# Mutating signed $\tau$-exceptional sequences

Aslak Bakke Buan  Department of Mathematical Sciences
Norwegian University of Science and Technology
7491 Trondheim
Norway
<EMAIL_ADDRESS>and  Bethany Rose Marsh  School of Mathematics
University of Leeds
Leeds, LS2 9JT
United Kingdom
<EMAIL_ADDRESS>Dedicated to the memory of Helmut Lenzing

###### Abstract.

We establish some properties of $\tau$-exceptional sequences for finite-
dimensional algebras. In an earlier paper we established a bijection between
the set of ordered support $\tau$-tilting modules and the set of complete
signed $\tau$-exceptional sequences. We describe the action of the symmetric
group on the latter induced by its natural action on the former. Similarly, we
describe the effect on a $\tau$-exceptional sequence obtained by mutating the
corresponding ordered support $\tau$-tilting module via a construction of
Adachi-Iyama-Reiten.

This work was supported by FRINAT grant number 301375 from the Norwegian
Research Council. The authors would like to thank the Isaac Newton Institute
for Mathematical Sciences, Cambridge, for support and hospitality during the
programme Cluster Algebras and Representation Theory where work on this paper
was undertaken. The first named author would also like to thank the Centre of
Advanced Study, Oslo for support and hospitality during the programme
Representation Theory: Combinatorial Aspects and Applications.

## Introduction

The usual notion of exceptional sequences in a module category over a finite-
dimensional algebra [8, 22] has some drawbacks. In particular, for some non-
hereditary algebras, complete exceptional sequences do not exist (see e.g. [6,
Introduction]). In [6], we introduced the notion of $\tau$-exceptional
sequences, motivated by $\tau$-tilting theory [2]. Such sequences can be
regarded as an alternative generalisation of exceptional sequences to the non-
hereditary case with the property that complete $\tau$-exceptional sequences
always exist. We also introduced signed $\tau$-exceptional sequences,
motivated by the concept of signed exceptional sequences for hereditary
algebras [16], and the link to picture groups [16, 17].

The aim of this paper is to establish further properties of (signed)
$\tau$-exceptional sequences, which we now proceed to discuss in more detail.
Recall that a subcategory of a module category is said to be a _wide
subcategory_ if it is closed under kernels, cokernels and extensions (and
therefore inherits an abelian structure). Let $\Lambda$ be the path algebra of
a quiver with $n$ vertices and $\mathop{\rm mod}\nolimits\Lambda$ the category
of finite-dimensional left $\Lambda$-modules. A $\Lambda$-module $X$ is said
to be _exceptional_ if $\mathop{\rm Ext}\nolimits^{1}(X,X)=0$. If $X$ is
exceptional, then the subcategory $X^{\perp_{0,1}}$ consisting of modules $Y$
such that $\mathop{\rm Hom}\nolimits(X,Y)=0$ and $\mathop{\rm
Ext}\nolimits^{1}(X,Y)=0$ is known as the _perpendicular category_ of $X$ [14,
§1]. By [14, Prop. 1.1], [23, Thm. 2.3], $X^{\perp_{0,1}}$ is a wide
subcategory of $\mathop{\rm mod}\nolimits\Lambda$ equivalent to the module
category of the path algebra of a quiver with $n-1$ vertices. An _exceptional
sequence_ is a sequence $(X_{1},\ldots,X_{r})$ where $X_{r}$ is an
indecomposable exceptional $\Lambda$-module and $(X_{1},\ldots,X_{r-1})$ is an
exceptional sequence in $X_{r}^{\perp_{0,1}}$.

If $r=n$, then an exceptional sequence $(X_{1},X_{2},\ldots,X_{r})$ is said to
be _complete_. Note that a complete exceptional sequence gives rise to a flag
of wide subcategories

$\mathcal{C}_{n}\subseteq\mathcal{C}_{n-1}\subseteq\cdots\subseteq\mathcal{C}_{0}=\mathop{\rm
mod}\nolimits\Lambda$

where $\mathcal{C}_{i}=(X_{1}\amalg X_{2}\amalg\cdots\amalg
X_{i})^{\perp_{0,1}}$.

The article [16] introduced the notion of a _signed exceptional sequence_. Let
$D^{b}(\Lambda)$ denote the bounded derived category of $\mathop{\rm
mod}\nolimits\Lambda$. Since $\Lambda$ is hereditary, every indecomposable
object in $D^{b}(\Lambda)$ is of the form $X[i]$, where $[i]$ denotes the
$i$th power of the shift and $X$ is an indecomposable $\Lambda$-module. We
write $|X[i]|=X$.

A signed exceptional sequence in $\mathop{\rm mod}\nolimits\Lambda$ is a
sequence $(X_{1},X_{2},\ldots,X_{r})$ of indecomposable objects in
$D^{b}(\Lambda)$ which are each of the form $Y[j]$ for $j=0$ or $j=1$ for some
$\Lambda$-module $Y$, where $X_{i}=Y[1]$ is allowed only if $|X_{i}|$ is
relatively projective in
$(|X_{i+1}|\amalg|X_{i+2}|\amalg\cdots\amalg|X_{r}|)^{\perp_{0,1}}$, and where
$(|X_{1}|,|X_{2}|,\ldots,|X_{r}|)$ is an exceptional sequence.

In [16], signed exceptional sequences were introduced in order to define the
_cluster morphism category_ of $\Lambda$, whose objects are the wide
subcategories of $\mathop{\rm mod}\nolimits\Lambda$. The morphisms are
described by the signed exceptional sequences. It is shown that the
classifying space of the cluster morphism category is a $K(\pi,1)$, where
$\pi$ is the _picture group_ [17] of $\Lambda$.

Now let $\Lambda$ be an arbitrary finite-dimensional algebra over a field.
Suppose that $\Lambda$ has $n$ simple modules. The linchpin of the definition
of $\tau$-exceptional sequence is the notion of a $\tau$-perpendicular
category [18, §1], which plays the role of the Geigle-Lenzing perpendicular
category in the general case. A $\Lambda$-module $X$ is said to be _$\tau$
-rigid_ if $\mathop{\rm Hom}\nolimits(X,\tau X)=0$. Then the
$\tau$-perpendicular category of $X$ is the subcategory $J(X)$ consisting of
modules $Y$ such that $\mathop{\rm Hom}\nolimits(X,Y)=0$ and $\mathop{\rm
Hom}\nolimits(Y,\tau X)=0$. By [4, Cor. 3.22], [10, Thm. 4.12] $J(X)$ is a
wide subcategory of $\mathop{\rm mod}\nolimits\Lambda$. By [18, Thm. 3.8]
$J(X)$ is equivalent to the module category of an algebra, which has $n-1$
simple modules if $X$ is indecomposable. A $\tau$-exceptional sequence in
$\mathop{\rm mod}\nolimits\Lambda$ is a sequence $(X_{1},X_{2},\ldots,X_{r})$
of $\Lambda$-modules where $X_{r}$ is $\tau$-rigid and
$(X_{1},\ldots,X_{r-1})$ is a $\tau$-exceptional sequence in $J(X_{r})$
(regarded as a module category). Signed $\tau$-exceptional sequences are then
defined in a similar way to signed exceptional sequences (see above, or
Section 1). A $\tau$-exceptional sequence, or signed $\tau$-exceptional
sequence, is said to be complete if it has $n$ terms. Clearly, complete
$\tau$-exceptional sequences and signed $\tau$-exceptional sequences exist for
any finite-dimensional algebra. In [21], an interesting interpretation in
terms of strandardly stratifying systems was given.

Also, $\tau$-exceptional sequences were used in [7] to define the morphisms in
the $\tau$-cluster morphism category of the module category of a
$\tau$-tilting-finite algebra, whose objects are the wide subcategories of the
module category. This was extended to an arbitrary finite-dimensional algebra
in [5]. In [15, Thm. 4.16], it was shown that, if $\Lambda$ is a Nakayama
algebra, the classifying space of the cluster morphism category is a
$K(\pi,1)$ for the picture group [17] of $\Lambda$.

In this paper, we study some properties of $\tau$-exceptional sequences. In
Section 2, we prove our first main result, which is restricted to the case of
$\tau$-tilting finite algebras, i.e algebras with a finite number of basic
$\tau$-tilting modules. Under this assumption, we show that if
$(X_{1},X_{2},\ldots,X_{i},\ldots X_{n})$ and
$(X_{1},X_{2},\ldots,X^{\prime}_{i},\ldots,X_{n})$ are complete
$\tau$-exceptional sequences, then $X^{\prime}_{i}\cong X_{i}$. We conjecture
that this result holds without this assumption.

Suppose now that $\Lambda$ is an arbitrary finite-dimensional algebra. In [6,
Thm. 5.4], it was shown that there is a bijection between complete signed
$\tau$-exceptional sequences in $\mathop{\rm mod}\nolimits\Lambda$ and ordered
support $\tau$-tilting objects in $\mathop{\rm mod}\nolimits\Lambda$. Here a
support $\tau$-tilting object is a pair $(P,M)$ where $P$ is projective, $M$
is $\tau$-rigid and $\mathop{\rm Hom}\nolimits(P,M)=0$, and an ordered support
$\tau$-tilting object is an ordering of the indecomposable summands of $P$ and
$M$ (retaining the information as to whether each object is a summand of $P$
or $M$). Thus the symmetric group acts naturally on the set of ordered support
$\tau$-tilting objects and hence, via the bijection, on the set of complete
signed $\tau$-exceptional sequences. In Section 3, we give an explicit
description of the action of a simple transposition.

Support $\tau$-tilting objects can be mutated (see [2, §2.3]), and thus so can
ordered $\tau$-tilting objects. In Section 4, we describe the effect on the
corresponding complete $\tau$-exceptional sequences, translated via the
bijection above. We also combine the action of the symmetric group and
mutations to give an action of the larger _mutation group_ considered in [19].

Since the braid group on $n$ strands acts transitively on the set of
exceptional sequences over a hereditary algebra [8, 22] with $n$ simple
modules up to isomorphism, a natural question is whether this braid group acts
transitively on the set of all signed $\tau$-exceptional sequences. In Section
5 we show that, for the Kronecker algebra, there is no transitive action of
the braid group on $2$ strands (i.e. the infinite cyclic group) on the set of
signed $\tau$-exceptional sequences which factors through the action of the
mutation group referred to above on the set of such sequences. We also give an
example showing that the obvious generalisation of the definition of the braid
action on exceptional sequences to the (signed) $\tau$-exceptional case does
not work, at least without substantial modification.

##  1\. Background

Let $\Lambda$ be a finite-dimensional basic algebra, and denote by
$\mathop{\rm mod}\nolimits\Lambda$ the category of finite-dimensional left
$\Lambda$-modules. We let $\tau$ denote the Auslander-Reiten translate on
$\mathop{\rm mod}\nolimits\Lambda$. We assume any subcategories $\mathcal{X}$
to be full and closed under isomorphism; we define
$\mathcal{X}^{\perp}=\\{Y\in\mathop{\rm mod}\nolimits\Lambda\mid\mathop{\rm
Hom}\nolimits(\mathcal{X},Y)=0\\}$ and define ${{}^{\perp}\mathcal{X}}$
dually.

Consider $\mathcal{C}(\mathop{\rm mod}\nolimits\Lambda)=\mathop{\rm
mod}\nolimits\Lambda\oplus\mathop{\rm mod}\nolimits\Lambda[1]$ as a full
subcategory of the bounded derived category $D^{b}(\mathop{\rm
mod}\nolimits\Lambda)$. For an indecomposable object $U$ in
$\mathcal{C}(\mathop{\rm mod}\nolimits\Lambda)$, we set $\left\lvert
U\right\rvert=U$ if $U$ is in $\mathop{\rm mod}\nolimits\Lambda$ and
$\left\lvert U\right\rvert=U[-1]$ if $U$ is in $\mathop{\rm
mod}\nolimits\Lambda[1]$. If $U$ in $\mathcal{C}(\mathop{\rm
mod}\nolimits\Lambda)$ is basic, we denote by $\mathop{\rm rk}\nolimits(U)$
the number of indecomposable summands of $U$.

We recall some notions from [2, §0] (in some cases stated slightly
differently, but equivalently). A $\Lambda$-module $M$ is called $\tau$-rigid
if $\mathop{\rm Hom}\nolimits(M,\tau M)=0$. A (usually assumed basic) object
$M\oplus P[1]$ in $\mathcal{C}(\mathop{\rm mod}\nolimits\Lambda)$ is said to
be a support $\tau$-rigid object if $M$ is a $\tau$-rigid $\Lambda$-module and
$P$ is a projective $\Lambda$-module with $\mathop{\rm
Hom}\nolimits_{\Lambda}(P,M)=0$. An object $U=M\oplus P[1]$ is said to be a
_support $\tau$-tilting object_ if $\mathop{\rm rk}\nolimits(U)=\mathop{\rm
rk}\nolimits(\Lambda)$. If, in addition, $P=0$, $U$ is said to be a _$\tau$
-tilting module_.

