# G-invariant Spin Structures on Spheres

Jordi Daura Serrano, Michael Kohn, and Marie-Amélie Lawn

###### Abstract

We examine which of the compact connected Lie groups that act transitively on
spheres of different dimensions leave the unique spin structure of the sphere
invariant. We study the notion of invariance of a spin structure and prove
this classification in two different ways; through examining the differential
of the actions and through representation theory.

This paper is dedicated to Lionel Bérard Bergery, who was one of the greatest
experts in homogeneous spaces, and whose enthusiasm and ability to communicate
his love to the subject remains unmatched.

keywords: Spin structures, Lie Groups, Homogeneous Spaces, Spheres .

2000 Mathematics Subject Classification: 53C27, 53C30, 57S15.

## Introduction

Given a Lie group $G$ acting on a manifold $M$, it is a natural question to
ask which structures of the manifold are preserved by $G$. For example, if $M$
is orientable, then connected Lie groups always preserve an orientation.
However, if we consider that $M$ admits spin structures (which can be regarded
as a refinement of orientation) the question of their preservation by $G$ is
more complicated. For example, even if the manifold has a unique spin
structure and the Lie group is connected, it is possible for the action to not
preserve it, as we will show in the main theorem.

The question is even more relevant in the case of homogeneous spaces $G/H$,
where the group action determines the manifold. Here, spin structures can be
nicely characterized in terms of lifts of the isotropy representation to the
group $\operatorname{Spin}(n)$ (see [AC19] and references therein).

The question is motivated by the utility of $G$-invariant spin structures; in
addition to their elegant form mentioned above, G-invariance allows to compute
easily the spinors which are invariant with respect to the connection induced
by the action of the Lie group. This fact is a further motivation to carefully
study the case of homogeneous spaces: Most examples of geometries admitting
special spinors are indeed homogeneous (see for example [Wan89], [Bär93] and
[Agr06]).

A particularly interesting case are the spheres. Most commonly viewed as the
homogeneous space $S^{n}=\operatorname{SO}(n+1)/\operatorname{SO}(n)$, they
can actually be realized, due to their great amount of symmetries, as various
homogeneous decompositions, according to the different groups acting
transitively and effectively on them. This groups were classified by
D.Montgomery and H.Samuelson (see [MS43b]). Part of the preliminaries and an
appendix at the end of this paper are dedicated to describing each of these
nine possible group actions, together with their isotropy representations, as
a comprehensible survey of these classical results is very difficult to find
in the literature. Understanding the features of these actions is of
particular importance in differential geometry; one of their most remarkable
properties being the well-known fact that the transitive effective Lie groups
on spheres appear as holonomy groups according to Berger’s classification
([Ber55]) on simply connected Riemannian manifolds, with the exception of
$\operatorname{Spin}(9)$ and $\operatorname{Sp}(n)\cdot\operatorname{U}(1)$.

Our main theorem studies which of these actions leave the unique spin
structure of spheres invariant. More precisely, we prove the following:

######  Main Theorem.

Let $G$ be a compact connected Lie group acting transitively and effectively
on a sphere of appropriate dimension. Then, its unique spin structure is
$G$-invariant if and only if $G$ is simply connected, or for $n$ odd,
$\operatorname{Sp}(n+1)\cdot\operatorname{U}(1)$ or
$\operatorname{Sp}(n+1)\cdot\operatorname{Sp}(1)$.

Perhaps the most surprising fact is that there are non-simply connected Lie
groups which preserve the spin structure.

We present two different methods to prove the main theorem. One is of more
differential geometric nature and based on the definition of the isotropy
representation using differentials, through the fact that the actions are
linear. The other approach uses the characterization of the isotropy
representation as a restriction of the adjoint representation of the
transitive Lie group, which enables the use of the tools of representation
theory. This method is more general as it is not necessary for the action to
be linear. It is important to point out that in general we do not need to
compute the whole isotropy representation, since it is enough to find the
image of loops whose classes generate the fundamental group of the stabiliser
$H$. By choosing the adequate loop representative we can reduce the complexity
of the computation.

The paper is divided as follows. First, we recall the basic concepts about the
theory of homogeneous spaces we need, focusing on describing the different
transitive group actions on spheres, and recall the concept of a $G$-invariant
spin structure, with special attention to the case of homogeneous spaces. In
the second section, we prove our main result, which we divide into lemmata for
each of the transitive group actions. We end the paper with a few closing
remarks and results. In particular we point out that our main theorem yields a
complete classification of $G$-invariant spin structures on spheres, with $G$
a compact connected Lie group acting transitively.

### Acknowledgements

The authors would like to thank Travis Schedler for fruitful discussions about
the representation theory used in this paper.

##  1 Preliminaries

###  1.1 Homogeneous spaces

In this section, we recall some well-known facts about homogeneous spaces. For
more details we refer to [Bes08, Chapter 7]. Let $M$ be an orientable
n-dimensional Riemannian manifold on which a connected Lie group $G$ acts
transitively from the left (i.e. a homogeneous space) and fix a point $o\in
M$. Then, it is well-known that $M\simeq G/H$, where
$H=\operatorname{Stab}(o)$. We denote by $\mu_{g^{\prime}}:M\rightarrow
M,\,gH\mapsto g^{\prime}gH$ the group action of $G$ on $M$. Additionally, we
assume that the isotropy subgroup $H$ is compact, where $o:=eH$. Then $M$
admits an invariant Riemannian metric, and the group $G$ acts by orientation-
preserving isometries.

We denote by
$\operatorname{Ad}^{G}:G\rightarrow\operatorname{GL}(\mathfrak{g})$ and
$\operatorname{Ad}^{H}:H\rightarrow GL(\mathfrak{h})$ the adjoint
representations of $G$ and $H$ respectively, where $\mathfrak{g}=T_{e}G$
denotes as usual the Lie algebra of $G$ and $\mathfrak{h}$ the Lie algebra of
$H$. Riemannian homogeneous spaces are reductive, i.e. there exists an
$H$-invariant vector space $\mathfrak{m}$ such that
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$. In this case, the tangent space
$T_{o}(G/H)$ can be identified with $\mathfrak{m}$.

By definition, the isotropy subgroup fixes the point $o$. Hence for all $h\in
H$ the differential $(d\mu_{h})_{o}:T_{o}(G/H)\rightarrow T_{h\cdot
o}(G/H)=T_{o}(G/H)$ yields a linear action of $H$ on the tangent space called
the isotropy representation
$\sigma:H\longrightarrow\operatorname{GL}(T_{o}(G/H))$, such that
$\sigma(h):=(d\mu_{h})_{o}$. As the group acts via orientation preserving
isometries, we may take the image to lie inside
$\operatorname{SO}(T_{o}(G/H))$. We will assume furthermore that $G$ acts
effectively and that therefore the isotropy representation is injective.

By the preceding paragraphs, the adjoint representation of $G$ restricted to
$H$ satisfies
$\operatorname{Ad}^{G}|_{H}\cong\operatorname{Ad}^{H}\oplus\sigma$. Therefore,
if $\mathfrak{m}$ is any $H$-subrepresentation of $\mathfrak{g}$ such that
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$, we have
$(\operatorname{Ad}^{G}|_{H},\mathfrak{m})\cong(\sigma,T_{o}(G/H))$.

It is important to describe precisely the structure of the orthonormal frame
bundle $FM$ of the homogeneous space $G/H$. By our previous considerations,
this is a $\operatorname{SO}(\mathfrak{m})$-bundle and we can consequently
identify $F_{o}M$ with $\operatorname{SO}(\mathfrak{m})$. Consider now the
$H$-bundle $p:G\times_{\sigma}\operatorname{SO}(\mathfrak{m})\rightarrow G/H$
associated to the bundle $G\rightarrow G/H$, where the right group action of
$H$ is given by $(g,A)\mapsto(gh,\sigma(h^{-1})A)=:[g,A]$. The map

$G\times_{\sigma}\operatorname{SO}(\mathfrak{m})\rightarrow
FM,\quad[g,A]\mapsto(gH,d\mu_{g}A)$

is a bundle isomorphism, with $d\mu_{g}:F_{o}M\rightarrow F_{gH}M$ being the
pushforward of the action by $G$.

###  1.2 Classification of group actions on spheres

We are interested in compact connected Lie groups acting effectively and
transitively on spheres. Note that the compactness condition enable us to work
with isometric actions, while we can always assume that the group is connected
by taking the identity component, which would also act transitively on the
homogeneous space. These groups were firstly classified by Montgomery and
Samuelson in [MS43b]. We will recall the actions of the classical Lie groups
below, as we will explicitly need them later. Since a comprehensive (and
comprehensible) survey of the description of these actions is somewhat hidden
in the literature, we added the exceptional cases in an appendix at the end of
the paper. We will also describe the loops whose class generate the
fundamental group of the connected Lie groups which are not simply connected.
For this purpose, we denote the usual rotation by

$R(t)=\left(\begin{matrix}\cos(2\pi t)&-\sin(2\pi t)\\\ \sin(2\pi t)&\cos(2\pi
t)\end{matrix}\right).$

#### The actions of $\operatorname{SO}(n+1)$, $\operatorname{U}(n+1)$,
$\operatorname{SU}(n+1)$ and $\operatorname{Sp}(n+1)$.

