambda^{+}_{c}\to K^{+}\bar{K}^{0}\Lambda$ by the ${\it BABAR}$
collaboration, where a coherent amplitude analysis of the $\Lambda a_{0}(980)$
and $K^{+}\Xi^{*0}(1690)$ productions was performed, the obtained mass and
width of $\Xi^{*}(1690)$ resonance are $M=1684.7\pm
1.3(\text{stat.})_{-1.6}^{+2.2}(\text{syst.})$ MeV,
$\Gamma=8.1_{-3.5}^{+3.9}(\text{stat.})_{-0.9}^{+1.0}(\text{syst.})$ MeV, and
its spin is consistent with $1/2$. Furthermore, the $\Xi^{*}(1690)$ resonance
has also been found in the hyperon-nucleon interactions [17, 18]. In most of
these experimental analysis the spin-parity of $\Xi^{*}(1690)$ resonance favor
$J^{P}=1/2^{-}$.

In 2019, the $\Xi^{*}(1620)$ resonance was observed in the $\pi^{+}\Xi^{-}$
invariant spectrum via the $\Xi_{c}^{+}\to\Xi^{-}\pi^{+}\pi^{+}$ decay.
Meanwhile, the evidence of the $\Xi^{*}(1690)$ resonance was also found with
the same data sample, and its significance is $4\sigma$ [19]. However, up to
present time the quantum numbers of $\Xi^{*}(1690)$ have not yet been
determined. Nevertheless, to fully understand the nature of $\Xi^{*}(1690)$
resonance, further experiments are certainly required.

On the theoretical side, within the constituent quark models, the first
excited state of $\Xi$ baryon is around $1800$ MeV [20, 21, 22], thus it is
difficult to treat $\Xi^{*}(1690)$ as a three-quark state. In Ref. [3], the
$\Xi^{*}(1690)$ was treated as a three-quark state and its spin-parity quantum
numbers are $J^{P}=1/2^{-}$. Note that the obtained mass with the constituent
quark model [3], is 1725 MeV, which is still 35 MeV higher than its nominal
mass. This implies that $\Xi^{*}(1690)$ might have some nontrivial structure
other than the usual three-quark state. In fact, using the chiral unitary
approach, the $\Xi^{*}(1690)$ can be interpreted as an $s$-wave meson-baryon
molecular state [1, 2, 23]. It couples strongly to the $\bar{K}\Sigma$ channel
[23], and its coupling to the $\pi\Xi$ channel is small. Thus its narrow width
can be naturally explained [23]. Furthermore, its spin-parity are $1/2^{-}$ in
the chiral unitary approach. In Ref. [24], the $\Xi^{*}(1690)$ was
investigated by means of the two-point QCD sum rules, where it was also
concluded that the $\Xi^{*}(1690)$ state, most probably, has quantum numbers
spin-parity $J^{P}=1/2^{-}$. However, the obtained width for the
$\Xi^{*}(1690)$ state is about $100$ MeV [24], which is much larger than the
experimental measurements [6].

The main property of $\Xi^{*}(1690)$ resonance is that its decay width is too
narrow compared with other baryon resonances that have similar mass [6]. In
spite of $\Xi^{*}(1690)$ state has a large phase space to decay to open
channels, such as $\pi\Xi$ and $\bar{K}\Lambda$, its width is just in the
order of $10$ MeV. In Ref. [25], using the chiral unitary approach, the
$\Xi^{*}(1690)$ state is dynamically generated in the pseudoscalar-baryon and
vector-baryon coupled channels 111Note that the transitions between
pseudoscalar-baryon and vector-baryon channels are crucial to obtain the pole
for $\Xi^{*}(1690)$ state. If those transitions were zero, the pole of
$\Xi^{*}(1690)$ disappears.. It was found that most of the properties of
$\Xi^{*}(1690)$, especially its narrow width, can be well explained [25]. In
that work, the $\Xi^{*}(1690)$ has strong couplings to $\bar{K}\Sigma$ and
$\eta\Xi$ channels, while its couplings to the $\bar{K}\Lambda$ and $\pi\Xi$
channels are small. Recently, the meson-baryon interaction in the neutral
strangeness $S=-2$ sector was studied using an extended Unitarized Chiral
Perturbation Theory, where the leading Weinberg-Tomozawa term, Born term, and
the next-to-leading order contributions are considered [26]. It was found that
both $\Xi^{*}(1620)$ and $\Xi^{*}(1690)$ states can be dynamically generated,
and the obtained properties of them are in reasonable agreement with the known
experimental data.

It has been shown that the weak decay of charmed baryons governed by $c\to s$
quark transition is beneficial to probe the strange baryons, some of them are
subjects of intense debate about their nature [27, 28, 29]. For instance, the
hyperon production from the $\Lambda^{+}_{c}\to K^{-}p\pi^{+}$ and
$\Lambda^{+}_{c}\to K^{0}_{S}p\pi^{0}$ decays were investigated within the
effective Lagrangian approach in Ref. [30]. The $\Xi^{*}(1620)$ and
$\Xi^{*}(1690)$ states were studied in the $\Xi_{c}\to\pi^{+}MB$ process in
Ref. [31], where $M$ and $B$ stand for mesons and baryons, respectively. It
was shown that these weak decays might be an ideal tool to study the
$\Xi^{*}(1620)$ and $\Xi^{*}(1690)$ resonances, which are dynamically produced
in the rescattering of $M$ and $B$ in the final states.

Therefore, in this work, we take advantage of those ideas and revisit the
$\Lambda_{c}^{+}\to\Lambda K^{+}\bar{K}^{0}$ decay, which requires the
creation of an $s\bar{s}$ quark pair. Following Refs. [23, 25, 31], in
addition to the contributions of the $\Xi^{*}(1690)$ resonance that is
dynamically generated from the final state interactions of
$\bar{K}^{0}\Lambda$ in $S$ wave, we will consider also the contributions of
the $a_{0}(980)$ and $N^{*}(1535)$ states, where they are dynamically produced
by the final state interactions of $K^{+}\bar{K}^{0}$ [32, 33] and $\Lambda
K^{+}$ [34, 35, 36, 37, 38, 39] in coupled channels, respectively.

This article is organized as follows. In the next section, we present the
theoretical formalism for studying the $\Lambda_{c}^{+}\to\Lambda
K^{+}\bar{K}^{0}$ decay. In Sec. III, theoretical numerical results and
discussions and presented, followed by a short summary in Sec. IV.

##  II Formalism

For the production of $\Xi^{*}(1690)$ in the weak decay process
$\Lambda^{+}_{c}\to K^{+}\bar{K}^{0}\Lambda$, we firstly take it as a
dynamically generated state in the final state interaction of $\bar{K}\Lambda$
in coupled channels. Secondly, we take $\Xi^{*}(1690)$ as a Breit-Wigner
resonance. The roles of $a_{0}(980)$ and $N^{*}(1535)$ are also investigated,
where the contribution of $a_{0}(980)$ state in encoded in the $s$-wave
$K\bar{K}$ final state interaction as done in Refs. [40, 41, 42, 43, 44, 45],
and the one of the $N^{*}(1535)$ resonance is in the $s$-wave $K\Lambda$ final
state interaction [34, 35, 36, 37, 38, 39].

Figure 1: The schematic diagram for the Cabibbo-favored process
$\Lambda_{c}^{+}\to\Lambda K^{+}\bar{K}^{0}$.

The schematic diagram for the Cabibbo-favored process
$\Lambda_{c}^{+}\to\Lambda K^{+}\bar{K}^{0}$ is presented in Fig. 1, where the
decay process proceeds in three parts:

  * •

In the first step, the $c$ quark in $\Lambda^{+}_{c}$ turns into an $s$ quark
via the weak decay.

  * •

Introduce a $q\bar{q}$ pair with the quantum numbers of the vacuum to form a
pseudoscalar-baryon ($PB$) or pseudoscalar-pseudoscalar ($PP$) pair.

  * •

The final state interactions are taken into account in coupled channels within
the chiral unitary approach, which will lead to the dynamical productions of
the $a_{0}(980)$, $N^{*}(1535)$, and $\Xi^{*}(1690)$ resonances.

###  II.1 Contributions of the $a_{0}(980)$, $N^{*}(1535)$, and
$\Xi^{*}(1690)$ resonances from final state interactions

Figure 2: The decay mechanisms for (a): $\Lambda_{c}^{+}\to\Lambda+(PP)^{+}$,
(b): $\Lambda_{c}^{+}\to\bar{K}^{0}+(PB)^{+}$

Following Refs. [31, 46, 47, 48, 49, 50], we assume two primary decay
mechanisms for the process $\Lambda_{c}^{+}\to\Lambda K^{+}\bar{K}^{0}$, which
are presented in Fig. 2. In the weak decay of charmed hadrons the diagrams are
classified in six different topologies [51, 52, 53, 54]: $W$ external
emission, $W$ internal emission, $W$-exchange, $W$-annihilation, horizontal
$W$-loop and vertical $W$-loop. Here, we consider the dominant $W$ external
emission and internal emission diagrams as shown in Fig. 2 (a) and (b),
respectively. The $c$ quark in $\Lambda^{+}_{c}$ turns into an $s$ quark and a
$W^{+}$ boson, followed by the $W^{+}$ boson decaying into a $u\bar{d}$ pair.
To get the final states $\Lambda K^{+}\bar{K}^{0}$, the $sud$ ($s\bar{d}$)
cluster forms the $\Lambda$ ($\bar{K}^{0}$), and the $u\bar{d}$ ($uud$),
together with the $q\bar{q}$ ($=u\bar{u}+d\bar{d}+s\bar{s}$) pair created from
the vacuum, hadronize into $K^{+}\bar{K}^{0}$ ($K^{+}\Lambda$). In the former
processes, the $ud$ diquark in the $\Lambda^{+}_{c}$ is always kept.
Therefore, the $\Lambda_{c}^{+}$ weak decay processes shown in Fig. 2 can be
formulated as following [46, 47, 48, 49, 50]:

$\displaystyle\Lambda_{c}^{+}$ $\displaystyle=\frac{1}{\sqrt{2}}\left|c(ud-
du)\right\rangle,$
$\displaystyle\to-\frac{\sqrt{6}}{3}V_{1}\left|\Lambda\right\rangle\left|u(u\bar{u}+d\bar{d}+s\bar{s})\bar{d}\right\rangle,$
(1)
$\displaystyle\to\frac{1}{\sqrt{2}}V_{2}\left|\bar{K}^{0}\right\rangle\left|u(u\bar{u}+d\bar{d}+s\bar{s})(ud-
du)\right\rangle,$ (2)

where $V_{1}$ and $V_{2}$ are the strength of the production vertices 222We
have assumed that these production vertices proceed through $S$-wave
interaction., which contain all the dynamical factors. According to these
results of Ref. [31], we then connect two degrees of freedom, the quarks and
the hadons. Then we can rewrite the intermediate states as

$\displaystyle\Lambda_{c}^{+}$ $\displaystyle\to
V_{1}\left|\Lambda\right\rangle\left|-\frac{2\sqrt{2}}{3}\pi^{+}\eta-\frac{\sqrt{6}}{3}K^{+}\bar{K}^{0}\right\rangle$
(3) $\displaystyle\to
V_{2}\left|\bar{K}^{0}\right\rangle\left|(\frac{\eta}{\sqrt{3}}+\frac{\pi^{0}}{\sqrt{2}})p+\pi^{+}n-\frac{\sqrt{6}}{3}K^{+}\Lambda\right\rangle,$
(4)

where we have omitted the $\eta^{\prime}$ terms as in Refs. [31, 23, 25],
because its mass threshold is located much higher in the energy and its
contribution should be small. After the production of the $PP$ or $PB$ pair,
the final state interactions between intermediate mesons and baryons come up,
which is shown in Fig. 3. And, we can obtain the explicit expressions of the
decay amplitude $\mathcal{M}_{i}$ ($i=a$, $b$, $c$, and $d$ according to these
diagrams shown in Fig. 3) using Eqs. (3) and (4):

$\displaystyle\mathcal{M}_{a}$ $\displaystyle=$ $\displaystyle\mathcal{M}^{\rm
Tree}=-\frac{\sqrt{6}}{3}(V_{1}+V_{2}),$ (5) $\displaystyle\mathcal{M}_{b}$
$\displaystyle=$ $\displaystyle\mathcal{M}^{\rm
FSI}_{K^{+}\bar{K}^{0}}=h_{K^{+}\bar{K}^{0}}G_{K^{+}\bar{K}^{0}}T_{K\bar{K}\to
K\bar{K}}^{I=1}$ (6) $\displaystyle-
h_{\pi^{+}\eta}G_{\pi^{+}\eta}T_{\pi\eta\to K\bar{K}}^{I=1},$
$\displaystyle\mathcal{M}_{c}$ $\displaystyle=$ $\displaystyle\mathcal{M}^{\rm
FSI}_{\bar{K}^{0}\Lambda}=h_{\bar{K}^{0}\Lambda}G_{\bar{K}^{0}\Lambda}T_{\bar{K}\Lambda\to\bar{K}\Lambda}^{I=1/2},$
(7) $\displaystyle\mathcal{M}_{d}$ $\displaystyle=$
$\displaystyle\mathcal{M}^{\rm
FSI}_{K^{+}\Lambda}=h_{K^{+}\Lambda}G_{K^{+}\Lambda}T_{K\Lambda\to
K\Lambda}^{I=1/2}$ (8)
$\displaystyle-(\sqrt{\frac{2}{3}}h_{\pi^{+}n}G_{\pi^{+}n}+\frac{1}{\sqrt{3}}h_{\pi^{0}p}G_{\pi^{0}p})T_{\pi
N\to K\Lambda}^{I=1/2}$ $\displaystyle+h_{\eta p}G_{\eta p}T_{\eta N\to
K\Lambda}^{I=1/2},$

where $G_{PP}$ and $G_{PB}$ are the loop functions of the $PP$ or the $PB$
propagator, respectively. The coefficients $h_{PP}$ and $h_{PB}$ for the terms
of final state interactions are

$\displaystyle h_{K^{+}\bar{K}^{0}}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{6}}{3}(V_{1}+V_{2}),~{}h_{\pi^{+}\eta}=-\frac{2\sqrt{2}}{3}V_{1},$
(9) $\displaystyle h_{\bar{K}^{0}\Lambda}$ $\displaystyle=$ $\displaystyle
h_{K^{+}\Lambda}=-\frac{\sqrt{6}}{3}(V_{1}+V_{2}),$ (10) $\displaystyle
h_{\pi^{+}n}$ $\displaystyle=$ $\displaystyle
V_{2},~{}h_{\pi^{0}p}=\frac{\sqrt{2}}{2}V_{2},~{}h_{\eta
p}=\frac{\sqrt{3}}{3}V_{2}.$ (11)

The analytic form of the loop function $G$, with dimensional regularization,
is given by

$\displaystyle G(s)=\frac{2M}{16\pi^{2}}$
$\displaystyle\Bigg{\\{}a_{\mu}+\text{ln}\frac{M^{2}}{\mu^{2}}+\frac{m^{2}-M^{2}+s}{2s}\text{ln}\frac{m^{2}}{M^{2}}$
$\displaystyle+\frac{p}{\sqrt{s}}\bigg{[}\text{ln}\left(s-\left(m^{2}-M^{2}\right)+2p\sqrt{s}\right)$
$\displaystyle+\text{ln}\left(s+\left(m^{2}-M^{2}\right)+2p\sqrt{s}\right)$
$\displaystyle-\text{ln}\left(-s+\left(m^{2}-M^{2}\right)+2p\sqrt{s}\right)$
$\displaystyle-\text{ln}\left(-s-\left(m^{2}-M^{2}\right)+2p\sqrt{s}\right)\bigg{]}\Bigg{\\}},$
(12)

