# An explicit central element of $\mathcal{U}_{q}(\mathfrak{so}_{5})$ and its
corresponding quantum Hamiltonian

Jeffrey Kuan

###### Abstract

A previous paper of the author developed a general method for producing
explicit central elements of quantized Lie algebras using Lusztig’s inner
product. This method had previously been applied for the type $C_{2},D_{3}$
and $D_{4}$ Lie algebras. The current paper repeats the calculation for the
type $B_{2}$ Lie algebra, which is actually isomorphic to the $C_{2}$ Lie
algebra. The explicit expression for the corresponding quantum Hamiltonian is
computed.

Accessibility Statement: An accessible version of this PDF, which meets the
guidelines of WCAG Level 2.1AA, can be found on the first author’s webpage at
this link:

https://www.math.tamu.edu/~jkuan/CentralElementB2_Accessible.pdf

##  1 Introduction

In mathematical physics, quantum Hamiltonians can be constructed from central
elements of quantization of Lie algebras. In order to compute the entries of
the Hamiltonian, these central elements need to be explicitly written in terms
of the generators. To this end, the author proved a formula in [Kua16] to
produce explicit expressions for central elements, using Lusztig’s inner
product. The method had found explicit central elements for the Lie algebras
with Dynkin diagrams $C_{2}$ and $D_{3},D_{4}$ [Kua16, KLLPZ20] (for the type
$A$ Lie algebras, the central element had already been found in [GZB91] and
used in [Kua17], but their method does not appear to apply here). In the
present paper, we carry about the calculation for $B_{2}$. Note that the
$B_{2}$ and $C_{2}$ Lie algebras are isomorphic, but whereas the $C_{2}$
calculation uses the four–dimensional representation ($\mathfrak{sp}_{4}$),
this paper uses the five–dimensional representation ($\mathfrak{so}_{5}$).
Indeed, by the Harish–Chandra isomorphism, the center of the $B_{2}\cong
C_{2}$ Lie algebra is generated by two elements. These two elements can be
taken to be the central elements of [Kua16] and the one here. See also
[KuaZha18] for an analog of the explicit central elements generating the
center in the case of the $A_{n}$ Lie algebras.

In the calculation done in this paper, we observe that not every element of
the dual basis needs to be calculated, and that the automorphisms of the Lie
algebra streamline some of the computations. This will slightly reduce
computational work in the future.

From the central element, the co–product of the quantized Lie algebra produces
a quantum Hamiltonian. If all the matrix entries of the quantum Hamiltonian
are non–negative, then a ground state conjugation will define a Markov
process, which can be interpreted as an interacting particle system. We will
also provide the matrix entries of the quantum Hamiltonian.

If the entries of the quantum Hamiltonian are non–negative, then a ground
state transformation will produce a Markov process. This ideas goes back to
[SchSan94] and [Sch97]. More recently it was generalized in [CGRS15] (see also
[CGRS16]). If some of the entries are negative, it is still sometimes possible
to remove such states, as in [Kua16] or [KLLPZ20].

##  2 Background

###  2.1 Description of type $B_{2}\cong C_{2}$ Lie algebra

Define the symplectic Lie algebra $\mathfrak{sp}_{4}$ to be the Lie algebra
consisting of $4\times 4$ matrices of the form (where $A,B,C,D$ are $2\times
2$ blocks)

$\left\\{\left(\begin{tabular}[]{cc}$A$&$B$\\\
$C$&$D$\end{tabular}\right):A=-D^{T},B=B^{T},C=C^{T}\right\\}$

Using the notation that $E_{ij}$ is a matrix with a $1$ in the $(i,j)$–entry
and $0$ everywhere else, set

$\displaystyle e_{1}$ $\displaystyle=E_{12}-E_{43},\quad
f_{1}=E_{21}-E_{34},\quad h_{1}=E_{11}-E_{22}-E_{33}+E_{44}$ $\displaystyle
e_{2}$ $\displaystyle=E_{24},\quad f_{2}=E_{42},\quad h_{2}=E_{22}-E_{44}$

Then $\mathfrak{sp}_{4}$ has a basis
$e_{1},e_{2},e_{1}e_{2},e_{2}e_{1},e_{1}e_{2}e_{1},f_{1},f_{2},f_{1}f_{2},f_{2}f_{1},f_{1}f_{2}f_{1},h_{1},h_{2}$.
Each $e_{i},f_{i},h_{i}$ generate a copy of $\mathfrak{sl}_{2},$ and the two
copies are related by

$[h_{1},e_{2}]=-2e_{2},\quad[h_{1},f_{2}]=2f_{2},\quad[2h_{2},e_{1}]=-2e_{1},\quad[2h_{2},f_{1}]=2f_{1},$

where we have written $2h_{2}$ to preserve the symmetry. As usual, these
relations can be summarized by describing the roots of $\mathfrak{sp}_{4}$.
Let $\mathfrak{h}$ denote the linear span of $h_{1},h_{2}$ and identify
$\mathfrak{h}\cong\mathbb{R}^{2}$ via $h_{1}\mapsto x_{1},2h_{2}\mapsto
x_{2}$. Define $\alpha_{1}(x_{1})=2,\alpha_{1}(x_{2})=-2$ and
$\alpha_{2}(x_{1})=-2,\alpha_{2}(x_{2})=4$ so that the above relations are

$[x_{i},e_{j}]=\alpha_{j}(x_{i})e_{j}\quad[x_{i},f_{j}]=-\alpha_{j}(x_{i})f_{j}$

If $\mathfrak{h}^{*}$ is identified with $\mathbb{R}^{2}$ then
$\alpha_{1}=(1,-1),\alpha_{2}=(0,2)$, so that
$\alpha_{j}(x_{i})=(\alpha_{i},\alpha_{j})$ where $(\cdot,\cdot)$ is the usual
Euclidean inner product.

Let $V$ be the natural four–dimensional representation of $\mathfrak{sp}_{4}$.
Using the identification of $\mathfrak{h}^{*}$ with $\mathbb{R}^{2}$, show
that $V$ has a basis $v_{1},v_{2},v_{4},v_{3}$ which are in the weight spaces
$V[(1,0)],V[(0,1)],V[(0,-1)],V[(-1,0)]$ respectively.

###  2.2 A five–dimensional representation

Let $V$ be the four–dimensional fundamental representation of
$\mathfrak{sp}_{4}$. The exterior power $V\wedge V$ is a representation of
dimension $\binom{4}{2}=6$. The action of $\mathfrak{sp}_{4}$ preserves the
subspace $\mathbb{C}v$ where $v=v_{1}\wedge v_{3}-v_{2}\wedge v_{4}$, and
therefore preserves the five–dimensional quotient space $W:=(V\wedge
V)/\mathbb{C}v$.

Define the basis $\\{w_{1},\ldots,w_{5}\\}$ by

$\displaystyle w_{1}$ $\displaystyle=v_{1}\wedge v_{2}$ $\displaystyle w_{2}$
$\displaystyle=v_{1}\wedge v_{4}$ $\displaystyle w_{3}$
$\displaystyle=v_{1}\wedge v_{3}+v_{2}\wedge v_{4}$ $\displaystyle w_{4}$
$\displaystyle=v_{2}\wedge v_{3}$ $\displaystyle w_{5}$
$\displaystyle={\color[rgb]{0,0,0}v_{4}\wedge v_{3}}$

Then (again using the identification of $\mathfrak{h}^{*}$ with
$\mathbb{R}^{2}$)

$\displaystyle W[(1,1)]$ $\displaystyle=\mathbb{C}w_{1}$ $\displaystyle
W[(1,-1)]$ $\displaystyle=\mathbb{C}w_{2}$ $\displaystyle W[(0,0)]$
$\displaystyle=\mathbb{C}w_{3}$ $\displaystyle W[(-1,1)]$
$\displaystyle=\mathbb{C}w_{4}$ $\displaystyle W[(-1,-1)]$
$\displaystyle=\mathbb{C}w_{5}$

The representation of $\mathfrak{sp}_{4}$ onto $W$ actually defines an
isomorphism $\mathfrak{sp}_{4}\cong\mathfrak{so}_{5}$, which can also be seen
in the Dynkin diagram.