Recall that a subcategory $\mathsf{W}$ of $\mathop{\rm mod}\nolimits\Lambda$
is said to be _wide_ if it is closed under kernels, cokernels and extensions.
If a wide subcategory $\mathsf{W}$ of $\mathop{\rm mod}\nolimits\Lambda$ is
equivalent to a module category $\mathop{\rm mod}\nolimits\Lambda^{\prime}$,
we set $\mathop{\rm rk}\nolimits\mathsf{W}\coloneqq\mathop{\rm
rk}\nolimits\Lambda^{\prime}$.

Objects which are $\tau$-rigid give rise to a particular class of wide
subcategories.

######  Definition 1.1.

[18, Defn. 3.3] For a support $\tau$-rigid object $U=M\oplus P[1]$ in
$\mathcal{C}(\mathop{\rm mod}\nolimits\Lambda)$ the category

$J(U)=(M\oplus P)^{\perp}\cap{{}^{\perp}{(\tau M)}}$

is called a $\tau$-perpendicular subcategory.

In the following Theorem, (a) is from [10, Thm. 4.12], [4, Cor. 3.22] and (c)
is from [18, Thm. 3.8]. For (b), see [11, Prop. 4.12] and [12, Lemma 4.7].

######  Theorem 1.2.

A $\tau$-perpendicular subcategory $J(U)$ of $\mathop{\rm
mod}\nolimits\Lambda$ is:

  * (a)

wide;

  * (b)

functorially finite;

  * (c)

equivalent to $\mathop{\rm mod}\nolimits\Lambda_{U}$ for some finite-
dimensional algebra $\Lambda_{U}$ with $\mathop{\rm
rk}\nolimits(\Lambda)=\mathop{\rm rk}\nolimits(U)+\mathop{\rm
rk}\nolimits(\Lambda_{U})$.

Let $\mathsf{W}$ be a $\tau$-perpendicular subcategory of $\mathop{\rm
mod}\nolimits\Lambda$. Since $\mathsf{W}$ is equivalent to a module category,
we can also consider the $\tau$-tilting theory of $\mathsf{W}$. Let
$\mathcal{C}(\mathsf{W})=\mathsf{W}\oplus\mathsf{W}[1]$, as a subcategory of
$D^{b}(\mathop{\rm mod}\nolimits\Lambda)$. Note that since $\mathsf{W}$ is an
exact subcategory of $\mathop{\rm mod}\nolimits\Lambda$, there is a canonical
isomorphism

$\mathop{\rm Hom}\nolimits_{D^{b}(\mathsf{W})}(X,Y[1])\simeq\mathop{\rm
Hom}\nolimits_{D^{b}(\mathop{\rm mod}\nolimits\Lambda)}(X,Y[1]),$

for modules $X,Y$ in $\mathsf{W}$, so we can also consider
$\mathcal{C}(\mathsf{W})$ as a subcategory of $D^{b}(\mathsf{W})$.

Note that in general $\tau_{\mathsf{W}}X\not\simeq\tau X$ for a module $X$ in
$\mathsf{W}$, and hence there in general there exist modules which are
$\tau$-rigid in $\mathsf{W}$ but not $\tau$-rigid in $\mathop{\rm
mod}\nolimits\Lambda$. But we do have the following. For a support
$\tau$-rigid object $V=N\oplus Q[1]$ in
$\mathcal{C}(\mathsf{W})\subset\mathcal{C}(\mathop{\rm mod}\nolimits\Lambda)$,
set

$J_{\mathsf{W}}(V)=(N\oplus
Q)^{\perp}\cap{{}^{\perp}{(\tau_{\mathsf{W}}N)}}\cap\mathsf{W}.$

The following useful Lemma follows from [1, Prop. 5.8] (see also [2, Prop.
1.2]).

######  Lemma 1.3.

Let $\mathsf{W}$ be a $\tau$-perpendicular subcategory of $\mathop{\rm
mod}\nolimits\Lambda$, and assume $X,Y$ lie in $\mathsf{W}$. Then the
following hold:

  * (a)

$\mathop{\rm Hom}\nolimits(Y,\tau_{\mathsf{W}}X)=0$ if and only if
$\mathop{\rm Ext}\nolimits^{1}(X,\mathsf{W}\cap\mathop{\rm Gen}\nolimits
Y)=0$.

  * (b)

$X$ is $\tau$-rigid in $\mathsf{W}$ if and only if $\mathop{\rm
Ext}\nolimits^{1}(X,\mathsf{W}\cap\mathop{\rm Gen}\nolimits X)=0$.

######  Lemma 1.4.

Let $\mathsf{W}^{\prime}\subseteq\mathsf{W}$ be $\tau$-perpendicular
subcategories of $\mathop{\rm mod}\nolimits\Lambda$, and let $X$ be an object
in $\mathsf{W}^{\prime}$.

  * (a)

If $X$ is $\tau$-rigid in $\mathsf{W}$, then $X$ is also $\tau$-rigid in
$\mathsf{W}^{\prime}$.

  * (b)

We have $J_{\mathsf{W}}(X)\subseteq J_{\mathsf{W}^{\prime}}(X)$.

###### Proof.

  * (a)

By Lemma 1.3, $X$ is $\tau$-rigid in $\mathsf{W}$ implies $\mathop{\rm
Ext}\nolimits^{1}(X,\mathsf{W}\cap\mathop{\rm Gen}\nolimits X)=0$. Hence also
$\mathop{\rm Ext}\nolimits^{1}(X,\mathsf{W}^{\prime}\cap\mathop{\rm
Gen}\nolimits X)=0$, and applying Lemma 1.3 again we obtain that $X$ is
$\tau$-rigid in $\mathsf{W}^{\prime}$.

  * (b)

If $Y$ is in $J_{\mathsf{W}}(X)$, then we have that $\mathop{\rm
Ext}\nolimits^{1}(X,\mathsf{W}\cap\mathop{\rm Gen}\nolimits Y)=0$ by Lemma
1.3. Hence also $\mathop{\rm
Ext}\nolimits^{1}(X,\mathsf{W}^{\prime}\cap\mathop{\rm Gen}\nolimits
Y)=\mathop{\rm Ext}\nolimits^{1}_{\mathsf{W}^{\prime}}(X,\mathop{\rm
Gen}\nolimits Y)=0$. We have $\mathop{\rm Hom}\nolimits(X,Y)=0$ by assumption.
Hence $Y$ lies in $J_{\mathsf{W}^{\prime}}(X)$.

∎

The following bijection is crucial. It was proved in [6], and can be seen as a
refinement of [18, Thm. 3.16].

######  Theorem 1.5.

[6, Prop. 5.6] Let $\mathsf{W}$ be a $\tau$-perpendicular subcategory of
$\mathop{\rm mod}\nolimits\Lambda$, and let $U$ be a support $\tau$-rigid
object in $\mathcal{C}(\mathsf{W})$. Then there is a bijection
$\mathcal{E}^{\mathsf{W}}_{U}$ from

$\\{X\in\mathop{\rm ind}\nolimits(\mathcal{C}(\mathsf{W}))\mid X\oplus U\text{
support }\tau\text{-rigid in }\mathcal{C}(\mathsf{W})\\}\setminus\mathop{\rm
ind}\nolimits U$

to

$\\{X\in\mathop{\rm ind}\nolimits(\mathcal{C}(J_{\mathsf{W}}(U))\mid\text{$X$
is support $\tau$-rigid in $\mathcal{C}(J_{\mathsf{W}}(U))$}\\}.$

We denote the inverse of $\mathcal{E}^{\mathsf{W}}_{U}$ by
$\mathcal{F}^{\mathsf{W}}_{U}$ and, when $\mathsf{W}=\mathop{\rm
mod}\nolimits\Lambda$, we denote the map in Theorem 1.5 and its inverse simply
by $\mathcal{E}_{U}$ and $\mathcal{F}_{U}$.

Using the bijection in Theorem 1.5, the following was proved in [7, Thms. 1.4,
1.7] for the $\tau$-tilting finite case. It was generalised in [5, Thms. 6.4,
6.12] to arbitrary finite-dimensional algebras.

######  Theorem 1.6.

[5, 7] Let $U\oplus V$ be a $\tau$-rigid object in $\mathcal{C}(\mathop{\rm
mod}\nolimits\Lambda)$

  * (a)

We have $J_{J(U)}\,(\mathcal{E}_{U}(V))=J(U\oplus V)$.

  * (b)

We have $\mathcal{E}_{U\oplus
V}=(\mathcal{E}^{J(U)}_{\mathcal{E}_{U}(V)})\,\mathcal{E}_{U}$.

We also recall the following.

######  Lemma 1.7.

[7, Lemma 4.5] Let $\mathsf{V}$ and $\mathsf{W}$ be wide subcategories of
$\mathop{\rm mod}\nolimits\Lambda$ with $\mathsf{V}\subseteq\mathsf{W}$. Then
$\mathsf{V}$ is a wide subcategory of $\mathsf{W}$.

##  2\. Uniqueness

Let $n$ be the number of simple $\Lambda$-modules. Recall that a complete
$\tau$-exceptional sequence in $\mathop{\rm mod}\nolimits\Lambda$ is a
sequence $(X_{1},X_{2},\ldots,X_{n})$ of indecomposable $\Lambda$-modules
where $X_{n}$ is $\tau$-rigid and $(X_{1},\ldots,X_{n-1})$ is a
$\tau$-exceptional sequence in $J(X_{n})$. Moreover, a sequence
$(X_{1},X_{2},\ldots,X_{n})$ of indecomposable objects in
$\mathcal{C}(\mathop{\rm mod}\nolimits\Lambda)$ is a signed $\tau$-exceptional
sequence, if (i) $X_{n}$ is either a $\tau$-rigid module or of the form $P[1]$
for some projective $\Lambda$-module $P$, and (ii)
$(X_{1},X_{2},\ldots,X_{n-1})$ is a signed $\tau$-exceptional sequence in
$J(\left\lvert X_{n}\right\rvert)$. Note that this means that $X_{n-1}$ is
either $\tau$-rigid in $J(\left\lvert X_{n}\right\rvert)$ (i.e. $\tau$-rigid
in the equivalent module category), or $X_{n-1}=P^{\prime}[1]$, where
$P^{\prime}$ is (relative) projective in $J(\left\lvert X_{n}\right\rvert)$,
and so on.

Recall that $\Lambda$ is said to be _$\tau$ -tilting finite_ if it only has a
finite number of indecomposable $\tau$-rigid modules. In this section, we
shall prove the following uniqueness result for $\tau$-exceptional sequences
over such algebras:

######  Theorem 2.1.

Let $\Lambda$ be a $\tau$-tilting finite algebra. Then the following hold.

  * (a)

Let $(A_{1},A_{2},\dots,A_{n})$ and $(B_{1},B_{2},\dots,B_{n})$ be complete
$\tau$-exceptional sequences in $\mathop{\rm mod}\nolimits\Lambda$. If, for
some $t\in\\{1,\dots,n\\}$, we have $A_{i}=B_{i}$ for all $i\neq t$, then also
$A_{t}=B_{t}$.

  * (b)

Let $(A_{1},A_{2},\dots,A_{n})$ and $(B_{1},B_{2},\dots,B_{n})$ be complete
signed $\tau$-exceptional sequences in $\mathcal{C}(\mathop{\rm
mod}\nolimits\Lambda)$. If, for some $t\in\\{1,\dots,n\\}$, we have
$\left\lvert A_{i}\right\rvert=\left\lvert B_{i}\right\rvert$ for all $i\neq
t$, then also $\left\lvert A_{t}\right\rvert=\left\lvert B_{t}\right\rvert$.

We first recall the following:

######  Theorem 2.2.

Let $\Lambda$ be a $\tau$-tilting finite algebra. Then the following hold for
any wide subcategory $\mathsf{W}$ of $\mathop{\rm mod}\nolimits\Lambda$.

  * (a)

[10, Theorem 4.18] We have $\mathsf{W}=J(U)$ for some support $\tau$-rigid
object $U$ in $\mathcal{C}(\mathop{\rm mod}\nolimits\Lambda)$.

  * (b)

[18, Thm. 3.8, Thm. 3.16] The wide subcategory $\mathsf{W}$ is $\tau$-tilting
finite.

We next make the following observation, which holds for all finite-dimensional
algebras:

######  Lemma 2.3.

If $(A_{1},A_{2},\dots,A_{n})$ is a complete signed $\tau$-exceptional
sequence, then $(\left\lvert A_{1}\right\rvert,\left\lvert
A_{2}\right\rvert,\dots,\left\lvert A_{n}\right\rvert)$ is a complete
(unsigned) $\tau$-exceptional sequence.