First we recall that

$\displaystyle\operatorname{SO}(n)$ $\displaystyle:=$
$\displaystyle\\{A\in\operatorname{O}(n):\det A=1\\},\textrm{ where
}\operatorname{O}(n):=\\{A\in\operatorname{GL}(n,\mathbb{R}):A^{t}A=AA^{t}=Id\\},$
$\displaystyle\operatorname{SU}(n)$ $\displaystyle:=$
$\displaystyle\\{A\in\operatorname{U}(n):\det A=1\\},\textrm{ where
}\operatorname{U}(n):=\\{A\in\operatorname{GL}(n,\mathbb{C}):A\bar{A}^{t}=\bar{A}^{t}A=Id\\},$
$\displaystyle\operatorname{Sp}(n)$ $\displaystyle:=$
$\displaystyle\\{A\in\operatorname{GL}(n,\mathbb{H}):A\bar{A}^{t}=\bar{A}^{t}A=Id\\}.$

The group $\operatorname{SO}(n+1)$ acts transitively on the unit sphere
$S^{n}:=\\{(v_{0},\dots,v_{n})\in\mathbb{R}^{n+1}:\sum|v_{i}|^{2}=1\\}$ by the
usual matrix multiplication. It is easy to see that the isotropy group at
$(1,0,\dots,0)^{T}$ is given in block from by the matrix
$\left(\begin{array}[]{c|c}1&0\\\ \hline\cr
0&\operatorname{SO}(n)\end{array}\right)$. In order to find the generator of
$\pi_{1}(\operatorname{SO}(n))=\mathbb{Z}_{2}$ for $n>2$, we use the fact that
$S^{n}$ is $2$-connected for $n>2$ ($\pi_{1}(S^{n})=\pi_{2}(S^{n})=0$),
together with the well-known long exact sequence of homotopy groups for the
principal bundle associated to the homogeneous space structure, to conclude
that the group morphism induced at the level of fundamental groups by the
inclusion of the isotropy subgroup is an isomorphism (or surjective when
$n=2$). Hence we can track back the generator of
$\pi_{1}(\operatorname{SO}(n))$ to $\pi_{1}(\operatorname{SO}(2))$. Thus, we
can write a generating loop $\alpha_{n}:I\longrightarrow\operatorname{SO}(n)$
of $\pi_{1}(\operatorname{SO}(n))$ by
$\alpha_{n}(t)=\left(\begin{array}[]{c|c}Id_{n-2}&0\\\ \hline\cr
0&R(t)\end{array}\right)$.

Similarly $\operatorname{U}(n+1)$ and $\operatorname{SU}(n+1)$ act by matrix
multiplication on the unit sphere in the complex $(n+1)$-dimensional space,
$S^{2n+1}:=\\{(v_{0},...,v_{n})\in\mathbb{C}^{n+1}:\sum|v_{i}|^{2}=1\\}$,
where $\mathbb{C}^{n+1}$ is identified with $\mathbb{R}^{2n+2}$, and
$\operatorname{Sp}(n+1)$ acts on the unit sphere in the quaternions
$S^{4n+3}=\\{(v_{0},...,v_{n})\in\mathbb{H}^{n+1}:\sum|v_{i}|^{2}=1\\}$. Here
$v_{i}=a_{i}+\textbf{i}b_{i}+\textbf{j}c_{i}+\textbf{k}d_{i}$ and
$|v_{i}|^{2}=a_{i}^{2}+b_{i}^{2}+c_{i}^{2}+d_{i}^{2}$ and again we can
identify $\mathbb{H}^{n+1}$ with $\mathbb{R}^{4n+4}$. The isotropy groups can
be computed as above and are $\operatorname{U}(n)$, $\operatorname{SU}(n)$ and
$\operatorname{Sp}(n)$ respectively.

We can use the same argument as above to find a loop whose class generates
$\pi_{1}(\operatorname{U}(n))=\mathbb{Z}$. Knowing that the loop
$\beta_{1}:I\longrightarrow\operatorname{U}(1)$ such that
$\beta_{1}(t)=e^{2\pi it}$ generates $\pi_{1}(\operatorname{U}(1))=\mathbb{Z}$
it is straightforward to see that the loop we seek is
$\beta_{n}(t)=\left(\begin{array}[]{c|c}Id_{n-1}&0\\\ \hline\cr 0&e^{2\pi
it}\end{array}\right)$.

#### The actions of $\operatorname{Sp}(n+1)\cdot\operatorname{U}(1)$ and
$\operatorname{Sp}(n+1)\cdot\operatorname{Sp}(1)$

The case $\operatorname{Sp}(n+1)\cdot\operatorname{U}(1)$ and
$\operatorname{Sp}(n+1)\cdot\operatorname{Sp}(1)$ are analogous, so we will
only explain the case $\operatorname{Sp}(n+1)\cdot\operatorname{U}(1)$ in
detail.

We have that
$\operatorname{Sp}(n+1)\cdot\operatorname{U}(1)=\operatorname{Sp}(n+1)\times\operatorname{U}(1)/\\{\pm(Id,1)\\}$,
so the group is doubly covered by
$\operatorname{Sp}(n+1)\times\operatorname{U}(1)$. We denote the elements of
$\operatorname{Sp}(n+1)\times\operatorname{U}(1)$ by $(A,z)$ and the elements
of $\operatorname{Sp}(n+1)\cdot\operatorname{U}(1)$ by $[A,z]$. Finally, we
need an embedding of $\operatorname{U}(1)\subset\mathbb{C}$ in
$\operatorname{Sp}(1)\subset\mathbb{H}$. We choose
$\iota:\operatorname{U}(1)\hookrightarrow\operatorname{Sp}(1)$ where
$\iota(a+ib)=a+\textbf{i}b+\textbf{j}0+\textbf{k}0$. Note that there are other
possible inclusions.

We define the action of $\operatorname{Sp}(n+1)\cdot\operatorname{U}(1)$ in
the following way. Given
$[A,z]\in\operatorname{Sp}(n+1)\cdot\operatorname{U}(1)$, we define
$\mu_{[A,z]}:S^{4n+3}\longrightarrow S^{4n+3}$ such that
$\mu_{[A,z]}(v)=Av(\iota(z))^{-1}$, where
$v(\iota(z))^{-1}=(v_{0}(\iota(z))^{-1},...,v_{n}(\iota(z))^{-1})$. Note that
this shows why we take $\operatorname{Sp}(n+1)\cdot\operatorname{U}(1)$
instead of $\operatorname{Sp}(n+1)\times\operatorname{U}(1)$ to get an
effective action. The action is moreover transitive, since the usual action of
$\operatorname{Sp}(n+1)$ on $S^{4n+3}$ is transitive. Now, we compute the
isotropy subgroup $H$ for $p=(1,0,...,0)$. If $[A,z]\in H$ then
$Ap=(\iota(z),0,...,0)$. From there, it is straightforward to see that

$H=\\{[\left(\begin{array}[]{c|c}\iota(z)&0\\\ \hline\cr
0&A\end{array}\right),z]:A\in\operatorname{Sp}(n),z\in\operatorname{U}(1)\\}\cong\operatorname{Sp}(n)\cdot\operatorname{U}(1).$

Moreover, the inclusion
$f_{n}:\operatorname{Sp}(n)\cdot\operatorname{U}(1)\longrightarrow\operatorname{Sp}(n+1)\cdot\operatorname{U}(1)$
fulfills

$f_{n}([A,z])=[\left(\begin{array}[]{c|c}\iota(z)&0\\\ \hline\cr
0&A\end{array}\right),z].$

Once again, all the inclusions of isotropy subgroups are isomorphisms at the
level of fundamental groups. Thus we can track back the generator of
$\pi_{1}(\operatorname{Sp}(n)\cdot\operatorname{U}(1))=\mathbb{Z}$ to the case
$n=0$. Then, we have that the generator the fundamental group of
$\operatorname{U}(1)/\mathbb{Z}_{2}$ is represented by the loop
$\gamma_{0}:I\longrightarrow\operatorname{U}(1)/\mathbb{Z}_{2}$ such that
$\gamma_{0}(t)=[e^{i\pi t}]$. Using the inclusions, we can see that the
generator of
$\pi_{1}(\operatorname{Sp}(n)\cdot\operatorname{U}(1))=\mathbb{Z}$ is
represented by a loop
$\gamma_{n}:I\longrightarrow\operatorname{Sp}(n)\cdot\operatorname{U}(1)$ such
that $\gamma_{n}(t)=[\iota(e^{i\pi t})Id,e^{i\pi t}]$.

The action of
$\operatorname{Sp}(n+1)\cdot\operatorname{Sp}(1)=\operatorname{Sp}(n+1)\times\operatorname{Sp}(1)/\\{\pm(Id,1)\\}$
on $S^{4n+3}$, as well as its isotropy group, are the same as
$\operatorname{Sp}(n)\cdot\operatorname{U}(1)$, just we replace the inclusion
$\iota(z)$ by $z$ and take $z$ in $\operatorname{Sp}(1)$. A loop whose class
generates the fundamental group of
$\pi_{1}(\operatorname{Sp}(n)\cdot\operatorname{Sp}(1))\cong\mathbb{Z}_{2}$ is
$\gamma_{n}^{\prime}(t)=[\iota(e^{i\pi t})Id,\iota(e^{i\pi t})]$.

###  1.3 Spin structures on homogeneous spaces

Let $M$ be an orientable smooth Riemannian manifold and let
$(FM,p,M,\operatorname{SO}(n))$ be its orthonormal frame bundle. Recall that
the group $\operatorname{Spin}(n)$ is the unique connected double covering of
$\operatorname{SO}(n)$, which we denote by $\lambda$. Then a spin structure is
a pair $(P,\Lambda)$, where $\Lambda:P\rightarrow FM$ is a 2-covering such
that we have a principal $\operatorname{Spin}(n)$-bundle
$(P,p^{\prime}=p\circ\Lambda,M,\operatorname{Spin}(n))$ and the following
diagram commutes;

${P\times\operatorname{Spin}(n)}$${P}$${X}$${FM\times\operatorname{SO}(n)}$${FM}$$\scriptstyle{\Phi_{\operatorname{Spin}}}$$\scriptstyle{\Lambda\times\lambda}$$\scriptstyle{\Lambda}$$\scriptstyle{p}$$\scriptstyle{\Phi_{\operatorname{SO}}}$$\scriptstyle{p}$

here $\Phi_{\operatorname{Spin}}$ and $\Phi_{\operatorname{SO}}$ denote the
action of these groups on the total space of their respective principal
bundles. In addition, two spin structures $(P_{1},\Lambda_{1})$ and
$(P_{2},\Lambda_{2})$ are said to be equivalent if there exists a principal
bundle isomorphism $f:P_{1}\longrightarrow P_{2}$ such that $\Lambda_{2}\circ
f=\Lambda_{1}$.