###  2.3 Quantum group

The quantization of a finite–dimensional simple Lie algebra $\mathfrak{g}$
depends on its Dynkin diagram. Rather than define
$\mathcal{U}_{q}(\mathfrak{g})$ most generally, here is the definition of
$\mathcal{U}_{q}(\mathfrak{sp}_{4})$. It is generated by
$\\{e_{i},f_{i},k_{i}\\},i=1,2$ with the Weyl relations

$[e_{i},f_{j}]=\delta_{ij}\frac{k_{i}-k_{i}^{-1}}{q_{i}-q_{i}^{-1}},\quad[k_{i},k_{j}]=0$
$k_{i}e_{j}=q^{(\alpha_{i},\alpha_{j})}e_{j}k_{i}\quad
k_{i}f_{j}=q^{-(\alpha_{i},\alpha_{j})}f_{j}k_{i},\quad 1\leq i,j\leq 2$

(where $q_{1}=q,q_{2}=q^{2}$) and the Serre relations

$\displaystyle e_{2}^{2}e_{1}-(q^{2}+q^{-2})e_{2}e_{1}e_{2}+e_{1}e_{2}^{2}$
$\displaystyle=0$ $\displaystyle
e_{1}^{3}e_{2}-(q^{2}+1+q^{-2})e_{1}^{2}e_{2}e_{1}+(q^{2}+1+q^{-2})e_{1}e_{2}e_{1}^{2}-e_{2}e_{1}^{3}$
$\displaystyle=0$ $\displaystyle
f_{2}^{2}f_{1}-(q^{2}+q^{-2})f_{2}f_{1}f_{2}+f_{1}f_{2}^{2}$ $\displaystyle=0$
$\displaystyle
f_{1}^{3}f_{2}-(q^{2}+1+q^{-2})f_{1}^{2}f_{2}f_{1}+(q^{2}+1+q^{-2})f_{1}f_{2}f_{1}^{2}-f_{2}f_{1}^{3}$
$\displaystyle=0$

The co–product is

$\Delta(e_{i})=e_{i}\otimes 1+k_{i}\otimes e_{i}\quad\Delta(f_{i})=1\otimes
f_{i}+f_{i}\otimes k_{i}^{-1},\quad\Delta(k_{i})=k_{i}\otimes k_{i},$

###  2.4 Central Element Construction

Here, we explain how to construct a central element of
$\mathcal{U}_{q}(\mathfrak{g})$.

Letting $\mathfrak{b}_{\pm}\subset\mathfrak{g}$ denote the Borel subalgebras
(that is, $\mathfrak{b}_{+}$ is generated by $e_{i},h_{i}$ and
$\mathfrak{b}_{-}$ is generated by $f_{i},h_{i}$), there is a bi–linear
pairing (see Proposition 6.12 of [J]) on
$\mathcal{U}_{q}(\mathfrak{b}_{-})\times\mathcal{U}_{q}(\mathfrak{b}_{+})$
defined on generators by

$\langle
k_{\alpha},k_{\beta}\rangle=q^{-(\alpha,\beta)_{\mathfrak{g}}},\quad\langle
f_{i},e_{j}\rangle=\frac{-\delta_{ij}}{q_{i}-q_{i}^{-1}},\quad\langle
k_{i},e_{j}\rangle=\langle f_{i},k_{j}\rangle=\langle 1,e_{i},\rangle=\langle
f_{j},1\rangle=0,\quad\langle 1,1\rangle=1$

and extended to all of
$\mathcal{U}_{q}(\mathfrak{b}_{-})\times\mathcal{U}_{q}(\mathfrak{b}_{+})$ by

$\langle y,xx^{\prime}\rangle=\langle\Delta(y),x^{\prime}\otimes
x\rangle,\quad\langle yy^{\prime},x\rangle=\langle y\otimes
y^{\prime},\Delta(x)\rangle,\quad\langle y\otimes y^{\prime},x\otimes
x^{\prime}\rangle=\langle y,x\rangle\langle y^{\prime},x^{\prime}\rangle.$ (1)

where $(\cdot,\cdot)_{\mathfrak{g}}$ is the invariant, non–degenerate
invariant symmetric bilinear form on $\mathfrak{h}^{*}$. Furthermore,
according to Lemma 6.16 of [J],

$\langle\omega(x),\omega(y)\rangle=\langle
y,x\rangle=\langle\tau(y),\tau(x)\rangle$ (2)

where $\omega$ is the automorphism and $\tau$ is the antiautomorphism defined
by

$\displaystyle\omega(e_{i})=f_{i},\quad\omega(f_{i})=e_{i},\quad\omega(k_{i})=k_{i}^{-1}$
$\displaystyle\tau(e_{i})=e_{i},\quad\tau(f_{i})=f_{i},\quad\tau(k_{i})=k_{i}^{-1}$

Choose an ordering $\geq$ on the weight space. Given $\nu\geq 0$, let $r(\nu)$
be the dimension of $U[\nu]$. Let $\\{u_{\nu}^{i}\\}_{1\leq i\leq r(\nu)}$ be
an arbitrary basis of $U[\nu]$, and let $\\{v_{\nu}^{i}\\}_{1\leq i\leq
r(\nu)}$ be the dual basis of $U[-\nu]$ under $\langle\cdot,\cdot\rangle$.

Let $V$ be a fundamentalrepresentation of $\mathfrak{g}$ and let $\rho$ be
half the sum of the positive roots of $\mathfrak{g}$, where a root $\alpha$ is
positive if $\alpha>0$. Explicitly, $\rho=(2,1)$ for $\mathfrak{sp}_{4}$.  Let
$\\{v_{\lambda}\\}$ be a basis of $V$ where each $v_{\lambda}\in V[\lambda]$,
and let $f_{\lambda}$ be a dual basis.

The next lemma is Lemma 2.1 of [Kua16], which constructs a central element of
$\mathcal{U}_{q}(\mathfrak{g})$. Recall that the root lattice of
$\mathfrak{g}$ is the lattice in $\mathfrak{h}^{*}$ spanned by the roots of
$\mathfrak{g}$. When $\mathfrak{g}=\mathfrak{sp}_{4}$, the root lattice can be
explicitly written as $\\{(x_{1},x_{2}):x_{1}+x_{2}\in
2\mathbb{Z}\\}\in\mathbb{Z}^{2}$.

######  Lemma 1.

If $q$ is not a root of unity and $2\mu$ is in the root lattice of
$\mathfrak{g}$ for all weights $\mu$ of $V$, then the element

$\sum_{\mu\geq\lambda}\sum_{i,j=1}^{r(\mu-\lambda)}q^{(\mu-\lambda,\mu)}q^{-(2\rho,\mu)}f_{\lambda}(v^{j}_{\mu-\lambda}u^{i}_{\mu-\lambda}v_{\lambda})v^{j}_{\mu-\lambda}k_{-\lambda-\mu}u^{i}_{\mu-\lambda}$
(3)

is central in $\mathcal{U}_{q}(\mathfrak{g})$, where the sum is taken over all
$\mu,\lambda$ such that $V[\mu]$ and $V[\lambda]$ are nonzero.