###### Proof.

We first claim that $(A_{1},A_{2},\dots,A_{n-1},\left\lvert
A_{n}\right\rvert)$ is a signed $\tau$-exceptional sequence. If $\left\lvert
A_{n}\right\rvert$ is projective, then $J(\left\lvert
A_{n}\right\rvert[1])=J(\left\lvert A_{n}\right\rvert)$. Hence the initial
claim follows from the definition of signed $\tau$-exceptional sequences. The
same argument gives that also $(A_{1},A_{2},\dots,A_{n-2},\left\lvert
A_{n-1}\right\rvert,\left\lvert A_{n}\right\rvert)$ is a $\tau$-exceptional
sequence, and so on. ∎

It is clear that Lemma 2.3 and Theorem 2.1 (a), imply Theorem 2.1 (b), so it
is enough to prove Theorem 2.1 (a).

In the remainder of this section, we will prove Theorem 2.1 (a). So, we assume
for the remainder of the section that $\Lambda$ is $\tau$-tilting finite.

We then have the following:

######  Lemma 2.4.

Let $\mathsf{W}$, $\mathsf{W}^{\prime}$ be wide subcategories of $\mathop{\rm
mod}\nolimits\Lambda$ with $\mathsf{W}^{\prime}\subseteq\mathsf{W}$. Then:

  * (a)

We have $\mathop{\rm rk}\nolimits\mathsf{W}^{\prime}\leq\mathop{\rm
rk}\nolimits\mathsf{W}$;

  * (b)

If $\mathop{\rm rk}\nolimits\mathsf{W}=\mathop{\rm
rk}\nolimits\mathsf{W}^{\prime}$ then $\mathsf{W}=\mathsf{W}^{\prime}$.

###### Proof.

For (a), we note that, by Lemma 1.7, $\mathsf{W}^{\prime}$ is a wide
subcategory of $\mathsf{W}$. By Theorem 2.2, $\mathsf{W}^{\prime}$ is of the
form $J_{\mathsf{W}}(U)$ for some $\tau$-rigid object $U$ in $\mathsf{W}$.
Hence, by Theorem 1.2, $\mathop{\rm
rk}\nolimits\mathsf{W}^{\prime}=\mathop{\rm
rk}\nolimits\mathsf{W}-r\leq\mathop{\rm rk}\nolimits\mathsf{W}$, where $r$ is
the number of non-isomorphic indecomposable direct summands of $U$.

For (b) suppose, in addition, that $\mathop{\rm
rk}\nolimits\mathsf{W}=\mathop{\rm rk}\nolimits\mathsf{W}^{\prime}$. Then
$r=0$ in the above, so $U=0$ and we have $\mathsf{W}=\mathsf{W}^{\prime}$ as
required. ∎

We give an alternative proof of (a) at the end of this section.

######  Lemma 2.5.

Let $\mathsf{W}$ and $\mathsf{W}^{\prime}$ be wide subcategories of
$\mathop{\rm mod}\nolimits\Lambda$, and assume that
$(A_{1},A_{2},\dots,A_{m})$ is a complete $\tau$-exceptional sequence in both
$\mathsf{W}$ and $\mathsf{W}^{\prime}$. Then we have
$\mathsf{W}=\mathsf{W}^{\prime}$.

###### Proof.

By assumption, $\mathop{\rm rk}\nolimits\mathsf{W}=m=\mathop{\rm
rk}\nolimits\mathsf{W}^{\prime}$ so, by Lemma 2.4 (a), we have $\mathop{\rm
rk}\nolimits(\mathsf{W}\cap\mathsf{W}^{\prime})\leq m$. By Lemma 1.4 (a), we
have that $A_{m}$ is $\tau$-rigid in $\mathsf{W}\cap\mathsf{W}^{\prime}$. We
then have $\mathop{\rm rk}\nolimits J_{\mathsf{W}}(A_{m})=m-1$ and
$\mathop{\rm rk}\nolimits J_{\mathsf{W}\cap\mathsf{W}^{\prime}}(A_{m})\leq
m-1$. By Lemma 1.4 (b), we have that $J_{\mathsf{W}}(A_{m})\subseteq
J_{\mathsf{W}\cap\mathsf{W}^{\prime}}(A_{m})$. Hence we must have $\mathop{\rm
rk}\nolimits J_{W}(A_{m})=\mathop{\rm rk}\nolimits
J_{\mathsf{W}\cap\mathsf{W}^{\prime}}(A_{m})$, and it follows that
$\mathop{\rm rk}\nolimits\mathsf{W}=\mathop{\rm rk}\nolimits
J_{W}(A_{m})+1=\mathop{\rm rk}\nolimits
J_{\mathsf{W}\cap\mathsf{W}^{\prime}}(A_{m})+1=\mathop{\rm
rk}\nolimits\mathsf{W}\cap\mathsf{W}^{\prime}$. By Lemma 2.4 (b), we have
$\mathsf{W}=\mathsf{W}\cap\mathsf{W}^{\prime}=\mathsf{W}^{\prime}$. ∎

######  Lemma 2.6.

If $(A_{1},A_{2},\dots,A_{n-1},A_{n})$ and
$(A_{1},A_{2},\dots,A_{n-1},A^{\prime}_{n})$ are complete signed
$\tau$-exceptional sequences in $\mathop{\rm mod}\nolimits\Lambda$, then
$A_{n}=A^{\prime}_{n}$.

###### Proof.

Let $\mathsf{W}=J(A_{n})$ and $\mathsf{W}^{\prime}=J(A^{\prime}_{n})$. Then
the statement follows as a direct consequence of [7, Proposition 10.7] and
Lemma 2.5. ∎

We can now complete the proof of Theorem 2.1 (a), and hence the main theorem
of this section.

###### Proof of Theorem 2.1 (a).

If $t=n$, this follows directly from Lemma 2.6. Assume
$t\in\\{1,\dots,n-1\\}$. Let $W^{A}_{n}=J(A_{n})$, and for
$j\in\\{t,\dots,n-1\\}$ define recursively
$\mathsf{W}^{A}_{j}=J_{\mathsf{W}^{A}_{j+1}}(A_{j})$. Define similarly
$\mathsf{W}^{B}_{n}=J(B_{n})$ and
$\mathsf{W}^{B}_{j}=J_{\mathsf{W}^{B}_{j+1}}(B_{j})$. Then
$\mathsf{W}^{A}_{t+1}=\mathsf{W}^{B}_{t+1}\coloneqq\mathsf{W}^{\prime}$, and
$\mathsf{W}^{\prime}\simeq\mathop{\rm mod}\nolimits\Lambda^{\prime}$ for a
finite-dimensional algebra $\Lambda^{\prime}$, and we have that
$(A_{1},A_{2},\dots,A_{t-1},A_{t})$ and
$(B_{1},B_{2},\dots,B_{t-1},B_{t})=(A_{1},A_{2},\dots,A_{t-1},B_{t})$ are
complete exceptional sequences in $\mathsf{W}^{\prime}$. Hence, we obtain that
$A_{t}=B_{t}$ by Lemma 2.6. ∎

We note that the uniqueness property of Theorem 2.1 also holds for arbitrary
finite-dimensional hereditary algebras, by [8, Lemma 2], and we conjecture
that the assumption on $\tau$-tilting finiteness should not be necessary.

######  Conjecture 2.7.

The statement of Theorem 2.1 holds for all finite-dimensional algebras.

An alternative proof of Lemma 2.4 (a) can be given using the theory of bricks.
Recall that a $\Lambda$-module $M$ is called a _brick_ if $\mathop{\rm
End}\nolimits(M)$ is a division algebra. A set of isoclasses of pairwise Hom-
orthogonal bricks is called a _semibrick_. Let $C$ be a full subcategory of
$\mathop{\rm mod}\nolimits\Lambda$. We denote by $T(C)$ the smallest torsion
class containing $C$, by $\mathop{\rm Gen}\nolimits(C)$ the collection of
modules obtained as quotients of finite direct sums of modules in $C$, and by
$\mathop{\rm Filt}\nolimits(C)$ the category of modules with filtrations by
modules in $C$. By the argument in [20, Lemma 3.1], $T(C)=\mathop{\rm
Filt}\nolimits(\mathop{\rm Gen}\nolimits(C))$. A semibrick, $S$, is called
_left finite_ [3, Definition 1.2] if $T(S)$ is functorially finite. Let
$n_{\Lambda}$ denote the number of isomorphism classes of simple $\Lambda$
modules. We recall:

######  Proposition 2.8.

[3, 1.10] If $S$ is a left finite semibrick, then $|S|\leq n_{\Lambda}$.

######  Proposition 2.9.

[9, 1.2] Let $A$ be a $\tau$-tilting finite algebra, and let $T$ be a torsion
class in $\mathop{\rm mod}\nolimits A$. Then $T$ is functorially finite.

###### Proof of Lemma 2.4 (a).

Let $S$ be the set of isoclasses of simple objects in $\mathsf{W}^{\prime}$.
Then $S$ is a semibrick in $\mathsf{W}$. Note that, by [18, Thm. 3.8],
$\mathsf{W}\simeq\mathop{\rm mod}\nolimits\Lambda^{\prime}$ for some finite-
dimensional algebra $\Lambda^{\prime}$. By [7, Prop. 4.2(b)],
$\Lambda^{\prime}$ is $\tau$-tilting finite. Hence, by Proposition 2.9, $S$ is
left-finite in $\mathsf{W}$. By Proposition 2.8, $\mathop{\rm
rk}\nolimits(\mathsf{W}^{\prime})=|S|\leq\mathop{\rm
rk}\nolimits(\mathsf{W})$. ∎

##  3\. Transposition

We now return to the general case, where $\Lambda$ is an arbitrary finite-
dimensional algebra. A sequence $(T_{1},T_{2},\dots,T_{r})$ of indecomposable
objects in $\mathcal{C}(\mathop{\rm mod}\nolimits\Lambda)$ is called an
ordered $\tau$-rigid object if $\oplus_{i=1}^{r}T_{i}$ is a $\tau$-rigid
object, and an ordered support $\tau$-tilting object if $r=n\colon=\mathop{\rm
rk}\nolimits\Lambda$. The symmetric group acts on the set of ordered support
$\tau$-tilting objects in a wide subcategory $\mathsf{W}$ of $\mathop{\rm
mod}\nolimits\Lambda$, by reordering. We recall the following theorem from
[6].

######  Theorem 3.1.

[6, Thm. 5.4] For each $\tau$-perpendicular subcategory $\mathsf{W}$ of
$\mathop{\rm mod}\nolimits\Lambda$, there are mutually inverse bijections

$\\{\text{ordered support $\tau$-tilting objects in $\mathsf{W}$ }\\}$
$\psi^{\mathsf{W}}\downarrow\text{ }\uparrow\phi^{\mathsf{W}}$
$\\{\text{complete signed $\tau$-exceptional sequences in $\mathsf{W}$}\\}$

In the case $\mathsf{W}=\mathop{\rm mod}\nolimits\Lambda$, we write $\psi$ for
$\psi^{\mathop{\rm mod}\nolimits\Lambda}$ and $\phi$ for $\phi^{\mathop{\rm
mod}\nolimits\Lambda}$. In this section, we will describe the action of the
symmetric group on complete signed $\tau$-exceptional sequences in a
$\tau$-perpendicular subcategory $\mathsf{W}$ of $\mathop{\rm
mod}\nolimits\Lambda$ induced by the bijections above.

######  Remark 3.2.

If $\psi(T_{1},T_{2},\dots,T_{n})=(A_{1},\dots,A_{n})$, let

$\displaystyle\mathsf{W}_{n-1}$ $\displaystyle=J(A_{n}),$
$\displaystyle\mathsf{W}_{n-2}$ $\displaystyle=J_{\mathsf{W}_{n-1}}(A_{n-1}),$
$\displaystyle\,\vdots$ $\displaystyle\mathsf{W}_{j}$
$\displaystyle=J_{\mathsf{W}_{j+1}}(A_{j+1}),$ $\displaystyle\,\vdots$
$\displaystyle\mathsf{W}_{1}$ $\displaystyle=J_{\mathsf{W}_{2}}(A_{2}).$

Then we have

$\displaystyle A_{n}$ $\displaystyle=T_{n},$ $\displaystyle A_{n-1}$
$\displaystyle=\mathcal{E}_{A_{n}}(T_{n-1}),$ $\displaystyle A_{n-2}$
$\displaystyle=\mathcal{E}^{\mathsf{W}_{n-1}}_{A_{n-1}}\mathcal{E}_{A_{n}}(T_{n-2}),$
$\displaystyle\,\vdots$ $\displaystyle A_{n-j}$
$\displaystyle=\mathcal{E}_{A_{n-j+1}}^{\mathsf{W}_{n-j+1}}\dots\mathcal{E}^{\mathsf{W}_{n-1}}_{A_{n-1}}\mathcal{E}_{A_{n}}(T_{n-j}),$
$\displaystyle\,\vdots$ $\displaystyle A_{1}$
$\displaystyle=\mathcal{E}_{A_{2}}^{\mathsf{W}_{2}}\dots\mathcal{E}^{\mathsf{W}_{n-1}}_{A_{n-1}}\mathcal{E}_{A_{n}}(T_{1}).$

######  Lemma 3.3.

With notation as above we have

$\mathsf{W}_{n-i}=J(T_{n}\oplus T_{n-1}\oplus\dots\oplus T_{n-{i+1}}),$

for $i=1,\dots,n-1$.