Although we fix a Riemannian metric to define a spin structure, it is well-
known that the obstruction to its existence is purely topological. More
precisely, an orientable Riemannian manifold $M$ is spin if and only if its
second Stiefel-Whitney class vanishes, $w_{2}(M)=0$. Then, there exists a one
to one correspondence between spin structures up to equivalence and cohomology
classes $c\in H^{1}(FM,\mathbb{Z}_{2})$ such that $i^{*}(c)\neq 0$, where $i$
is the inclusion of a fiber. Moreover, the number of spin structures is
precisely $|H^{1}(M,\mathbb{Z}_{2})|$ (see [Fri00, Chapter 2]).

It is immediate to conclude from the above discussion the well known fact that
$S^{n}$ is spin, with a unique spin structure.

The first step to study the relation between spin structures and group actions
is to understand how an orientation preserving isometry $f:M\longrightarrow M$
transforms spin structures on a spin manifold $M$. Recall that each
orientation preserving isometry induces an isomorphism on the frame bundle of
the manifold, which we also denote by $f:FM\longrightarrow FM$, such that
$f(p,(v_{1},...,v_{n}))=(f(p),(df_{|p}v_{1},...,df_{|p}v_{n}))$.

We would like to extend this map to a map to a spin structure $(P,\Lambda)$.
More precisely, we seek a map $\tilde{f}:P\longrightarrow P$ such that

${P}$${P}$${FM}$${FM}$$\scriptstyle{\tilde{f}}$$\scriptstyle{\Lambda}$$\scriptstyle{\Lambda}$$\scriptstyle{f}$

that must also preserve the fibers and be equivariant with respect
$\operatorname{Spin}(n)$ right action on $P$. Since the spin structure is
represented by a cohomology class $c\in H^{1}(FM,\mathbb{Z}_{2})$ such that
$i^{*}(c)\neq 0\in H^{1}(\operatorname{SO}(n),\mathbb{Z}_{2})$, then it can be
seen that the map $f:FM\longrightarrow FM$ can be lifted to a map
$\tilde{f}:P\longrightarrow P$ between structures if and only if $f^{*}(c)=c$
by using Čech cohomology (see [Chi72]). It is important to remark that there
are always two possible lifts.

Now, we can introduce the concept of $G$-invariant spin structure. Let $G$ be
Lie group acting by orientation preserving isometries on a spin manifold $M$
and let $\phi_{g}:M\longrightarrow M$ be the isometry induced by the action of
an element $g\in G$ (we use $\phi$ instead of $\mu$ to remark that the action
is not necessarily transitive). As above, we also denote by $\phi_{g}$ the
unique bundle isomorphism induced in $FM$. Then:

######  Definition 1.1.

We say that a spin structure $(P,\Lambda)$ is $G$-invariant if there exists an
action of $G$ on $P$ covering the action of $G$ on $FM$. This means that for
each $g\in G$ there is an isomorphism $\tilde{\phi}_{g}:P\longrightarrow P$
such that $\Lambda\circ\tilde{\phi}_{g}=\phi_{g}\circ\Lambda$.

######  Remark 1.2.

Let $G^{\prime}\subset G$ be Lie groups acting on a spin manifold $M$. If a
fixed spin structure is $G$-invariant, then it is also $G^{\prime}$-invariant.
Thus, if the spin structure is not $G^{\prime}$ invariant, then it is not
$G$-invariant.

Since the action of a connected Lie group is homotopically trivial, the above
cohomological condition is always fulfilled. This implies that if a connected
Lie group acts on a spin manifold $M$, then either $G$ or a $2$-covering
$\tilde{G}$ acts in a way that preserves the spin structure (see [Chi72]).
Note that the action of $\tilde{G}$ on $M$ is not effective. Finally, if $G$
is simply connected the only possibility for $\tilde{G}$ is
$G\times\mathbb{Z}_{2}$. This implies that every spin structure is
$G$-invariant when $G$ is simply connected.

For the case of Riemannian homogeneous space, the action of $G$ on its frame
bundle $G\times_{\sigma}\operatorname{SO}(n)$ is simply
$\mu_{g}([g^{\prime},A])=[gg^{\prime},A]$. Assume now that there exists a lift
of the isotropy representation to the spin group, that is a group morphism
$\tilde{\sigma}:H\longrightarrow\operatorname{Spin}(n)$ such that
$\sigma=\lambda\circ\tilde{\sigma}$. Note that by using covering space theory
and the fact that $\operatorname{Spin}(n)$ is simply connected for $n>2$, it
is straightforward to see that this lift exists if and only if
$\sigma_{*}:\pi_{1}(H)\longrightarrow\pi_{1}(\operatorname{SO}(n))$ is
trivial. Then, it is a known fact that we can construct a spin structure of
the form $G\times_{\tilde{\sigma}}\operatorname{Spin}(n)$ with a 2-covering
$\Lambda([g,x])=[g,\lambda(x)]$. The next proposition shows that the existence
of a lift and the existence of a $G$-invariant spin structure are equivalent.
One of the implications is well-known (see [AC19, CGT93]), while the converse
statement and its proof are not clearly presented in the literature (for
example, it is only mentioned in [Bär92]). Since it is a key step to prove the
main theorem we provide a detailed proof of this fact.

######  Proposition 1.3.

Let $M=G/H$ be a spin homogeneous space where $H$ is connected. Then there
exists a $G$-invariant spin structure if and only if the isotropy
representation lifts to a map
$\tilde{\sigma}:H\longrightarrow\operatorname{Spin}(n)$.

###### Proof.

Assume that the isotropy representation lifts to a group morphism
$\tilde{\sigma}:H\longrightarrow\operatorname{Spin}(n)$, then
$\Lambda:G\times_{\tilde{\sigma}}\operatorname{Spin}(n)\longrightarrow
G\times_{\sigma}\operatorname{SO}(n)$ is a spin structure. We consider the
action of $G$ on $G\times_{\tilde{\sigma}}\operatorname{Spin}(n)$ such that
$\tilde{\mu}_{g}:G\times_{\tilde{\sigma}}\operatorname{Spin}(n)\longrightarrow
G\times_{\tilde{\sigma}}\operatorname{Spin}(n)$ fulfils that
$\tilde{\mu}_{g}([g^{\prime},x])=[gg^{\prime},x]$ for every $g\in G$. It is
clear that $\Lambda\circ\tilde{\mu}_{g}=\mu_{g}\circ\Lambda$, hence the spin
structure is $G$-invariant.

Now, we assume that there exists a $G$-invariant spin structure $(P,\Lambda)$
and that the isotropy representation does not lift in order to reach a
contradiction. With these conditions we can find a loop $\gamma(t)$ in $H$
with $\gamma(0)=\gamma(1)=e$ such that
$\sigma_{*}[\gamma(t)]=1\in\mathbb{Z}_{2}$, therefore we have a path
$\tilde{\gamma}(t)$ in $\operatorname{Spin}(n)$ such that
$\tilde{\gamma}(0)=1$, $\tilde{\gamma}(1)=-1$ and
$\lambda(\tilde{\gamma}(t))=\sigma(\gamma(t))$. Since $\mu_{\gamma(0)}$ and
$\mu_{\gamma(1)}$ are the identity on $G\times_{\sigma}\operatorname{SO}(n)$,
thus $\tilde{\mu}_{\gamma(0)}$ and $\tilde{\mu}_{\gamma(1)}$ shall be the
identity on $P$. Our aim is to see that if $\tilde{\mu}_{\gamma(0)}$ is the
identity then $\tilde{\mu}_{\gamma(1)}$ can not the identity.

We consider the path $\tilde{\mu}_{\gamma(t)}(y)$ in $P$, where $y$ is an
element such that $\Lambda(y)=[e,Id]$. Note that
$\tilde{\mu}_{\gamma(0)}(y)=y$. Then we have that
$\Lambda(\tilde{\mu}_{\gamma(t)}(y))=\mu_{\gamma(t)}[e,Id]=[\gamma(t),Id]=[e,\sigma(\gamma(t))]$.
Thus the path in $P$ descends to a loop inside a fiber of the orthonormal
frame bundle. Since $\Lambda$ preserves the fibers, we have that
$\tilde{\mu}_{\gamma(t)}(y)$ is in a fiber of the
$\operatorname{Spin}(n)$-principal bundle. Therefore, there exists a path
$\gamma^{\prime}(t)$ in $\operatorname{Spin}(n)$ such that
$\gamma^{\prime}(0)=1$ and $\tilde{\mu}_{\gamma(t)}(y)=y\gamma^{\prime}(t)$.
This implies that
$[e,\sigma(\gamma(t))]=\Lambda(\tilde{\mu}_{\gamma(t)}(y))=\Lambda(y\gamma^{\prime}(t))=\Lambda(y)\lambda(\gamma^{\prime}(t))=[e,\lambda(\gamma^{\prime}(t))]$.
Therefore, we have that $\sigma(\gamma(t))=\lambda(\gamma^{\prime}(t))$ and
since $\gamma^{\prime}(0)=\tilde{\gamma}(0)=1$, we conclude
$\gamma^{\prime}(t)=\tilde{\gamma}(t)$. Hence,
$\tilde{\mu}_{\gamma(1)}(y)=-y$, which implies that
$\tilde{\mu}_{\gamma(t)}\neq Id_{P}$, reaching a contradiction. Thus the claim
follows. ∎

If the lift exists then it is unique since $H$ is connected. In this case, a
$G$-invariant spin structure $(P,\Lambda)$ on $G/H$ is equivalent to the spin
structure $G\times_{\tilde{\sigma}}\operatorname{Spin}(n)$ given by the map
$f:G\times_{\tilde{\sigma}}\operatorname{Spin}(n)\longrightarrow P$ such that
$f([g,x])=\tilde{\mu}_{g}(p_{o}x)$, where $\Lambda(p_{o})=[e,Id]$. This leads
to the next corollary:

######  Corollary 1.4.

If there exists a $G$-invariant spin structure on $G/H$ with $H$ connected,
then it is unique.