##  3 Main Result and Proof

######  Theorem 1.

The element $C$ defined by

$q^{-2}\left(q-q^{-1}\right)^{2}\left((1-q^{2})f_{1}f_{2}f_{1}f_{2}+(q^{4}-q^{-2})f_{2}f_{1}^{2}f_{2}+(1-q^{2})f_{2}f_{1}f_{2}f_{1}+(1-q^{2})f_{1}f_{2}^{2}f_{1}\right)k_{(0,0)}\\\
\times\left((1-q^{2})e_{1}e_{2}e_{1}e_{2}+(q^{4}-q^{-2})e_{2}e_{1}^{2}e_{2}+(1-q^{2})e_{2}e_{1}e_{2}e_{1}+(1-q^{2})e_{1}e_{2}^{2}e_{1}\right)\\\
+{\left(q-q^{-1}\right)^{4}f_{1}f_{1}k_{(0,0)}e_{1}e_{1}}\\\
+\left(q-q^{-1}\right)^{2}\left((1+q^{2})f_{1}f_{2}f_{1}-f_{2}f_{1}f_{1}-q^{2}f_{1}f_{1}f_{2}\right)k_{(0,2)}\left((1+q^{2})e_{1}e_{2}e_{1}-e_{1}e_{1}e_{2}-q^{2}e_{2}e_{1}e_{1}\right)\\\
+\left(q-q^{-1}\right)^{2}\left(q+q^{-1}\right)\left(f_{1}f_{2}q^{2}-f_{2}f_{1}\right)k_{(1,1)}\left(e_{2}e_{1}q^{2}-e_{1}e_{2}\right)\\\
+\left(q^{2}-q^{-2}\right)^{2}q^{4}f_{2}k_{(2,0)}e_{2}\\\
+q^{-4}\left(q-q^{-1}\right)^{2}\left((1+q^{2})f_{1}f_{2}f_{1}-f_{1}f_{1}f_{2}-q^{2}f_{2}f_{1}f_{1}\right)k_{(0,-2)}\left((1+q^{2})e_{1}e_{2}e_{1}-e_{2}e_{1}e_{1}-q^{2}e_{1}e_{1}e_{2}\right)\\\
+q^{-4}\left(q-q^{-1}\right)^{2}\left(q+q^{-1}\right)\left(q^{2}f_{2}f_{1}-f_{1}f_{2}\right)k_{(-1,-1)}\left(q^{2}e_{1}e_{2}-e_{2}e_{1}\right)+q^{-4}\left(q^{2}-q^{-2}\right)^{2}f_{2}k_{(-2,0)}e_{2}\\\
+\left(q-q^{-1}\right)^{2}\left(q+q^{-1}\right)f_{1}k_{(-1,1)}e_{1}+\left(q-q^{-1}\right)^{2}\left(q+q^{-1}\right)f_{1}k_{(1,-1)}e_{1}\\\
+q^{6}k_{(2,2)}+q^{-6}k_{(-2,-2)}+q^{2}k_{(2,-2)}+q^{-2}k_{(-2,2)}+k_{(0,0)}$

is central.

######  Corollary 1.

The action of

$\left(\frac{1}{q^{5}}-\frac{1}{q^{3}}-q^{3}+q^{5}\right)^{-1}\left(C-\left(1+\frac{1}{q^{10}}+\frac{1}{q^{6}}+q^{6}+q^{10}\right)\mathrm{Id}\right)$

on the tensor power of two $4$–dimensional representations is given by the
following $16\times 16$ matrix:

###  3.1 Proof

We wish to apply Lemma 1. The first step is to find the dual elements. Most of
them were found in Lemma 2.2 of [Kua16]. There is an additional dual element
that needs to be found, due to our use of the five–dimensional representation
rather than the four–dimensional representation. The remaining one is stated
in the following lemma.

######  Lemma 2.

1\. The set $\\{e_{2}^{2}e_{1},e_{1}e_{2}^{2}\\}$ is a basis of $U[(1,3)]$,
and the dual of $e_{2}^{2}e_{1}$ is

$(q^{-2}-q^{2})(q^{2}+q^{-2})^{-1}(e_{2}e_{1})^{*}f_{2}.$

2\. The (ordered) set
$\mathcal{B}=\\{e_{1}e_{2}e_{1}e_{2},e_{2}e_{1}^{2}e_{2},e_{1}e_{2}^{2}e_{1},e_{2}e_{1}e_{2}e_{1}\\}$
is a basis of $U[(2,2)]$. Its dual basis is the (ordered) set

$\mathcal{B}^{*}=\\{-(e_{1}e_{1}e_{2})^{*}f_{2}+f_{2}(e_{1}e_{1}e_{2})^{*},q^{-2}(e_{1}e_{1}e_{2})^{*}f_{2}-q^{2}f_{2}(e_{1}e_{1}e_{2})^{*},\\\
(q+q^{-1})^{-1}((e_{2}e_{2}e_{1})^{*}f_{1}-q^{2}f_{1}(e_{2}e_{2}e_{1})^{*}),-f_{2}(e_{2}e_{1}e_{1})^{*}+(e_{2}e_{1}e_{1})^{*}f_{2}\\},$

where

$f_{2}(e_{2}e_{1}e_{1})^{*}=\frac{q-q^{-1}}{q+q^{-1}}\left((q^{2}-q^{4})f_{2}f_{1}f_{2}f_{1}-f_{2}f_{1}^{2}f_{2}+q^{2}f_{1}f_{2}^{2}f_{1}\right)$
$(e_{2}e_{1}e_{1})^{*}f_{2}=\frac{q-q^{-1}}{q+q^{-1}}\left((1-q^{-2})f_{1}f_{2}f_{1}f_{2}+f_{1}f_{2}^{2}f_{1}-q^{2}f_{2}f_{1}^{2}f_{2}\right)$

and applying $\tau$

$f_{2}(e_{1}e_{1}e_{2})^{*}=\frac{q-q^{-1}}{q+q^{-1}}\left((1-q^{-2})f_{2}f_{1}f_{2}f_{1}+f_{1}f_{2}^{2}f_{1}-q^{2}f_{2}f_{1}^{2}f_{2}\right)$
$(e_{1}e_{1}e_{2})^{*}f_{2}=\frac{q-q^{-1}}{q+q^{-1}}\left((q^{2}-q^{4})f_{1}f_{2}f_{1}f_{2}-f_{2}f_{1}^{2}f_{2}+q^{2}f_{1}f_{2}^{2}f_{1}\right)$

###### Proof.

1\. By Serre’s relations, $U[(1,3)]$ is two–dimensional (rather than
three–dimensional). It is immediate that

$\langle(e_{2}e_{1})^{*}f_{2},e_{1}e_{2}^{2}\rangle=\langle
f_{2}(e_{1}e_{2})^{*},e_{2}^{2}e_{1}\rangle=0.$

because $e_{2}e_{1}$ cannot appear in the left tensor power of
$\Delta(e_{1}e_{2}e_{2})$. One the needs only compute that

$\displaystyle\langle(e_{2}e_{1})^{*}f_{2},e_{2}^{2}e_{1}\rangle$
$\displaystyle=\langle(e_{2}e_{1})^{*}\otimes
f_{2},\Delta(e_{2}^{2}e_{1})\rangle$
$\displaystyle=\langle(e_{2}e_{1})^{*}\otimes f_{2},e_{2}k_{2}e_{1}\otimes
e_{2}+k_{2}e_{2}e_{1}\otimes e_{2}\rangle$
$\displaystyle=(1+q^{4})\langle(e_{2}e_{1})^{*}\otimes
f_{2},e_{2}k_{2}e_{1}\otimes e_{2}\rangle$
$\displaystyle=(1+q^{4})q^{-2}\langle(e_{2}e_{1})^{*}\otimes
f_{2},e_{2}e_{1}k_{2}\otimes e_{2}\rangle$
$\displaystyle=(q^{2}+q^{-2})(q^{-2}-q^{2})^{-1}.$

2\. By Serre’s relations, $U[(2,2)]$ is four–dimensional (rather than
six–dimensional).