###### Proof.

This follows from repeated use of Theorem 1.6 (a). ∎

Let $\Lambda$ be an algebra of rank $n$, and let $\mathcal{T}_{o}^{\Lambda}$
be the set of ordered basic support $\tau$-tilting objects in
$\mathcal{C}(\mathop{\rm mod}\nolimits\Lambda)$. There is a natural action of
the symmetric group $S_{n}$ on $T_{o}^{\Lambda}$, given by
$\pi_{i}(T_{1},\dots,T_{n})=(T_{1},\dots,T_{i-1},T_{i+1},T_{i},T_{i+2},\dots,T_{n})$,
where $\pi_{i}$ denotes the simple transposition $(i\ i+1)$.

######  Theorem 3.4.

  * (a)

If $\SS=(A_{1},\dots,A_{n})$ is a signed $\tau$-exceptional sequence, then for
$i\in\\{1,\dots,n-1\\}$ we have that

$\widetilde{\pi_{i}}(\SS)\coloneqq(A_{1},\dots,A_{i-1},\mathcal{E}^{(i)}(A_{i+1}),\mathcal{F}^{(i)}(A_{i}),A_{i+2},\dots,A_{n})$

is a signed $\tau$-exceptional sequence, where

$\mathcal{F}^{(i)}\coloneqq\mathcal{F}_{A_{i+1}}^{\mathsf{W}_{i+1}}$

and

$\mathcal{E}^{(i)}\coloneqq\mathcal{E}^{\mathsf{W}_{i+1}}_{\mathcal{F}^{(i)}(A_{i})}$

for $i\in\\{1,\dots,n-2\\}$ and where
$\mathcal{F}^{(n-1)}\coloneqq\mathcal{F}_{A_{n}}$ and
$\mathcal{E}^{(n-1)}\coloneqq\mathcal{E}_{\mathcal{F}^{(n-1)}(A_{n-1})}$.

  * (b)

For each $i\in\\{1,\dots,n-1\\}$ we have
$\widetilde{\pi_{i}}\psi=\psi\pi_{i}$.

######  Lemma 3.5.

Let $(B,C)$ be an ordered $\tau$-rigid object in $\mathcal{C}(\mathop{\rm
mod}\nolimits\Lambda)$. Then we have that
$\mathcal{E}^{J(C)}_{\mathcal{E}_{C}(B)}\mathcal{E}_{C}=\mathcal{E}^{J(B)}_{\mathcal{E}_{B}(C)}\mathcal{E}_{B}$.

###### Proof.

By applying Theorem 1.6 (b) twice, we obtain

$\mathcal{E}^{J(C)}_{\mathcal{E}_{C}(B)}\mathcal{E}_{C}=\mathcal{E}_{C\oplus
B}=\mathcal{E}_{B\oplus
C}=\mathcal{E}^{J(B)}_{\mathcal{E}_{B}(C)}\mathcal{E}_{B}.$

∎

######  Proposition 3.6.

Let $(T_{1},\dots,T_{n})$ be an ordered support $\tau$-tilting object in
$\mathcal{C}(\mathop{\rm mod}\nolimits\Lambda)$, and assume that
$\psi(T_{1},\dots,T_{n})=(A_{1},\dots,A_{n})$. Then

$\psi(\pi_{n-1}(T_{1},\dots,T_{n}))=(A_{1},\dots
A_{n-2},\mathcal{E}_{\mathcal{F}_{A_{n}}(A_{n-1})}(A_{n}),\mathcal{F}_{A_{n}}(A_{n-1})).$

###### Proof.

Let $\
(B_{1},B_{2},\dots,B_{n})=\psi(\pi_{n-1}(T_{1},\dots,T_{n}))=\psi(T_{1},\dots,T_{n-2},T_{n},T_{n-1})$.
We need to show that

  * (i)

$B_{n}=\mathcal{F}_{A_{n}}(A_{n-1})$,

  * (ii)

$B_{n-1}=\mathcal{E}_{\mathcal{F}_{A_{n}}(A_{n-1})}(A_{n})$, and

  * (iii)

$B_{j}=A_{j}$, for $1\leq j\leq n-2$.

Note that, by Remark 3.2, we have that $B_{n}=T_{n-1}$, $A_{n}=T_{n}$ and that
$A_{n-1}=\mathcal{E}_{T_{n}}(T_{n-1})$. Hence, we have
$\mathcal{F}_{T_{n}}(A_{n-1})=\mathcal{F}_{T_{n}}\mathcal{E}_{T_{n}}(T_{n-1})=T_{n-1}=B_{n}$,
which proves claim (i). Moreover, it also follows from Remark 3.2 that
$B_{n-1}=\mathcal{E}_{T_{n-1}}(T_{n})=\mathcal{E}_{\mathcal{F}_{T_{n}}(A_{n-1})}(A_{n})$,
which proves claim (ii).

It remains to prove that $B_{j}=A_{j}$ for $j\leq n-2$. Apply Lemma 3.5, with
$B=T_{n-1}$ and $C=T_{n}$ to obtain that

$\mathcal{E}^{J(T_{n})}_{\mathcal{E}_{T_{n}}(T_{n-1})}\mathcal{E}_{T_{n}}=\mathcal{E}^{J(T_{n-1})}_{\mathcal{E}_{T_{n-1}}(T_{n})}\mathcal{E}_{T_{n-1}}.$

It now follows directly, from Remark 3.2 and Lemma 3.3, that $B_{j}=A_{j}$ for
$j\leq n-2$. ∎

###### Proof of Theorem 3.4.

By Proposition 3.6, it follows that both (a) and (b) hold for $i=n-1$. Assume
$i<n-1$. Then $(A_{1},\dots,A_{i-1})$ is a complete signed $\tau$-exceptional
sequence for the $\tau$-perpendicular subcategory $\mathsf{W}_{i-1}$, as
defined in Remark 3.2. Finally, Proposition 3.6 implies that both (a) and (b)
hold also in this case. ∎

##  4\. Mutation

For a fixed positive integer $n$, consider the group
$G_{n}=\langle\mu_{1},\dots,\mu_{n}\mid\mu_{i}^{2}=e\rangle$ (as in [19, §1]).
Let $\Lambda$ be a fixed algebra of rank $n$.

Mutation of support $\tau$-tilting objects (as in [2, Thm. 2.18]) induces a
mutation on the set $\mathcal{T}^{\Lambda}_{o}$ of ordered basic support
$\tau$-tilting objects in $\mathcal{C}(\mathop{\rm mod}\nolimits\Lambda)$,
which can be regarded as an action of $G_{n}$ on $\mathcal{T}^{\Lambda}_{o}$
and hence, via the bijections in Theorem 3.1, an action on the set of complete
signed $\tau$-exceptional sequences.

The following result follows from [2, Thm. 2.18].

######  Proposition 4.1.

[2, Thm. 2.18] Let $T=(T_{1},\dots,T_{n})$ be an ordered support
$\tau$-tilting object in $\mathcal{C}(\mathop{\rm mod}\nolimits\Lambda)$. Let
$i\in\\{1,\dots,n\\}$. Then there is a unique indecomposable object
$T_{i}^{\ast}$ in $\mathcal{C}(\mathop{\rm mod}\nolimits\Lambda)$ such that
$T(i)=(T_{1},\dots,T_{i-1},T_{i}^{\ast},T_{i+1},\dots,T_{n})$ is an ordered
support $\tau$-tilting object with $T_{i}^{\ast}\not\simeq T_{i}$.

With $T$ and $T(i)$ as above, we set $\mu_{i}(T)=T(i)$. This defines a
$G$-action on $\mathcal{T}^{\Lambda}_{o}$. We now describe the corresponding
action on the set of complete signed $\tau$-exceptional sequences.

For a complete signed $\tau$-exceptional sequence $\SS=(A_{1},\dots,A_{n})$,
let

$s_{1}(\SS)=(\widetilde{A_{1}},A_{2},\dots,A_{n}),$

where

$\widetilde{A_{1}}=\begin{cases}A_{1}[1]&\text{ if }A_{1}\in\mathop{\rm
mod}\nolimits\Lambda;\\\ A_{1}[-1]&\text{ if }A_{1}\in\mathop{\rm
mod}\nolimits\Lambda[1].\end{cases}$

Moreover, for $j>1$, let
$s_{j}(\SS)=\widetilde{\pi}_{j-1}\widetilde{\pi}_{j-2}\dots\widetilde{\pi_{1}}s_{1}\widetilde{\pi}_{1}\dots\widetilde{\pi}_{j-2}\widetilde{\pi}_{j-1}(\SS)$.

We make the following observation:

######  Lemma 4.2.

Let $\Lambda$ be an algebra with a unique simple module. Then the projective
cover of the simple module is the unique indecomposable $\tau$-rigid
$\Lambda$-module.

###### Proof.

Let $S$ be the unique simple $\Lambda$-module, and suppose that $X$ is a non-
projective indecomposable $\tau$-rigid $\Lambda$-module. Then $\mathop{\rm
Ext}\nolimits^{1}(X,M)\not=0$ for some $\Lambda$-module $M$. Since $M$ must be
constructed from $S$ by repeated extensions with $S$, it follows that
$\mathop{\rm Ext}\nolimits^{1}(X,S)\not=0$. Since $X$ is also constructed from
$S$ by repeated extensions with $S$, we have that $S$ is a factor of $X$, so
$\mathop{\rm Ext}\nolimits^{1}(X,\mathop{\rm Gen}\nolimits X)\not=0$. By Lemma
1.3, $X$ is not $\tau$-rigid. ∎

We now have:

######  Lemma 4.3.

If $\SS=(A_{1},A_{2},\dots,A_{n})$ is a signed $\tau$-exceptional sequence,
then so is $s_{j}(\SS)$ for each $j=1,\dots,n$. Moreover $\SS$ and
$s_{1}(\SS)$ are the only signed $\tau$-exceptional sequences of the form
$(X,A_{2},\dots,A_{n})$ for some object $X$ in $\mathcal{C}(\mathop{\rm
mod}\nolimits\Lambda)$.

###### Proof.

Define $W^{A}_{j}$, for $j=1,\ldots,n$, as in the proof of Theorem 2.1 (a). By
repeated application of Theorem 1.2, $W^{A}_{j}$ is equivalent to a module
category over a finite-dimensional algebra of rank $j-1$ for $j=1,\ldots,n$.
Hence $\mathsf{W}^{A}_{2}$ is equivalent to the module category of a finite-
dimensional algebra with a unique simple module. By Lemma 4.2,
$\mathsf{W}^{A}_{2}$, regarded as a module category, has a unique
indecomposable $\tau$-rigid module, given by the unique indecomposable
projective module. This proves that $s_{1}(\SS)$ is a signed
$\tau$-exceptional sequence, and also that $\SS$ and $s_{1}(\SS)$ are the only
signed $\tau$-exceptional sequences of the form $(X,A_{2},\dots,A_{n})$, i.e.
the claim for $j=1$.

The claim for $j>1$ follows by combining this with Theorem 3.4 (a). ∎

######  Proposition 4.4.

With notation as above $s_{i}\psi=\psi\mu_{i}$ holds for all $i=1,\dots,n$.

###### Proof.

Consider first the case $i=1$, and assume that
$\psi(M_{1},M_{2},\dots,M_{n})=(A_{1},A_{2},\dots,A_{n})$. We have
$\mu_{1}(M_{1},M_{2},\dots,M_{n})=(M_{1}^{\ast},M_{2},\dots,M_{n})$, where
$M^{\ast}_{1}\not\simeq M_{1}$, and so
$\psi\mu_{1}(M_{1},M_{2},\dots,M_{n})=\psi(M_{1}^{\ast},M_{2},\dots,M_{n})=(X,A_{2},\dots,A_{n})$,
for some object $X$. The claim for $i=1$ now follows from Lemma 4.3.

For $j>1$, we first note that
$\mu_{j}=\pi_{j-1}\pi_{j-2}\dots\pi_{1}\mu_{1}\pi_{1}\dots\pi_{j-2}\pi_{j-1}$.