######  Remark 1.5.

Let $G$ be a connected Lie group, seen as the homogeneous spaces $G\cong
G/\\{e\\}$. Then, the isotropy representation lifts trivially and we have a
$G$-invariant spin structure, which is corresponds to the trivial bundle
$G\times\operatorname{Spin}(n)$. This spin structure is the only $G$-invariant
spin structure on $G$. This fact is especially relevant when $G$ has multiple
inequivalent spin structures, like $T^{n}$, since it gives a preferred spin
structure to work with.

In the spirit of the main theorem, we can also ask which connected Lie groups
$G$ acting transitively on other Lie groups $M$, seen as a manifold, leave its
trivial spin structure invariant. There are cases, like $M=T^{n}$, where the
only group acting transitively on it is the group itself, thus the answer is
trivial. However, if we pick $M=\operatorname{SU}(2)\cong S^{3}$, then
$\operatorname{SO}(4)$ also acts transitively and effectively on $M$, but as
we will see later, the unique spin structure of $S^{3}$ is not
$\operatorname{SO}(4)$-invariant.

##  2 Main Result

The groups $\operatorname{SU}(n)$, $\operatorname{Sp}(n)$,
$\operatorname{G}_{2}$, $\operatorname{Spin}(7)$ and $\operatorname{Spin}(9)$
are simply connected, so in these cases the isotropy representations always
lift and the spin structure is invariant. Hence, we only need to compute the
cases $\operatorname{SO}(n)$, $\operatorname{U}(n)$,
$\operatorname{Sp}(n)\cdot\operatorname{U}(1)$ and
$\operatorname{Sp}(n)\cdot\operatorname{Sp}(1)$.

We present two different approaches to prove the theorem for these groups. Our
first approach is to compute $\sigma(\gamma(t))=(d\mu_{\gamma(t)})_{o}$, where
$\gamma(t)$ is a loop whose class generates the fundamental group of the group
we study. In the second approach we use representation theory to find $\sigma$
by using that $\operatorname{Ad}^{G}|_{H}=\operatorname{Ad}^{H}\oplus\sigma$,
and then compute $\sigma(\gamma(t))$.

For each of the following lemmata corresponding to the four cases above, we
provide first the proof using the first approach and then the more
representation theoretical technique. Note that for the complex and
quaternionic group, we will be using complex representations which are more
convenient to deal with. One can uniquely recover, up to isomorphism, the
isotropy representation from its complexification: if $\rho,\rho^{\prime}$ are
real finite-dimensional representations of any group, then
$\rho\cong\rho^{\prime}$ if and only if the complexifications
$(\rho\otimes\mathbb{C})\cong(\rho^{\prime}\otimes\mathbb{C})$ are isomorphic.
This follows because
$(\rho\otimes\mathbb{C})^{\mathbb{R}}\cong\rho\otimes(\mathbb{C})^{\mathbb{R}}\cong\rho\oplus\rho$.
Here $\psi^{\mathbb{R}}$ denotes the underlying real representation of a
complex one $\psi$.

######  Lemma 2.1.

The spin structure of $S^{n}$ is not $\operatorname{SO}(n+1)$-invariant.

###### Proof.

Recall that in this case, $S^{n}=\operatorname{SO}(n+1)/\operatorname{SO}(n)$
and we take $o=(1,0,...,0)$. Then the action is by restriction of a linear
action, hence the differential
$(d\mu_{\alpha_{n}(t)})_{o}:T_{o}\mathbb{R}^{n+1}\longrightarrow
T_{o}\mathbb{R}^{n+1}$ is the matrix $f_{n}(\alpha_{n}(t))$, where
$\alpha_{n}$ is the generating loop described in Section 1.2. If we restrict
it to $T_{p}S^{n}=\\{v\in\mathbb{R}^{n+1}:\langle
v,(1,0,...,0)\rangle=0\\}=\langle e_{2},...,e_{n+1}\rangle$, we obtain that
$\sigma(\alpha_{n}(t))=\alpha_{n}(t)\in\operatorname{SO}(n)$. Since
$[\alpha_{n}(t)]$ is the generator of $\pi_{1}(\operatorname{SO}(n))$, we can
conclude that the isotropy representation does not lift, which implies that
the spin structure is not $\operatorname{SO}(n+1)$-invariant.

Let us now denote by $\lambda_{n}$ the standard representation of
$\operatorname{SO}(n)$, which is given by the left multiplication on
$\mathbb{R}^{n}$. It is a well-known fact that
$\operatorname{Ad}^{\operatorname{SO}(n)}\cong\bigwedge^{2}\lambda_{n}$.
Recall that
$\sigma=\operatorname{Ad}^{\operatorname{SO}(n+1)}|_{\operatorname{SO}(n)}:\operatorname{SO}(n)\longrightarrow\operatorname{SO}(\mathfrak{m})$
and that we have the chain of isomorphisms
$\operatorname{Ad}^{\operatorname{SO}(n+1)}|_{\operatorname{SO}(n)}\cong\bigwedge^{2}(\lambda_{n+1}|_{\operatorname{SO}(n)})\cong\bigwedge^{2}(\lambda_{n}\oplus
1)\cong\bigwedge^{2}\lambda_{n}\oplus\lambda_{n}$, where $1$ denotes the
trivial representation. The first summand is the adjoint representation of the
isotropy subgroup $\operatorname{SO}(n)$, which means that
$\sigma\cong\lambda_{n}$. This implies, as we have already seen, that
$\sigma(\alpha_{n}(t))=\alpha_{n}(t)$ and hence the isotropy representation
does not lift. ∎

The next case we discuss is the action of $\operatorname{U}(n)$.

######  Lemma 2.2.

The spin structure of $S^{2n+1}$ is not $\operatorname{U}(n+1)$-invariant.

###### Proof.

We proceed in the same way for
$S^{2n+1}=\operatorname{U}(n+1)/\operatorname{U}(n)$. Let again $\beta_{n}$ be
the generating loop described in Section 1.2. In order to compute
$\sigma(\beta_{n}(t))$, we view the action as a linear action on
$S^{2n+1}\subset\mathbb{C}^{n+1}\subset\mathbb{R}^{2n+2}$. We can decomplexify
the matrices of $\operatorname{U}(n+1)$ in order to see them in
$\operatorname{SO}(2n+2)$. Consequently, the isotropy representation is given
by the natural inclusions
$\operatorname{U}(n)\subset\operatorname{SO}(2n)\subset\operatorname{SO}(2n+1)$.
Therefore, we obtain that
$\sigma(\beta_{n}(t))=\left(\begin{array}[]{c|c}Id_{2n-1}&0\\\ \hline\cr
0&R(t)\end{array}\right)\in\operatorname{SO}(2n+1)$. Like in the above case,
this means that the isotropy representation does not lift and the spin
structure is not $\operatorname{U}(n)$-invariant.

We now look at the second approach. Let $\mu_{n}$ be the standard complex
representation of $\operatorname{U}(n)$ as a matrix acting on
$\mathbb{C}^{n}$. Then, the complexified adjoint representation satisfies
$\operatorname{Ad}^{\operatorname{U}(n+1)}\otimes\mathbb{C}\cong\mu_{n+1}\otimes_{\mathbb{C}}\mu_{n+1}^{*}\cong\mu_{n+1}\otimes_{\mathbb{C}}\bar{\mu}_{n+1}$.
If we restrict it to $\operatorname{U}(n)$, we obtain that
$\operatorname{Ad}^{\operatorname{U}(n+1)}\otimes\mathbb{C}|_{\operatorname{U}(n)}\cong(\mu_{n}\oplus
1)\otimes_{\mathbb{C}}(\bar{\mu}_{n}\oplus
1)\cong(\mu_{n}\otimes\bar{\mu}_{n})\oplus\bar{\mu}_{n}\oplus\mu_{n}\oplus 1$.
Therefore, the complexified isotropy representation is isomorphic to
$\bar{\mu}_{n}\oplus\mu_{n}\oplus 1$. Note that this is isomorphic to the
complexification of $\mu_{n}^{\mathbb{R}}\oplus 1$ (where now $1$ is the real
trivial representation). So the real isotropy representation is isomorphic to
this.

If we apply the isotropy representation to $\beta_{n}(t)$, we obtain
$[R(t)\oplus 1\oplus...\oplus 1]$. So we will obtain a rotation matrix that
generates the fundamental group of $\operatorname{SO}(2n+1)$. ∎

For the sake of completeness, we also compute the (complexified) isotropy
representation for the Lie groups $\operatorname{SU}(n)$ and
$\operatorname{Sp}(n)$. In the first case, the sphere is the same as for
$\operatorname{U}(n)$, with a restricted action, so we simply restrict the
previous isotropy representation to $\operatorname{SU}(n)$.

For the second case, let $\nu_{n}$ be the standard complex representation of
$\operatorname{Sp}(n)$ (of complex dimension $2n$). Then,
$\operatorname{Ad}^{\operatorname{Sp}(n+1)}\otimes\mathbb{C}=S^{2}(\nu_{n+1})$,
where $S^{2}$ denotes the second complex linear symmetric power. Thus,
$\operatorname{Ad}^{\operatorname{Sp}(n+1)}\otimes\mathbb{C}|_{\operatorname{Sp}(n)}\cong
S^{2}(\nu_{n}\oplus 1\oplus 1)\cong S^{2}(\nu_{n})\oplus(S^{1}(\nu_{n})\otimes
S^{1}(1\oplus 1))\oplus S^{2}(1\oplus 1)\cong
S^{2}(\nu_{n})\oplus\nu_{n}\oplus\nu_{n}\oplus 1\oplus 1\oplus 1$. This
implies that the complexified isotropy representation is isomorphic to the
last sum after removing the first summand. Note that this is isomorphic to the
complexification of $\nu_{n}^{\mathbb{R}}\oplus 1\oplus 1\oplus 1$. Thus the
real isotropy representation is isomorphic to this.