Finding the first two elements amounts to showing that

$\displaystyle\langle-(e_{1}e_{1}e_{2})^{*}f_{2}+f_{2}(e_{1}e_{1}e_{2})^{*},e_{1}e_{2}e_{1}e_{2}\rangle$
$\displaystyle=1$
$\displaystyle\langle-(e_{1}e_{1}e_{2})^{*}f_{2}+f_{2}(e_{1}e_{1}e_{2})^{*},e_{2}e_{1}^{2}e_{2}\rangle$
$\displaystyle=0$
$\displaystyle\langle-(e_{1}e_{1}e_{2})^{*}f_{2}+f_{2}(e_{1}e_{1}e_{2})^{*},e_{1}e_{2}^{2}e_{1}\rangle$
$\displaystyle=0$
$\displaystyle\langle-(e_{1}e_{1}e_{2})^{*}f_{2}+f_{2}(e_{1}e_{1}e_{2})^{*},e_{2}e_{1}e_{2}e_{1}\rangle$
$\displaystyle=0$ $\displaystyle\langle
q^{-2}(e_{1}e_{1}e_{2})^{*}f_{2}-q^{2}f_{2}(e_{1}e_{1}e_{2})^{*},e_{1}e_{2}e_{1}e_{2}\rangle$
$\displaystyle=0$ $\displaystyle\langle
q^{-2}(e_{1}e_{1}e_{2})^{*}f_{2}-q^{2}f_{2}(e_{1}e_{1}e_{2})^{*},e_{2}e_{1}^{2}e_{2}\rangle$
$\displaystyle=1$ $\displaystyle\langle
q^{-2}(e_{1}e_{1}e_{2})^{*}f_{2}-q^{2}f_{2}(e_{1}e_{1}e_{2})^{*},e_{1}e_{2}^{2}e_{1}\rangle$
$\displaystyle=0$ $\displaystyle\langle
q^{-2}(e_{1}e_{1}e_{2})^{*}f_{2}-q^{2}f_{2}(e_{1}e_{1}e_{2})^{*},e_{2}e_{1}e_{2}e_{1}\rangle$
$\displaystyle=0$

Observe that the third, fourth, seventh, and eighth lines are immediate, since

$\langle(e_{1}e_{1}e_{2})^{*},e_{1}e_{2}e_{1}\rangle=\langle(e_{1}e_{1}e_{2})^{*},e_{2}e_{1}e_{1}\rangle=0.$

For example,

$\langle(e_{1}e_{1}e_{2})^{*}f_{2},e_{2}e_{1}e_{2}e_{1}\rangle=\langle(e_{1}e_{1}e_{2})^{*}\otimes
f_{2},\Delta\left(e_{2}e_{1}e_{2}e_{1}\right)\rangle$

equals zero because the term $e_{1}e_{1}e_{2}$ cannot appear in
$\Delta\left(e_{2}e_{1}e_{2}e_{1}\right)$.

This leaves four pairings to compute, which are

$\displaystyle\langle(e_{1}e_{1}e_{2})^{*}f_{2},e_{1}e_{2}e_{1}e_{2}\rangle$
$\displaystyle=\langle(e_{1}e_{1}e_{2})^{*}\otimes
f_{2},\Delta(e_{1}e_{2}e_{1}e_{2})\rangle$
$\displaystyle=\langle(e_{1}e_{1}e_{2})^{*}\otimes
f_{2},e_{1}k_{2}e_{1}e_{2}\otimes e_{2}\rangle$
$\displaystyle=q^{2}\langle(e_{1}e_{1}e_{2})^{*},(e_{1}e_{1}e_{2})k_{2}\rangle(q^{-2}-q^{2})^{-1}$
$\displaystyle=q^{2}(q^{-2}-q^{2})^{-1}$

and

$\displaystyle\langle f_{2}(e_{1}e_{1}e_{2})^{*},e_{1}e_{2}e_{1}e_{2}\rangle$
$\displaystyle=\langle
f_{2}\otimes(e_{1}e_{1}e_{2})^{*},\Delta(e_{1}e_{2}e_{1}e_{2})\rangle$
$\displaystyle=\langle
f_{2}\otimes(e_{1}e_{1}e_{2})^{*},k_{1}e_{2}k_{1}k_{2}\otimes
e_{1}e_{1}e_{2}\rangle$ $\displaystyle=\langle
f_{2},k_{1}e_{2}k_{1}k_{2}\rangle$ $\displaystyle=q^{-2}\langle
f_{2},e_{2}k_{1}k_{1}k_{2}\rangle$ $\displaystyle=q^{-2}(q^{-2}-q^{2})^{-1}$

and

$\displaystyle\langle(e_{1}e_{1}e_{2})^{*}f_{2},e_{2}e_{1}^{2}e_{2}\rangle$
$\displaystyle=\langle(e_{1}e_{1}e_{2})^{*}\otimes
f_{2},\Delta(e_{2}e_{1}^{2}e_{2})\rangle$
$\displaystyle=\langle(e_{1}e_{1}e_{2})^{*}\otimes
f_{2},k_{2}e_{1}^{2}e_{2}\otimes e_{2}\rangle$
$\displaystyle=\langle(e_{1}e_{1}e_{2})^{*},(e_{1}e_{1}e_{2})k_{2}\rangle(q^{-2}-q^{2})^{-1}$
$\displaystyle=(q^{-2}-q^{2})^{-1}$

and

$\displaystyle\langle f_{2}(e_{1}e_{1}e_{2})^{*},e_{2}e_{1}^{2}e_{2}\rangle$
$\displaystyle=\langle
f_{2}\otimes(e_{1}e_{1}e_{2})^{*},\Delta(e_{2}e_{1}^{2}e_{2})\rangle$
$\displaystyle=\langle
f_{2}\otimes(e_{1}e_{1}e_{2})^{*},e_{2}k_{1}k_{1}k_{2}\otimes
e_{1}e_{1}e_{2}\rangle$ $\displaystyle=\langle
f_{2},e_{2}k_{1}k_{1}k_{2}\rangle$ $\displaystyle=(q^{-2}-q^{2})^{-1}.$

Finding the fourth element is easy by applying $\tau$ to the first four
elements above. One gets

$\displaystyle\langle-
f_{2}(e_{2}e_{1}e_{1})^{*}+(e_{2}e_{1}e_{1})^{*}f_{2},e_{2}e_{1}e_{2}e_{1}\rangle$
$\displaystyle=1$ $\displaystyle\langle-
f_{2}(e_{2}e_{1}e_{1})^{*}+(e_{2}e_{1}e_{1})^{*}f_{2},e_{2}e_{1}^{2}e_{2}\rangle$
$\displaystyle=0$ $\displaystyle\langle-
f_{2}(e_{2}e_{1}e_{1})^{*}+(e_{2}e_{1}e_{1})^{*}f_{2},e_{1}e_{2}^{2}e_{1}\rangle$
$\displaystyle=0$ $\displaystyle\langle-
f_{2}(e_{2}e_{1}e_{1})^{*}+f(e_{2}e_{1}e_{1})^{*}f_{2},e_{1}e_{2}e_{1}e_{2}\rangle$
$\displaystyle=0$

Now, finding the third element amounts to showing that

$\displaystyle\langle(e_{2}e_{2}e_{1})^{*}f_{1}-q^{2}f_{1}(e_{2}e_{2}e_{1})^{*},e_{2}e_{1}e_{2}e_{1}\rangle$
$\displaystyle=0$
$\displaystyle\langle(e_{2}e_{2}e_{1})^{*}f_{1}-q^{2}f_{1}(e_{2}e_{2}e_{1})^{*},e_{1}e_{2}^{2}e_{1}\rangle$
$\displaystyle=q+q^{-1}$
$\displaystyle\langle(e_{2}e_{2}e_{1})^{*}f_{1}-q^{2}f_{1}(e_{2}e_{2}e_{1})^{*},e_{2}e_{1}^{2}e_{2}\rangle$
$\displaystyle=0$
$\displaystyle\langle(e_{2}e_{2}e_{1})^{*}f_{1}-q^{2}f_{1}(e_{2}e_{2}e_{1})^{*},e_{1}e_{2}e_{1}e_{2}\rangle$
$\displaystyle=0$

The third and fourth lines follow from similar reasoning as above. This leaves
two pairings to compute, which are