Next we note that, by repeated applications of Theorem 3.4, we have that

$\widetilde{\pi_{1}}\dots\widetilde{\pi_{j}}\psi=\psi\pi_{1}\dots\pi_{j}.$

Combining this with the above we obtain

$\displaystyle s_{j}\psi$
$\displaystyle=(\widetilde{\pi}_{j-1}\widetilde{\pi}_{j-2}\dots\widetilde{\pi}_{1}s_{1}\widetilde{\pi}_{1}\dots\widetilde{\pi}_{j-2}\widetilde{\pi}_{j-1})\psi$
$\displaystyle=(\widetilde{\pi}_{j-1}\widetilde{\pi}_{j-2}\dots\widetilde{\pi}_{1}s_{1})(\widetilde{\pi}_{1}\dots\widetilde{\pi}_{j-2}\widetilde{\pi}_{j-1}\psi)$
$\displaystyle=(\widetilde{\pi}_{j-1}\widetilde{\pi}_{j-2}\dots\widetilde{\pi}_{1}s_{1})(\psi\pi_{1}\dots\pi_{j-2}\pi_{j-1})$
$\displaystyle=(\widetilde{\pi}_{j-1}\widetilde{\pi}_{j-2}\dots\widetilde{\pi}_{1})(s_{1}\psi)(\pi_{1}\dots\pi_{j-2}\pi_{j-1})$
$\displaystyle=(\widetilde{\pi}_{j-1}\widetilde{\pi}_{j-2}\dots\widetilde{\pi}_{1})(\psi\mu_{1})(\pi_{1}\dots\pi_{j-2}\pi_{j-1})$
$\displaystyle=(\widetilde{\pi}_{j-1}\widetilde{\pi}_{j-2}\dots\widetilde{\pi}_{1}\psi)(\mu_{1}\pi_{1}\dots\pi_{j-2}\pi_{j-1})$
$\displaystyle=(\psi\pi_{j-1}\pi_{j-2}\dots\pi_{1})(\mu_{1}\pi_{1}\dots\pi_{j-2}\pi_{j-1})$
$\displaystyle=\psi(\pi_{j-1}\pi_{j-2}\dots\pi_{1}\mu_{1}\pi_{1}\dots\pi_{j-2}\pi_{j-1})$
$\displaystyle=\psi\mu_{j}.$

∎

King and Pressland [19, Defn. 1.2] consider the following group:

######  Definition 4.5.

[19] Let $M_{n}=S_{n}\ltimes G_{n}$ be the _mutation group_ of degree $n$,
where $S_{n}$ acts on $G_{n}$ via $\sigma(\mu_{i})=\mu_{\sigma(i)}$.

They show that this group acts naturally on labelled (i.e. ordered) seeds in a
cluster algebra [13] via permutation and mutation. The mutation of
support-$\tau$-tilting objects in [2, §2.3] can be regarded as a
generalisation of cluster mutation, so it is natural to consider the action of
the mutation group in this context. Note that, for an ordered support
$\tau$-tilting object $T$, we have $\sigma(\mu_{i}T)=\mu_{\sigma(i)}(\sigma
T)$ for a permutation $\sigma$ and mutation $\mu_{i}$, so $M_{n}$ acts on the
set of all ordered support $\tau$-tilting objects,
$\mathcal{T}_{o}^{\Lambda}$.

We get an induced action of the mutation group on the set of signed
$\tau$-exceptional sequences.

######  Theorem 4.6.

Let $\SS=(A_{1},A_{2},\dots,A_{n})$ be a signed $\tau$-exceptional sequence.
The operations

$\widetilde{\pi_{i}}(\SS)\coloneqq(A_{1},\dots,A_{i-1},\mathcal{E}^{(i)}(A_{i+1}),\mathcal{F}^{(i)}(A_{i}),A_{i+2},\dots,A_{n}),$
$s_{1}(\SS)=(\widetilde{A_{1}},A_{2},\dots,A_{n}),$

and, for $j=2\ldots,n$,

$s_{j}(\SS)=\widetilde{\pi_{j-1}}\widetilde{\pi_{j-2}}\dots\widetilde{\pi_{1}}s_{1}\widetilde{\pi_{1}}\dots\widetilde{\pi_{j-2}}\widetilde{\pi_{j-1}}(\SS),$

define an action of the mutation group $M_{n}$ on the set of signed
$\tau$-exceptional sequences.

###### Proof.

As already noted, $M_{n}$ acts on the set of ordered support $\tau$-tilting
objects, and the result hence follows directly from combining Theorem 3.4 and
Proposition 4.4 with the fact that $\psi$ is a bijection between the set of
ordered support $\tau$-tilting objects and the set of signed
$\tau$-exceptional sequences (Theorem 3.1). ∎

##  5\. Examples relating to braid actions

Note that the braid group, $B_{n}$, on $n$ strands, has the symmetric group
$S_{n}$ as a quotient. Since $S_{n}$ is a subgroup of the mutation group
$M_{n}=S_{n}\ltimes G_{n}$, it follows that $B_{n}$ acts naturally on the set
of all ordered support $\tau$-tilting objects $\mathcal{T}_{o}^{\Lambda}$ and
thus on the set of all complete signed $\tau$-exceptional sequences for
$\Lambda$, by Theorem 3.1. However, this action is highly non-transitive in
general, since the braid group is only permuting the possible orderings of
each support $\tau$-tilting object.

It is therefore natural to ask whether there is a transitive action. In the
first part of this section, we give an example to show that, at least via the
mutation group, this is not possible: we give an algebra for which there is no
transitive action of $B_{2}$ which factors through the action of $M_{2}$ on
$\mathcal{T}_{o}^{\Lambda}$.

Let $Q$ be the Kronecker quiver
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2}$,
and let $\Lambda$ be the corresponding path algebra. Let $P_{i}$
(respectively, $I_{i}$), for $i=1,2$ be the indecomposable projective
(respectively, injective) $\Lambda$-modules corresponding to the vertices of
$Q$. Then the $\tau$-tilting (equivalently, tilting) $\Lambda$-modules are the
modules $\tau^{-r}P_{1}\amalg\tau^{-r}P_{2}$,
$\tau^{-r}P_{1}\amalg\tau^{-(r+1)}P_{2}$, $\tau^{r}I_{1}\amalg\tau^{r}I_{2}$
and $\tau^{r+1}I_{1}\amalg\tau^{r}I_{2}$, for $r=0,1,2,\ldots$. The support
$\tau$-tilting (equivalently, support tilting) objects over $\Lambda$ which
are not $\tau$-tilting are $I_{1}\amalg P_{2}[1]$, $P_{1}[1]\oplus P_{2}$ and
$P_{1}[1]\oplus P_{2}[1]$. In particular, note that $\Lambda$ is not
$\tau$-tilting finite.

The mutation group (see Definition 4.5) is $M_{2}=S_{2}\ltimes G_{2}$, where
$S_{2}=\\{1,\sigma\\}$ is the symmetric group of degree $2$ and
$G_{2}=\langle\mu_{1},\mu_{2}\ :\ \mu_{1}^{2}=\mu_{2}^{2}=e\rangle$. We have
$\sigma\mu_{1}=\mu_{2}\sigma$ and $\sigma\mu_{2}=\mu_{1}\sigma$. The action of
$M_{2}$ on the set $\mathcal{T}_{o}^{\Lambda}$ of ordered basic support
$\tau$-tilting objects is shown in Figure 1.

######  Proposition 5.1.

There is no transitive action of the braid group on two strands on
$\mathcal{T}_{o}^{\Lambda}$ which factors through the action of $M_{2}$ on
$\mathcal{T}_{o}^{\Lambda}$.

###### Proof.

Note that the braid group $B_{2}$ on two strands is isomorphic to the infinite
cyclic group. If there was a transitive action of $B_{2}$ on
$\mathcal{T}_{o}^{\Lambda}$ factoring through the action of $M_{2}$ on
$\mathcal{T}_{o}^{\Lambda}$, then there would be a subgroup of $M_{2}$ which
is a quotient of $B_{2}$ which acted transitively on
$\mathcal{T}_{o}^{\Lambda}$. Such a subgroup would have to be cyclic.

The elements of $M_{2}$ are of the form

$\sigma^{\varepsilon}\mu_{i}\mu_{\sigma(i)}\mu_{i}\cdots\mu_{i}$

and

$\sigma^{\varepsilon}\mu_{i}\mu_{\sigma(i)}\mu_{i}\cdots\mu_{\sigma(i)},$

where $i\in\\{1,2\\}$ and $\varepsilon\in\\{0,1\\}$. It is easy to check from
the description of the action of $M_{2}$ (see Figure 1) that, for each of
these elements, the (infinite) cyclic group it generates does not act
transitively on $\mathcal{T}_{o}^{\Lambda}$. ∎

######  Corollary 5.2.

There is no transitive action of the braid group on two strands on the set of
all signed $\tau$-exceptional sequences which factors through the action of
$M_{2}$ on the set of all such sequences.

$\textstyle{\vdots}$$\textstyle{\tau^{-1}P_{1},\tau^{-1}P_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\scriptstyle{\mu_{2}}$$\scriptstyle{\sigma}$$\textstyle{\tau^{-1}P_{2},\tau^{-1}P_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}}$$\scriptstyle{\mu_{1}}$$\textstyle{P_{1},\tau^{-1}P_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}}$$\scriptstyle{\sigma}$$\textstyle{\tau^{-1}P_{2},P_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\textstyle{P_{1},P_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\scriptstyle{\sigma}$$\textstyle{P_{2},P_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}}$$\textstyle{P_{1}[1],P_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}}$$\scriptstyle{\sigma}$$\textstyle{P_{2},P_{1}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\textstyle{P_{1}[1],P_{2}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\scriptstyle{\sigma}$$\textstyle{P_{2}[1],P_{1}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}}$$\textstyle{I_{1},P_{2}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}}$$\scriptstyle{\sigma}$$\textstyle{P_{2}[1],I_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\textstyle{I_{1},I_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\scriptstyle{\sigma}$$\textstyle{I_{2},I_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}}$$\textstyle{\tau
I_{1},I_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}}$$\scriptstyle{\sigma}$$\textstyle{I_{2},\tau
I_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\textstyle{\tau
I_{1},\tau
I_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\scriptstyle{\sigma}$$\textstyle{\tau
I_{2},\tau
I_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}}$$\textstyle{\vdots}$
Figure 1. The action of $M_{2}$ on $T_{o}^{\Lambda}$ for the Kronecker algebra
$\Lambda$.

The transitive braid group action on the set of complete exceptional sequences
for a hereditary algebra arises in the following way [8, Lemma 9,Theorem],
[22, §5,§7]. Write the braid group $B_{n}$ on $n$ strands in the usual way as

$B_{n}=\left\langle\sigma_{1},\ldots,\sigma_{n-1}\,:\,\begin{array}[]{cc}\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},&|i-j|>1;\\\
\sigma_{i}\sigma_{j}\sigma_{i}=\sigma_{j}\sigma_{j}\sigma_{i},&|i-j|=1\end{array}\right\rangle.$

Given a complete exceptional sequence $(X_{1},\ldots,X_{n})$ and $1\leq i\leq
n$, there is a unique complete exceptional sequence of the form
$(X_{1},\ldots,X_{i-1},X_{i+1},Y,X_{i+2},\ldots,X_{n})$ for some exceptional
indecomposable module $Y$. The left version of the action of the braid group
is given by:

$\sigma_{i}(X_{1},\ldots,X_{n})=(X_{1},\ldots,X_{i-1},X_{i+1},Y,X_{i+2},\ldots,X_{n}).$

If $n=2$, this means, in particular, that if $(X_{1},X_{2})$ is a complete
exceptional sequence then there is an exceptional sequence of the form
$(X_{2},Y)$.

However, if $(X_{1},X_{2})$ is a complete $\tau$-exceptional sequence, it can
be the case that there is no $\tau$-exceptional sequence of the form
$(X_{2},Y)$. This is not surprising, since the requirement on $X_{2}$ in the
first case is that it is a $\tau$-rigid module, while the requirement in the
second case is that it is a $\tau$-rigid object in the $\tau$-perpendicular
category of $Y$. In the hereditary case, where a module $M$ is $\tau$-rigid if
and only if $Ext^{1}(M,M)=0$, a module in $J(Y)=Y^{\perp_{0,1}}$ is
$\tau$-rigid in $J(Y)$ if and only if it is $\tau$-rigid in $\mathop{\rm
mod}\nolimits\Lambda$, but this is not true in general (see [6, §1]). We
illustrate this point with the following example.