######  Lemma 2.3.

The spin structure of $S^{4n+3}$ is invariant by the transitive actions of
$\operatorname{Sp}(n+1)\cdot\operatorname{U}(1)$ and
$\operatorname{Sp}(n+1)\cdot\operatorname{Sp}(1)$ if and only if $n$ is odd.

###### Proof.

Both case are analogous, so we focus mainly in the case
$S^{4n+3}=\operatorname{Sp}(n+1)\cdot\operatorname{U}(1)/\operatorname{Sp}(n+1)\cdot\operatorname{U}(1)$.
Like in the previous cases, we start by computing
$\sigma([A,z])=(d\mu_{f_{n}[A,z]})_{o}$. In order to do this, we will see that
$\mu_{f_{n}[A,z]}$ is a linear map acting on
$S^{4n+3}\subset\mathbb{R}^{4n+4}$. We have that
$\mu_{\gamma_{n}(t)}(v)=\iota(e^{i\pi t})v\iota(e^{i\pi
t})^{-1}=(C_{\iota(e^{i\pi t})}v_{0},...,C_{\iota(e^{i\pi t})}v_{n})$, where
$C_{w}:\operatorname{Sp}(1)\longrightarrow\operatorname{Sp}(1)$ is the
conjugation by an element $w\in\operatorname{Sp}(1)$. Then, a tedious
computation shows that the conjugation induces a linear map on
$\mathbb{R}^{4}$ (which we can restrict to $S^{3}$) given by a $4\times 4$
matrix

$\left(\begin{array}[]{c|c}Id&0\\\ \hline\cr 0&R(t)\end{array}\right),$

Therefore, we have that $\mu_{f_{n}(\gamma_{n}(t))}\in\operatorname{SO}(4n+4)$
is

$\left(\begin{array}[]{c|c|c}\begin{array}[]{c|c}Id&0\\\ \hline\cr
0&R(t)\end{array}&\cdots&0\\\ \hline\cr\vdots&\ddots&\vdots\\\ \hline\cr
0&\cdots&\begin{array}[]{c|c}Id&0\\\ \hline\cr 0&R(t)\end{array}\\\
\end{array}\right).$

Now, we use that $T_{p}S^{4n+3}=\\{v\in\mathbb{R}^{4n+4}:\langle
v,(1,0,...,0)\rangle=0\\}=\langle e_{2},...,e_{4n+4}\rangle$. This means that
we have that

$(d\mu_{f_{n}(\gamma_{n}(t))})_{o}=\left(\begin{array}[]{c|c|c}\begin{array}[]{c|c}1&0\\\
\hline\cr 0&R(t)\end{array}&\cdots&0\\\ \hline\cr\vdots&\ddots&\vdots\\\
\hline\cr 0&\cdots&\begin{array}[]{c|c}Id&0\\\ \hline\cr 0&R(t)\end{array}\\\
\end{array}\right)\in\operatorname{SO}(4n+3).$

Since, we have $n+1$ rotations, we can conclude that
$\sigma_{*}[\gamma_{n}]=n+1\mod 2$. Hence, the isotropy representation lifts
when $n$ is odd. In other words, the spin structure of $S^{8m-1}$ is invariant
by the action of $\operatorname{Sp}(2m+2)\cdot\operatorname{U}(1)$ and
$\operatorname{Sp}(2m+2)\cdot\operatorname{Sp}(1)$.

In order to use our second method, we need again to find the isotropy
representation first. We note that
$G=\operatorname{Sp}(n+1)\cdot\operatorname{Sp}(1)$ and
$\operatorname{Sp}(n+1)\times\operatorname{Sp}(1)$ both have the same
(complexified) Lie algebra
$\mathfrak{g}^{\mathbb{C}}\cong\mathfrak{sp_{\mathbb{C}}}(2n+2)\oplus\mathfrak{sp_{\mathbb{C}}}(2)$
(note that here the dimension is not the quaternionic but the complex
dimension). Moreover
$\mathfrak{g}^{\mathbb{C}}=\mathfrak{m}^{\mathbb{C}}\oplus\mathfrak{h}^{\mathbb{C}}$,
where $\mathfrak{m}^{\mathbb{C}}$ is isomorphic, as an
$\mathfrak{h}^{\mathbb{C}}$-representation, to the complexified isotropy
representation.

Since by Section 1.2, $H=\\{[\left(\begin{array}[]{c|c}\iota(z)&0\\\ \hline\cr
0&A\end{array}\right),z]:A\in\operatorname{Sp}(n),z\in\operatorname{Sp}(1)\\}\cong\operatorname{Sp}(n)\cdot\operatorname{Sp}(1)$,
we have that

$\mathfrak{h}^{\mathbb{C}}=\\{(\left(\begin{array}[]{c|c}d\iota(\xi)&0\\\
\hline\cr
0&\mathfrak{a}\end{array}\right),\xi):\mathfrak{a}\in\mathfrak{sp_{\mathbb{C}}}(2n),\xi\in\mathfrak{sp}_{\mathbb{C}}(2)\\}\cong\mathfrak{sp_{\mathbb{C}}}(2n)\oplus\mathfrak{sp_{\mathbb{C}}}(2)^{\prime}\subseteq\mathfrak{g}_{\mathbb{C}},$

where
$\mathfrak{sp_{\mathbb{C}}}(2)^{\prime}=\\{(\left(\begin{array}[]{c|c}d\iota(\xi)&0\\\
\hline\cr 0&0\end{array}\right),\xi):\xi\in\mathfrak{sp}_{\mathbb{C}}(2)\\}$.

Note that for every direct sum of Lie algebras
$\mathfrak{g}=\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}$, the first summand is a
subrepresentation of $\mathfrak{g}$ under the adjoint action; on
$\mathfrak{g}_{1}$, $\mathfrak{g}_{1}$ acts by its adjoint representation, and
$\mathfrak{g}_{2}$ acts trivially. Consequently if $\mathfrak{h}$ is a Lie
subalgebra of $\mathfrak{g}$, its action on $\mathfrak{g}_{1}$ is via the
composition
$\displaystyle\mathfrak{h}\to\mathfrak{g}\twoheadrightarrow\mathfrak{g}_{1}\mathop{\to}^{\operatorname{Ad}}\operatorname{GL}(\mathfrak{g}_{1})$.

Under the projection to the first summand $\mathfrak{sp}_{\mathbb{C}}(2n+2)$,
$\mathfrak{h}^{\mathbb{C}}$ maps isomorphically to
$\mathfrak{sp}_{\mathbb{C}}(2)\oplus\mathfrak{sp}_{\mathbb{C}}(2n)$. By the
preceding paragraph, the action of $\mathfrak{h}^{\mathbb{C}}$ on the first
summand $\mathfrak{sp_{\mathbb{C}}}(2n+2)$ of $\mathfrak{g}^{\mathbb{C}}$ is
the restriction of the adjoint representation of
$\mathfrak{sp_{\mathbb{C}}}(2n+2)$ to
$\mathfrak{sp_{\mathbb{C}}}(2n)\oplus\mathfrak{sp_{\mathbb{C}}}(2)\subseteq\mathfrak{sp_{\mathbb{C}}}(2n+2)$.

Now, a complement to $\mathfrak{h}^{\mathbb{C}}$ in
$\mathfrak{g}^{\mathbb{C}}$ can be obtained as the complement to
$\mathfrak{sp}_{\mathbb{C}}(2n)$ in the first factor
$\mathfrak{sp}_{\mathbb{C}}(2n+2)$. Namely, as we computed previously,

$\mathfrak{sp}_{\mathbb{C}}(2n+2)\cong S^{2}(\mathbb{C}^{2n+2})\cong
S^{2}\mathbb{C}^{2n}\oplus(\mathbb{C}^{2n}\otimes_{\mathbb{C}}\mathbb{C}^{2})\oplus
S^{2}\mathbb{C}^{2},$

with the first summand $S^{2}\mathbb{C}^{2n}$ corresponding to
$\mathfrak{sp}_{\mathbb{C}}(2n)$. So we can identify
$\mathfrak{m}^{\mathbb{C}}$ with
$(\mathbb{C}^{2n}\otimes_{\mathbb{C}}\mathbb{C}^{2})\oplus\mathfrak{sp}_{\mathbb{C}}(2)$.
In terms of matrices this is:

$\mathfrak{m}^{\mathbb{C}}=\\{(\left(\begin{array}[]{c|c}*&*\\\
\hline\cr*&0\end{array}\right),0)\\}\in\mathfrak{sp}_{\mathbb{C}}(2n+2)\oplus\mathfrak{sp}_{\mathbb{C}}(2).$

As a result we can realize the complexified isotropy representation as
$(\nu_{n}\boxtimes_{\mathbb{C}}\nu_{1})\oplus(\operatorname{Ad}^{\operatorname{Sp}(1)}\otimes\mathbb{C})$,
where $\nu_{n}$ is the standard representation of $\operatorname{Sp}(n)$. Here
the action on the second summand is via the projection
$H=\operatorname{Sp}(n)\cdot\operatorname{Sp}(1)\to\operatorname{Sp}(1)/\pm
I$. This representation is isomorphic to the complexification of
$\tilde{\nu}_{n}^{\mathbb{R}}\oplus\operatorname{Ad}^{\operatorname{Sp}(1)}$.
Here, $\tilde{\nu}_{n}^{\mathbb{R}}$ denotes the real representation
corresponding to the first summand, which is nothing but $\mathbb{H}^{n}$ with
the action described in Section 1.2 (note that $\operatorname{Sp}(1)$ does not
act trivially). Note that for the second summand, the action of
$\operatorname{Sp}(n)\cdot\operatorname{Sp}(1)$ factors through the projection
$\operatorname{Sp}(n)\cdot\operatorname{Sp}(1)\twoheadrightarrow\operatorname{Sp}(1)/\pm
I$ (whose adjoint representation is the same as that of $\operatorname{Sp}(1)$
itself).