Figure 3: The schematic diagram for the final state interaction for the
$\Lambda_{c}^{+}\to\Lambda K^{+}\bar{K}^{0}$ decay. (a): tree level diagram,
(b): $K^{+}\bar{K}^{0}$ rescattering, (c): $\bar{K}^{0}\Lambda$ rescattering,
and (d): $K^{+}\Lambda$ rescattering.

where $\mu$ is a scale of dimensional regularization, and $a_{\mu}$ is the
subtraction constant. Any change of $\mu$ can be reabsorbed by a change in
$a_{\mu}$. In this work, we choose $\mu=630$ MeV, while $a_{\mu}$ will be
determined from the experimental data, and they will be discussed in the
following. $m$ and $M$ are the masses of the meson and baryon in the loop,
respectively. $p$ represents the magnitude of the three-momentum of one
particle in the meson-meson or meson-baryon rest frame,

$\displaystyle p$
$\displaystyle=\frac{\lambda^{1/2}(s,m^{2},M^{2})}{2\sqrt{s}},$ (13)
$\displaystyle\lambda^{1/2}(x,y,z)$
$\displaystyle=\sqrt{x^{2}+y^{2}+z^{2}-2xy-2yz-2xz}.$ (14)

The two body scattering amplitudes $T_{PP\to PP}$ and $T_{PB\to PB}$ are
obtained by solving the Bethe-Salpeter equation within the chiral unitary
approach [33, 55, 56]

$T=[1-VG]^{-1}V,$ (15)

where $V$ is the transition potential between the involved channels, which are
given explicitly in Refs. [33, 39, 25, 57]. The scattering amplitude element
$T$ in Eq. (15) in the particle basis can be related to the one in the isospin
basis, and we get

$\displaystyle T_{K^{+}\bar{K}^{0}\to K^{+}\bar{K}^{0}}$ $\displaystyle=$
$\displaystyle T_{K\bar{K}\to K\bar{K}}^{I=1},$ (16) $\displaystyle
T_{\pi^{+}\eta\to K^{+}\bar{K}^{0}}$ $\displaystyle=$ $\displaystyle-
T_{\pi\eta\to K\bar{K}}^{I=1},$ (17) $\displaystyle
T_{\bar{K}^{0}\Lambda\to\bar{K}^{0}\Lambda}$ $\displaystyle=$ $\displaystyle
T_{\bar{K}\Lambda\to\bar{K}\Lambda}^{I=1/2},$ (18) $\displaystyle
T_{K^{+}\Lambda\to K^{+}\Lambda}$ $\displaystyle=$ $\displaystyle
T_{K\Lambda\to K\Lambda}^{I=1/2},$ (19) $\displaystyle T_{\pi^{+}n\to
K^{+}\Lambda}$ $\displaystyle=$ $\displaystyle-\sqrt{\frac{2}{3}}T_{\pi N\to
K\Lambda}^{I=1/2},$ (20) $\displaystyle T_{\pi^{0}p\to K^{+}\Lambda}$
$\displaystyle=$ $\displaystyle-\sqrt{\frac{1}{3}}T_{\pi N\to
K\Lambda}^{I=1/2},$ (21) $\displaystyle T_{\eta p\to K^{+}\Lambda}$
$\displaystyle=$ $\displaystyle T_{\eta N\to K\Lambda}^{I=1/2}.$ (22)

Although the form of these $K\bar{K}$, $K\Lambda$, and $\bar{K}\Lambda$
interactions have been detailed elsewhere [33, 36, 58, 39, 25, 57], we briefly
revisit here the $\bar{K}\Lambda$ case. This will allow us to review the
general procedure of calculating the two-body amplitudes entering the total
decay amplitude of $\Lambda^{+}_{c}\to\Lambda\bar{K}^{0}K^{+}$ reaction.

In order to describe the $\Xi^{*}(1690)$ state, a coupled channel analysis was
performed. In the isospin $1/2$ sector, there are four coupled channels, which
are: $\pi\Xi$, $\bar{K}\Lambda$, $\bar{K}\Sigma$, and $\eta\Xi$. These
channels are labeled by the indices $j=1$, …, $4$. For the transition
potential $V_{ij}$, it is expressed as [58, 25]

$\displaystyle V_{ij}$ $\displaystyle=$
$\displaystyle-\frac{C_{ij}}{4f_{i}f_{j}}(2\sqrt{s}-M_{i}-M_{j})$ (23)
$\displaystyle\times\sqrt{\frac{(M_{i}+E_{i})(M_{j}+E_{j})}{4M_{i}M_{j}}},$

where $f_{i}$ is the meson decay constant of the $i$th channel, $f_{\pi}=92.1$
MeV, $f_{K}=1.2f_{\pi}$ and $f_{\eta}=1.3f_{\pi}$. $E_{i}$ and $E_{j}$ are the
initial and final baryon energies, and
$E_{i}=\sqrt{M_{i}^{2}+|\vec{p}_{i}|^{2}}$,
$|\vec{p}_{i}|=\frac{\lambda^{1/2}(s,m_{i}^{2},M_{i}^{2})}{2\sqrt{s}}\theta(\sqrt{s}-M_{i}-m_{i})$,
with $s$ the invariant mass squared of the meson-baryon system, and the meson
and baryon masses $m_{i}$ and $M_{i}$ in the $i$th channel, respectively. The
factor $C_{ij}$ is symmetric and its value is listed in Table 1.

Table 1: Coefficients $C_{ij}$ for $\pi\Xi$, $\bar{K}\Lambda$, $\bar{K}\Sigma$, and $\eta\Xi$ coupled channels in the isospin $I=1/2$ basis. | $\pi\Xi$ | $\bar{K}\Lambda$ | $\bar{K}\Sigma$ | $\eta\Xi$
---|---|---|---|---
$\pi\Xi$ | $2$ | $-3/2$ | $-1/2$ | $0$
$\bar{K}\Lambda$ |  | $0$ | $0$ | $-3/2$
$\bar{K}\Sigma$ |  |  | $2$ | $3/2$
$\eta\Xi$ |  |  |  | $0$

Then one can solve the Bethe-Salpeter equation as shown in Eq. 15 with the on-
shell factorized potential and, thus, the $S$ wave scattering $T_{ij}$ matrix
can be easily obtained. Then one can also look for poles of the scattering
amplitude $T_{ij}$ on the complex plane of $\sqrt{s}$. The pole, $Z_{R}$, on
the second Riemann sheet could be associated with the $\Xi^{*}(1690)$
resonance. The real part of $Z_{R}$ is associated with the mass
$M_{\Xi^{*}(1690)}$ of $\Xi^{*}(1690)$ resonance, and the imaginary part of
$Z_{R}$ is associated with one half of its width $\Gamma_{\Xi^{*}(1690)}$.

###  II.2 Contribution of $\Xi^{*}(1690)$ resonance as a Breit-Wigner
resonance

On the other hand, the decay process
$\Lambda_{c}^{+}\to\Xi^{*}(1690)K^{+}\to\Lambda\bar{K}^{0}K^{+}$ can also
proceed through $\Xi^{*}(1690)$ as a Breit-Wigner resonance and decaying into
$\bar{K}^{0}\Lambda$, which is shown in Fig. 4. In this case, the
$\Xi^{*}(1690)$ state is formed with $ud\bar{d}ss$ as shown in Fig. 4 (a), and
it decays into $\bar{K}^{0}\Lambda$ in the final state. The hadron level
diagram for the decay of $\Lambda_{c}^{+}\to\Xi^{*}(1690)K^{+}\to\Lambda
K^{+}\bar{K}^{0}$ is also shown in Fig. 4 (b), where the propagator of
$\Xi^{*}(1690)$ resonance is parametrized as the Breit-Wigner form.

Then, the general decay amplitude for the process
$\Lambda_{c}^{+}\to\Xi^{*}(1690)^{0}K^{+}\to\bar{K}^{0}\Lambda K^{+}$ can be
expressed as:

$\mathcal{M}_{e}=\frac{\Gamma_{\Xi^{*}(1690)}}{2}\frac{V_{3}}{M_{\bar{K}^{0}\Lambda}-M_{\Xi^{*}(1690)}+i\frac{\Gamma_{\Xi^{*}(1690)}}{2}},$
(24)

where $M_{\bar{K}^{0}\Lambda}$ is the invariant mass of the
$\bar{K}^{0}\Lambda$ system, and we take $M_{\Xi^{*}(1690)}=1690$ MeV and
$\Gamma_{\Xi^{*}(1690)}=20$ MeV for the Breit-Wigner mass and width for
$\Xi^{*}(1690)$ resonance, respectively, which are quoted in the PDG [6]. The
model free parameter $V_{3}$ will be determined by the experimental data.

Figure 4: (a) Quark level diagram for the $\Lambda_{c}^{+}\to
K^{+}\Xi^{*}(1690)$ decay and (b) hadron level diagram for the
$\Lambda^{+}_{c}\to K^{+}\Xi^{*}(1690)\to K^{+}\bar{K}^{0}\Lambda$ decay.

Before going further, we emphasize that the $ud\bar{d}ss$ component of
$\Xi^{*}(1690)$ resonance cannot be guaranteed from the decay process shown in
Fig. 4. Indeed, the $\Xi^{*}(1690)$ resonance can also be produced from the
$W$-exchange diagram [59, 15, 60, 61], where $cd$ transitions firstly into
$su$ with the weak interaction, and then with a $s\bar{s}$ pair from the
vacuum, the $u\bar{s}$ forms the $K^{+}$, while the $\Xi^{*}(1690)^{0}$ is
constructed from the $uss$ cluster and then it decays into
$\bar{K}^{0}\Lambda$. Nevertheless, this kind of contributions can be absorbed
into the model parameter $V_{3}$.

###  II.3 Invariant mass distributions of the $\Lambda^{+}_{c}\to\Lambda
K^{+}\bar{K}^{0}$ decay

With the ingredients obtained in the previous sections, we can write the
double invariant mass distributions for the $\Lambda_{c}^{+}\to
K^{+}\bar{K}^{0}\Lambda$ decay as

$\frac{d^{2}\Gamma}{dM_{K^{+}\bar{K}^{0}}dM_{\bar{K}^{0}\Lambda}}=\frac{M_{\Lambda}M_{K^{+}\bar{K}^{0}}M_{\bar{K}^{0}\Lambda}}{16\pi^{3}M_{\Lambda_{c}^{+}}^{2}}|\mathcal{M}|^{2},$
(25)

where ${\cal M}$ is the total decay amplitude. We will take two models for
${\cal M}$:

$\displaystyle\mathcal{M}^{\rm I}$ $\displaystyle=$
$\displaystyle\mathcal{M}_{a}+\mathcal{M}_{b}+\mathcal{M}_{d}+\mathcal{M}_{c},$
(26) $\displaystyle\mathcal{M}^{\rm II}$ $\displaystyle=$
$\displaystyle\mathcal{M}_{a}+\mathcal{M}_{b}+\mathcal{M}_{d}+e^{i\theta}\mathcal{M}_{e},$
(27)

where a relative phase $\theta$ is added for Model II, and it is a free
parameter. In addition, another relative phase, $\phi$, between $V_{1}$ and
$V_{2}$ is also taken into account. We will replace $V_{2}$ with
$e^{i\phi}V_{2}$ in the following fitting process. Furthermore, it is worth to
mention that $V_{1}$, $V_{2}$ and $V_{3}$ have dimension ${\rm MeV}^{-1}$.

Then the invariant $\bar{K}^{0}\Lambda$ and $K^{+}\bar{K}^{0}$ mass
distributions can be obtained by integrating over the other invariant mass in
Eq. (25). For a given value of $M_{\bar{K}^{0}\Lambda}$, the invariant mass
$M_{K^{+}\bar{K}^{0}}$ satisfies the following relation

$\displaystyle(M_{K^{+}\bar{K}^{0}}^{\text{min}})^{2}$
$\displaystyle=(E_{K^{+}}+E_{\bar{K}^{0}})^{2}$ (28)
$\displaystyle-(\sqrt{E_{K^{+}}^{2}-m_{K^{+}}^{2}}+\sqrt{E_{\bar{K}^{0}}^{2}-m_{\bar{K}^{0}}^{2}})^{2},$
$\displaystyle(M_{K^{+}\bar{K}^{0}}^{\text{max}})^{2}$
$\displaystyle=(E_{K^{+}}+E_{\bar{K}^{0}})^{2}$
$\displaystyle-(\sqrt{E_{K^{+}}^{2}-m_{K^{+}}^{2}}-\sqrt{E_{\bar{K}^{0}}^{2}-m_{\bar{K}^{0}}^{2}})^{2},$

where $E_{K^{+}}$ and $E_{\bar{K}^{0}}$ are the particle energies in the
$\bar{K}^{0}\Lambda$ rest frame, which can be expressed explicitly

$\displaystyle E_{K^{+}}$
$\displaystyle=\frac{M_{\Lambda_{c}^{+}}^{2}-M_{\bar{K}^{0}\Lambda}^{2}-M_{K^{+}}^{2}}{2M_{\bar{K}^{0}\Lambda}},$
(29) $\displaystyle E_{\bar{K}^{0}}$
$\displaystyle=\frac{M_{\bar{K}^{0}\Lambda}^{2}+M_{\bar{K}^{0}}^{2}-M_{\Lambda}^{2}}{2M_{\bar{K}^{0}\Lambda}}.$

Note that the invariant $K^{+}\Lambda$ mass distribution can be obtained by
substituting $M_{K^{+}\bar{K}^{0}}$ with $M_{K^{+}\Lambda}$.

##  III Results and Discussion

With the above formalism, we perform the $\chi^{2}$ fits to the experimental
data on the invariant mass distributions of the process
$\Lambda^{+}_{c}\to\Lambda K^{+}\bar{K}^{0}$. There are a total of 79 data
points. For Model I, there are five free parameters: $V_{1}$, $V_{2}$,
$a_{\bar{K}\Sigma}$, $a_{\eta\Xi}$, and $\phi$. For both Model II, there are
also five free parameters: $V_{1}$, $V_{2}$, $V_{3}$, $\phi$, and $\theta$.
Note that we have fixed $a_{\pi\Xi}=a_{\bar{K}\Lambda}=-2$ as their natural
value, and $a_{\bar{K}\Sigma}$ and $a_{\eta\Xi}$ are free parameters in this
work. It is found that the pole position of $\Xi^{*}(1690)$ state is not
sensitive to the values of $a_{\pi\Xi}$ and $a_{\bar{K}\Lambda}$, as found in
Refs. [58, 23].