Let $\Gamma$ be the algebra given by the path algebra of the quiver
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$,
subject to the relation $\beta\alpha=0$. The $\tau$-exceptional sequences over
$\Gamma$ were given in [6, §6.2]. They are:

$\left(1,\begin{smallmatrix}2\\\ 1\\\
2\end{smallmatrix}\right),\left(2,\begin{smallmatrix}1\\\
2\end{smallmatrix}\right),\left(\begin{smallmatrix}2\\\
1\end{smallmatrix},2\right),\left(\begin{smallmatrix}1\\\
2\end{smallmatrix},1\right).$

We make the following observation.

######  Lemma 5.3.

For the algebra $\Gamma$, which has complete $\tau$-exceptional sequences of
length $2$, there is a $\tau$-rigid indecomposable module $X$ such that there
is no complete $\tau$-exceptional sequence of the form $(X,Y)$ for some
$\tau$-rigid module $Y$.

###### Proof.

This can be seen from the list of $\tau$-exceptional sequences in $\mathop{\rm
mod}\nolimits\Gamma$ above, since the module $P_{2}=\begin{smallmatrix}2\\\
1\\\ 2\end{smallmatrix}$, despite being $\tau$-rigid, does not occur as the
first term in any of the complete $\tau$-exceptional sequences in the list. ∎

Note that it follows that $P_{2}$ also does not occur as the first term in a
complete signed $\tau$-exceptional sequence.

The phenomenon described above also occurs with the right version [8, 22] of
the action of the braid group. Given a complete exceptional sequence
$(X_{1},\ldots,X_{n})$ and $1\leq i\leq n$, there is a unique complete
exceptional sequence of the form
$(X_{1},\ldots,X_{i-1},Y,X_{i+1},X_{i+2},\ldots,X_{n})$ for some exceptional
indecomposable module $Y$. The right version of the action of the braid group
is given by:

$\sigma_{i}(X_{1},\ldots,X_{n})=(X_{1},\ldots,X_{i-1},Y,X_{i+1},X_{i+2},\ldots,X_{n}).$

If $n=2$, this means, in particular, that if $(X_{1},X_{2})$ is a complete
exceptional sequence then there is an exceptional sequence of the form
$(Y,X_{1})$.

We can see from the list above that, although $(\begin{smallmatrix}2\\\
1\end{smallmatrix},2)$ is a complete $\tau$-exceptional sequence for $\Gamma$,
there is no such sequence of the form $(Y,\begin{smallmatrix}2\\\
1\end{smallmatrix})$. This is because, although $\begin{smallmatrix}2\\\
1\end{smallmatrix}$ is $\tau$-rigid in $J(2)$, it is not a $\tau$-rigid
$\Gamma$-module.

## Acknowledgement

The first named author would like to thank Eric Hanson for insightful
conversations related to this work.

## References

  * [1] M. Auslander and S. O. Smalø, _Almost split sequences in subcategories_ , J. Algebra 69 (1981), 426–454. Addendum: J. Algebra 71 (1981), 592–594.
  * [2] T. Adachi, O. Iyama and I. Reiten, _$\tau-$ tilting theory_, Compos. Math. 150 (2015), no. 3, 415–452
  * [3] S. Asai, _Semibricks_ , Int. Math. Res. Not. (2020) no. 16, 4993–5054.
  * [4] T. Brüstle, D. Smith and H. Treffinger, _Wall and chamber structure for finite-dimensional algebras_ , Adv. Math. 354 (2019), 106746, 31 pp.
  * [5] A. B. Buan and E. Hanson, _$\tau$ -perpendicular wide subcategories_, Preprint arXiv: 2107.01141 [math.RT], 2021.
  * [6] A. B. Buan and B. R. Marsh, _$\tau$ -exceptional sequences_, Journal of Algebra 585 (2021), 36–68.
  * [7] A. B. Buan and B. R. Marsh, _A category of wide subcategories_ , Int. Math. Res. Not. (2021) No. 13, 10278–10338.
  * [8] W. Crawley-Boevey, _Exceptional sequences of representations of quivers_ , Proceedings of the Sixth International Conference on Representations of Algebras (Ottawa, ON, 1992), 7 pp., Carleton-Ottawa Math. Lecture Note Ser., 14, Carleton Univ., Ottawa, ON, 1992.
  * [9] L. Demonet, O. Iyama, G. Jasso, _$\tau$ -tilting finite algebras, bricks, and g-vectors_, Int. Math. Res. Not. (2019), no. 3, 852–892.
  * [10] L. Demonet, O. Iyama, N. Reading, I. Reiten, H. Thomas, _Lattice theory of torsion classes_ , Preprint arXiv:1711.01785v2 [math.RT], 2017; updated 2018.
  * [11] H. Enomoto, _Rigid modules and ICE-closed subcategories in quiver representations_ , J. Algebra 594 (2022), 364–388.
  * [12] H. Enomoto, A. Sakai, _ICE-closed subcategories an wide $\tau$-tilting modules_, Math. Z. 300 (2022), no. 1, 541–577.
  * [13] S. Fomin and A. Zelevinsky, _Cluster algebras. I. Foundations_ , J. Amer. Math. Soc. 15 (2002), no. 2, 497–529.
  * [14] W. Geigle and H. Lenzing, _Perpendicular categories with applications to representations and sheaves_ , J. Algebra 144 (1991), no. 2, 273–343.
  * [15] E. J. Hanson and K. Igusa, _$\tau$ -cluster morphism categories and picture groups_, Comm. Algebra 49 (2021), no. 10, 4376–4415.
  * [16] K. Igusa and G. Todorov, _Signed exceptional sequences and the cluster morphism category_ , Preprint arXiv:1706.02041 [math.RT], 2017.
  * [17] K. Igusa, G. Todorov and J. Weyman, _Picture groups of finite type and cohomology in type $A_{n}$_, Preprint arXiv:1609.02636 [math.RT], 2016.
  * [18] G. Jasso, _Reduction of $\tau$-tilting modules and torsion pairs_, Int. Math. Res. Not. (2015), no. 16, 7190–7237.
  * [19] A. King and M. Pressland, _Labelled seeds and the mutation group_ , Math. Proc. Cambridge Philos. Soc. 163 (2017), no. 2, 193–217.
  * [20] F. Marks and J. Šťovíček, _Torsion classes, wide subcategories and localisations_ , Bull. London Math. Soc. 49 (2017), Issue 3, 405–416.
  * [21] H. O. Mendoza and H. Treffinger, _Stratifying systems through $\tau$-tilting theory_, Doc. Math. 25 (2020), 701–720.
  * [22] C. M. Ringel, _The braid group action on the set of exceptional sequences of a hereditary Artin algebra_ , Abelian group theory and related topics (Oberwolfach, 1993), 339–352, Contemp. Math., 171, Amer. Math. Soc., Providence, RI, 1994.
  * [23] A. Schofield, _Semi-invariants of quivers_ , J. London Math. Soc. (2) 43 (1991), no. 3, 385–395.


# The Pedagogical Representation of Mass Functions with LEGO and their Origin

Stefan J. Kautsch1, Dimitri Veras2,3, Kyle K. Hansotia1 1 Department of
Chemistry and Physics, Halmos College of Arts and Sciences, Nova Southeastern
University, Fort Lauderdale, FL 33314, USA  2 Centre for Exoplanets and
Habitability, University of Warwick, Coventry CV4 7AL, UK  3 Department of
Physics, University of Warwick, Coventry CV4 7AL, UK<EMAIL_ADDRESS>
###### Abstract

We promote the teaching of mass functions as an integral part of an
interdisciplinary science education. Mass functions characterize the frequency
distributions of objects with different masses on all cosmic scales. We intend
to enhance experiential learning of this concept with a creative LEGO brick
experiment for a diverse student audience. To our surprise, the LEGO mass
function is not only qualitatively but also quantitatively comparable to mass
functions found across the Universe. We also discuss the relation between
gravitation and mass distributions as a possible explanation for the
continuity of the universal mass function.

  * This is the version of the article before submission and peer review. The published version is Kautsch, Veras, & Hansotia 2021, European Journal of Physics, 42, 035605 (`https://doi.org/10.1088/1361-6404/abe75c`). The published version has the same content and results, and includes more educational references. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it.

Keywords: Astronomy data visualization (1968), Stellar mass function (1612),
Initial mass function (796), Interdisciplinary astronomy (804), Physics &
astronomy education (2165), Pareto distribution (1897), Mass ratio (1012)

##  1 Introduction

The teaching of the concept of mass functions aims to broaden the
understanding of mass distributions in the cosmos. The target audiences are
students of introductory physics and astronomy courses in higher education. We
also want to engage students in general, undergraduate STEAM (science,
technology, engineering, arts, math) courses to embrace this topic, because
similar distribution concepts can be found in other natural sciences,
mathematics, social sciences, arts, applied sciences, etc. In this article, we
describe an inexpensive experiment which can improve the comprehension of this
concept and can be performed in a variety of creative ways. We also explain
the potential origin of mass functions in the cosmos.

Mass is one of the most important physical properties of objects in space. For
instance, it determines the lifetime of stars, the dark matter content of
galaxies, and determines how planets can harbor life as we know it, or simply
how heavy objects are. Therefore, number and frequency distributions of masses
are of great interest for astrophysical research. If the mass distributions
are known, then they help to explain structure formation and the subsequent
evolution of matter in the Universe.

Astronomical objects (planetary objects, stars, galaxies, dark matter halos,
etc.) can have a wide variety of individual masses. A fundamental observation
is that objects with low masses are far more abundant than massive ones within
a unit of space and of a given kind. For instance, our Milky Way galaxy
contains many more low-mass stars compared to only very few massive stars.
Moreover, the same principle can be seen when comparing objects of different
kinds (e.g., more asteroids than planets, more planets than stars, more stars
than galaxies).

This number frequency distribution of masses of many objects can be
mathematically quantified with a mass function. Thus, the mass function is a
fundamental law of mass distribution in the Universe. Despite its importance,
this concept is rarely presented in introductory courses and is barely
discussed in popular physics and astronomy textbooks, and is even lesser known
in other sciences. A few advanced, general astronomy textbooks (e.g., [1],
[2]) do actually cover mass functions but these are limited to the context of
stellar astrophysics.

We believe that the concept of mass function is important because of its
universality. This topic should be an integral part of the introductory
physics, astronomy, as well as the STEAM curriculum. Moreover, we also think
that this concept has philosophical implications for how everything is
connected in the cosmos and can inspire students and their educators to
advance their understanding of the Universe. We present an experiment using
LEGO bricks in order to promote easy visualization and application for
undergraduate university and college lectures and classes, laboratory courses,
and independent studies. The mass function is introduced in chapter 1.1 below.
In chapters 2 and 3 we present the LEGO visualization experiment; and further
discuss the concept in chapter 4.

###  1.1 Mass Function

A mass function is mathematically a power law, also known as scaling law. It
is the mathematical model $\Phi(m)$ that fits the number of objects $n$ in
different ranges of mass between $m$ and $m+\rmd m$. We use the formalism for
mass functions by Hopkins [3]. Some publications use other Greek letters like
$\xi$ or $\chi$ instead of $\Phi$. The mass function in linear units is:

$\frac{\rmd n}{\rmd m}=\Phi(m)=km^{\alpha}$ (1)

$k$ is a constant, i.e., the coefficient of that function. $\alpha$ is the
power index, i.e., the slope of the function. The slope determines how quickly
the number of high-mass objects is decreasing and low-mass objects is
increasing while the proportion of high-to-low mass objects remains constant.

Integration of (1) leads to the useful formula (2) to compute the number of
objects with masses between ${m_{1}}$ and ${m_{2}}$ if $k$ and $\alpha$ are
known:

$n=\int_{m_{1}}^{m_{2}}\Phi(m)\rmd m=k\int_{m_{1}}^{m_{2}}m^{\alpha}\rmd
m=\frac{k}{\alpha+1}(m_{2}^{\alpha+1}-m_{1}^{\alpha+1})$ (2)

Traditionally, mass functions $\Phi(m)$ (1) have been applied to the
distribution of stars for decades. Salpeter [4] [5] was one of the first who
realized the importance of stellar mass functions for understanding stellar
evolution. Since then, the goal of such studies is to find the initial mass
function (IMF), i.e., the proportion of stars of various masses at the moment
of their birth within a unit of space volume and compare it with the present
day mass function (PDMF, i.e., the mass distribution of stars at present). The
PDMF usually lacks massive stars compared to the IMF, because they have short
life spans. Modern IMF studies (see the reviews [6] [7] [8] [3] and references
therein) are very important for understanding star formation history at
different look-back times and environments by comparing the IMF to the PDMF
(e.g., [9] [10] [11] [12]).

Individual stars are not the only possible target. Mass functions can be
applied to unresolved stellar populations [13] [14]. Moreover, mass functions
are being studied for almost all other types of astronomical objects, such as
galaxies and galaxy halos including the enigmatic dark matter [15] [16] [17]
[18]. Recently, planet hunters have focused on mass functions of planetary
bodies [19] [20] [21].