$\displaystyle\langle(e_{2}e_{2}e_{1})^{*}f_{1},e_{1}e_{2}^{2}e_{1}\rangle$
$\displaystyle=\langle(e_{2}e_{2}e_{1})^{*}\otimes
f_{1},\Delta(e_{1}e_{2}^{2}e_{1})\rangle$
$\displaystyle=\langle(e_{2}e_{2}e_{1})^{*}\otimes
f_{1},k_{1}e_{2}^{2}e_{1}\otimes e_{1}\rangle$
$\displaystyle=q^{-2}\langle(e_{2}e_{2}e_{1})^{*},(e_{2}e_{2}e_{1})k_{1}\rangle(q^{-1}-q)^{-1}$
$\displaystyle=q^{-2}(q^{-1}-q)^{-1}$

and

$\displaystyle\langle f_{1}(e_{2}e_{2}e_{1})^{*},e_{1}e_{2}^{2}e_{1}\rangle$
$\displaystyle=\langle
f_{1}\otimes(e_{2}e_{2}e_{1})^{*},\Delta(e_{1}e_{2}^{2}e_{1})\rangle$
$\displaystyle=\langle
f_{1}\otimes(e_{2}e_{2}e_{1})^{*},e_{1}k_{2}k_{2}k_{1}\otimes
e_{2}e_{2}e_{1}\rangle$ $\displaystyle=\langle
f_{1},e_{1}k_{2}k_{2}k_{1}\rangle$ $\displaystyle=(q^{-1}-q)^{-1}.$

This finishes the proof of the lemma. ∎

The next step is to write the representation matrices.

######  Lemma 3.

$f_{2}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\ 1&0&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\\\ 0&0&0&1&0\\\ \end{array}\right),\quad
f_{1}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&1&0&0&0\\\
0&0&q+q^{-1}&0&0\\\ 0&0&0&0&0\\\ \end{array}\right),\quad
e_{2}=\left(\begin{array}[]{ccccc}0&1&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&1\\\ 0&0&0&0&0\\\ \end{array}\right),\\\
e_{1}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\ 0&0&q+q^{-1}&0&0\\\
0&0&0&1&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ \end{array}\right),\quad
k_{2}=\left(\begin{array}[]{ccccc}q^{2}&0&0&0&0\\\ 0&q^{-2}&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&q^{2}&0\\\ 0&0&0&0&q^{-2}\\\ \end{array}\right),\quad
k_{1}=\left(\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&q^{2}&0&0&0\\\ 0&0&1&0&0\\\
0&0&0&q^{-2}&0\\\ 0&0&0&0&1\\\ \end{array}\right)$

###### Proof.

By using the five–dimensional representation and extending to the quantum
group, the result follows. One can also check the relations directly. ∎

By applying the previous two lemmas and applying Lemma 1, this computes every
term in the central element except for the first one. Because the terms
$v_{3},v_{4}$ were not explicitly found, the term is

$q^{-2}\left(q-q^{-1}\right)^{2}\left((1-q^{2})f_{1}f_{2}f_{1}f_{2}+(q^{4}-q^{-2})f_{2}f_{1}^{2}f_{2}+(1-q^{2})f_{2}f_{1}f_{2}f_{1}+(1-q^{2})f_{1}f_{2}^{2}f_{1}\right)k_{(0,0)}\\\
\times\left((1-q^{2})e_{1}e_{2}e_{1}e_{2}+(q^{4}-q^{-2})e_{2}e_{1}^{2}e_{2}+Ae_{2}e_{1}e_{2}e_{1}+Be_{1}e_{2}^{2}e_{1}\right)$

for some $A,B$. Since the bases were arbitrary, by using (2) with the
automorphism $\omega$, it can be seen that $A=B=(1-q^{2})$. Another way to
find $A$ and $B$ is to use the following explicit representation.

One can check that the action on $\mathbb{C}^{4}\otimes\mathbb{C}^{4}$ is
given by the $16\times 16$ matrices

$\displaystyle e_{1}$
$\displaystyle=E_{12}+E_{34}+E_{67}+E_{10,12}-E_{59}-E_{8,11}-E_{13,14}-E_{15,16}$
$\displaystyle+qE_{13}+q^{-1}E_{24}+qE_{58}-qE_{6,10}-q^{-1}E_{7,12}+q^{-1}E_{9,11}-qE_{13,15}-q^{-1}E_{14,16}$
$\displaystyle e_{2}$
$\displaystyle=E_{25}+E_{48}+E_{7,13}+E_{12,15}+E_{3,6}+q^{2}E_{47}+q^{-2}E_{8,13}+E_{11,14}$
$\displaystyle f_{1}$
$\displaystyle=E_{31}+E_{42}+E_{85}-E_{10,6}-E_{12,7}+E_{11,9}-E_{15,13}-E_{16,14}$
$\displaystyle+q^{-1}E_{21}+qE_{43}+q^{-1}E_{76}+qE_{12,10}-q^{-1}E_{95}-qE_{11,8}-q^{-1}E_{14,13}-qE_{16,15}$
$\displaystyle f_{2}$
$\displaystyle=E_{63}+E_{74}+E_{13,8}+E_{14,11}+E_{52}+q^{-2}E_{84}+q^{2}E_{13,7}+E_{15,12}$
$\displaystyle k_{(a,b)}$
$\displaystyle=q^{2a}E_{11}+q^{a+b}(E_{22}+E_{33})+q^{2b}E_{44}+q^{a-b}(E_{55}+E_{66})+E_{77}+E_{88}+E_{99}+E_{10,10}$
$\displaystyle+q^{b-a}(E_{11,11}+E_{12,12})+q^{-2b}E_{13,13}+q^{-b-a}(E_{14,14}+E_{15,15})+q^{-2a}E_{16,16}$

and that the action of the element only commutes with the other generators
when $A=B=1-q^{2}$. This action only provides a way to double–check that the
answer is correct.

Note that one could also check that the action on the four and
five–dimensional fundamental representations is a multiple of the identity,
but this would not determine the values of $A$ and $B$.

Finally, the corollary follows by explicitly calculating the action of $C$ on
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}$, using the above representation.

## References

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# Understanding Social Immunity in Ants: A Markovian Approach to Collective
Cleaning Strategies

Isabella Bueno1∗ Gabriel R. Palma2,3 Idemauro A. R. Lara4 Rafael A. Moral2,3
Italo Delalibera Jr1 Wesley A. C. Godoy1

1\. Department of Entomology and Acarology, University of São Paulo,
Piracicaba, São Paulo, Brazil;

2\. Hamilton Institute, Maynooth University, Maynooth, Ireland;

3\. Department of Mathematics and Statistics, Maynooth University, Maynooth,
Ireland;

4\. Department of Mathematics and Statistics, University of São Paulo,
Piracicaba, São Paulo, Brazil;

$\ast$ Corresponding author; e-mail<EMAIL_ADDRESS>

Keywords: Multi-state models, Cleaning behavior, Self-grooming, Allogrooming,
Entomopathogenic fungi, Leaf-cutting ants, _Atta sexdens_

Manuscript type: Research paper.

Prepared using the suggested LaTeX template for Am. Nat.

## Abstract

Understanding social immunity mechanisms in ant colonies remains crucial to
comprehending the evolution of defense strategies in eusocial organisms. This
study assumes the absence of the role of memory in the ants’ defense strategy,
considering that they can make a new exploration of collective cleaning
behaviors. We investigate how worker interactions and previous behaviors
influence the evolution of social immunity strategies in response to the
presence of pathogens and vulnerable colony members. In this context, with the
application of Markov transition models it was possible to describe changes in
cleaning behavior over time influenced by the presence of vulnerable members
and treatment with an entomopathogenic fungus. We found a significant effect
on ant behavior when exposed to Metarhizium anisopliae when a fungus garden
fragment and one larva were present. Our findings confirm a link between
prophylactic cleaning and the presence of vulnerable members. Remarkably,
distinct behaviors and transition times vary among treatments, revealing
workers’ adaptability to threats. Allogrooming shows adaptive changes when
exposed to pathogens, potentially affecting pathogen transmission. In
addition, our study elucidates the intricate interaction between internal and
external factors shaping worker behavior. The influence of environmental
context on decision-making principles emerges, emphasizing the importance of
both intrinsic colony organization and external threats. By using a Markov
model to understand ant hygiene behavior, we offer insights into social
immunity mechanisms in eusocial organisms. Deciphering collective cleaning
strategies can aid in understanding disease dynamics and decision-making
processes in complex societies.