The fitted parameters and the corresponding $\chi^{2}/{\rm dof}$ are listed in
table 2. One can find that both of the two fits in table 2 show reasonably
small $\chi^{2}/{\rm dof}$. With the fitted parameters $a_{\bar{K}\Sigma}$ and
$a_{\eta\Xi}$, the pole position of $\Xi^{*}(1690)$ state is
$M_{R}=1685.83+i2.76$ MeV, and the corresponding mass and width are:
$M_{\Xi^{*}(1690)}=1685.83$ MeV, $\Gamma_{\Xi^{*}(1690)}=5.52$ MeV. The
obtained mass of $\Xi^{*}(1690)$ state is sightly below the mass threshold
(1689 MeV) of the $\bar{K}\Sigma$ channel, and close to the averaged value
quoted in the PDG. However, the obtained width is rather narrow, which is a
common conclusion of the chiral unitary approach [23, 25].

Table 2: Values of some of the parameters used or determined in this work. The values of $a_{\pi\Xi}$ and $a_{\bar{K}\Lambda}$ in Model $\mathrm{I}$ are fixed as the natural value $-2$. The values of the mass and width for the $\Xi^{*}(1690)$ state are obtained from the fitted parameters of Model I. The mass and width of $\Xi^{*}(1690)$ resonance, for Model II, are taken as $M_{\Xi^{*}}=1690$ MeV and $\Gamma_{\Xi^{*}}=20$ MeV as quoted in PDG [6]. | Model $\mathrm{I}$ | Model $\mathrm{II}$
---|---|---
$V_{1}$ (${\rm MeV}^{-1}$) | 3.60$\pm$0.55 | 0.45$\pm$0.83
$V_{2}$ (${\rm MeV}^{-1}$) | -0.89$\pm$0.67 | -4.61$\pm$1.16
$V_{3}$ (${\rm MeV}^{-1}$) | $-$ | 1.81$\pm$0.18
$a_{\bar{K}\Sigma}$ | -1.99$\pm$0.09 | $-$
$a_{\eta\Xi}$ | -3.53$\pm$0.29 | $-$
$\phi$ | 0.60$\pm$0.45 | 3.20$\pm$1.16
$\theta$ | $-$ | 0.67$\pm$1.02
$\chi^{2}/{\rm dof}$ | 1.51 | 1.56
$M_{\Xi^{*}(1690)}$ (MeV) |  $1685.83$ (output) |  $1690$ (input)
$\Gamma_{\Xi^{*}(1690)}$ (MeV) |  $5.52$ (output) |  $20$ (input)

Figure 5: (Color online) The invariant mass distributions for the
$\Lambda_{c}^{+}\to\Lambda K^{+}K^{0}_{S}$ reaction: (a) $\Lambda K^{0}_{S}$
invariant mass distributions; (b) $K^{+}K^{0}_{S}$ invariant mass
distribution; (c) $\Lambda K^{+}$ invariant mass distribution.

The fitted invariant mass distributions 333To compare with the experimental
measurements, we take
$\left|K^{0}\right\rangle=\frac{1}{\sqrt{2}}(\left|K^{0}_{S}\right\rangle+\left|K^{0}_{L}\right\rangle)$
and
$\left|\bar{K}^{0}\right\rangle=\frac{1}{\sqrt{2}}(\left|K^{0}_{S}\right\rangle-\left|K^{0}_{L}\right\rangle)$,
where we have ignored the small effect of $CP$ violation. of the
$\Lambda^{+}_{c}\to\Lambda K^{+}\bar{K}^{0}$ reaction are shown in Fig. 5 for
Model I and II. It is found that both Model I and II can describe these
experimental measurements fairly well. In Fig. 5 (a), the peak of
$\Xi^{*}(1690)$ resonance in the $\Lambda K^{0}_{S}$ invariant mass
distributions is well reproduced. 444It is worth pointing out that we have
added an extra factor $5/8$ to the theoretical results for the
$\bar{K}^{0}\Lambda$ invariant mass distributions, since the experimental
events were accumulated for a bin width of $5$ MeV, while the events for the
invariant $K^{+}K^{0}_{S}$ and $K^{+}\Lambda$ mass distributions are obtained
in a bin width of $8$ MeV. And, it is clearly seen that the line shapes of
$\Xi^{*}(1690)$ resonance are much different for Model I and Model II. The
peak produced from the Model I is higher but narrower, and there is a sharp
decrease around the mass threshold of the $\bar{K}\Sigma$ channel. It is
expected that more precise data, around the energy of
$M_{\bar{K}^{0}\Lambda}=1.69$ GeV, could be used to clarify this issue.
Furthermore, one can see that without the contributions of $\Xi^{*}(1690)$
(${\cal M}_{c}$ for Model I; ${\cal M}_{e}$ for Model II), the experimental
data cannot be described.

In Figs. 5 (b) and (c), we show the $K^{0}_{S}K^{+}$ and $\Lambda K^{+}$
invariant mass distributions, respectively. It is found that the theoretical
results of Model I and II are very similar. This may indicate that the
contributions of $\Xi^{*}(1690)$ resonance, from Model I and II, to the
$K^{0}_{S}K^{+}$ and $\Lambda K^{+}$ invariant mass distributions also are
similar, though they give very different line shapes for $\Xi^{*}(1690)$
state. Once again, one can see that the contributions from the $K\bar{K}$ or
$K\Lambda$ final state interactions in $S$-wave, as shown in Fig. 3, are
crucial to reproduce the experimental measurements.

Next, we study the branching fractions of the $\Lambda^{+}_{c}\to
K^{+}\Xi^{*}(1690)\to K^{+}\bar{K}^{0}\Lambda$ decay to see the effect of the
$\Xi^{*}(1690)$ resonance in the process of $\Lambda^{+}_{c}\to\Lambda
K^{+}\bar{K}^{0}$. With the full amplitude of Eqs. (26) and (27) and the
fitted parameters shown in table 2, the numerical result for the branching
fraction of the production of $\Xi^{*}(1690)$ resonance can be calculated as
follows,

$\frac{\mathcal{B}(\Lambda_{c}^{+}\to K^{+}\Xi^{*}(1690)\to
K^{+}\bar{K}^{0}\Lambda)}{\mathcal{B}(\Lambda_{c}^{+}\to\Lambda
K^{+}\bar{K}^{0})}=\begin{cases}0.28\\\ 0.07\end{cases},$ (30)

for Model I and II, respectively. The above result of Model I is in good
agreement with the experimental data of Belle collaboration: $0.26\pm 0.08\pm
0.03$ [15], while the result of the Model II is much smaller. We hope that the
calculations here can be checked by future precise experiments.

##  IV Summary

In this work, we have performed an analysis of the $\Lambda_{c}^{+}\to\Lambda
K^{+}\bar{K}^{0}$ decay via two cabibbo-favored mechanism. By considering the
hadronization process and the final state interaction in the pseudoscalar-
baryon and pseudoscalar-pseudoscalar channel with chiral unitary approach, the
role of $\Xi^{*}(1690)$, $a_{0}(980)$, and $N^{*}(1535)$ resonances are
investigated. Taking into account the contributions of these three above
resonances, we have calculated the $K^{+}K_{S}^{0}$, $\Lambda K_{S}^{0}$ and
$\Lambda K^{+}$ invariant mass distributions. Up to five free parameters, we
have done $\chi^{2}$-fit to the experimental measurements. It was found that
the reaction proposed here can describe the experimental data of $BABAR$
collaboration [16]. Especially, the clear signal for the $\Xi^{*}(1690)$ state
can be well reproduced in the $\Lambda K^{0}_{S}$ invariant mass spectrum.

Within the fitted model parameters, we have further calculated the ratio of
branching fractions $\mathcal{B}(\Lambda_{c}^{+}\to K^{+}\Xi^{*}(1690)\to
K^{+}\bar{K}^{0}\Lambda)/\mathcal{B}(\Lambda_{c}^{+}\to\Lambda
K^{+}\bar{K}^{0})$ to see the contribution of $\Xi^{*}(1690)$ resonance in the
$\Lambda^{+}_{c}\to\Lambda K^{+}\bar{K}^{0}$ decay. The obtained value 0.28
for Model I, where the $\Xi^{*}(1690)$ is dynamically generated from the
meson-baryon final state interactions, is consistent with the experimental
value by Belle collaboration [15]. It is expected that the $\Xi^{*}(1690)$
state can be analyzed from the precise measurements of the $\Lambda^{+}_{c}\to
K^{+}\bar{K}^{0}\Lambda$ decay by BESIII, BelleII, and LHCb collaborations in
the future [62].

## Acknowledgments

We would like to thank Profs. Wei-Hong Liang and Eulogio Oset for useful
discussions. This work is partly supported by the National Natural Science
Foundation of China under Grant No. 12075288. It is also supported by the
Youth Innovation Promotion Association CAS.