Zwicky [22] [23] hypothesized that most of the objects in the cosmos follow a
single distribution and are therefore universally linked (through a common
origin of all mass in the Universe). This idea motivated Binggeli & Hascher
[24] to investigate if a single mass function for all astronomical objects
exists: the universal mass function. They studied nearly all hierarchies in
space, which span 36 orders of magnitude in mass, including asteroids,
planets, stars and their remnants, open and globular star clusters, molecular
gas and dust clouds, galaxies, galaxy groups and clusters, and simulated cold
dark matter halos. Their surprising result is that all these objects follow a
mass function with a constant slope of about $\alpha=-2$ with the simple form
$\Phi(m)\propto m^{-2}$ (with an exception for planetary objects with a lower
absolute value of the power index and therefore flatter distribution). Because
this value is less than $-1$, that confirms that objects with low masses are
much more common than massive objects on all cosmic scales. Moreover, the
proportion of low-to-high mass objects is nearly constant across all objects
in space.

##  2 Methods

LEGO toy sets naturally exhibit the basic properties to construct a mass
function because LEGO pieces, the bricks, come in a variety of masses. Most
LEGO sets contain many pieces of low masses and only a few massive bricks,
making this toy ideal to show mass frequency distributions [25] [26]. We
decided to use the set _LEGO 70656 NINJAGO garmadon, Garmadon, GARMADON!_
`https://www.lego.com/en-us/product/garmadon-garmadon-garmadon-70656` for our
experiment, because it contains a shark model like the mascot of our
university.

In our experiment, the LEGO bricks of the set were ordered by shape (i.e.,
mold) and color. The number of bricks per mold and color are classified in the
instruction manual of the set. Therefore, we only measured the mass of one
brick per mold and color and assumed that the mass difference between
identical bricks are negligible for our experiment. Each mass measurement was
done with a lab-grade scale and repeated three times. The values are then
averaged in order to minimize measurement errors. We defined equidistant mass
interval bins and counted the number of bricks within each interval. The
number of intervals was naturally limited due to low numbers of massive
bricks. Thus, we created tables with four, five, six, and seven bins of mass.

We use the data and function-plotting program gnuplot [27], which uses a
nonlinear least-squares Marquardt-Levenberg algorithm for data fitting. The
data were fit with the mass function power law (1). The fitting algorithm
calculated the final set of values for the constant $k$ and the slope $\alpha$
after a few automatic iterations. No initial values for the parameters were
provided for the fitting process.

##  3 Results

The best fit was produced using the data in six mass intervals. The other mass
bin configurations exhibited similar results, but the fitting of six mass bins
had the lowest standard errors of the parameters and the best visualization of
the mass function. Figure 1 shows the mass function as a line in a histogram
with the mass of the bricks in grams on the $x$-axis and the numbers of bricks
on the $y$-axis. The fitting result values are $\alpha=-2.13\pm 0.16$; and
$k=266.21\pm 1.21$. This is also shown in the legend of figure 1.

Figure 1: The histogram shows the result of our measurements and fitting
procedure. The LEGO bricks are sorted into six bins of increasing mass, and
fitted with the mass function power law as shown in the legend.

Astronomers have the habit of illustrating mass functions in logarithmic
plots. These plots have the advantage of including large ranges of numbers and
masses. So, we converted the mass function (1) by taking advantage of the
properties of logarithms. We can also test the quality of our fit and confirm
the fitting parameters. We use base-ten logarithms, $\log_{10}$, but it can be
done for any logarithmic base. The mass function in its logarithmic form is

$\displaystyle\log_{10}\Phi(m)$ $\displaystyle=\log_{10}(k~{}m^{\alpha})$ (3)
$\displaystyle=\log_{10}k+\log_{10}m^{\alpha}$ (4)
$\displaystyle=\alpha\log_{10}m+\log_{10}k$ (5)

(5) is the equation of a straight line in a $\log_{10}-\log_{10}$ graph.
$\alpha$ is still the slope of the line. $\log_{10}k$ is the $y$ intercept of
the line at location $\log_{10}m=0$. This conversion enables us to fit the
mass function with a linear regression line. Figure 2 shows the result of
$\log_{10}$(number of bricks) versus $\log_{10}$(mass of bricks in grams). The
best-fit parameters are $\alpha=-2.13\pm 0.16$ and $\log_{10}k=2.43\pm 0.08$
and are shown in the legend of figure 2.

Figure 2: Same as figure 1 but converted into a linear relationship. This
$\log_{10}-\log_{10}$ plot has the advantage to visualize the regression with
a straight line. The linear regression line and the best-fit values are shown
in the legend.

The fitting algorithm also provided us with the variance of residuals, i.e.,
the reduced chisquare. It should be around unity if the number of data points
is large enough and the fit is excellent. If this value is lower, then the
model function may be too general. Although it is 0.02 in our case, we
consider the fit to be good after visual inspection of figure 2. It might be
possible to get a value closer to one if our mass distribution is fit with
different slopes for different mass intervals. Such broken power laws are
commonly used in studies of stellar IMFs. The absolute slope value tends to be
larger at the high-mass end of stars. The distribution is flatter—aka bottom-
light—for low-mass stars, i.e., the slope has a lower absolute value [28] [29]
[10]. We also see slightly different slopes for the low-mass and high-mass
intervals in our LEGO fit in figure 2.

##  4 Discussion

This project can be designed as a classroom demonstration, semester project,
an independent study, or a lab experiment with a wide range of creative
approaches. We suggest the following main steps: 1) select a random play set
of bricks, discuss selection biases; 2) measure the mass of the individual
bricks, discuss measurement uncertainties; 3) select mass intervals, discuss
difficulties of interval selections; 4) distribute the bricks into the
appropriate mass intervals; 5) count the number of bricks in each interval; 6)
plot the data in a diagram; 7) fit the data with function (1); 8) compare the
results to the universal mass function and its slope; 9) find similar
distributions and discuss potential origins.

###  4.1 The Importance of the Slope $\alpha$

The most important part in the mass function is the slope. It determines the
shape of the function. It quantitatively describes the fraction of high-to-low
mass objects and how fast the number of low-mass objects increases. Therefore,
many modern studies focus to find the value of this slope. Many IMF studies
find $\alpha=-2$ on average for the majority of stars in many galaxies [4]
[29] [10] [30] [31]; although it is highly debated why and whether or not the
IMF slope is universally similar [7] [3].

Binggeli & Hascher [24] also found a slope of $\alpha=-2$ for their combined
mass function, which covers almost all objects in space. They conclude that
the function is continuous and the slope is the nearly same for the individual
distribution functions as well for the grand total. Everything from stars to
galaxy clusters can be fitted with the same universal function and slope
$\alpha=-2$.

It is very surprising that the slope of our LEGO mass function has a similar
value, $\alpha=-2.13$. This might be a fortunate coincidence of this
particular LEGO set. It needs further investigation of more bricks in other
sets to claim that LEGO fits directly into the universal mass function. It
shows, however, that LEGO represents the universal mass function not only
qualitatively, but also quantitatively. Figure 3 shows the visualization of a
slope value of $-2$ for LEGO bricks (left) and stars (right). It means that
for each star with one solar mass, four stars with $\frac{1}{2}$ the mass of
the Sun exist, and 16 stars with $\frac{1}{4}$ of the mass of the Sun exist
and so on. The LEGO distribution is the same but for bricks instead of stars.

Figure 3: This graphic visualizes the meaning of the slope in a mass function
and the constant proportionality of mass distributions. We represent the
distribution of stars and LEGO bricks from our set using the universal mass
function slope of $\alpha=-2$.

###  4.2 Other Power Law-like Distributions

Comprehensive reviews [32] [33, and references therein] show that power law-
like distributions—similar to mass functions—can be found in many components
of nature, such as structure distribution of stellar flares, black holes,
planetary surface geometry, galactic structures, in particular impact crater
sizes [34], lake sizes [35], earthquake magnitudes [36], and many more. This
frequency distribution behavior can be also observed in chemistry (e.g., the
magnitude distribution of (organic) molecules (Alexenberg & Layman, private
communication), in biology (e.g., the mass distributions of mammals and other
animals on Earth: lots of mice, few elephants and whales), in social contexts
(e.g., the distribution of wealth, Gini, and Pareto distributions), or just in
mass distributions of pieces of broken porcelain, glass, or rocks. Aschwanden
et al[32] [33] conclude that these distribution functions follow the concept
of self-organizing critical processes (cf. [37] [38]). Applied to the mass
function it could mean the ability of complex systems to self organize (based
on the laws of gravitation) on many different mass scales in the Universe.

###  4.3 The Origin of Mass Functions in Astronomy

What is the driving force behind masses to generate the observed mass
functions? The origins of individual mass functions are highly debated. For
instance, IMFs alone depend on complex interactions between dynamical,
chemical, and other environmental properties of star-forming regions [10]
[39]. Furthermore, the origin of the universal mass function seems to be even
more mysterious, because it includes such a wide range of objects in space.
Additionally, the general formation processes of different object types
(planetary objects, stars, gas and dust clouds, galaxies, galaxy clusters,
etc.) are often opposite in nature. For example, stars form by top-down
fragmentation of molecular clouds during their gravitationally-driven
collapse, while galaxies grow bottom-up via hierarchical gravitational
clustering [24].

Mass is the common denominator here. Masses—no matter if considering regular
matter, anti matter, or dark matter—interact with each other by the force of
gravitation. Therefore, Binggeli & Hascher [24] and Kroupa et al[7] conclude
that gravitation is responsible for the continuous universal mass function.

One could imagine this in the following gedankenexperiment: The gravitational
force ($F=G~{}m_{1}m_{2}/r^{2}$, $G$…gravitational constant) between two point
masses ($m_{1}$ and $m_{2}$ separated by the displacement $r$) decreases by
the factor of $\frac{1}{r^{2}}$. So, during fragmentation, this $r^{-2}$
dependency of the force means that it is more difficult for fragments to break
up from a mass when they are close to a gravitational center, e.g., inside a
molecular cloud. But weaker gravitational attraction (and more volume at the
outer parts of the cloud) helps to create many more low-mass objects when they
are more distant and less attracted from that center. Similarly, clustering
affects masses more strongly, when they are close to a gravitational seed and
start clumping more effectively compared to loosely attracted, distant masses.
Only when mass scales become small, then dynamical, chemical (and
electromagnetic) processes start to compete more effectively with gravitation.
This is why we see a changing mass function slope typically for low-mass
objects.

Consequently, gravitation could explain both, the distribution of mass and the
common slope in the universal mass function. The slope of the mass function
determines the observed frequency distributions shaped by gravitation. This
might be reflected in a connection of the indices in $r^{\alpha}$ of the
universal law of gravitation and $m^{\alpha}$ of the universal mass function
with the common value of $\alpha=-2$.

##  5 Conclusions

Our experiment provides a pedagogical tool for presenting the concept of mass
functions. We also discuss the potential origin of mass functions in space.
LEGO bricks are ideal to qualitatively visualize this concept. Moreover, our
experiment also reflects the quantitative distributions of masses in the
Universe. Therefore, this experiment can inspire students to learn more how
the cosmos works and how structures are assembled in the Universe.

This project was generously funded by the President’s Faculty Research and
Development Grant 335510 of Nova Southeastern University. DV gratefully
acknowledges the support of the STFC via an Ernest Rutherford Fellowship
(grant ST/P003850/1). We would like to thank Prof. Dr. D. Castano (Nova
Southeastern University, U.S.A.) for assisting us in the work process through
intellectual conversations. The plots were made with gnuplot,
`http://www.gnuplot.info`. The Sun image is from NASA. The comet, head, brick,
and shuttle visuals are from `https://Piktochart.com`. LEGO is a trademark of
The LEGO Group, `https://www.lego.com/en-us/aboutus/lego-group/`.