## 1 Introduction

Leafcutter ants are part of the group of fungus-farming ants Wetterer et al.,
(1998); Ward, (2014) that exclusively utilize leaves cut by workers to
cultivate their symbiotic fungus Weber, (1966). Cultivating fungi for
sustenance represents an ecological relationship that originated among a
particular group of ants in South America approximately $53.6$ to $66.7$
million years ago Ješovnik et al., (2016). These ants, recognized as
ecologically dominant herbivores, form expansive ”superorganisms” comprising
millions of individuals Branstetter et al., (2017).

Atta Fabricius is one of the three extant genera of leafcutter ants situated
within the subtribe Attina (Hymenoptera: Formicidae: Myrmicinae) Sosa-Calvo et
al., (2017). Due to their communal lifestyle, this group forms densely
populated colonies of closely related individuals, rendering them susceptible
to parasites Schmid-Hempel, (1998); Boomsma et al., (2005). Various parasites,
notably entomopathogenic fungi, exhibit close associations with social insects
Schmid-Hempel, (1998). In the context of leafcutter ants, defending against
pathogens is a critical factor for sustaining their symbiotic fungus.

The mutualistic relationship of attine ants with the fungus Leucoagaricus
gongylophorus Weber, (1966); Mehdiabadi and Schultz, (2010) increases the
complexity of interactions between this group of ants and microorganisms. In
addition to safeguarding their species from microbial threats, worker ants
must also defend the fungus garden. This mutualism is mandatory for both the
ants and the fungus. The ants rely on the fungus for shelter, serving as the
exclusive food source for their larvae and providing partial nourishment for
the workers Quinlan and Cherrett, (1979); Bass and Cherrett, (1995).
Simultaneously, the fungus depends on the ants for care and nutrient supply
Hölldobler and Wilson, (1990); Holldobler and Wilson, (2011).

This pronounced interdependence between the fungal garden and the ants,
coupled with the clonal nature of the gardens, causes the fungus to rely
heavily on the workers for defense against microorganisms. Among the defense
strategies ants employ is promoting genetic diversity through the horizontal
transfer of fungal cultivars between attine species Mueller et al., (2018).
Additionally, they manage the garden by isolating it in chambers to prevent
contamination, monitoring, weeding, removing pathogens, and applying
antibiotics to suppress invasive microorganisms, among other techniques
Mueller et al., (2005).

To protect themselves and vital colony members like the queen, offspring, and
the fungus garden, worker ants possess an array of physiological and
behavioral components to combat pathogens. A primary physiological defense
involves the production of antimicrobial compounds by the metapleural gland
Fernández-Marín et al., (2006, 2015). Workers utilize cleaning behaviors to
selectively disperse the antimicrobial compounds produced by the metapleural
glands throughout the colony, especially when facing pathogenic challenges
Fernández-Marín et al., (2006). Through specific leg movements, workers clean
the openings of the metapleural glands, collect the secretions, and transfer
them to their mouthparts, allowing the antibiotic substances to be distributed
in the colony’s vicinity Fernández, (2003); Fernández-Marín et al., (2006).

Cleaning behaviors, as observed in worker ants, constitute a pivotal strategy
for managing pathogenic microorganisms, assisting in addressing infections
impacting the workers and the fungal garden Currie and Stuart, (2001); Jaccoud
et al., (1999). These behaviors intensify particularly when confronted with a
fungal infection Currie and Stuart, (2001); Jaccoud et al., (1999); Nilsson-
Møller et al., (2018); Morelos-Juárez et al., (2010). Furthermore, in response
to the presence of pathogens, not only do ants engage in reactive cleaning
behaviors but also exhibit proactive actions aimed at protecting vulnerable
colony members (e.g., offspring) and the fungal garden, serving to prevent
contaminations Morelos-Juárez et al., (2010).

Additionally, these social insects deploy collective defenses alongside
individual physiological mechanisms to combat pathogens. Notably, mutual
cleaning behaviors such as allogrooming exemplify the effectiveness of
communal pathogen removal and the promotion of collective immunity Cremer et
al., (2007, 2018). Termed ”social immunity,” this suite of defenses
significantly reduces group mortality Cremer et al., (2007).

The extensively studied phenomenon of social immunity Cremer and Sixt, (2009);
Ugelvig et al., (2010); Cremer et al., (2018); Meunier, (2015); Pull et al.,
(2018); Stroeymeyt et al., (2018) coupled with individual defense strategies
Schluns and Crozier, (2009); Konrad et al., (2012); Tragust et al., (2013);
Nilsson-Møller et al., (2018) reveals the comprehensive exploration in this
field. Experiments involving the recording of time-related response
categories, such as insect behavior, is a common approach to verifying
treatment effects and forecasting behavioral transitions. Analyzing
longitudinal categorical data necessitates employing models that accommodate
the correlation structure within dependent data. In this context, multi-state
models are a particular class of models that allows for describing the changes
of response categories over time, conditional to the effects of time and
treatments. These models are based on stochastic processes and help predict
behavioral transitions Meira-Machado et al., (2009); Lara et al., (2020).

This study aimed to apply the multi-state Markov model to explain the
decision-making process of cleaning behaviors in leaf-cutter ant workers under
different experimental conditions. These models are based on continuous
stochastic process Ross, (1995) and they are instrumental in analyzing
behavior over time de Lara et al., (2017). We carried out an experiment to
measure the cleaning behavioral response of ants when exposed to the
entomopathogenic fungus Metarhizium anisopliae, both in the presence and
absence of vulnerable colony elements, such as brood and the fungus garden.

## 2 Methods

### 2.1 Species studied

The Atta sexdens ants used in the study were obtained from colonies collected
by excavating the soil in a teak plantation (Tectona grandis) in Anhembi,
State of São Paulo, Brazil (22°40’S 48°10’W). After collection, colonies were
kept in the laboratory under controlled temperature and light conditions. Five
colonies of approximately two years of age were used to supply the workers.

### 2.2 Fungal suspension

The Metarhizium anisopliae fungus, isolate ESALQ E9, is deposited in the
Entomopathogen Collection ”Prof. Sérgio Batista Alves” from the Luiz de
Queiroz College of Agriculture – University of São Paulo (ESALQ-USP), Brazil.
Seven days before the assay, the Petri dishes were replicated from samples
kept in a $-80$ºC freezer in cryotubes. The fungus was cultivated in PDA
medium (potato dextrose agar, KASVI®). The Petri dishes (8 x 1,5cm) were kept
in BOD chambers with a controlled temperature ($25$ºC) and a photoperiod of
$12$ hours for seven days. Suspensions were prepared using conidia scraped
from the Petri dishes and added to 10ml of Tween $80$º surfactant. The
concentration used was $10^{8}$ conidia/mL.

### 2.3 Behavioral experiment

To study the grooming behaviors of leaf-cutting ants under different
conditions, an experiment was set up using a randomized complete block design,
with five blocks. Each block corresponded to the particular colony from which
workers were removed. Each block was repeated twice, totalling 10 replicates
of four treatments: T1 = five workers exposed to Metarhizium anisopliae; T2 =
five workers exposed to Tween $0.05\%$; T3 = five workers exposed to
Metarhizium anisopliae with the presence of a fungus garden fragment and one
larva; and T4 = five workers exposed to Tween $0.05\%$ with the presence of a
fungus garden fragment and one larva. Each experimental unit consisted of a
$9$ cm diameter round Petri dish lined with a plaster.

The worker ants used in the experiment were medium-sized, and painted in
different colors with an insect marking pen, to allow for distinguishing each
one. The painted workers were added to Petri dishes $24$ hours before the
experiment and kept for acclimation. The experiment was carried out under
controlled conditions of temperature ($25$ºC) and humidity ($70\%$). The
workers were anaesthetized with CO2 before the application of the treatments.
Exposure to treatments was performed by placing workers on a $10$-minute walk
on filter paper with 1 ml of fungal suspension or Tween.