## References

  * Garcia-Recio  _et al._ [2004] C. Garcia-Recio, M. F. M. Lutz, and J. Nieves, Quark mass dependence of s wave baryon resonances, Phys. Lett. B 582, 49 (2004), arXiv:nucl-th/0305100 .
  * Gamermann  _et al._ [2011] D. Gamermann, C. Garcia-Recio, J. Nieves, and L. L. Salcedo, Odd Parity Light Baryon Resonances, Phys. Rev. D 84, 056017 (2011), arXiv:1104.2737 [hep-ph] .
  * Pervin and Roberts [2008] M. Pervin and W. Roberts, Strangeness -2 and -3 baryons in a constituent quark model, Phys. Rev. C 77, 025202 (2008), arXiv:0709.4000 [nucl-th] .
  * Pavon Valderrama  _et al._ [2012] M. Pavon Valderrama, J.-J. Xie, and J. Nieves, Are there three $\Xi(1950)$ states?, Phys. Rev. D 85, 017502 (2012), arXiv:1111.2218 [hep-ph] .
  * Nishibuchi and Hyodo [2022] T. Nishibuchi and T. Hyodo, Nature of excited $\Xi$ baryons with threshold effects, EPJ Web Conf. 271, 10002 (2022), arXiv:2208.14608 [hep-ph] .
  * Zyla  _et al._ [2020] P. A. Zyla _et al._ (Particle Data Group), Review of Particle Physics, PTEP 2020, 083C01 (2020).
  * Nagae [2005] T. Nagae, The J-PARC project, AIP Conf. Proc. 773, 6 (2005).
  * Guo  _et al._ [2007] L. Guo _et al._ , Cascade production in the reactions $\gamma p\to K^{+}K^{+}(X)$ and $\gamma p\to K^{+}K^{+}\pi^{-}(X)$, Phys. Rev. C 76, 025208 (2007), arXiv:nucl-ex/0702027 .
  * Schumacher [2010] R. Schumacher (CLAS), Strangeness physics with CLAS at Jefferson Lab, AIP Conf. Proc. 1257, 100 (2010).
  * Lutz  _et al._ [2009] M. F. M. Lutz _et al._ (PANDA), Physics Performance Report for PANDA: Strong Interaction Studies with Antiprotons, (2009), arXiv:0903.3905 [hep-ex] .
  * Ahn and Nam [2018] J. K. Ahn and S.-i. Nam, $\Xi(1690)^{-}$ production in the $K^{-}p\to K^{+}K^{-}\Lambda$ reaction process near threshold, Phys. Rev. D 98, 114012 (2018), arXiv:1809.07765 [hep-ph] .
  * Dionisi  _et al._ [1978] C. Dionisi _et al._ (Amsterdam-CERN-Nijmegen-Oxford), An Enhancement at the $\Sigma\bar{K}$ Threshold 1680-MeV Observed in $K^{-}p$ Reactions at 4.2-GeV/$c$, Phys. Lett. B 80, 145 (1978).
  * Adamovich  _et al._ [1998] M. I. Adamovich _et al._ (WA89), First observation of the $\Xi^{-}\pi^{+}$ decay mode of the $\Xi^{0}(1690)$ hyperon, Eur. Phys. J. C 5, 621 (1998), arXiv:hep-ex/9710024 .
  * Aubert  _et al._ [2008] B. Aubert _et al._ (BaBar), Measurement of the Spin of the $\Xi(1530)$ Resonance, Phys. Rev. D 78, 034008 (2008), arXiv:0803.1863 [hep-ex] .
  * Abe  _et al._ [2002] K. Abe _et al._ (Belle), Observation of Cabibbo suppressed and W exchange $\Lambda+_{c}$ baryon decays, Phys. Lett. B 524, 33 (2002), arXiv:hep-ex/0111032 .
  * Aubert  _et al._ [2006] B. Aubert _et al._ (BaBar), Measurement of the Mass and Width and Study of the Spin of the $\Xi(1690)$ 0 Resonance from $\Lambda^{+}_{c}\to\Lambda\bar{K}^{0}K^{+}$ Decay at Babar, in  _33rd International Conference on High Energy Physics_ (2006) arXiv:hep-ex/0607043 .
  * Biagi  _et al._ [1981] S. F. Biagi _et al._ , Production of Hyperons and Hyperon Resonances in $\Xi^{-}N$ Interactions at 102-GeV/$c$ and 135-GeV/$c$, Z. Phys. C 9, 305 (1981).
  * Biagi  _et al._ [1987] S. F. Biagi _et al._ , $\Xi^{*}$ Resonances in $\Xi^{-}$ Be Interactions. 1. Diffractive Production in the $\Lambda K^{-}$ and $\Xi^{-}\pi^{+}\pi^{-}$ Channels, Z. Phys. C 34, 15 (1987).
  * Sumihama  _et al._ [2019] M. Sumihama _et al._ (Belle), Observation of $\Xi(1620)^{0}$ and evidence for $\Xi(1690)^{0}$ in $\Xi_{c}^{+}\rightarrow\Xi^{-}\pi^{+}\pi^{+}$ decays, Phys. Rev. Lett. 122, 072501 (2019), arXiv:1810.06181 [hep-ex] .
  * Chao  _et al._ [1981] K.-T. Chao, N. Isgur, and G. Karl, Strangeness -2 and -3 Baryons in a Quark Model With Chromodynamics, Phys. Rev. D 23, 155 (1981).
  * Capstick and Isgur [1986] S. Capstick and N. Isgur, Baryons in a relativized quark model with chromodynamics, Phys. Rev. D 34, 2809 (1986).
  * Xiao and Zhong [2013] L.-Y. Xiao and X.-H. Zhong, $\Xi$ baryon strong decays in a chiral quark model, Phys. Rev. D 87, 094002 (2013), arXiv:1302.0079 [hep-ph] .
  * Sekihara [2015] T. Sekihara, $\Xi(1690)$ as a $\bar{K}\Sigma$ molecular state, PTEP 2015, 091D01 (2015), arXiv:1505.02849 [hep-ph] .
  * Aliev  _et al._ [2018] T. M. Aliev, K. Azizi, and H. Sundu, Analysis of the structure of $\Xi(1690)$ through its decays, Eur. Phys. J. C 78, 396 (2018), arXiv:1804.02656 [hep-ph] .
  * Khemchandani  _et al._ [2018] K. P. Khemchandani, A. Martínez Torres, A. Hosaka, H. Nagahiro, F. S. Navarra, and M. Nielsen, Why $\Xi(1690)$ and $\Xi(2120)$ are so narrow?, Phys. Rev. D 97, 034005 (2018), arXiv:1608.07086 [nucl-th] .
  * Feijoo  _et al._ [2023] A. Feijoo, V. Valcarce, and V. K. Magas, The $\Xi(1620)$ and $\Xi(1690)$ molecular states from $S=-2$ meson-baryon interaction up to next-to-leading order, (2023), arXiv:2303.01323 [hep-ph] .
  * Oset  _et al._ [2016a] E. Oset _et al._ , Weak decays of heavy hadrons into dynamically generated resonances, Int. J. Mod. Phys. E 25, 1630001 (2016a), arXiv:1601.03972 [hep-ph] .
  * Oset  _et al._ [2016b] E. Oset _et al._ , Study of reactions disclosing hidden charm pentaquarks with or without strangeness, Nucl. Phys. A 954, 371 (2016b).
  * Yu and Hsiao [2021] Y. Yu and Y.-K. Hsiao, Cabibbo-favored $\Lambda_{c}^{+}\to\Lambda a_{0}(980)^{+}$ decay in the final state interaction, Phys. Lett. B 820, 136586 (2021), arXiv:2012.14575 [hep-ph] .
  * Ahn  _et al._ [2019] J. K. Ahn, S. Yang, and S.-I. Nam, Hyperon production in $\Lambda_{c}^{+}\to K^{-}p\pi^{+}$ and $\Lambda_{c}^{+}\to K^{0}_{S}p\pi^{0}$, Phys. Rev. D 100, 034027 (2019), arXiv:1907.04475 [hep-ph] .
  * Miyahara  _et al._ [2017] K. Miyahara, T. Hyodo, M. Oka, J. Nieves, and E. Oset, Theoretical study of the $\Xi$(1620) and $\Xi$(1690) resonances in $\Xi_{c}\to\pi^{+}MB$ decays, Phys. Rev. C 95, 035212 (2017), arXiv:1609.00895 [nucl-th] .
  * Nieves and Ruiz Arriola [1999] J. Nieves and E. Ruiz Arriola, Bethe-Salpeter approach for meson meson scattering in chiral perturbation theory, Phys. Lett. B 455, 30 (1999), arXiv:nucl-th/9807035 .
  * Oller and Oset [1997] J. A. Oller and E. Oset, Chiral symmetry amplitudes in the S wave isoscalar and isovector channels and the $\sigma$, $f_{0}$(980), $a_{0}$(980) scalar mesons, Nucl. Phys. A 620, 438 (1997), [Erratum: Nucl.Phys.A 652, 407–409 (1999)], arXiv:hep-ph/9702314 .
  * Kaiser  _et al._ [1995] N. Kaiser, P. B. Siegel, and W. Weise, Chiral dynamics and the S11 (1535) nucleon resonance, Phys. Lett. B 362, 23 (1995), arXiv:nucl-th/9507036 .
  * Nieves and Ruiz Arriola [2001] J. Nieves and E. Ruiz Arriola, The S11-N(1535) and -N(1650) resonances in meson baryon unitarized coupled channel chiral perturbation theory, Phys. Rev. D 64, 116008 (2001), arXiv:hep-ph/0104307 .
  * Inoue  _et al._ [2002] T. Inoue, E. Oset, and M. J. Vicente Vacas, Chiral unitary approach to S wave meson baryon scattering in the strangeness S = O sector, Phys. Rev. C 65, 035204 (2002), arXiv:hep-ph/0110333 .
  * Bruns  _et al._ [2011] P. C. Bruns, M. Mai, and U. G. Meissner, Chiral dynamics of the S11(1535) and S11(1650) resonances revisited, Phys. Lett. B 697, 254 (2011), arXiv:1012.2233 [nucl-th] .
  * Khemchandani  _et al._ [2013] K. P. Khemchandani, A. Martinez Torres, H. Nagahiro, and A. Hosaka, Role of vector and pseudoscalar mesons in understanding $1/2^{-}N^{*}$ and $\Delta$ resonances, Phys. Rev. D 88, 114016 (2013), arXiv:1307.8420 [nucl-th] .
  * Garzon and Oset [2015] E. J. Garzon and E. Oset, Mixing of pseudoscalar-baryon and vector-baryon in the $J^{P}=1/2^{-}$ sector and the $N^{*}$(1535) and $N^{*}$(1650) resonances, Phys. Rev. C 91, 025201 (2015), arXiv:1411.3547 [hep-ph] .
  * Xie  _et al._ [2015] J.-J. Xie, L.-R. Dai, and E. Oset, The low lying scalar resonances in the $D^{0}$ decays into $K^{0}_{S}$ and $f_{0}(500)$, $f_{0}(980)$, $a_{0}(980)$, Phys. Lett. B 742, 363 (2015), arXiv:1409.0401 [hep-ph] .
  * Ling  _et al._ [2021] X.-Z. Ling, M.-Z. Liu, J.-X. Lu, L.-S. Geng, and J.-J. Xie, Can the nature of $a_{0}(980)$ be tested in the $D_{S}^{+}\to\pi^{+}\pi^{0}\eta$ decay?, Phys. Rev. D 103, 116016 (2021), arXiv:2102.05349 [hep-ph] .
  * Wang  _et al._ [2021] J.-Y. Wang, M.-Y. Duan, G.-Y. Wang, D.-M. Li, L.-J. Liu, and E. Wang, The $a_{0}(980)$ and $f_{0}(980)$ in the process $D_{S}^{+}\to K^{+}K^{-}\pi^{+}$, Phys. Lett. B 821, 136617 (2021), arXiv:2105.04907 [hep-ph] .
  * Molina  _et al._ [2020] R. Molina, J.-J. Xie, W.-H. Liang, L.-S. Geng, and E. Oset, Theoretical interpretation of the $D^{+}_{S}\to\pi^{+}\pi^{0}\eta$ decay and the nature of $a_{0}(980)$, Phys. Lett. B 803, 135279 (2020), arXiv:1908.11557 [hep-ph] .
  * Duan  _et al._ [2020] M.-Y. Duan, J.-Y. Wang, G.-Y. Wang, E. Wang, and D.-M. Li, Role of scalar $a_{0}(980)$ in the single Cabibbo suppressed process $D^{+}\rightarrow\pi^{+}\pi^{0}\eta$, Eur. Phys. J. C 80, 1041 (2020), arXiv:2008.10139 [hep-ph] .
  * Zhu  _et al._ [2022] X. Zhu, H.-N. Wang, D.-M. Li, E. Wang, L.-S. Geng, and J.-J. Xie, Further understanding the nature of $a_{0}(1710)$ in the $D^{+}_{S}\to\pi^{0}K^{+}K^{0}_{S}$ decay, (2022), arXiv:2210.12992 [hep-ph] .
  * Xie and Geng [2016] J.-J. Xie and L.-S. Geng, The $a_{0}(980)$ and $\Lambda(1670)$ in the $\Lambda^{+}_{c}\to\pi^{+}\eta\Lambda$ decay, Eur. Phys. J. C 76, 496 (2016), arXiv:1604.02756 [nucl-th] .
  * Xie and Geng [2017a] J.-J. Xie and L.-S. Geng, Role of the $N^{*}(1535)$ in the $\Lambda^{+}_{c}\to\bar{K}^{0}\eta p$ decay, Phys. Rev. D 96, 054009 (2017a), arXiv:1704.05714 [hep-ph] .
  * Xie and Geng [2017b] J.-J. Xie and L.-S. Geng, $\Sigma^{*}_{1/2^{-}}(1380)$ in the $\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$ decay, Phys. Rev. D 95, 074024 (2017b), arXiv:1703.09502 [hep-ph] .
  * Wang  _et al._ [2022] G.-Y. Wang, N.-C. Wei, H.-M. Yang, E. Wang, L.-S. Geng, and J.-J. Xie, Roles of $a_{0}(980)$, $\Lambda$(1670), and $\Sigma$(1385) in the $\Lambda_{c}^{+}\to\eta\Lambda\pi^{+}$ decay, Phys. Rev. D 106, 056001 (2022), arXiv:2206.01425 [hep-ph] .
  * Pavao  _et al._ [2018] R. Pavao, S. Sakai, and E. Oset, Production of $N^{*}(1535)$ and $N^{*}(1650)$ in $\Lambda_{c}\rightarrow\bar{K}^{0}\eta p$ $(\pi N)$ decay, Phys. Rev. C 98, 015201 (2018), arXiv:1802.07882 [nucl-th] .
  * Chau and Cheng [1987] L.-L. Chau and H.-Y. Cheng, Analysis of Exclusive Two-Body Decays of Charm Mesons Using the Quark Diagram Scheme, Phys. Rev. D 36, 137 (1987), [Addendum: Phys.Rev.D 39, 2788–2791 (1989)].
  * Chau and Cheng [1990] L. L. Chau and H.-Y. Cheng, Nonresonant three-body decays of charmed mesons, Phys. Rev. D 41, 1510 (1990).
  * Chau  _et al._ [1996] L.-L. Chau, H.-Y. Cheng, and B. Tseng, Analysis of two-body decays of charmed baryons using the quark diagram scheme, Phys. Rev. D 54, 2132 (1996), arXiv:hep-ph/9508382 .
  * Qin  _et al._ [2022] Q. Qin, J.-L. Qiu, and F.-S. Yu, Diagrammatic analysis of hidden- and open-charm tetraquark production in $B$ decays, (2022), arXiv:2212.03590 [hep-ph] .
  * Oller  _et al._ [1998] J. A. Oller, E. Oset, and J. R. Pelaez, Nonperturbative approach to effective chiral Lagrangians and meson interactions, Phys. Rev. Lett. 80, 3452 (1998), arXiv:hep-ph/9803242 .
  * Oller  _et al._ [1999] J. A. Oller, E. Oset, and J. R. Pelaez, Meson meson interaction in a nonperturbative chiral approach, Phys. Rev. D 59, 074001 (1999), [Erratum: Phys.Rev.D 60, 099906 (1999), Erratum: Phys.Rev.D 75, 099903 (2007)], arXiv:hep-ph/9804209 .
  * Garzon and Oset [2016] E. J. Garzon and E. Oset, Mixing of Pseudoscalar-Baryon and Vector-Baryon in the $J^{P}=1/2^{-}$ Sector and the $N^{*}(1535)$ and $N^{*}(1650)$ resonances , JPS Conf. Proc. 10, 022004 (2016).
  * Ramos  _et al._ [2002] A. Ramos, E. Oset, and C. Bennhold, On the spin, parity and nature of the $\Xi(1620)$ resonance, Phys. Rev. Lett. 89, 252001 (2002), arXiv:nucl-th/0204044 .
  * Avery  _et al._ [1993] P. Avery _et al._ (CLEO), Study of the decays $\Lambda_{c}^{+}\to\Xi^{0}K^{+}$, $\Lambda_{c}^{+}\to\Sigma^{+}K^{+}K^{-}$ and $\Lambda_{c}^{+}\to\Xi^{-}K^{+}\pi^{+}$, Phys. Rev. Lett. 71, 2391 (1993).
  * Ablikim  _et al._ [2018] M. Ablikim _et al._ (BESIII), Measurements of absolute branching fractions for $\Lambda^{+}_{c}\to\Xi^{0}K^{+}$ and $\Xi(1530)^{0}K^{+}$, Phys. Lett. B 783, 200 (2018), arXiv:1803.04299 [hep-ex] .
  * Gronau  _et al._ [2018] M. Gronau, J. L. Rosner, and C. G. Wohl, Overview of $\Lambda_{c}$ decays, Phys. Rev. D 97, 116015 (2018), [Addendum: Phys.Rev.D 98, 073003 (2018)], arXiv:1808.03720 [hep-ph] .
  * Ablikim  _et al._ [2023] M. Ablikim _et al._ (BESIII), Measurement of branching fractions of $\Lambda_{c}^{+}$ decays to $\Sigma^{+}K^{+}K^{-}$, $\Sigma^{+}\phi$ and $\Sigma^{+}K^{+}\pi^{-}(\pi^{0})$, (2023), arXiv:2304.09405 [hep-ex] .


# Representation Stability for the Ordered Configuration Space of Star Graphs

Nicholas Wawrykow

###### Abstract

We study the ordered configuration space of star graphs. Inspired by the
representation stability results of Church–Ellenberg–Farb for the ordered
configuration space of a manifold and the edge stabilization maps of
An–Drummond-Cole–Knudsen for the unordered configuration space of a graph, we
show that the ordered configuration space of a star graph with $k$ leaves
satisfies a notion of representation stability. Namely, we prove that its
homology has an FIk,o-module structure, and we show that as such first
homology is finitely generated in degree $4$ for $k=3$, degree $3$ for $k=4$,
and degree $2$ for $k\geq 5$.

##  1 Introduction

Configuration spaces of particles in manifolds have been studied for nearly a
century, though it wasn’t until the work of Abrams [Abr00] and Ghrist [Ghr01]
in the late 1990s and early 2000s, that configuration spaces of graphs became
a topic of focus for the mathematical community.

Recall that given a space $X$, the _$n^{\text{th}}$ -ordered configuration
space of particles in $X$_ is

$F_{n}(X):=\big{\\{}(x_{1},\dots,x_{n})\in X^{n}|x_{i}\neq x_{j}\mbox{ if
}i\neq j\big{\\}},$

i.e., the space of ways of embedding $n$ distinct labeled points in $X$. The
_$n^{\text{th}}$ -unordered configuration space of particles in $X$_, denoted
$C_{n}(X)$, is the quotient of $F_{n}(X)$ by the natural $S_{n}$-action.

There has been a surge of interest in the topology of configuration spaces in
the last decade. This is especially true in the case $X=\Gamma$ is a graph,
that is a $1$-dimensional cellular complex, whose $0$-cells are called
_vertices_ , and whose $1$-cells are call _edges_. In particular, there has
been considerable effort to understand how graph configurations behave both as
the number of particles in the configuration change and as the underlying
graphs change. The papers [ADCK20, ADCK22, AK22, AM21, Che16, CL18, Lüt17a,
Lüt17b, Ram20] are among the many works in these directions. We continue these
efforts, seeking to determine to how the homology of the ordered configuration
space of _star graphs_ behave as the number of particles in the configuration
increases.

###  1.1 Stability in the Particle Direction

A common technique for determining the structure of the configuration space of
particles in $X$ is to study how the configuration space of $X$ behaves as the
number of particles in the configuration increases. In the case $X$ is a
connected non-compact manifold of dimension at least $2$, McDuff [McD75] and
Segal [Seg79] noted that for $n$ sufficiently large with respect to $i$, there
is an isomorphism

$H_{i}\big{(}C_{n}(X)\big{)}\to H_{i}\big{(}C_{n+1}(X)\big{)}.$

This _homological stability_ arises from stabilization maps

$i:C_{n}(X)\hookrightarrow C_{n+1}(X)$

that “add a particle at infinity” to the configuration.

In the case of unordered graph configuration spaces, the analogous operation
of adding a particle at a valence $1$ vertex, i.e., a _leaf_ , induces similar
maps on configuration space, though such maps rarely yield homological
stability results. To ameliorate this, An–Drummond-Cole–Knudsen [ADCK20]
introduced _edge stabilization_ maps that add a new particle at the interior
of an edge. These maps induce a $\mathbb{Z}[E]$-module structure on
$H_{i}\big{(}C_{\bullet}(\Gamma)\big{)}$, where $E$ is the set of edges of
$\Gamma$. They proved that $H_{i}\big{(}C_{\bullet}(\Gamma)\big{)}$ is
finitely generated as a $\mathbb{Z}[E]$-module [ADCK20, Theorem 1.1], and they
proved that the rank of $H_{i}\big{(}C_{n}(\Gamma)\big{)}$ is a polynomial in
$n$ of degree determined by a connectivity invariant of $\Gamma$ [ADCK20,
Theorem 1.2].