In the case of $\operatorname{Sp}(n)\cdot\operatorname{U}(1)$, including
$\operatorname{U}(1)$ into $\operatorname{Sp}(1)$ as above, we must restrict
the factor for $\operatorname{Sp}(1)$ to that of $\operatorname{U}(1)$. This
yields the complexified representation
$\nu_{n}\boxtimes\nu_{1}|_{\operatorname{U}(1)}\oplus\operatorname{Ad}^{\operatorname{Sp}(1)}|_{\operatorname{U}(1)}\otimes\mathbb{C})$.

Note that $\nu_{1}|_{\operatorname{U}(1)}$ and
$\operatorname{Ad}^{\operatorname{Sp}(1)}|_{\operatorname{U}(1)}\otimes\mathbb{C}$
are reducible, as they are complex representations of the abelian group
$\operatorname{U}(1)$ of finite dimension greater than one. It is well-known
how to decompose these into one-dimensional representations (e.g., by viewing
$\operatorname{U}(1)$ as the maximal torus inside $\operatorname{Sp}(1)$ and
then decomposing the standard and complex adjoint representations of
$\operatorname{Sp}(1)$, or alternatively of the complexified Lie algebra,
isomorphic to $\mathfrak{sl}_{\mathbb{C}}(2)$). The one-dimensional
representations of $\operatorname{U}(1)$ are of the form
$\rho_{m}:z\mapsto(z^{m})$ for $m\in\mathbb{Z}$. We have
$\nu_{1}|_{\operatorname{U}(1)}\cong\rho_{1}\oplus\rho_{-1}$ and
$\operatorname{Ad}^{\operatorname{Sp}(1)}|_{\operatorname{U}(1)}\otimes\mathbb{C}\cong\rho_{2}\oplus\rho_{0}\oplus\rho_{-2}$,
where $\rho_{0}=1$ is the trivial representation. Putting this together, the
complexified isotropy representation decomposes as:

$\nu_{n}\boxtimes(\rho_{-1}\oplus\rho_{1})\oplus\rho_{2}\oplus
1\oplus\rho_{-2}.$

This is isomorphic to the complexification of
$\tilde{\nu}_{n}^{\mathbb{R}}|_{\operatorname{Sp}(n)\cdot
U(1)}\oplus\rho_{2}^{\mathbb{R}}\oplus 1$. Here $\rho_{2}^{\mathbb{R}}$ is a
representation of $U(1)$ which factors through the quotient
$\operatorname{U}(1)/\pm I$, so its overall representation of
$\operatorname{Sp}(n)\cdot U(1)$ is given as the composition
$\displaystyle\operatorname{Sp}(n)\cdot\operatorname{U}(1)\twoheadrightarrow\operatorname{U}(1)/\pm
I\mathop{\to}^{\rho_{2}}\operatorname{GL}(1,\mathbb{C})$. So this sum is
isomorphic to the real isotropy representation.

Plugging in the generator calculated above, one obtains:

$\left(\begin{array}[]{c|c|c}\begin{array}[]{c|c}Id&0\\\ \hline\cr
0&R(t)\end{array}&\cdots&0\\\ \hline\cr\vdots&\ddots&\vdots\\\ \hline\cr
0&\cdots&\begin{array}[]{c|c}Id&0\\\ \hline\cr 0&R(t)\end{array}\\\
\end{array}\right)\oplus(R(t))\oplus 1.$

We note that this matrix contains $n+1$ generators of the fundamental group of
$\operatorname{SO}(4n+3)$ on the diagonal, and can thus be expressed as the
product of $n+1$ generators of the fundamental group $\mathbb{Z}/2$. This is
equal to the parity of $n+1$, so the isotropy representation lifts for $n+1$
even. ∎

We summarise the information of the transitive actions on spheres in Table 1.

Lie group | Manifold | Isotropy Subgroup | Isotropy Representation |  $G$-invariant spin ?
---|---|---|---|---
$\operatorname{SO}(n+1)$ | $S^{n}$ | $\operatorname{SO}(n)$ | $\lambda_{n}$ | No
$\operatorname{U}(n+1)$ | $S^{2n+1}$ | $\operatorname{U}(n)$ | $\mu_{n}^{\mathbb{R}}\oplus 1$ | No
$\operatorname{SU}(n+1)$ | $S^{2n+1}$ | $\operatorname{SU}(n)$ | $\mu_{n}^{\mathbb{R}}\oplus 1$ | Yes
$\operatorname{Sp}(n+1)$ | $S^{4n+3}$ | $\operatorname{Sp}(n)$ | $\nu_{n}^{\mathbb{R}}\oplus 1\oplus 1\oplus 1$ | Yes
$\operatorname{Sp}(n+1)\operatorname{Sp}(1)$ | $S^{4n+3}$ | $\operatorname{Sp}(n)\operatorname{Sp}(1)$ | $\tilde{\nu}_{n}^{\mathbb{R}}\oplus\operatorname{Ad}^{\operatorname{Sp}(1)}$ |  $n$ odd
$\operatorname{Sp}(n+1)\operatorname{U}(1)$ | $S^{4n+3}$ | $\operatorname{Sp}(n)\operatorname{U}(1)$ | $\tilde{\nu}_{n}^{\mathbb{R}}\oplus\operatorname{Ad}^{\operatorname{Sp}(1)}|_{\operatorname{U}(1)}$ |  $n$ odd
$\operatorname{G}_{2}$ | $S^{6}$ | $\operatorname{SU}(3)$ | $\mu_{3}^{\mathbb{R}}$ | Yes
$\operatorname{Spin}(7)$ | $S^{7}$ | $\operatorname{G}_{2}$ | $\zeta$ | Yes
$\operatorname{Spin}(9)$ | $S^{15}$ | $\operatorname{Spin}(7)$ | $\lambda_{7}\oplus\Delta_{7}$ | Yes
Table 1: Invariance of the spin structure on the sphere by transitive and
effective Lie group actions.

###  2.1 Closing remarks

As we have shown in the main theorem, there are cases where the spin structure
of the sphere is not $G$-invariant, but we can then take a double covering
which does preserve the spin structure. In all of the non-lifting cases, there
is in fact a unique connected double covering up to isomorphism. The next
remark focus on them.

######  Remark 2.4.

For $\operatorname{SO}(n+1)$ acting on $S^{n}$, the double covering is
precisely $\operatorname{Spin}(n)$ acting non-effectively on $S^{n}$. More
explicitly, the action is given by $\mu_{x}(v)=\lambda(x)v$ for
$x\in\operatorname{Spin}(n+1)$ and $v\in S^{n}$.

For $\operatorname{U}(n+1)$ acting on $S^{2n+1}$, the double covering is
$\operatorname{MU}(n+1):=\\{(A,z)\in U(n+1)\times\mathbb{C}^{\times}\mid\det
A=z^{2}\\}$ (sometimes called the metaunitary group).

For $\operatorname{Sp}(n+1)\cdot\operatorname{Sp}(1)$ and
$\operatorname{Sp}(n+1)\cdot U(1)$ the double coverings are
$\operatorname{Sp}(n+1)\times\operatorname{Sp}(1)$ and
$\operatorname{Sp}(n+1)\times U(1)$, respectively.

Now, given any connected group $G$ acting transitively on a sphere by
isometries via $\alpha:G\to\operatorname{Isom}(S^{n})$, then
$G/\operatorname{Ker}(\alpha)$ is one of the groups in Table 1. Then there
exists a $G$-invariant spin structure on $S^{n}$ if and only if either the
last column is “Yes” or the map $G\twoheadrightarrow
G/\operatorname{Ker}(\alpha)$ factors through the aforementioned connected
double covering. This classifies all groups $G$ acting transitively on a
sphere by isometries for which the sphere admits a $G$-invariant spin
structure.

The next proposition finds a nice relationship between $G$-invariant spin
structures on $G/H$ and finite subgroups of $G$:

######  Proposition 2.5.

Let $M=G/H$ be a spin homogeneous space where $H$ is connected. Then, a spin
structure in $G/H$ is $G$-invariant if and only if the spin structure is
invariant by all subgroups of $G$ isomorphic to $\mathbb{Z}_{2}$.

###### Proof.

One direction is trivial, since if the spin structure is $G$-invariant, then
it is also invariant by all its subgroups. Thus, we only need to proof the
converse. Assume that there is a spin structure which is not $G$-invariant,
then there exists $\alpha\in\pi_{1}(H)$ such that $\sigma_{*}(\alpha)\neq 0$.
Firstly, we want to find an adequate loop $a:S^{1}\longrightarrow H$, such
that $[a]=\alpha$. Let $i:T^{r}\hookrightarrow H$ be the inclusion of a
maximal torus, then the induced map
$i_{*}:\mathbb{Z}^{r}\cong\pi_{1}(T,e)\longrightarrow\pi_{1}(H,e)$ is
surjective [BTD13, Theorem 7.1]. Alternatively, one can see that $G/T$ is
simply connected by arguing that $G/T$ is a complex manifold with a CW-
structure with only even dimensional cells, and then use the long exact
sequence of homotopy.