Each experimental unit was filmed for 30 minutes after the ants walked in the
Petri dish. The captured images were taken with a Sony Handycam camcorder and
indirect artificial lighting. The camera was placed perpendicularly above the
Petri dish at an angle of $90$º. The grooming behaviors (allo and self-
grooming) were recorded and quantified using the Behavioral Observation
Research Interactive Software (Boris) program, version 7.13.8 Friard and
Gamba, (2016). The observed behaviors were based on the descriptions presented
by Nilsson-Møller et al., (2018): allogrooming was considered when one or more
ants were close to each other, with physical contact. The ‘cleaner’ ant moved
slightly to cover the receiving ant’s body area; self-grooming was recorded
when the worker cleaned its antenna, the antenna cleaners, and later the ant
cleaned the legs, taking the legs to the mouthparts and removing particles and
pathogens with the glossa.

Other behaviors were rare or sporadic; therefore, we classified them as
”other” when a particular worker was not engaged in self-grooming or
allogrooming, thus consisting a categorical response variable with three
categories. Most of the “other” behaviors involved walking and staying at the
edges of the Petri dish. After the end of the recording sessions, the workers
were kept in their respective Petri dishes inside humid chambers ($25$ºC and
$90\%$), and the mortality was counted every two days. The dead workers were
removed, washed in $70$º alcohol, and later in distilled water, and kept in a
humid chamber for 12 days for spore growth and confirmation of death by the
applied entomopathogenic fungus.

### 2.4 Statistical analysis

We considered multi-state models Meira-Machado et al., (2009) to jointly model
the behavior and time associated with each ant action. Our objective with this
modeling approach was to describe changes in response categories (behavior
transitions), time spent in each state, and the effects of treatments on the
process. These models are based on continuous-time stochastic processes Ross,
(1995). The process is represented by the set $\\{Y(t),t\in\tau\\}$, where
$Y(t)\in S=\\{1,2,3\\}$, that represents the categories of ant behavior (1:
allogrooming, 2: self-grooming and antenna cleaning, or 3: other) and
$\tau=[0,30)$ the observation time (Figure 1). To improve model convergence
and favor parsimony, we assumed that the behavior transition probabilities are
given by the Markovian property:

$\displaystyle\pi_{ab}(t)=P(Y_{(t+s)}=b\mid Y_{(s)}=a),\hskip
5.69046pt\forall\hskip 5.69046pts<t\in\tau\hskip 5.69046pt\mbox{and}\hskip
5.69046pta,b\in S.$ (1)

The transition intensities are such that

$\theta_{ab}=\left\\{\begin{array}[]{ll}-\theta_{a}&\mbox{if}\hskip
8.5359ptk=k^{\prime}\\\ \theta_{a}\pi_{ab}(t)&\mbox{if}\hskip 8.5359ptk\neq
k^{\prime},\\\ \end{array}\right.$

where $k$ is the dimension of the square intensity matrix $\mathbf{Q}$. For
our experiment, $k=3$, and therefore we have

$\displaystyle\mathbf{Q}=\left(\begin{array}[]{ccc}-(\theta_{1}+\theta_{2})&\theta_{1}&\theta_{2}\\\
\theta_{3}&-(\theta_{3}+\theta_{4})&\theta_{4}\\\
\theta_{5}&\theta_{6}&-(\theta_{5}+\theta_{6})\\\ \end{array}\right).$ (5)

The transition intensities are the parameters associated with time, such that
$\displaystyle{-\frac{1}{\theta_{a}}}$ is the mean time spent in behavior $a$.
To consider the effect of experimental design (blocks and treatments), the
multi-state model considered was the Cox (proportional hazards model), given
by:

$\displaystyle\theta_{ab}(\mbox{\boldmath{$x$}})=\theta_{ab}^{(0)}\exp\\{\mbox{\boldmath{$\beta$}}_{a}^{\top}\mbox{\boldmath{$x$}}\\}$
(6)

where $\theta_{ab}^{(0)}$ is the initial estimate for the transition
intensities matrix, $x$ is the vector that describes the experimental design
and $\mbox{\boldmath{$\beta$}}_{a}$ is the associated vector of parameters,
that depends on $a\in S$.

The Cox model assumption was tested by the hazard proportionality test
proposed by Grambsch and Therneau, (1994). Also, the parameters are estimated
through iterative processes, where the intensity matrix is updated at each
step. Once convergence is obtained, the probabilities given by (1) are
estimated from (6), yielding the estimated transition probability matrix

$\mbox{\boldmath{$\hat{P}$}}(s,t)=\exp[(t-s)\mbox{\boldmath{$\hat{Q}$}}].$

These models were fit and assumptions were tested using the msm Jackson,
(2011) and survival packages Therneau, (2023), available for R software R Core
Team, (2022). To allow for full reproducibility of our results, the data and R
scripts are available at
https://github.com/GabrielRPalma/UnderstandingComplexBehavior.

Figure 1: Illustrative diagram of the transition probability matrix
representing the probability $\pi_{ab}$ of changing from behavior $a$ to
behavior $b$.

## 3 Results

The most common behaviors observed in were self-grooming, allogrooming, and
walking. Other behaviors such as offspring care, fungus grooming and weeding
were rare. As other behaviors were rare, we delimited the category ’others’
for any behavior not defined as self-grooming or allogrooming. A total of
3897.2 minutes of behavior were recorded, with $5.76\%$ being allogrooming,
$14.61\%$ self-grooming, and $79.63\%$ classified as ’others’. With respect to
changes in the behavior of ants, we observed 2,858 first-order transitions:
214 from allogrooming, 1,216 from self-grooming, and 1,428 from ‘other’
behaviors. The distribution of time spent in each behavior was skewed,
especially for ‘other’ behaviors.

We fail to reject the null hypothesis that hazard rates are not proportional
(LR = $3.43$, df = $3$, $p=0.330$). Therefore, the assumption of the Cox model
is adequate. The time point estimates and $95\%$ confidence interval
constructed using the fitted multi-state model that insects have spent in each
behavior given the treatment effect are shown in Figure 2.

Figure 2: Time point estimates with their respective $95\%$ confidence
interval based on the multi-state model for each behavior and treatments (T1 =
5 workers exposed to Metarhizium anisopliae; T2 = 5 workers exposed to Tween
$5\%$; T3 = 5 workers exposed to Metarhizium anisopliae with the presence of a
fungus garden fragment and one larva per plate; and T4 = 5 workers exposed to
Tween $5\%$ with the presence of a fungus garden fragment and one larva per
plate). Figure 3: Estimated transition probability matrices for each treatment
of the experiment according to the fitted multi-state model for each of three
types of ant behavior: 1 – Allogrooming, 2 – self-grooming and antenna
cleaning or 3 – other behaviors. T1 = Metarhizium anisopliae; T2 = Tween
$0.05\%$; T3 = Metarhizium anisopliae with the presence of a fungus garden
fragment and one larva per plate; and T4 = Tween $0.05\%$ with the presence of
a fungus garden fragment and one larva per plate. The color gradient varies
with the increasing probability of transitioning from one behavior to another.
Warmer colors correspond to a higher transition probability, while lighter
colors correspond to lower transition probabilities.

We found a significant effect of treatments (LR = $97.52$, df = $18$,
$p<0.01$) according to the fitted multi-state model. In general, the estimated
transition matrix shows how the choice of behavior is influenced by the
behavior performed by the same individual one time step before and how the
treatment influences this transition probability. In T1, composed only of
workers exposed to the entomopathogenic fungus (Fig. 3), the highest
transition probability found was that workers engaged in self-grooming would
remain in the same behavior, followed by staying in the behavior of
allogrooming. The lowest transition probability in this treatment was that
workers engaged in self-grooming would shift to allogrooming behavior. In
Treatment 2, workers were exposed only to the control treatment (Fig. 3), the
highest transition probability found was that workers engaged in self-grooming
would remain in the same behavior. The lowest transition probability in this
treatment was that workers engaged in self-grooming would stay on the same
behavior.