In the case of ordered configuration spaces, the labeling of the particles
precludes a notion of homological stability. However, this labeling induces an
$S_{n}$-representation structure on $H_{i}\big{(}F_{n}(X)\big{)}$.
Church–Ellenberg–Farb proved that if one takes this structure into account and
$X$ is a connected non-compact finite-type manifold of dimension at least $2$,
then $\Big{\\{}H_{i}\big{(}F_{n}(X)\big{)}\Big{\\}}_{n\geq 0}$ stabilizes as a
sequence of symmetric group representations, as it is a finitely generated
module over the combinatorial category FI [CEF15, Theorem 6.5.3]. Moreover,
this _representation stability_ also arises from the map on configuration
space that “adds a particle at infinity.”

Inspired by the work of An–Drummond-Cole–Knudsen and Church–Ellenberg–Farb, we
approach the problem of representation stability for the one of the simplest
families of graph: _star graphs_. The star graph $\Gamma_{k}$ consists of $1$
central vertex, and $k$ surrounding vertices, each connected by a single edge
to the central vertex, see Figure 1 for an example. An argument analogous to
An–Drummond-Cole–Knudsen’s proves that for each $n$ the $k$ maps that push the
particles in a configuration towards the central vertex and add a labeled
particle at a leaf

$i_{n,j}:F_{n}(\Gamma_{k})\hookrightarrow F_{n+1}(\Gamma_{k})$

introduce maps on homology that give
$H_{i}\big{(}F_{\bullet}(\Gamma_{k})\big{)}$ the structure of a module over a
combinatorial category. We formalize this through the notion of FId,o-modules,
a relative of the FId-modules studied by Ramos in [Ram17], and we prove that
it is finitely generated as an FIk,o-module and calculate its generation
degree.

######  Theorem 1.1.

For $k\geq 3$, the sequence of homology groups
$H_{i}\big{(}F_{\bullet}(\Gamma_{k})\big{)}$ has the structure of a finitely
generated FIk,o-module, generated in degree $1$ for $i=0$, degree $0$ for
$i>1$, and degree $4$ for $k=3$, degree $3$ for $k=4$, and degree $2$ for
$k\geq 5$.

Notably, Theorem 1.1 proves that for $n=3,4$, there are classes in
$H_{1}\big{(}F_{n}(\Gamma_{3})\big{)}$ that cannot be viewed as a sum of a
product of a class of $H_{1}\big{(}F_{n-1}(\Gamma_{3})\big{)}$ with a copy of
the fundamental of $H_{0}\big{(}F_{1}(\mathbb{R})\big{)}$. This differs from
the case of the $1^{\text{st}}$-homology of the unordered configuration space
of $\Gamma_{3}$, which An–Drummond-Cole–Knudsen’s results show is freely
generated by the fundamental class of $H_{1}\big{(}C_{2}(\Gamma_{3})\big{)}$
and the $\mathbb{Z}[E]$-module structure on
$H_{1}\big{(}C_{\bullet}(\Gamma_{3})\big{)}$. Thus we see that ordered
configuration spaces of graphs are more complex than their unordered
counterparts, even in the simplest cases.

Figure 1: The star graph $\Gamma_{3}$.

###  1.2 Structure of the Paper

In section 2 we recall the categories FI and FI-mod, as well as the
representation stability results of Church–Ellenberg–Farb for ordered
configuration spaces of points in manifolds. We introduce the category FId,o,
an ordered version of the category FId, that we will use to formalize our
representation stability results. Next, in section 3, we recall the definition
of a cubical complex defined by Lügehetmann that is homotopy equivalent to the
ordered configuration space of a graph. After that, we recall the
Mayer–Vietoris spectral sequence in section 4, which, along with the cube
complex of section 3, will be central to our stability argument. Finally, in
section 5, we prove Theorem 1.1 by finding a “good” cover for configuration
space and analyzing the first row of the Mayer–Vietoris spectral sequence with
respect to this cover.

###  1.3 Acknowledgements

The author would like to Ben Knudsen, Andrew Snowden, and Jennifer Wilson for
insightful conversations.

##  2 FId,o

Church–Ellenberg–Farb used FI-modules to state their representation stability
results for connected non-compact finite-type manifolds of dimension at least
$2$. In this section we recall the definitions of the category FI and of FI-
modules, and introduce generalizations, which we will call FId,o and
FId,o-modules that we will use to state our representation stability results.

Let _FB_ denote the category whose objects are finite sets and whose morphisms
are injections. An _FB-module_ over a commutative ring $R$ is a covariant
functor from FB to the category of $R$-modules, and an _FB-space_ is a
covariant functor from FB to the category of spaces. Since every finite set
$A$ is isomorphic to a set of the form $[n]:=\\{1,\dots,n\\}$ for some
$n\in\mathbb{Z}_{\geq 0}$, it follows that FB is equivalent to its skeleton
where there is one object $[n]$ for each $n\in\mathbb{Z}_{\geq 0}$. This
allows us to interpret an FB-module $U$ as a sequence
$(U_{n})_{n\in\mathbb{Z}_{\geq 0}}$ of symmetric group representations.

If we allow injections between sets of different cardinalities in FB, we get
the category _FI_ of finite sets and injections. An _FI-module_ over a
commutative ring $R$ is a covariant functor from FI to the category of
$R$-modules, and an _FI-space_ is a covariant function from FI to the category
of topological spaces. As is the case for FB, the category FI is equivalent to
its skeleton with one object $[n]$ for each $n\in\mathbb{Z}_{\geq 0}$.

######  Example 2.1.

If $M$ is a non-compact finite-type manifold of dimension at least $2$, then
$F_{\bullet}(M)$ is an FI-space, where the standard inclusion
$[n]\hookrightarrow[n+1]$ induces a map that sends $F_{n}(M)$ to $F_{n+1}(M)$
by “adding a particle labeled $n+1$ at infinity,” see Figure 2. We also have
that for all $i$, $H_{i}\big{(}F_{\bullet}(M)\big{)}$ has the structure of an
FI-module where the inclusion map $[n]\hookrightarrow[n+1]$ induces a map from
$H_{i}\big{(}F_{n}(M)\big{)}$ to $H_{i}\big{(}F_{n+1}(M)\big{)}$ that tensors
a class in $H_{i}\big{(}F_{n}(M)\big{)}$ with the fundamental class of
$H_{0}\big{(}F_{1}(\mathbb{R}^{2})\big{)}$.

Figure 2: The action of the inclusion map on a point in $F_{3}(T^{\circ})$,
where $T^{\circ}$ is the once-punctured torus. The map retracts the torus away
from the puncture and adds a new particle near the puncture away from the
image of the retraction.

We say that an FI-module $V$ is _generated_ by a set $S\subseteq\sqcup_{n\geq
0}V_{n}$ if $V$ is the smallest FI-submodule containing $S$. If there is some
finite set $S$ that generates $V$, then $V$ is _finitely generated_ , and if
$V$ is generated by $\sqcup_{0\leq n\leq m}V_{n}$, then $V$ is said to be
_generated in degree at most $m$_.

Church–Ellenberg–Farb proved that the homology groups of connected non-compact
manifolds of dimension at least $2$ stabilize in a representation theoretic
sense as they are finitely generated FI-modules.

######  Theorem 2.2.

(Church–Ellenberg–Farb [CEF15, Theorem 6.4.3] for oriented manifolds,
Miller–Wilson [MW19, 1.1] for the general case) Let $M$ be a connected non-
compact manifold of dimension at least $2$, then
$H_{i}\big{(}F_{\bullet}(M)\big{)}$ has the structure of a finitely generated
FI-module generated in degree at most $2i$.

The homology of the ordered configuration space of particles on a graph rarely
has the structure of a finitely generated FI-module. Unlike the configuration
space of a manifold of dimension at least $2$, the order in which we add
particles “at infinity” to the configuration space of a graph matters.
Moreover, for ordered configurations of particles on graphs the “infinity” or
leaf of a graph to which one adds new particles matters, whereas for connected
manifolds of dimension at least $2$, all infinities yield the same span of
classes. Therefore, we need a new category that takes into account the order
in which we add particles and the various places we can add them.

For fixed $k\geq 1$, let $\emph{FI}_{d}$ denote the category whose objects are
finite sets and whose morphisms $(f,c)$ are injections $f$ with a $d$-coloring
$c$ of the complement of the image of $f$. This category, which was studied by
Ramos in [Ram17], allows one to take into account the different ways (edges)
in which we add a new particle; however, it does not record their order. To
include this data, we introduce a new combinatorial category.

######  Definition 2.3.

Let _FI d,o_ denote the category whose objects are finite sets and whose
morphisms $(f,c,o)$ are injections $f$ with a $d$-coloring $c$ of the
complement of the image of $f$, and an ordering $o$ for each set of elements
of the same color. We can compose FId,o-morphisms in the following way: if the
domain of an injection $f$ is the codomain of an injection $f^{\prime}$, we
compose $(f,c,o):[m]\to[n]$ and $(f^{\prime},c^{\prime},o^{\prime}):[n]\to[l]$
by setting

$(f^{\prime},c^{\prime},o^{\prime})\circ(f,c,o)=(f^{\prime}\circ
f,c^{\prime\prime},o^{\prime\prime}),$

where for $x\notin\text{Im}f^{\prime}\circ f$

$c^{\prime\prime}(x)=\begin{cases}c(x)&\mbox{if }x\in\text{Im}f^{\prime}\\\
c^{\prime}(x)&\mbox{if
}x\notin\text{Im}f^{\prime}\end{cases}\indent\text{and}\indent
o^{\prime\prime}(x)=\begin{cases}o(x)&\mbox{if }x\in\text{Im}f^{\prime}\\\
o^{\prime}(x)+(n-m)&\mbox{if }x\notin\text{Im}f^{\prime}.\end{cases}$

See Figure 3 for an example.

Additionally, an _FI d,o-module_ over a ring $R$ is a functor from FId,o to
$R$-mod.

Figure 3: The composition of two FI2,o-morphisms. Here the ordering is denoted
by the subscripts.

Note that every morphism $(f,c,o):[n]\to[m]$ can be written

$(f,o,c)=\sigma\circ(i_{m-1},c_{j_{m}},o_{j_{m}})\circ\cdots\circ(i_{n},c_{j_{n+1}},o_{j_{n+1}}),$

where $\sigma\in S_{n+m}$, $i_{l}:[l]\to[l+1]$ is the standard inclusion of
$[l]$ into $[l+1]$, $c_{j_{l+1}}$ colors $l+1$ color $j_{l+1}$ and
$o_{j_{l+1}}$ orders $l+1$ first among the elements colored $j_{l+1}$ as it is
the only element in the complement.

Like FB and FI, the category FId,o is equivalent to its skeleton where there
is one object $[n]$ for each $n\in\mathbb{Z}_{\geq 0}$, and we will think of
FId,o as this skeleton. Given an FId,o-module $W$, we write $W_{n}$ for
$W\big{(}[n]\big{)}$. The set of FId,o-morphisms from $[n]$ to itself can be
identified with the symmetric group $S_{n}$, making $W_{n}$ an
$S_{n}$-representation.

Given $W$, an _FI d,o-submodule_ $W^{\prime}$ of $W$ is a sequence of
symmetric group representations $W^{\prime}_{n}\subseteq W_{n}$ that is closed
under the induced actions of the FId,o-morphisms. We say that $W$ is
_generated_ by a set $S\subseteq\bigsqcup_{n\geq 0}W_{n}$ if $W$ is the
smallest FId,o-submodule containing $S$. If there is some finite set $S$ that
generated $W$, then we say that $W$ is _finitely generated_. If $W$ is
generated by $S\subseteq\bigsqcup_{0\leq n\leq m}W_{n}$, we say that $W$ is
_generated in degree at most $m$_.

Next we provide a pair of necessary and sufficient conditions for an FB-space
$X_{\bullet}$ with $d$ maps $i_{n,j}:X_{n}\to X_{n+1}$ for all $n$ to be an
FId,o-space. We will use this proposition to prove that
$H_{i}\big{(}F_{\bullet}(\Gamma_{k})\big{)}$ has the structure of an
FIk,o-module.

Given an FB-space $X$ with $d$ _insertion maps_ $i_{n,j}:X_{n}\to X_{n+1}$ for
every $n$, we say that _insertion maps commute with permutations_ if for every
$j$ and $n$ and every $\sigma\in S_{n}$, we have

$i_{n,j}\circ\sigma=\tilde{\sigma}\circ i_{n,j}$

where $\tilde{\sigma}\in S_{n+1}$ is the image in $S_{n+1}$ of $\sigma$ under
the standard inclusion $[n]\hookrightarrow[n+1]$. Additionally, if for every
$j\neq l$ and every $n$, we have

$(n+1\,n+2)\circ i_{n+1,l}\circ i_{n,j}=i_{n+1,j}\circ i_{n,l},$

we say that _distinct insertions are unordered_.

######  Proposition 2.4.

Let $X$ be an FB-space such that for all $n$ there exists $d$ insertion maps
$i_{n,j}:X_{n}\to X_{n+1}$. If these insertion maps commute with permutations
and distinct insertions are unordered, then $X$, along with these insertion
maps, has the structure of an FId,o-space.

###### Proof.

For each morphism $(f,c,o):[m]\to[n]$ in FId,o, we need to define a map
$(f,c,o)_{*}:X_{m}\to X_{n}$. When $m=n$, we have $f=\sigma$ for some
$\sigma\in S_{m}$, and $c=\cdot$ and $o=\cdot$ are the trivial coloring and
ordering, respectively. In this case $(f,c,o)_{*}=\sigma$, where we also use
$\sigma$ to denote its action on $X_{m}$. The insertion maps $i_{l,j_{l+1}}$
describe what happens when $f$ is the standard inclusion
$[l]\hookrightarrow[l+1]$, $c_{j_{l+1}}$ colors $l+1$ color $j_{l+1}$, and
$o_{j_{l+1}}$ orders $l+1$ the first among the elements color $j=j_{l+1}$,
i.e., $(f,c,o)=(i_{l},c_{j_{l+1}},o_{j_{l+1}})$ and $(f,c,o)_{*}=[i_{l,j}]$.