Then, there exists $x=(x_{1},...,x_{r})\in\mathbb{Z}^{r}$ such that
$i_{*}(x)=\alpha$. Note that we may assume without lost of generality that $x$
is primitive in $\mathbb{Z}^{r}$ (i.e there is no $y\in\mathbb{Z}^{r}$ and
$\lambda\in\mathbb{Z}$ with $|\lambda|>1$ such that $x=\lambda y$). Thus, we
have a map $\tilde{a}:\mathbb{R}\longrightarrow\mathbb{R}^{r}$ such that
$\tilde{a}(t)=xt$, which induces a loop $a:S^{1}\longrightarrow T^{s}$ whose
class is $(x_{1},...,x_{r})$. Moreover, this map is an injective group
morphism. Therefore, we have seen that we can choose the representative of
$\alpha\in\pi_{1}(H)$ to be an injective group morphism
$a:S^{1}\longrightarrow H$ (note that if we choose a non primitive element,
the map may not be injective).

We want to see that the involution $a(\tfrac{1}{2})$ does not preserve the
spin structure. Because $\sigma_{*}(\alpha)\neq 0$, we can deduce that the
spin structure is not $S^{1}$-invariant, thus we have the short exact sequence

$1\longrightarrow\mathbb{Z}_{2}\longrightarrow S^{\prime}\longrightarrow
S_{1}\longrightarrow 1,$

where $S^{\prime}\cong S^{1}$ is the double covering which preserves the spin
structure of $G/H$. Then, the lift of $\mathbb{Z}_{2}$ is a subgroup of
$S^{1}$, which implies that it is $\mathbb{Z}_{4}$. Consequently, the
involution action $\mathbb{Z}_{2}=\\{Id,a(\tfrac{1}{2})\\}$ does not leave the
spin structure invariant. ∎

######  Example 2.6.

We can find explicitly a subgroup $\mathbb{Z}_{2}$ in the case of $S^{n}$ with
the group action of $\operatorname{SO}(n)$. By the above proposition, we can
pick the involution $f=\mu_{\alpha_{n}(\tfrac{1}{2})}$. The isometry
$f:S^{n}\longrightarrow S^{n}$ is given by restricting the linear map
$A:\mathbb{R}^{n+1}\longrightarrow\mathbb{R}^{n+1}$, where
$A\in\operatorname{SO}(n+1)$ is the matrix

$A=\left(\begin{array}[]{c|c}Id_{n-1}&0\\\ \hline\cr
0&-Id_{2}\end{array}\right).$

We claim that the spin structure of $S^{n}$ is not invariant under this
action. Indeed, the induced map by $f$ on the frame bundle
$FS^{n}=\operatorname{SO}(n+1)$,
$f_{A}:\operatorname{SO}(n+1)\longrightarrow\operatorname{SO}(n+1)$, is simply
the matrix multiplication $X\mapsto AX$. Thus it lifts to a map
$\tilde{f}:\operatorname{Spin}(n+1)\longrightarrow\operatorname{Spin}(n+1)$
which fulfils that $\tilde{f}(x)=(e_{n}\cdot e_{n+1})\cdot x$, where $\cdot$
is the Clifford multiplication (see the appendix for details on Clifford
algebras). Since $e_{n}\cdot e_{n+1}\cdot e_{n}\cdot e_{n+1}=-1$, the map
$\tilde{f}^{2}(x)=-x$. Hence, the lift of the action on the spin structure is
given by the group $\mathbb{Z}_{4}$, which implies that the spin structure is
not invariant.

Finally, it would be interesting to study the same question of G-invariant
spin structures on other homogeneous spaces. The starting point of this
problem has two difficult conditions to check; firstly,given a homogeneous
space $G/H$, we need to know which Lie groups acts transitively and
effectively on it and secondly we need to know if it admits a spin structure.
The first condition has been studied in [HS68, Oni68], while we refer to
[CGT93, AC19] for the study of spin structures on some classes of homogeneous
spaces.

For example, we can consider the complex projective space $\mathbb{C}P^{m}$,
which is known to be spin if and only if $m$ is odd. Then, if $m=2n+1$, the
only groups that act effectively and transitively on them are
$\operatorname{SU}(2n+2)$ and $\operatorname{Sp}(n+1)$ (see [Oni68]).
Therefore, its unique spin structure is invariant by both group actions.

## Appendix: Exceptional group actions and isotropy representations

Since the octonions are a crucial tool in understanding the three exceptional
Lie group actions, we will recall their properties quickly. Let $\mathbb{O}$
be the eight dimensional octonion algebra over $\mathbb{R}$, with basis
$\\{1=i_{0},i_{1},\dots,i_{7}\\}$ satisfying the usual relations:

$\displaystyle 1i_{k}$ $\displaystyle=$ $\displaystyle i_{k}1=i_{k},\textrm{
for all k in }\\{0,\dots 7\\}$ $\displaystyle i_{k}^{2}$ $\displaystyle=$
$\displaystyle-1,\textrm{ for all k in}\\{1,\dots 7\\}$ $\displaystyle
i_{k}i_{k+1}$ $\displaystyle=$ $\displaystyle
i_{k+3},\,i_{k+1}i_{k+3}=i_{k},\,i_{k+3}i_{k}=i_{k+1},\textrm{ for all k
in}\\{1,\dots 7\\},$

where the last indices are interpreted modulo 7. Any element of $\mathbb{O}$
can be written as $z=\sum_{k=0}^{7}a_{k}i_{k}$, for some real numbers $a_{k}$.
The octonionic conjugation is given by
$\bar{z}:=a_{0}i_{0}-\sum_{k=1}^{7}a_{k}i_{k}$ and the real and imaginary part
of $z$ by $\textrm{Re}z:=\frac{z+\bar{z}}{2}$ and
$\textrm{Im}z:=\frac{z-\bar{z}}{2}$ respectively. The inner product is defined
for all $x,y$ in $\mathbb{O}$ by $\langle x,y\rangle=\textrm{Re}(x\bar{y})$,
which is the usual scalar product on $\mathbb{R}^{8}$. Any triple of unit
imaginary octonions $(e_{1},e_{2},e_{3})$, such that $e_{1}$ anticommutes with
$e_{2}$, and $e_{3}$ anticommutes with $e_{1},\,e_{2}$ and $e_{1}e_{2}$, known
as a Cayley triple, generates the entire octonions.

#### The action of $\operatorname{G}_{2}$ on $S^{6}$:

The group $\operatorname{G}_{2}$ is the automorphism group

$\operatorname{G}_{2}:=\\{\phi\in\operatorname{GL}_{\mathbb{R}}(\mathbb{O}):\phi(xy)=\phi(x)\phi(y),\,\forall
x,y\in\mathbb{O}\\}$

of the 8-dimensional real vector space $\mathbb{O}\cong\mathbb{R}^{8}$. Since
$\phi(1)=1$, and $\phi$ preserves the inner product
$\langle\cdot,\cdot\rangle$ for all $\phi$ in $\operatorname{G}_{2}$,
$\operatorname{G}_{2}$ acts on the imaginary octonions
$\textrm{Im}(\mathbb{O})\cong\mathbb{R}^{7}$, and
$\operatorname{G}_{2}\leq\operatorname{SO}(\textrm{Im}(\mathbb{O}))\cong\operatorname{SO}(7)$.
We call this seven dimensional irreducible representation
$\zeta:G_{2}\rightarrow\operatorname{SO}(\textrm{Im}(\mathbb{O}))$.

The action restricts to a left action $\operatorname{G}_{2}\times
S^{6}\rightarrow S^{6},\,(\phi,v)\mapsto\phi(v)$ on the unit imaginary
octonions
$S^{6}:=\\{(v_{0},...,v_{n})\in\textrm{Im}(\mathbb{O})\cong\mathbb{R}^{7}:\sum|v_{i}|^{2}=1\\}$.
Since, given any two Cayley triples, there exists a unique automorphism of
$\mathbb{O}$ mapping the first to the second, this last action is transitive.

We now want to compute the isotropy subgroup ${\operatorname{G}_{2}}_{i_{1}}$
stabilizing the element $i_{1}$. Note that the field of complex numbers lives
naturally inside of $\mathbb{O}$ as $\mathbb{C}=span\\{1,i_{1}\\}$. Any
element $z$ in $\mathbb{O}$ can be written as
$z=(a_{0}+a_{1}i_{1})+(a_{2}+a_{4}i_{1})i_{2}+(a_{3}+a_{7}i_{1})i_{3}+(a_{5}+a_{6}i_{1})i_{5}=:z_{0}+z_{1}i_{2}+z_{2}i_{3}+z_{3}i_{5}$,
and we get an isomorphism
$f:\mathbb{O}\rightarrow\mathbb{C}\oplus\mathbb{C}^{3},\,z\mapsto
z_{0}+(z_{1},z_{2},z_{3})^{T}$. It is easy to see that if $\phi\in G_{2}$ is
an automorphism fixing $i_{1}$, then
$\phi(z)=z_{0}+z_{1}\phi(i_{2})+z_{2}\phi(i_{3})+z_{3}\phi(i_{5})$. It
therefore fixes $\mathbb{C}$, preserves $\mathbb{C}^{3}$, and induces a
$\mathbb{C}$-linear transformation of $\mathbb{C}^{3}$ given by
$f\circ\phi(z)=z_{0}+A(z_{1},z_{2},z_{3})^{T}$, with
$A=(\phi(i_{2}),\phi(i_{3}),\phi(i_{5}))$. Since $\phi$ is invertible, $A$ is
in $\operatorname{GL}(3,\mathbb{C})$. Moreover the inner product
$\langle,\rangle$ on $\mathbb{O}$ induces the standard real inner product on
$\mathbb{R}^{6}\cong\mathbb{C}^{3}$ which is preserved by $\phi$, hence $A$ is
in
$\operatorname{U}(3)=\operatorname{SO}(6)\cap\operatorname{GL}(3,\mathbb{C})$.