## References

## References

  * [1] Kutner M L 2003 Astronomy, A Physical Perspective (Cambridge: Cambridge University Press)
  * [2] Carroll B W & Ostlie D A 2017 Introduction to Modern Astrophysics 2nd edn. (Cambridge: Cambridge University Press)
  * [3] Hopkins A M 2018 Publications of the Astron. Soc. of Australia 35 39
  * [4] Salpeter E E 1955 Astrophysical Journal 121 161
  * [5] Kroupa P & Jerabkova T 2019 Nature Astronomy 3 482
  * [6] Corbelli E, Palle F, & Zinnecker H 2005 The Initial Mass Function 50 Years Later (in Astrophysics and Space Science Library) vol 327 (Berlin: Springer)
  * [7] Kroupa P et al2013 Planets, Stars and Stellar Systems vol 5 (Berlin: Springer)
  * [8] Krumholz M R 2014 Physics Reports 539 49
  * [9] Scalo J M 1986 Fundamentals of Cosmic Physics 11 1
  * [10] Chabrier G 2003 Publications of the Astronomical Society of the Pacific 115 763
  * [11] Bovy J 2017 Monthly Notices of the Royal Astronomical Society 470 2
  * [12] Sollima A 2019 Monthly Notices of the Royal Astronomical Society 489 2377
  * [13] Kennicutt Jr R C 1983 Astrophysical Journal 272 54
  * [14] Westera P et al2007 Astronomy and Astrophysics 465 417
  * [15] Press W H & Schechter P 1974 Astrophysical Journal 187 425
  * [16] Schechter P 1976 Astrophysical Journal 203 297
  * [17] Moffett A J et al2016 Monthly Notices of the Royal Astronomical Society 457 1308
  * [18] Pengfei L et al2019 Astrophysical Journal 886 L11
  * [19] Malhorta R 2015 Astrophysical Journal 808 71
  * [20] Bruegger N et al2018 Astronomy & Astrophysics 619 174
  * [21] Wagner K et al2019 Astrophysical Journal 877 46
  * [22] Zwicky F 1942 Phys. Rev. 61 489
  * [23] Zwicky F 1957 Morphological Astronomy (Berlin: Springer)
  * [24] Binggeli B & Hascher T 2007 Publications of the Astronomical Society of the Pacific 119 592
  * [25] Kautsch S J & Hansotia K K 2020 Bulletin of the American Astronomical Society 52 3
  * [26] Hansotia K K & Kautsch S J 2020 Research Notes of the American Astronomical Society 4 134
  * [27] Williams T, Kelley D, et al2020 gnuplot 5.2 An Interactive Plotting Program (`http://www.gnuplot.info`)
  * [28] Miller G E & Scalo J M 1979 Astrophysical Journal Supplement Series 41 513
  * [29] Kroupa P 2001 Monthly Notices of the Royal Astronomical Society 322 231
  * [30] Bastian N et al2010 Annual Review of Astronomy and Astrophysics 48 339
  * [31] Weidner C et al2013 Monthly Notices of the Royal Astronomical Society 436 3309
  * [32] Aschwanden, M J et al2016 Space Science Reviews 198 47
  * [33] Aschwanden, M J et al2018 Space Science Reviews 214 55
  * [34] Zaninetti L & Scholkmann F 2016 Progress in Physics 12 34
  * [35] Seekell D A et al2013 Geophysical Research Letters 40 517
  * [36] Pasari S & Dikshit O 2015 Earth, Planets and Space 67 129
  * [37] Frigg R 2003 Studies in History and Philosophy of Science 34 613
  * [38] Watkins N et al2016 Space Science Review 198 3
  * [39] Offner S S R et al2014 Protostars and Planets VI (Tucson: University of Arizona Press)


# The KM3NeT Open Science System

Jutta Schnabel1, Tamas Gal1, and Zineb Aly2

###### Abstract

The KM3NeT neutrino detectors are currently under construction at two
locations in the Mediterranean Sea, aiming to detect the Cherenkov light
generated by high-energy relativistic charged particles in sea water. The
KM3NeT collaboration will produce scientific data valuable both for the
astrophysics and neutrino physics communities as well as for the Earth and Sea
science community. An Open Science Portal and infrastructure are under
development to provide public access to open KM3NeT data, software and
services. In this contribution, the current architecture, interfaces and usage
examples are presented.

1Erlangen Centre for Astroparticle Physics (ECAP), Erlangen, Germany;
<EMAIL_ADDRESS><EMAIL_ADDRESS>
2 Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France

##  1 Introduction

The KM3NeT collaboration has committed to support open science in its
research. In order to facilitate the sharing of FAIR data, the demonstrator of
an open science system has been developed by the collaboration. It includes
not only a platform to provide science-ready data, but also offers open
software and tutorials and integrates the offered products with widely-used
repositories and platforms. For this and further developments, test data sets
and example analysis workflows are crucial to shape data format and access
options according to the needs of the scientists. Therefore, example use cases
have been provided to provide KM3NeT research results to the fields of
astrophysics, neutrino physics and accross domain boundaries.

##  2 The KM3NeT detectors

The Cubic Kilometre Neutrino Telescope, KM3NeT, is a European neutrino
research infrastructure located in the Mediterranean Sea. KM3NeT operates
water-Cherenkov detectors from two locations: Toulon, in France, hosts the
Oscillation Research with Cosmics in the Abyss (ORCA) detector at a depth of
2475m, while Portopalo di Capo Passero, Sicily, in Italy, hosts the
Astroparticle Research with Cosmics in the Abyss (ARCA) detector at a depth of
3400m.

ARCA is characterized by wider spacing of a 3D array of Digital Optical
Modules, DOMs, at a spacing optimised for the detection of high-energy cosmic
neutrinos in the TeV to PeV energy range. Its forseen 230 strings are 650m
long, spaced 90m apart. On the other hand, ORCA is more compact to optimise
the detection of atmospheric neutrinos in the GeV range. ORCA will consist of
115 strings in a 20m triangular grid, with a 9m vertical spacing between the
DOMs. Currently, there are 6 Detection Units, DUs, operational for ORCA and
one DU for ARCA.

At the shore stations of both ORCA and ARCA, computer farms perform the first
data filtering and event triggering to search for the signal of neutrinos,
prior to streaming data to central KM3NeT data centres for storage and further
analysis by KM3NeT scientists.

###  2.1 Scientific goal

Beyond the detection of astrophysical neutrinos and neutrino oscillation
studies as primary objectives, KM3NeT detectors have a wide range of
capabilities. In fact, KM3NeT data can be used for Supernova bursts studies,
searches for physics beyond the standard model, acoustic neutrino detection
and for Earth and Sea science applications such as bioluminescence activity,
whale songs or dolphin clicks. More details on the motivations behind these
objectives can be found in (Adrián-Martínez & al. 2016).

###  2.2 Data sets and data formats

The basic data structure produced from the neutrino detectors is an event,
which can be reduced to an array of values per event including particle
arrival time, direction, energy, classification, processing parameters as well
as uncertainties on derived quantites. However, as the format definition for
public data sets are guided by community standards or are developed in
exchange with the prospective users, the format varies depending on the
scientific context.

For astrophysics data, the available standards set by the IVOA are applied,
and data from a point-source search with the ANTARES detector (Illuminati &
the ANTARES Collaboration 2019) is offered as testdata set through a KM3NeT-
hosted server with the DaCHS software (Demleitner et al. 2014). This allows to
perform e.g. a Simple Cone Search, or retrieve the data set as VOTable. For
particle events with high significance for multi-messenger analysis, the data
is offered as VOEvent through the KM3NeT alert system, see (Lincetto 2021).

However, for event data to be used beyond the Virtual Observatory environment,
the KM3NeT Open Data Centre (ODC, has been set up, a platform providing event
tables as HDF5 or FITS files and the associated meta data through a REST-API.
As a test data sample, all events detected by the ORCA detector within one
week with 4 DUs were processed and made available as reference point for
further format developments.

In the ODC, also additional high-level science data like event distributions
from simulations is offered to support the interpretation of the event tables.
It also serves as interface to acoustic data samples measured with a
hydrophone hosted in the ORCA detector as an example for environmental data
useable for sea science.

###  2.3 Analysis examples

Analysis examples employing a python interface software and test data samples
are served in the form of Jupyter notebooks. As example for a search for
neutrino point sources, the ANTARES neutrino sample is combined with
supplementary services providing a background event estimate and the detector
acceptance.

For the ORCA test data set, four analysis examples are offered. In the first
three examples, the continuity of data-taking and the event distribution in
local coordinates is analysed, the use of the reconstructed direction and
quality parameters is used to select events with increased probability to be
of astrophyical origin, and the event sample is converted to galactic
coordinates. In the fourth example, the galactic coordinates are used to
search for events coincident in space and time with a Gravitational Wave event
provided by the Gravitational-wave Candidate Event Database of LIGO.

In order to promote a quick understanding of how to analyse KM3NeT data, an
education portal offering self-paced online courses has been set up. It offers
information on KM3NeT science and a detailed introduction to the use of
software analysis tools, facilitating the integration of KM3NeT data in user-
defined analyses.

##  3 Data access and sharing

###  3.1 The KM3NeT software landscape

KM3NeT uses a self-hosted GitLab instance as the main platform to develop
software. The continuous integration (CI) feature is a powerful automation
tool and is utilised to generate consistently up-to-date test reports,
documentation and software releases for a large variety of software and
firmware including scientific analyses and even documents like research
papers. The CI utilises Docker containers, and KM3NeT adopts this solution for
operating-system-level virtualisation and provides Docker and Singularity
containers of all major software to facilitate easy software use in versatile
contexts.

KM3NeT develops open-source software for accessing and working with data taken
by the detector, produced in simulations or in other analysis pipelines e.g.
event reconstructions. All data produced by the detector and consecutive
processing pipelines which are presented to the public are readable using
Python-based tools, which are released each time a new version is tagged. The
choice of Python is mainly motivated by its ease of use and massively
increased popularity in a wide range of scientific research areas during the
past years. Each release is uploaded to the Python Package Index (PyPI) and
can be installed via the pip-package installer. As examples, the km3pipe and
km3io packages allow handling of various data formats used in the KM3NeT
experiment and include a provenance demonstrator for high-level data
processing. Public data from the ODC can be accessed using the openkm3
package. It interlinks with the ODC REST-API and allows to query metadata of
the resources and collections.

All projects which are declared as open source and are hosted on the KM3NeT
GitLab server are visible to the public. However, only KM3NeT members are able
to log in and create content on the GitLab server. To enable the interaction
with users outside the KM3NeT collaboration, open source software is mirrored
to GitHub. The GitHub organisation “KM3NeT” has been created to collect all
the related projects. It is also planned to link KM3NeT software to the ESCAPE
Open Science and Software Repository.

###  3.2 Interlinking to repositories and registries

Registering the KM3NeT open science products with well-used platforms and
assigning global identifiers such as DOIs is a key to findability of the data.
In the VO, the KM3NeT server is registered as a registry of resources, making
KM3NeT data fully findable within the Virtual Observatory, and each resource
being identifiable through the naming of the individual endpoint of the
service within the registry.

Although the VO does support the integration of a DOI in their resource
metadata, it does not provide an authority to assign DOIs. To this end, Zenodo
was chosen as well-established data repository in the physics community to
obtain a DOI through mirroring the data to the repository. For this
demonstrator, the event sample from the KM3NeT ORCA use case will be
registered with Zenodo.

##  4 Linking to the EOSC

The current setup foremost serves as a starting point for further developments
to improve data sharing with observatories and scientists. Being involved in
the ESCAPE project, the KM3NeT collaboration pursues a deepened integration
and technology sharing of the open science system with the wider scientfic
community in both astroparticle and particle physics.

This includes efforts to enhance the understanding of how to integrate
neutrino data in the Virtual Observatory environment to increase its use in
multimessenger analyses, especially regarding the necessity to include
theoretical predictions for neutrino measurements with the detectors for
neutrino source models. Sharing of e.g. detector sensitivity estimates for the
full KM3NeT detector is aimed for not only in the astrophysics context, but
also for the area of particle physics, and the specifications to make also
this simulation-derived data FAIR are currently investigated.

The ESCAPE project also offers the platform to further pursue standardiziation
of data processing, metadata assignment and common format development.
Regarding data processing, a sound provenance scheme for the complex event
processing workflow is developed drawing on the experience of and in exchange
with ESCAPE partners, and the standardization of data formats beyond the VO as
well as common simulation of airshowers especially in cooperation with the CTA
Observatory is pursued. The development of the EOSC services will also help to
pick up these solutions for data storage, sharing, software development and
workflow integration in the KM3NeT context.

##  5 Summary

With the KM3NeT Open Data Centre, Virtual Observatory server, Gitlab software
development platform and Open Science and Education Portals in place, the
KM3NeT collaboration has developed an architecture covering all major
requirements for open science and data sharing and is dedicated to improving
open science by building on the architecture presented here.

All relevant links can be found at http://openscience.km3net.de.

## References

  * Adrián-Martínez & al. (2016) Adrián-Martínez, S., & al. 2016, Journal of Physics G: Nuclear and Particle Physics, 43, 084001. URL https://doi.org/10.1088%2F0954-3899%2F43%2F8%2F084001
  * Demleitner et al. (2014) Demleitner, M., Neves, M. C., Rothmaier, F., & Wambsganss, J. 2014, Astronomy and Computing, 7, 27. 1408.5733
  * Illuminati & the ANTARES Collaboration (2019) Illuminati, G., & the ANTARES Collaboration 2019, PoS(ICRC2019)920
  * Lincetto (2021) Lincetto, M. 2021, in ADASS XXX, edited by J.-E. Ruiz, & F. Pierfederici (San Francisco: ASP), vol. TBD of ASP Conf. Ser., 999 TBD