Given an arbitrary morphism $(f,c,o):[m]\to[n]$ in FId,o we can write
$f:[m]\to[n]$ as $\sigma^{\prime}\circ i_{n-1}\circ\cdots\circ i_{m}$ for
$\sigma^{\prime}\in S_{n}$. Moreover, there is a unique such $\sigma^{\prime}$
such that $\sigma^{\prime}$ sends $m+1$ to the first element in complement of
the image of $f$ colored $j_{m+1}$, $m+2$ to the fist element colored
$j_{m+2}$ complement of the image $f$ if $j_{m+2}\neq j_{m+1}$ and the second
element colored $j_{m+1}$ if $j_{m+2}=j_{m+1}$, so on and so forth. Then, we
have

$(f,c,o)=(\sigma^{\prime},\cdot,\cdot)\circ(i_{n-1},c_{j_{n}},o_{j_{n}})\circ\cdots\circ(i_{m},c_{j_{m+1}},o_{j_{m+1}}).$

It follows that we should define

$(f,c,o)_{*}=\sigma^{\prime}\circ[i_{n-1,j_{n}}]\circ\cdots\circ[i_{m,j_{m+1}}].$

To check functoriality, we must show that if we have another sequence of
permutations and insertions that compose to $(f,c,o)$ in FId,o, then the
corresponding maps on the various spaces $X_{l}$ compose to $(f,c,o)_{*}$.
Given an arbitrary sequence of permutations and insertion maps, the property
that insertion maps commute with permutations implies that we can push all the
permutations to the left of the insertion maps, without changing the
composition map, obtaining a composition of insertions followed by a
composition of permutations. Since there is an $S_{n}$-action on $X_{n}$, we
can replace the composition of permutations by a single permutation.
Therefore, it suffices to show that if

$(f,c,o)=(\sigma^{\prime\prime},\cdot,\cdot)\circ(i_{n-1},c_{j^{\prime}_{n}},o_{j^{\prime}_{n}})\circ\cdots\circ(i_{m},c_{j^{\prime}_{m+1}},o_{j^{\prime}_{m+1}}),$

then

$\sigma^{\prime\prime}\circ[i_{n-1,j^{\prime}_{n}}]\circ\cdots\circ[i_{m,j^{\prime}_{m+1}}]=\sigma^{\prime}\circ[i_{n-1,j_{n}}]\circ\cdots\circ[i_{m,j_{m+1}}].$

Note that $\sigma^{\prime}$ and $\sigma^{\prime\prime}$ take the same values
on $[n]$ as $f$, we can write
$\sigma^{\prime\prime}=\sigma^{\prime}\circ\tilde{\sigma}$ where
$\tilde{\sigma}$ only permutes $[m]\backslash[n]$. Therefore, after canceling
$\sigma^{\prime}$ from both sides it suffices to show that

$\tilde{\sigma}\circ[i_{m-1,j^{\prime}_{m}}]\circ\cdots\circ[i_{n,j^{\prime}_{n+1}}]=[i_{m-1,j_{m}}]\circ\cdots\circ[i_{n,j_{n+1}}].$

This follows from the fact that distinct insertions are unordered. We see this
by induction on $m-n$. If $m-n=1$, this is trivial. Otherwise, let
$m+k=(\tilde{\sigma})^{-1}(m+1)$, i.e., in the alternate composition, element
$m+k$ gets inserted with color $j^{\prime}_{m+k}=j_{m+1}$ and then
$\tilde{\sigma}$ changes its number to $m+1$. Since distinct insertions are
unordered if $j^{\prime}_{m+k}\neq j^{\prime}_{m+k-1}$, we can write

$[i_{m+k-1,j^{\prime}_{m+k}}]\circ[i_{m+k-2,j^{\prime}_{m+k-1}}]=(m+k\,m+k-1)\circ[i_{m+k-1,j^{\prime}_{m+k-1}}]\circ[i_{m+k-2,j^{\prime}_{m+k}}].$

Moreover, we need not worry about the case
$j^{\prime}_{m+k}=j^{\prime}_{m+k-1}$ as this would imply that the two
compositions have a different first element of color $j^{\prime}_{m+k}$ in the
ordering given by $o$, a contradiction. By applying the fact that insertion
maps commute with permutations, we can move all the transpositions to the left
to make the composition

$\tilde{\sigma}\circ(m+k\,m+k-1)\circ\cdots\circ(m+2\,m+1),$

which is equal to

$\tilde{\sigma}\circ(m+k\,m+k-1\,\cdots\,m+2\,m+1),$

which fixes $m+1$. Denote this composition by $\tilde{\sigma}^{\prime}$. It
follows that

$\tilde{\sigma}\circ[i_{n-1,j^{\prime}_{n}}]\circ\cdots\circ[i_{m,j^{\prime}_{m+1}}]=\tilde{\sigma}^{\prime}\circ[i_{n-1,j^{\prime}_{n}}]\circ\cdots\circ[i_{m+k,j^{\prime}_{m+k+1}}]\circ[i_{m+k-2,j^{\prime}_{m+k-1}}]\circ\cdots\circ[i_{m,j^{\prime}_{m+1}}]\circ[i_{m+k-1,j^{\prime}_{m+k}}],$

and $j^{\prime}_{m+k}=j_{m+1}$. By the induction hypothesis we have

$\tilde{\sigma}^{\prime}\circ[i_{n-1,j^{\prime}_{n}}]\circ\cdots\circ[i_{m+k,j^{\prime}_{m+k+1}}]\circ[i_{m+k-2,j^{\prime}_{m+k-1}}]\circ\cdots\circ[i_{m,j^{\prime}_{m+1}}]=[i_{n-1,j_{n}}]\circ\cdots\circ[i_{m+1,j_{m+2}}]$

so by composing with $[i_{m,j_{m+1}}]$ on the right we get the desired
equality. ∎

Note that there exist $k$ insertion maps $i_{n,j}:F_{n}(\Gamma_{k})\to
F_{n+1}(\Gamma_{k})$ that give $F_{\bullet}(\Gamma_{k})$ the structure of an
FIk,o-space.

######  Lemma 2.5.

Let

$i_{n,j}:F_{n}(\Gamma_{k})\to F_{n+1}(\Gamma_{k})$

be the map that sends a configuration of $n$ particles in $\Gamma_{k}$ to the
configuration of $n+1$ particles in $\Gamma_{k}$ where the distance of all the
particles from the central vertex is scaled by $\frac{4}{5}$, and then adds a
particle labeled $n+1$ on the $j^{\text{th}}$-leaf of $\Gamma_{k}$. See Figure
4. Then, for all $k\geq 3$, the ordered configuration space of points in the
star graph with $k$ leaves, along with these insertion maps, has the structure
of an FIk,o-space.

###### Proof.

By Proposition 2.4 it suffices to check that our insertion maps $i_{n,j}$ are
unordered and commute with permutations. This follows immediately from the
definition of the $i_{n,j}$. ∎

Figure 4: The action of $i_{4,1}$ on a point in $F_{4}(\Gamma_{3})$.

Since homology with $R$-coefficients is a covariant functor from the category
of topological spaces to the category $R$-modules, the following corollary is
immediate.

######  Corollary 2.6.

For all $k\geq 3$ and all $i$, $H_{i}\big{(}F_{\bullet}(\Gamma_{k})\big{)}$
has the structure of an FIk,o-module.

This structure comes from noting that $i_{n,j}$ induces a map

$(i_{n,j})_{*}:H_{i}\big{(}F_{n}(\Gamma_{k})\big{)}\to
H_{i}\big{(}F_{n+1}(\Gamma_{k})\big{)}$

that tensors a class in $H_{i}\big{(}F_{n}(\Gamma_{k})\big{)}$ with the
fundamental class of $H_{0}\big{(}F_{1}(\mathbb{R})\big{)}$ along the
$j^{\text{th}}$ edge.

While we will not need the material in the rest of this section, we include it
as it consists of standard definitions and results for modules over a
combinatorial category.

The _free_ FId,o-module generated in degree $m$ is

$M^{\text{FI}_{d,o}}(m)_{\bullet}:=\text{Hom}_{\text{FI}_{d,o}}\big{(}[m],\bullet\big{)},$

i.e.,
$M^{\text{FI}_{d,o}}(m)_{n}:=\text{Hom}_{\text{FI}_{d,o}}\big{(}[m],[n]\big{)}$.
If $W_{m}$ is an $S_{m}$-representation we set

$M^{\text{FI}_{d,o}}(W_{m})_{\bullet}=W_{m}\otimes_{S_{m}}M^{\text{FI}_{d,o}}(m)_{\bullet}.$

We extend this definition to FB-modules $W$, by setting

$M^{\text{FI}_{d,o}}(W)_{\bullet}=\bigoplus_{m=0}M^{\text{FI}_{d,o}}(W_{m})_{\bullet}.$

Note that there are $(n)\cdots(n-m+1)$ injections
$\phi:[m]\hookrightarrow[n]$. For the remaining $n-m$ elements of the image of
$[n]\backslash\phi\big{(}[m]\big{)}$ and each weak $d$-composition
$(a_{1},\dots,a_{d})$ of $n-m$ there are $\binom{n-m}{a_{1}!\cdots a_{d}!}$
ways of coloring $a_{1}$ of the elements color $1$, coloring $a_{2}$ of the
elements $2$, so on and so forth. There are $a_{1}!$ ways of ordering the
color $1$ elements, $a_{2}!$ ways of ordering the color $2$ elements, etc.
Since there are $\binom{n-m+d-1}{d-1}$ weak $d$-compositions of $n-m$, we get
the following proposition.

######  Proposition 2.7.

For $n\geq m$

$\dim\big{(}M^{\text{FI}_{d,o}}(m)_{n}\big{)}=n!\binom{n-m+d-1}{d-1}$

In fact, more can be said: As $S_{n}$-representations

$M^{\text{FI}_{d,o}}(W_{m})_{n}=\bigoplus_{(a_{1},\dots,a_{d})}\big{(}\text{Ind}^{S_{n}}_{S_{m}\times
S_{a_{1}}\times\cdots\times
S_{a_{d}}}W_{m}\boxtimes\text{Reg}_{a_{1}}\boxtimes\cdots\boxtimes\text{Reg}_{a_{d}}\big{)},$

where $\text{Reg}_{a_{i}}$ is the regular representation of $S_{a_{i}}$.

######  Remark 2.8.

The results in this section could be restated in the language of twisted
algebras. In particular, an FId,o-module can be thought of as module over the
twisted algebra version of the $d$-fold tensor product of the tensor algebra
on a $1$-dimensional vector space, i.e., the twisted algebra generated by the
tensor product of $d$ copies of the twisted tensor algebra generated by the
trivial representation in degree $1$ viewed as an FB-module where the product
is the Day convolution. For more on this interpretation see [SS12].

Having proved that $H_{i}\big{(}F_{n}(\Gamma_{k})\big{)}$ is an FIk,o-module,
we turn to calculating the generation degree of
$H_{i}\big{(}F_{n}(\Gamma_{k})\big{)}$ as an FIk,o-module. In order to do so,
we recall the definition of a cubical complex homotopy defined by Lütgehetmann
equivalent to $F_{n}(\Gamma_{k})$.

##  3 $F_{n}(\Gamma)$ and $K_{n}(\Gamma)$

The focus of this paper is the first homology of the ordered configuration
space of particles in the star graph $\Gamma_{k}$. Instead of directly
studying $F_{n}(\Gamma_{k})$, we study a combinatorial model of graph
configuration spaces due to Lütgehetmann [Lüt14], which generalizes a model
for unordered configuration space due to Świątkowski [Świ01]. This
combinatorial model has several nice properties that we will leverage through
a spectral sequence argument.

Given a graph $\Gamma$, we describe the cube complex $K_{n}(\Gamma)$. The
$0$-cells of $K_{n}(\Gamma)$ correspond to configurations of $n$ particles in
$\Gamma$, such that every particle in the configuration is either on a vertex
of $\Gamma$, or the interior of an edge of $\Gamma$ with the following caveat:
if $e$ is an edge of $\Gamma$ connected to a leaf $v$, then a particle can
only be on $e$ if one is already on $v$, i.e., we ensure the outermost
particle on $e$ is moved to the leaf if there is not already a particle on the
leaf.

The $1$-cells of $K_{n}(\Gamma)$ correspond to the movement of the closest
particle on an edge of $\Gamma$ to a vertex to that vertex, if the vertex is
unoccupied. Such a $1$-cell connects the two $0$-cells where this particle is
at the vertex and where this particle is the particle on that edge closest to
that vertex. Additionally, if $v$ is a leaf and there is no particle on the
edge connecting $v$ to the rest of the graph and no particle at the
neighboring vertex, there is $1$ cell in $K_{n}(\Gamma)$ connecting the
corresponding $0$-cells.

In general, the $m$-cells of $K_{n}(\Gamma)$ correspond to the movement of $m$
particles in $\Gamma$, closest on their edge to the vertex they are moving to,
to unoccupied vertices. Moreover, no two particles go to the same vertex. See
Figure 5 for an example of a $K_{n}(\Gamma)$.

We recall the following theorem of Lütgehetmann, which will allow us to study
$K_{n}(\Gamma)$ in place of $F_{n}(\Gamma)$.

Figure 5: The cubical complex $K_{2}(\Gamma_{3})$, which is an equivariant
deformation retract of $F_{2}(\Gamma_{3})$.

######  Theorem 3.1.

([Lüt14, Theorem 2.3]) Let $\Gamma$ be a locally finite graph, then the
finite-dimensional cube complex $K_{n}(\Gamma)$ is an equivariant deformation
retract of $F_{n}(\Gamma)$. If $\Gamma$ is finite, then $K_{n}(\Gamma)$
consists of finitely many cells.

In particular, if $\Gamma=\Gamma_{k}$ it follows from the definition of the
cube complex that $K_{n}(\Gamma_{k})$ is a $1$-dimensional complex, i.e., a
graph; we parametrize it so that its edges have length $1$. We state for
convenience the following well-known result, that will limit our work to two
cases.

######  Proposition 3.2.

For $i\neq 0,1$, $H_{i}\big{(}F_{n}(\Gamma_{k})\big{)}=0$.

Next, we state a result due to Ghrist [Ghr01] describing the structure of
$\pi_{1}\big{(}F_{n}(\Gamma_{k})\big{)}$. We will use an immediate corollary
of this proposition to find a bound on the generation degree of
$H_{1}\big{(}F_{\bullet}(\Gamma_{k})\big{)}$ as an FIk,o-module.

######  Proposition 3.3.

([Ghr01, Proposition 4.1]) The braid group
$\pi_{1}\big{(}F_{n}(\Gamma_{k})\big{)}$ is isomorphic to a free group on $Q$
generators, where

$Q=1+(nk-2n-k+1)\frac{(n+k-2)!}{(k-1)!}$

This proposition immediately implies that

$\text{rk}\Big{(}H_{1}\big{(}F_{n}(\Gamma_{k})\big{)}\Big{)}=1+(nk-2n-k+1)\frac{(n+k-2)!}{(k-1)!}.$

In the next section we recall the Mayer–Vietoris spectral sequence which we
will apply to $K_{n+1}(\Gamma_{k})$ to get our representation stability
results.

##  4 The Mayer–Vietoris Spectral Sequence

We recall the Mayer–Vietoris spectral sequence that we will use to prove our
representation stability results. After choosing a sufficiently nice cover of
our configuration spaces, the vanishing of the $E^{2}_{1,0}$ entry of this
spectral sequence will correspond to the
$H_{1}\big{(}F_{n+1}(\Gamma_{k})\big{)}$ being generated by the classes of
$H_{1}\big{(}F_{n}(\Gamma_{k})\big{)}$ tensored with the fundamental class of
$H_{0}\big{(}F_{1}(\mathbb{R})\big{)}$ in $k$ different ways, i.e.,
$H_{1}\big{(}F_{n+1}(\Gamma_{k})\big{)}$ is generated by
$H_{1}\big{(}F_{n}(\Gamma_{k})\big{)}$ and the FIk,o-module structure of
$H_{1}\big{(}F_{\bullet}(\Gamma_{k})\big{)}$.