Treatments T3 and T4, with workers exposed to the entomopathogenic fungus and
the control treatment, respectively, in addition to the presence of the fungus
garden and brood, showed similar results regarding the transition probability
between behaviors. The highest probabilities of behavioral transition from
one-time step before to the next were that workers engaged in self-grooming
would remain in that behavior. Next, the highest transition probability found
was that workers engaged in the behavior categorized as ”other” would
transition to self-grooming behavior. The lowest transition probability in
this treatment was that workers engaged in self-grooming would remain in the
same behavior.

## 4 Discussion

Although memory is expected to play a significant role in the social immunity
mechanisms of ant colonies Goes et al., (2020), this study considers ants as
”Markovian organisms”, meaning without memory Lumsden, (1982). Consequently,
it is possible to model the cleaning behavior of ants to predict which
collective cleaning strategies may have evolved to achieve optimal social
immunity, considering solely the influence of worker interactions among
themselves and with the environment, as well as the occurrence of behaviors at
a previous time step.

Our results confirm the assumption that prophylactic cleaning occurs in the
presence of the most vulnerable members of the colony and increases as some
pathogen is encountered. Furthermore, our model demonstrated the differences
in the transition time of each behavior in different treatments. For both
allogrooming and self-grooming behaviors, the longest transition time was
found in treatment T2, indicating that when workers are not in the presence of
the fungus garden and brood or entomopathogenic fungus, they are more inclined
to remain in the same behavior. Concerning self-grooming, the shortest
transition time was found in treatment T3, where the fungus garden/brood and
entomopathogenic fungus were present, meaning each worker spent less time
engaged in a specific task before switching to another. Regarding
allogrooming, the two shortest transition times were in treatments T3 and T4,
where the fungus garden and brood were present, with and without the pathogen.
We can infer that when the threat or vulnerable component is present, the
transition time to another activity decreases, indicating that each worker
needs to alternate behavior, possibly to increase cleaning efficiency.

Despite a few occurrences of allogrooming, this is an important behavior that
allows workers to clean each other in hard-to-reach places. Despite its
importance, allogrooming may have a disease-spreading effect. Workers exposed
to the pathogen tend to increase self-grooming while decreasing allogrooming
adaptively to contain the pathogen spread Theis et al., (2015). We found that
in treatment T2, where there was no fungus garden or pathogen, workers spent
more time continuously performing allogrooming before moving to another
behavior. We can infer that in the absence of vulnerability or threat, workers
tend to remain in the same cleaning behavior for allogrooming and self-
grooming. As it involves two or more workers, the allogrooming rate and
duration can alter the colony’s social contact networks. Apart from the
intrinsic organization of social insects to avoid harmful microorganisms Hart
and Ratnieks, (2001), behavioral modulation in the face of a pathogen affects
interaction dynamics, impacting the risk of pathogen exposure for other
workers and disease transmission Theis et al., (2015). Colonies exposed to
pathogens have their social contact network altered to minimize disease
transmission, including protection for valuable colony components such as the
queen Baracchi and Cini, (2014); Stroeymeyt et al., (2018).

The transition probability matrix showed significant differences across the
four treatments. This matrix demonstrates the likelihood of the current
behavior being influenced by the behavior one time step before. Additionally,
these probabilities are influenced by the treatment. In the two treatments
where vulnerable members were present (T3 and T4), the lowest probabilities
were found for a worker engaged in allogrooming to remain in that state, as
well as the lowest probabilities for workers engaged in the other two
behaviors to transition to allogrooming. This lower transition probability to
allogrooming in the presence of vulnerable members may be directly related to
the potential role of allogrooming in disease dissemination Theis et al.,
(2015), as mentioned earlier.

Despite the matrix presenting significant differences across all treatments, a
pattern can be observed. In all four treatments, the highest probability found
was for individuals engaged in self-grooming to remain in self-grooming, while
the lowest probability found was for individuals engaged in self-grooming to
transition to allogrooming. Moreover, the transition probabilities were
generally considerably higher for persisting in a state than for switching
states, even across different treatments. This might indicate fixed
mechanistic principles governing individual worker decision-making, albeit
with varying intensities depending on the environmental context.

Although cleaning behaviors are partially individual, the decision-making
process that leads to social immunity involves several factors. These factors
can be both external, such as the presence of microorganisms Jaccoud et al.,
(1999); Currie and Stuart, (2001), and internal, such as the number of present
workers Hughes et al., (2002), presence of the queen Keiser et al., (2018),
presence of other vulnerable colony members Morelos-Juárez et al., (2010),
among others. The presence of valuable colony members increases the disease
cost; therefore, an optimization of defense strategy is expected. In this
regard, an ant colony follows a system of stigmergic decision-making Friedman
et al., (2021). Essentially, stigmergy describes a mechanism of indirect
communication among individuals, where the actions of one organism alter the
environment and influence the future behavior of other individuals in the same
shared environment. Thus, stigmergy involves decentralized coordination,
allowing for the coordination of complex activities among eusocial organisms
Heylighen, (2016).

Insect societies can serve as models for studying disease spread in large and
complex societies Naug and Camazine, (2002); Theis et al., (2015) due to their
allowance for experimental manipulations Baracchi and Cini, (2014); Richardson
and Gorochowski, (2015); Ulrich and Schmid-Hempel, (2015); Stroeymeyt et al.,
(2018). However, there are significant limitations in groups with millions of
individuals in underground colonies, as with leaf-cutter ants. Additionally,
this group has an additional factor, the fungus garden, which is a fundamental
part of these interactions. Mathematical models become especially interesting
in these cases as they allow inferences for large colonies based on laboratory
data.

Recent studies have associated Markovian processes with decision-making in ant
colonies related to foraging Baddeley et al., (2019); Friedman et al., (2021);
Baltiansky et al., (2023). These models have been instrumental as they not
only analyze the behavior itself and its consequences for the colony but also
allow the association of which environmental elements trigger stimuli and the
potential for these stimuli to affect the final response, which could lead to
different strategies. This is the first work that applies this model to
cleaning behaviors. Understanding how decision-making occurs in the presence
of a pathogenic agent can help comprehend the mechanistic principles that
unify the evolution of social immunity in eusocial organisms.

## 5 Conclusion

This study modeled the behavioral responses of ants challenged with an
entomopathogenic fungus. Our results highlight how workers’ interactions and
past behaviors predict collective cleaning strategies. They reveal a strong
link between cleaning and vulnerable limbs or pathogens, clarifying distinct
behaviors and their transition times in different conditions. We discuss the
importance of allogrooming, the potential implications for disease
transmission, and possible changes to social contact networks. The conclusions
highlight fixed decision-making principles in workers’ behavior, highlighting
the complex interaction between internal and external factors that influence
social immunity strategies. This was the first study that applied Markov
processes to understand ant grooming behavior, providing insights into
mechanisms of social immunity and decision-making in eusocial organisms.

## 6 Acknowledgments

This publication has emanated from research conducted with the financial
support of Brazilian Foundation CAPES, Brazilian Foundation CNPq, and Science
Foundation Ireland under Grant 18/CRT/6049. The opinions, findings and
conclusions or recommendations expressed in this material are those of the
authors and do not necessarily reflect the views of the funding agencies.

## 7 Declarations

Competing interests The authors declare no conflicts of interest.

Authors’ contributions All authors conceived and designed the research. C.T.
collected the data and provided insights into the discussion of results.
G.R.P. and R.A.M. created the feature engineering methodology and analysed the
data. G.R.P. led the writing of the manuscript. All authors contributed to the
overall writing.

Funding This work had the support from Science Foundation Ireland under grant
number 18/CRT/6049; Brazilian Foundation CAPES process number
88887.716582/2022-00; Brazilian Foundation CNPq process number 141049/2020-0

Availability of data and materials All datasets and scripts are made available
at https://github.com/GabrielRPalma/UnderstandingLearningWithML

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