Let $J$ be a countable index set, and let $\mathcal{U}=\\{U_{j}\\}_{j\in J}$
be an open cover of a space $X$. The Mayer–Vietoris spectral sequence has
$E^{1}$-page

$E^{1}_{p,q}=\bigoplus_{\\{i_{0},\dots,i_{p}\\}}H_{q}(U_{i_{0}}\cap\cdots\cap
U_{i_{p}})\implies H_{p+q}(X)$

with boundary map $d_{1}:E^{1}_{p,q}\to E^{1}_{p-1,q}$ given by the
alternating sum of the face maps induced by

$U_{i_{0}}\cap\cdots\cap U_{i_{p}}\hookrightarrow
U_{i_{0}}\cap\cdots\cap\widehat{U_{i_{j}}}\cap\cdots\cap U_{i_{p}}.$

${3}$${\bigoplus_{\\{i_{0}\\}}H_{3}(U_{i_{0}})}$${\bigoplus_{\\{i_{0},i_{1}\\}}H_{3}(U_{i_{0}}\cap
U_{i_{1}})}$${\bigoplus_{\\{i_{0},i_{1},i_{2}\\}}H_{3}(U_{i_{0}}\cap
U_{i_{1}}\cap
U_{i_{2}})}$${\bigoplus_{\\{i_{0},\dots,i_{3}\\}}H_{3}(U_{i_{0}}\cap\cdots\cap
U_{i_{3}})}$${2}$${\bigoplus_{\\{i_{0}\\}}H_{2}(U_{i_{0}})}$${\bigoplus_{\\{i_{0},i_{1}\\}}H_{2}(U_{i_{0}}\cap
U_{i_{1}})}$${\bigoplus_{\\{i_{0},i_{1},i_{2}\\}}H_{2}(U_{i_{0}}\cap
U_{i_{1}}\cap
U_{i_{2}})}$${\bigoplus_{\\{i_{0},\dots,i_{3}\\}}H_{2}(U_{i_{0}}\cap\cdots\cap
U_{i_{3}})}$${1}$${\bigoplus_{\\{i_{0}\\}}H_{1}(U_{i_{0}})}$${\bigoplus_{\\{i_{0},i_{1}\\}}H_{1}(U_{i_{0}}\cap
U_{i_{1}})}$${\bigoplus_{\\{i_{0},i_{1},i_{2}\\}}H_{1}(U_{i_{0}}\cap
U_{i_{1}}\cap
U_{i_{2}})}$${\bigoplus_{\\{i_{0},\dots,i_{3}\\}}H_{1}(U_{i_{0}}\cap\cdots\cap
U_{i_{3}})}$${0}$${\bigoplus_{\\{i_{0}\\}}H_{0}(U_{i_{0}})}$${\bigoplus_{\\{i_{0},i_{1}\\}}H_{0}(U_{i_{0}}\cap
U_{i_{1}})}$${\bigoplus_{\\{i_{0},i_{1},i_{2}\\}}H_{0}(U_{i_{0}}\cap
U_{i_{1}}\cap
U_{i_{2}})}$${\bigoplus_{\\{i_{0},\dots,i_{3}\\}}H_{0}(U_{i_{0}}\cap\cdots\cap
U_{i_{3}})}$ ${0}$${1}$${2}$${3}$ Figure 6: The $E^{1}$-page of the
Meyer–Vietoris spectral sequence.

The $E^{2}_{1,0}$ entry of this spectral sequence will be of great interest to
us. As such, we relate it to a simplicial complex arising from an open cover
of configuration space.

Given a set of indices $J$ and $\mathcal{U}=\\{U_{j}\\}_{j\in J}$ a family of
sets, the _nerve complex_ $N(\mathcal{U})$ of $\mathcal{U}$ is the abstract
simplicial complex whose $p$-simplices are the intersections of $p+1$ distinct
elements of $\mathcal{U}$:

$U_{i_{0}}\cap\cdots\cap U_{i_{p}}.$

By ordering $J$, we can define a boundary map on this simplicial complex given
by

$d(U_{i_{0}}\cap\cdots\cap
U_{i_{p}})=\sum_{j=0}^{p}(-1)^{k-1}U_{i_{0}}\cap\cdots\cap\widehat{U_{i_{j}}}\cap\cdots\cap
U_{i_{p}}.$

The following proposition immediately follows from the definitions, and
relates the Mayer–Vietoris spectral sequence to the homology of the nerve
complex given that the cover is sufficiently nice.

######  Proposition 4.1.

Let $J$ be a countable index set and $\mathcal{U}=\\{U_{j}\\}_{j\in J}$ an
open cover of $X$ such that each intersection $U_{i_{0}}\cap\cdots\cap
U_{i_{p}}$ is either empty or connected. Then, the $E^{2}_{p,0}$-entry of the
Mayer–Vietoris spectral sequence is the $p^{\text{th}}$-homology of
$N(\mathcal{U})$.

For more on the Meyer–Vietoris spectral sequence see, for example, [Bro12,
Chapter 7.4].

In the next section we will find a good cover of $F_{n+1}(\Gamma_{k})$ and
calculate the first homology of its nerve complex. This will allow us to use
Proposition 4.1 to determine when the first homology of $F_{n+1}(\Gamma_{k})$
is governed by the $E^{1}_{0,1}$ term, a condition equivalent to
$H_{1}\big{(}F_{n+1}(\Gamma_{k})\big{)}$ being generated by
$H_{1}\big{(}F_{n}(\Gamma_{k})\big{)}$ and the FIk,o-module structure of
$H_{1}\big{(}F_{\bullet}(\Gamma_{k})\big{)}$.

##  5 Representation Stability for
$H_{i}\big{(}F_{\bullet}(\Gamma_{k})\big{)}$

We prove that the homology of $F_{\bullet}(\Gamma_{k})$ satisfies a notion of
representation stability, namely that
$H_{i}\big{(}F_{\bullet}(\Gamma_{k})\big{)}$ is a finitely generated
FIk,o-module, and calculate its generation degree. When $i\neq 1$, this
problem is easy. Proposition 3.2 shows that if $i>1$, then
$H_{i}\big{(}F_{\bullet}(\Gamma_{k})\big{)}$ is the trivial FIk,o-module.
Since $F_{n}(\Gamma_{k})$ is connected and adding a new point is nontrivial in
homology, it follows that $H_{0}\big{(}F_{\bullet}(\Gamma_{k})\big{)}$ is
finitely generated in degree $1$ as an FIk,o-module. We recall the statement
of our representation stability results for all $i$ and spend the rest of the
section proving the case $i=1$.

######  Theorem 1.1.

For $k\geq 3$, the sequence of homology groups
$H_{i}\big{(}F_{\bullet}(\Gamma_{k})\big{)}$ has the structure of a finitely
generated FIk,o-module, generated in degree $1$ for $i=0$, degree $0$ for
$i>1$, and degree $4$ for $k=3$, degree $3$ for $k=4$, and degree $2$ for
$k\geq 5$.

In order to prove Theorem 1.1 for the case $i=1$ we take advantage of the
Mayer–Vietoris spectral sequence to compute the first homology of
$F_{n+1}(\Gamma_{k})$ by covering $F_{n+1}(\Gamma_{k})$ with $k(n+1)$ copies
of $F_{n}(\Gamma_{k})$, each of which corresponds to adding a point at one of
the $k$ leaves of $\Gamma_{k}$. We prove that the vanishing of the
$E^{2}_{1,0}$ entry of this spectral sequence for $n$ sufficiently large is
equivalent to $H_{1}\big{(}F_{\bullet}(\Gamma_{k})\big{)}$ being a finitely
generated FIk,o-module. Throughout this section we will use
$F_{n}(\Gamma_{k})$ interchangeably with its $1$-dimensional cube complex
$K_{n}(\Gamma_{k})$.

######  Proposition 5.1.

For $i\in[k]$ and $j\in[n+1]$, let $U_{i,j}$ be the subspace of
$F_{n+1}(\Gamma_{k})$ consisting of configurations where particle $j$ is the
outermost particle on edge $i$. Fix $0<\epsilon<\frac{1}{2}$, and extend
$U_{i,j}$ to $U_{i,j}^{\prime}$ by taking the open $\epsilon$-ball around
$U_{i,j}$. Then $U_{i,j}$ is a deformation retract of $U_{i,j}^{\prime}$, and
$U_{i,j}$ is homotopy equivalent to $F_{n}(\Gamma_{k})$.

###### Proof.

That $U_{i,j}$ is a deformation retract of $U_{i,j}^{\prime}$ is immediate
from the fact that $F_{n+1}(\Gamma_{k})$ can be viewed as a graph with edges
of length $1$.

To see that $U_{i,j}$ is homotopy equivalent to $F_{n}(\Gamma_{k})$, note that
forgetting the particle labeled $j$ in $U_{i,j}$, yields a configuration space
on $n$ points in $\Gamma_{k}$. Since particle $j$ is the outermost particle on
its edge we may assume that it is at the leaf. Forgetting it yields a space
homeomorphic to $\Gamma_{k}$, implying that $U_{i,j}$ is homotopy equivalent
to $F_{n}(\Gamma_{k})$. ∎

Note that the $U_{i,j}$ correspond to the image of $F_{n}(\Gamma_{k})$ in
$F_{n+1}(\Gamma_{k})$ under the insertion map $i_{n,i}$ and an action of
$S_{n+1}$. See Figure 7 for an example of a $U_{i,j}$ in $F_{2}(\Gamma_{3})$.

Figure 7: The green part of $F_{2}(\Gamma_{3})$ is the region $U_{1,1}$, where
blue particle can be thought of as having label $1$. Note that this region is
homotopy equivalent to $F_{1}(\Gamma_{3})$.

######  Proposition 5.2.

Let $U_{i,j}^{\prime}$ be the open sets described in Proposition 5.1. Then,
for $n\geq 1$, the collection
$\mathcal{U}_{k,n+1}:=\\{U^{\prime}_{i,j}\\}_{i\in[k],j\in[n+1]}$ is an open
cover of $F_{n+1}(\Gamma_{k})$.

###### Proof.

Since $n\geq 1$, every point in $F_{n+1}(\Gamma_{k})$ corresponds to a
configuration where a particle is on some edge of $\Gamma_{k}$ as only one
particle can be at the central vertex. Given such an edge $i$, there is an
outermost particle $j$ on $i$. This configuration is in $U^{\prime}_{i,j}$, so
$\mathcal{U}$ is an open cover of $F_{n+1}(\Gamma_{k})$. ∎

From now on we will write $E^{r}_{p,q}[\Gamma_{k}](n+1)$ for the
Mayer–Vietoris spectral sequence coming from the open cover
$\mathcal{U}_{k,n+1}$ of $F_{n+1}(\Gamma_{k})$.

Note that $N(\mathcal{U}_{k,n+1})$ is a simplical complex, so we may view its
$1$-skeleton $N(\mathcal{U}_{k,n+1})^{(1)}$ as a graph. Moreover, every non-
trivial intersection $U_{i_{0},j_{0}}\cap\cdots\cap U_{i_{m},j_{m}}$ of unique
$U_{i,j}$ is connected, as this corresponds to the set of configurations where
$m+1$ distinct particles are the outermost particles on $m+1$ distinct edges,
and the remaining $n-m$ particles may be anywhere else in $\Gamma_{k}$ as long
as they are not past any of the particles $j_{l}$, i.e.,
$U_{i_{0},j_{0}}\cap\cdots\cap U_{i_{m},j_{m}}$ is homotopy equivalent to
$F_{n-m}(\Gamma_{k})$, which is contractible. It follows that
$N(\mathcal{U}_{k,n+1})^{(1)}$ is a _simple_ graph, i.e., there is at most one
edge between any two vertices and there are no loops. With that in mind, we
note the following.

######  Proposition 5.3.

For all $k\geq 3$ and all $n\geq 2$, any length $3$ cycle in
$N(\mathcal{U}_{k,n+1})^{(1)}$ is contractible in $N(\mathcal{U})$.

###### Proof.

For $(i,j)\neq(i^{\prime},j^{\prime})$, intersection of $U_{i,j}^{\prime}$ and
$U_{i^{\prime},j^{\prime}}$ is non-trivial if and only if $i\neq i^{\prime}$
and $j\neq j^{\prime}$, as if $i=i^{\prime}$, then the outermost particle on
edge $i$ of $\Gamma_{k}$ would have to be labeled both $j$ and $j^{\prime}$,
and if $j=j^{\prime}$, then the particle labeled $j$ would have to be the
outermost particle on edges $i$ and $i^{\prime}$. Therefore, a length $3$
cycle in $N(\mathcal{U}_{k,n+1})^{(1)}$ must be of the form

$U_{i,j}^{\prime}\cap
U_{i^{\prime},j^{\prime}}^{\prime}-U_{i^{\prime},j^{\prime}}^{\prime}\cap
U_{i^{\prime\prime},j^{\prime\prime}}^{\prime}+U_{i^{\prime\prime},j^{\prime\prime}}^{\prime}\cap
U_{i,j}^{\prime},$

where $i\neq i^{\prime}\neq i^{\prime\prime}$ and $j\neq j^{\prime}\neq
j^{\prime\prime}$. Since $i\neq i^{\prime}\neq i^{\prime\prime}$ and $j\neq
j^{\prime}\neq j^{\prime\prime}$, it follows that $U_{i,j}^{\prime}\cap
U_{i^{\prime},j^{\prime}}^{\prime}\cap
U_{i^{\prime\prime},j^{\prime\prime}}^{\prime}\neq\emptyset$, as the set of
configurations where particle $j$ is the outermost particle on edge $i$,
particle $j^{\prime}$ is the outermost particle on edge $i^{\prime}$, and
particle $j^{\prime\prime}$ is the outermost particle on edge
$i^{\prime\prime}$ is non-empty, see Figure 8. Moreover,

$U_{i,j}^{\prime}\cap
U_{i^{\prime},j^{\prime}}^{\prime}-U_{i^{\prime},j^{\prime}}^{\prime}\cap
U_{i^{\prime\prime},j^{\prime\prime}}^{\prime}+U_{i^{\prime\prime},j^{\prime\prime}}^{\prime}\cap
U_{i,j}^{\prime}=d(U_{i,j}^{\prime}\cap U_{i^{\prime},j^{\prime}}^{\prime}\cap
U_{i^{\prime\prime},j^{\prime\prime}}^{\prime}),$

in $N(\mathcal{U}_{k,n+1})$, so this length $3$ $1$-cycle is contractible in
$N(\mathcal{U}_{k,n+1})$. ∎

Figure 8: The $2$-cell in $N(\mathcal{U}_{k,n+1})$ that makes length $3$
$1$-cycles contractible.

Next, we note that there are only $5$ types of length $3$ paths in
$N(\mathcal{U}_{k,n+1})^{(1)}$ that are not immediately homotopy equivalent to
a length $2$ path in $N(\mathcal{U}_{k,n+1})$. We will show that these paths
are in fact homotopy equivalent to paths of length $2$, allowing us to apply
an inductive argument to show that for large enough $n$ all $1$-cycles are
contractible.

######  Proposition 5.4.

For all $k\geq 3$ there are five types of paths of length $3$ in
$N(\mathcal{U}_{k,n+1})^{(1)}$ such that no three consecutive distinct
vertices correspond to open sets $U_{i,j}$, $U_{i^{\prime},j^{\prime}}$, and
$U_{i^{\prime\prime},j^{\prime\prime}}$ with non-trivial intersection
$U_{i,j}\cap U_{i^{\prime},j^{\prime}}\cap
U_{i^{\prime\prime},j^{\prime\prime}}$.

The five types of paths of length 3 in $N(\mathcal{U}_{k,n+1})^{(1)}$ that are
not immediately homotopy equivalent to a path of length $2$ in
$N(\mathcal{U}_{k,n+1})$ are of the following forms:

  1. 1.

The third vertex $U_{i_{3},j_{3}}$ is such that $j_{3}=j_{1}$ and the fourth
vertex $U_{i_{4},j_{4}}$ is such that $j_{4}=j_{2}$, and $i_{3}\neq i_{1}\neq
i_{2}$ and $i_{4}\neq i_{3}\neq i_{2}$.

  2. 2.

The third vertex $U_{i_{3},j_{3}}$ is such that $j_{3}=j_{1}$ and the fourth
vertex $U_{i_{4},j_{4}}$ is such that $j_{4}\neq j_{1}=j_{3}\neq j_{2}$, and
$i_{3}\neq i_{2}\neq i_{1}$ and $i_{4}=i_{2}$.

  3. 3.

The third vertex $U_{i_{3},j_{3}}$ is such that $j_{3}\neq j_{1}$ and the
fourth vertex $U_{i_{4},j_{4}}$ is such that $j_{4}=j_{2}$, and $i_{3}=i_{1}$
and $i_{4}\neq i_{1}=i_{3}\neq i_{2}$.

  4. 4.

The third vertex $U_{i_{3},j_{3}}$ is such that $j_{3}\neq j_{1}$ and the
fourth vertex $U_{i_{4},j_{4}}$ is such that $j_{4}=j_{1}$, and $i_{3}=i_{1}$
and $i_{4}=i_{2}$.

  5. 5.

The third vertex $U_{i_{3},j_{3}}$ is such that $j_{3}\neq j_{1}$ and the
fourth vertex $U_{i_{4},j_{4}}$ is such that $j_{4}\neq j_{1}\neq j_{2}\neq
j_{3}$, and $i_{3}=i_{1}$ and $i_{4}=i_{2}$.

See Figure 9.

Figure 9: The $5$ possible paths of length $3$ in
$N(\mathcal{U}_{k,n+1})^{(1)}$ that are not immediately homotopy equivalent to
a path of length $2$ in $N(\mathcal{U}_{k,n+1})$. There are several more types
of paths of length $3$ in $N(\mathcal{U}_{k,n+1})^{(1)}$, though in each type
either the first $3$ vertices or the last $3$ vertices are in the same
$2$-simplex in $N(\mathcal{U}_{k,n+1})$, yielding a shorter homotopy
equivalent path immediately.

Next, we determine for what values of $k$ and $n$ can paths of length $3$ in
$N(\mathcal{U}_{k,n+1})$ be shortened to paths of length $2$. This will show
us when $1$-cycles in $N(\mathcal{U}_{k,n+1})$ are contractible.

######  Proposition 5.5.

For $k=3$ and $n\geq 4$, $k=4$ and $n\geq 3$, and $k\geq 5$ and $n\geq 2$,
every path $\gamma$ of length $3$ in $N(\mathcal{U}_{k,n+1})$ is homotopy
equivalent to a path of length $2$ in $N(\mathcal{U}_{k,n+1})$.

###### Proof.

We only need to consider paths $\gamma$ of length $3$ whose vertices are
distinct such that for any three consecutive vertices corresponding to sets
$U_{i,j}$, $U_{i^{\prime},j^{\prime}}$, and
$U_{i^{\prime\prime},j^{\prime\prime}}$ we have that $U_{i,j}\cap
U_{i^{\prime},j^{\prime}}\cap
U_{i^{\prime\prime},j^{\prime\prime}}=\emptyset$, as if $U_{i,j}\cap
U_{i^{\prime},j^{\prime}}\cap
U_{i^{\prime\prime},j^{\prime\prime}}\neq\emptyset$ then Proposition 5.3
proves we can shorten our path.

Thus we only need to consider the $5$ types of paths following Proposition
5.4. We prove that for each such path there is a path $\gamma^{\prime}$ of
length $2$ in $N(\mathcal{U}_{k,n+1})$ such that $\gamma-\gamma^{\prime}$ is
the boundary of a $2$-chain in $N(\mathcal{U}_{k,n+1})$. We do this for $k=3$
and $n\geq 4$ in Figure 10, for $k=4$ and $n\geq 3$ in Figure 11, and for
$k\geq 5$ and $n\geq 2$ in Figure 12.

Figure 10: The collections of $2$-cells in $N(\mathcal{U}_{3,5})$ that allow
us to shorten the $5$ paths of Proposition 5.4 when $k=3$ and $n\geq 4$.
Figure 11: The collections of $2$-cells in $N(\mathcal{U}_{4,n})$ that allow
us to shorten the $5$ paths of Proposition 5.4 when $k=4$ and $n\geq 3$.
Figure 12: The collections of $2$-cells in $N(\mathcal{U}_{k,n})$ that allow
us to shorten the $5$ paths of Proposition 5.4 when $k\geq 5$ and $n\geq 3$.

∎

We use Proposition 5.5 to provide conditions for $N(\mathcal{U}_{k,n+1})$ to
have trivial first homology. This will allow us to use Proposition 4.1 to show
that the $E^{2}_{1,0}[\Gamma_{k}](n+1)$ of the Mayer–Vietoris spectral
sequence is $0$, proving our representation stability results.

######  Lemma 5.6.

For $k=3$ and $n\geq 4$, $k=4$ and $n\geq 3$, and $k\geq 5$ and $n\geq 2$,
$H_{1}\big{(}N(\mathcal{U}_{k,n+1})\big{)}=0$.

###### Proof.

We proceed by induction on the length of a $1$-cycle in
$N(\mathcal{U}_{k,n+1})$. There are no $1$-cycles of length $1$ in
$N(\mathcal{U}_{k,n+1})$, and all $1$-cycles of length $2$ are immediately
trivial. By Proposition 5.3 all $1$-cycles of length $3$ in
$N(\mathcal{U}_{k,n+1})$ are contractible.

Assume all $1$-cycles of length at most $m$ are contractible in
$N(\mathcal{U}_{k,n+1})$. Let $C$ be a $1$-cycle of length $m+1$ in
$N(\mathcal{U}_{k,n+1})$. We may assume that $C$ does not self intersect as if
it does, we can decompose $C$ into $2$ strictly shorter $1$-cycles, which are
contractible by assumption. Let $P$ be a subpath of $C$ of length $3$. By
Proposition 5.5 $P$ is homotopy equivalent to a path $P^{\prime}$ of length
$2$, implying that $C$ is homotopy equivalent to a $1$-cycle $C^{\prime}$ of
length $m$. By the induction hypothesis $C^{\prime}$ is contractible, so $C$
is contractible and $H_{1}\big{(}N(\mathcal{U}_{k,n+1})\big{)}=0$. ∎

Since the $E^{2}_{1,0}[\Gamma_{k}](n+1)$ entry of the Mayer–Vietoris spectral
sequence is the first homology of $N(\mathcal{U})_{k,n+1}$ we get the
following corollary.

######  Corollary 5.7.

For $k=3$ and $n\geq 4$, $k=4$ and $n\geq 3$, and $k\geq 5$ and $n\geq 2$, the
$E^{2}_{1,0}[\Gamma_{k}](n+1)$ entry of the Mayer–Vietoris spectral sequence
is $0$.

Finally, we use this to complete the proof of Theorem 1.1.

###### Proof.

To determine the generation degree of
$H_{1}\big{(}F_{\bullet}(\Gamma_{k})\big{)}$ as an FIk,o-module, it suffices
to determine when all the classes in $H_{1}\big{(}F_{n+1}(\Gamma_{k})\big{)}$
can be written as sums of classes in $H_{1}\big{(}F_{n+1}(\Gamma_{k})\big{)}$
that arise from the $k$ ways of tensoring classes in
$H_{1}\big{(}F_{n}(\Gamma_{k})\big{)}$ with the fundamental class of
$H_{0}\big{(}F_{1}(\mathbb{R})\big{)}$. By the choice of our cover
$\mathcal{U}_{k,n+1}$ of $F_{n+1}(\Gamma_{k})$, this is the same as showing
that $H_{1}\big{(}F_{n+1}(\Gamma_{k})\big{)}$ is generated by
$E^{1}_{0,1}[\Gamma_{k}](n+1)$. Since the Mayer–Vietoris spectral sequence is
trivial for $p,q<0$, this is equivalent to determining when
$E^{2}_{1,0}[\Gamma_{k}](n+1)=0$.

By Corollary 5.7, the $E^{2}_{1,0}[\Gamma_{k}](n+1)=0$ for $n\geq 4$ and
$k=3$, for $n\geq 3$ and $k=4$, and for $n\geq 2$ and $k\geq 5$. It follows
that as an FIk,o-module $H_{1}\big{(}F_{\bullet}(\Gamma_{k})\big{)}$ is
finitely generated in degree at most $4$ for $k=3$, degree at most $3$ for
$k=4$, and degree at most $2$ for $k\geq 5$.

To see that these bounds are strict we show that
$E^{2}_{1,0}[\Gamma_{k}](n+1)\neq 0$ for $k=3$ and $n=3$, $k=4$ and $n=2$, and
$k\geq 5$ and $n=1$ by using the formula for the rank of
$H_{1}\big{(}F_{n}(\Gamma_{k})\big{)}$ following from Proposition 3.3. If
$k\geq 5$ and $n=1$ this is immediate from the fact
$E^{1}_{0,1}[5](2)=H_{1}\big{(}F_{1}(\Gamma_{k})\big{)}^{\oplus 2k}=0$,
whereas $H_{2}\big{(}F_{2}(\Gamma_{k})\big{)}\neq 0$, so
$E^{2}_{1,0}[k](2)\neq 0$. If $k=4$ and $n=2$ one may check that
$E^{1}_{1,0}[4](3)=H_{1}\big{(}F_{2}(\Gamma_{4})\big{)}^{\oplus
12}\cong\mathbb{Z}^{60}$, but
$H_{1}\big{(}F_{3}(\Gamma_{4})\big{)}\cong\mathbb{Z}^{61}$, so
$E^{2}_{1,0}[4](3)\not\cong 0$. Finally, if $k=3$ and $n=3$, note that
$N(\mathcal{U}_{3,4})$ is a $2$-dimensional simplicial complex with $12$
$0$-cells, $36$ $1$-cells, and $24$ $2$-cells. It follows from Betti number
considerations that $H_{1}\big{(}N(\mathcal{U}_{3,4})\big{)}\neq 0$, i.e.,
$E^{2}_{1,0}[3](4)\neq 0$, so $H_{1}\big{(}F_{\bullet}(\Gamma_{3})\big{)}$
cannot be generated as an FI3,o-module in degree less than $4$.111In fact
$H_{1}\big{(}N(\mathcal{U}_{3,4})\big{)}\cong\mathbb{Z}^{2}$, a calculation
performed by the author with John Wiltshire Gordon’s homology calculator
(https://jwiltshiregordon.github.io/homology/index.html). This was done by
labeling the 2-simplices of $N(\mathcal{U}_{3,4})$ by triplets of elements,
one from each of the three sets $\\{1,2,3,4\\}$, $\\{5,6,7,8\\}$, and
$\\{9,10,11,12\\}$, such that no two elements in a triplet are equal mod $4$.
The $3$ sets correspond to the three edges of $\Gamma_{3}$ and the elements of
each set mod 4 correspond the 4 labeled particles in the configuration. ∎

We have shown that $H_{i}\big{(}F_{\bullet}(\Gamma_{k})\big{)}$ has the
structure of a finitely generated FIk,o-module and determined its generation
degree as such, proving that it satisfies a notion of representation
stability. The case of general graphs is significantly more complicated. When
$\Gamma$ has edges not connected to leaves the operation of adding particles
at the leaves is in general insufficient to generate all classes in homology.
Moreover, a direct analogue of the edge stabilization maps of [ADCK20] is not
available due to the impossibility of such a map existing at the level of
configuration space. However, this does not preclude the possibility of such
maps existing at the level of homology, an idea we plan to consider in the
future.

## References

  * [Abr00] Aaron David Abrams, _Configuration spaces and braid groups of graphs_ , Ph.D. thesis, University of California, Berkeley, 2000.
  * [ADCK20] Byung Hee An, Gabriel C. Drummond-Cole, and Ben Knudsen, _Edge stabilization in the homology of graph braid groups_ , Geometry & Topology 24 (2020), no. 1, 421–469.
  * [ADCK22] , _Asymptotic homology of graph braid groups_ , Geometry & Topology 26 (2022), no. 4, 1745–1771.
  * [AK22] Byung Hee An and Ben Knudsen, _On the second homology of planar graph braid groups_ , Journal of Topology 15 (2022), no. 2, 666–691.
  * [AM21] Byung Hee An and Tomasz Maciazek, _Geometric presentations of braid groups for particles on a graph_ , Communications in Mathematical Physics 384 (2021), no. 2, 1109–1140.
  * [Bro12] Kenneth S. Brown, _Cohomology of groups_ , vol. 87, Springer Science & Business Media, 2012.
  * [CEF15] Thomas Church, Jordan S. Ellenberg, and Benson Farb, _FI-modules and stability for representations of symmetric groups_ , Duke Mathematical Journal 164 (2015), no. 9, 1833–1910.
  * [Che16] Safia Chettih, _Dancing in the stars: topology of non-k-equal configuration spaces of graphs_ , Ph.D. thesis, University of Oregon, 2016.
  * [CL18] Safia Chettih and Daniel Lütgehetmann, _The homology of configuration spaces of trees with loops_ , Algebraic & Geometric Topology 18 (2018), no. 4, 2443–2469.
  * [Ghr01] Robert Ghrist, _Knots, braids, and mapping class groups–papers dedicated to Joan S. Birman_ , vol. 24, ch. Configuration spaces and braid groups on graphs in robotics, pp. 29–40, American Mathematical Society, 2001.
  * [Lüt14] Daniel Lütgehetmann, _Configuration spaces of graphs_ , Master’s thesis, Freie Universität Berlin, 2014.
  * [Lüt17a] Daniel Lütgehetmann, _Representation stability for configuration spaces of graphs_ , Ph.D. thesis, Freie Universität Berlin, 2017.
  * [Lüt17b] , _Representation stability for configuration spaces of graphs_ , arXiv preprint arXiv:1701.03490 (2017).
  * [McD75] Dusa McDuff, _Configuration spaces of positive and negative particles_ , Topology 14 (1975), no. 1, 91–107.
  * [MW19] Jeremy Miller and Jennifer Wilson, _Higher-order representation stability and ordered configuration spaces of manifolds_ , Geometry & Topology 23 (2019), no. 5, 2519–2591.
  * [Ram17] Eric Ramos, _Generalized representation stability and FI d-modules_, Proceedings of the American Mathematical Society 145 (2017), no. 11, 4647–4660.
  * [Ram20] , _An application of the theory of FI-algebras to graph configuration spaces_ , Mathematische Zeitschrift 294 (2020), no. 1-2, 1–15.
  * [Seg79] Graeme Segal, _The topology of spaces of rational functions_ , Acta Mathematica 143 (1979), 39–72.
  * [SS12] Steven V. Sam and Andrew Snowden, _Introduction to twisted commutative algebras_ , arXiv preprint arXiv:1209.5122 (2012).
  * [Świ01] Jacek Świątkowski, _Estimates for homological dimension of configuration spaces of graphs_ , Colloquium Mathematicum 89 (2001), 69–79.