Next we claim that two vectors $v,w\in\mathbb{C}^{3}$ are orthonormal for the
standard Hermitian form if and only if $(i_{1},v,w)$ is a Cayley triple.
Indeed, given two octonions $v$ and $w$ are orthonormal for the Hermitian form
if and only if $(v,i_{1}v,w,i_{1}w)$ are orthonormal for the standard real
form. Since they are orthogonal to $i_{1}$ itself, the result follows. Since
$\operatorname{G}_{2}$ acts simply transitively on Cayley triples,
${\operatorname{G}_{2}}_{i_{1}}$ acts simply transitively on Cayley triples of
the form $(i_{1},v,w)$. We conclude that ${\operatorname{G}_{2}}_{i_{1}}$ acts
simply transitively on the pairs of orthonormal vectors in $\mathbb{C}^{3}$.

We claim that ${\operatorname{G}_{2}}_{i_{1}}\subseteq\operatorname{SU}(3)$;
to do this, it remains to show that every element $A$ as above has determinant
one. Since $A\in\operatorname{U}(3)$, by the spectral theorem there exists an
orthonormal eigenbasis for $A$. By the preceding paragraph, there is an
element $\psi\in{\operatorname{G}_{2}}_{i_{1}}$ such that $B:=\psi^{-1}A\psi$
has $i_{2},i_{3}$ as eigenvectors: let the eigenvalues be
$\lambda_{1},\lambda_{2}$ in $\mathbb{(}O)$. Now we compute that
$B(i_{5})=B(i_{2}i_{3})=B(i_{2})B(i_{3})=(\lambda_{1}i_{2})(\lambda_{2}i_{3})=\overline{\lambda_{1}\lambda_{2}}i_{5}$.
Thus the determinant of $A$ is one, as required.

Since ${\operatorname{G}_{2}}_{i_{1}}\subseteq\operatorname{SU}(3)$ and
${\operatorname{G}_{2}}_{i_{1}}$ acts transitively on the pairs of orthonormal
vectors in $\mathbb{C}^{3}$, it follows that
${\operatorname{G}_{2}}_{i_{1}}=\operatorname{SU}(3)$, as a transformation in
$\operatorname{SU}(3)$ is determined by the image of the first two basis
vectors of $\mathbb{C}^{3}$.

As a result the real isotropy representation is just the standard
6-dimensional representation $\mu_{3}^{\mathbb{R}}$ of $SU(3)$.

#### The action of $\operatorname{Spin}(7)$ on $S^{7}$:

The spin group $\operatorname{Spin}(n)$ is the connected double covering of
$\operatorname{SO}(n)$. In order to understand the action of
$\operatorname{Spin}(7)$ and $\operatorname{Spin}(9)$ on spheres, we need to
study the real spin representation of these groups. Explicitly,
$\operatorname{Spin}(n)$ can be realised in the following way. Let
$\\{e_{i}\\}_{i=1}^{n}$ be an orthonormal basis of
$(\mathbb{R}^{n},\langle\cdot,\cdot\rangle)$ (with the usual scalar product).
Recall that the Clifford algebra $Cl_{n}$ is the algebra over $\mathbb{R}$
generated by the vectors $\\{e_{i}\\}_{i=1}^{n}$ satisfying the relations
$e_{i}\cdot e_{j}+e_{j}\cdot e_{i}=-2\delta_{ij}.$ Then
$\operatorname{Spin}(n)$ is the subgroup of $Cl_{n}$ given by

$\operatorname{Spin}(n):=\\{x_{1}\cdot x_{2}\dots
x_{2k}:x_{i}\in\mathbb{R}^{n},\|x_{i}\|=1,i\in(1,\dots 2k)\\}\subset Cl_{n}$

Note that the double covering is given by
$\lambda:\operatorname{Spin}(n)\rightarrow\operatorname{SO}(n),\,s\mapsto\lambda(s),\lambda(s)x=sxs^{-1}.$

The spin representation
$\Delta_{n}:\operatorname{Spin}(n)\rightarrow\operatorname{GL}(\Sigma_{n})$ is
obtained by restricting an irreducible representation of $Cl_{n}$ to
$\operatorname{Spin}(n)$. In the case $n=1\mod 4$ (and hence for $n=9$), the
Clifford algebra $Cl_{4k+1}$ has a single irreducible representation, which
splits under $\operatorname{Spin}(4k+1)$ into two equivalent representations,
either one of which defines $\Sigma_{n}$.

In the case $n=3\mod 4$ (and hence for $n=7$), the representation of the
Clifford algebra $Cl_{4k+3}$ is a sum of two inequivalent irreducible
representations. The restriction to $\operatorname{Spin}(4k+3)$ is a unique
representation, independent of which irreducible representation of $Cl_{4k+3}$
is used. In particular, in dimension $n=7\mod 8$,
$Cl_{8k+7}=\mathbb{R}(2^{3+4k})\oplus\mathbb{R}(2^{3+4k})$ and
$\Sigma_{n}=\mathbb{R}^{2^{3+4k}}$. From this we get immediately the action of
$\operatorname{Spin}(7)$ on $\mathbb{R}^{8}\cong\mathbb{O}$ and hence on the
unit sphere $S^{7}\cong\\{v\in\mathbb{O}\cong\mathbb{R}^{8}:\|v\|=1\\}$.

We claim that the action of $\operatorname{Spin}(7)$ on $S^{7}$ is transitive
with isotropy group $\operatorname{G}_{2}$. Here, $\operatorname{G}_{2}$ is
realised inside of $\operatorname{Spin}(7)$ as follows:
$\operatorname{G}_{2}<\operatorname{SO}(7)$ as above and since
$\operatorname{G}_{2}$ is simply connected, this inclusion lifts to
$\operatorname{G}_{2}<\operatorname{Spin}(7)$.

To prove the claim, we first show that $\operatorname{Spin}(7)$ acts
transitively on $S^{7}$. To see this we realise the spin representation
explicitly in the following known way. There is an action of
$Cl(\mathbb{R}^{7})\cong Cl(\textrm{Im}(\mathbb{O}))$ on $\mathbb{O}$ given by
the left multiplication $\textrm{Im}(\mathbb{O})\times\mathbb{O}\to\mathbb{O}$
of imaginary octonions. The fact that this defines an action of
$Cl(\mathbb{R}^{7})$ follows from explicitly checking the Clifford relations—a
consequence of the multiplication table of $\mathbb{O}$. As this is a
nontrivial module, it must be one of the irreducible modules of dimension
eight of $Cl(\mathbb{R}^{7})$, therefore its restriction to
$\operatorname{Spin}(7)$ is the spin representation. The Lie algebra
$\mathfrak{so}(7)\subseteq Cl(\mathbb{R}^{7})$ is the span of commutators of
the generators under Clifford multiplication. Let us compute
$\mathfrak{so}(7)\cdot 1$, where $1\in\mathbb{O}$ is considered a unit vector,
hence in $S^{7}$. For $v,w\in\mathbb{R}^{7}$, so $v\wedge
w\in\mathfrak{so}(7)$, we have $(v\wedge w)\cdot 1=vw-wv$, the commutator in
the octonions. We see from the multiplication table that commutators of
imaginary octonions span the imaginary octonions again. Thus,
$\mathfrak{so}(7)\cdot 1=\mathbb{R}^{7}$, the imaginary octonions. Therefore,
the orbit of $1$ under $\operatorname{Spin}(7)$ has dimension $7$. Since
$\operatorname{Spin}(7)$ is compact, the orbit is a closed submanifold of
$S^{7}$, hence all of $S^{7}$. This implies transitivity.

We now compute the isotropy. Note that
$\operatorname{G}_{2}<\operatorname{Spin}(7)$ acts on the spin representation
$\mathbb{R}^{8}$. Since the only nontrivial irreducible representation of
$\operatorname{G}_{2}$ of dimension $\leq 8$ has dimension $7$, and all
finite-dimensional representations of $\operatorname{G}_{2}$ are completely
reducible, it follows that there is a trivial
$\operatorname{G}_{2}$-subrepresentation of $\mathbb{R}^{8}$. That means there
is a fixed vector, hence a unit fixed vector $v\in S^{7}$. Let $H$ be the
isotropy group at $v$ then we have $\operatorname{G}_{2}<H$ by definition of
$v$. Since $\operatorname{Spin}(7)$ acts transitively on $S^{7}$, we have
$\dim H=21-7=14=\dim\operatorname{G}_{2}$. Thus, $\operatorname{G}_{2}<H$ is
an open subgroup, and as $G_{2}$ is compact, it is the connected component of
the identity. Finally, since $\operatorname{Spin}(7)/H\cong S^{7}$ is simply
connected, by the long exact sequence on homotopy groups, $|\pi_{0}(H)|=1$, so
$H=\operatorname{G}_{2}$ as desired.

Since the isotropy representation is a nontrivial seven-dimensional
representation of $G_{2}$, it has to be isomorphic to $\zeta$.

#### The action of $\operatorname{Spin}(9)$ on $S^{15}$.

In dimension $n=1\mod 8$, we have $Cl_{8k+1}=\mathbb{C}(2^{4k})$, which after
restricting to $\operatorname{Spin}(8k+1)$ yields
$\Sigma_{n}=\mathbb{R}^{2^{4k}}$. Therefore in dimension 9,
$\Sigma_{9}=\mathbb{R}^{16}$ and $\operatorname{Spin}(9)$ acts via the spin
representation on the unit sphere
$S^{15}\subset\mathbb{R}^{16}\cong\mathbb{O}^{2}$. Moreover, we identify:

$\mathbb{R}^{9}\cong\mathbb{R}\oplus\mathbb{O}=\Biggl{\\{}i\begin{pmatrix}r&L_{z}\\\
L_{\bar{z}}&-r\end{pmatrix}:r\in\mathbb{R},z\in\mathbb{O}\Biggr{\\}}\subseteq\textrm{End}_{\mathbb{C}}(\mathbb{C}\otimes\mathbb{O}^{2}),$

where $L_{z}$ denotes the map associated to the left multiplication by $z$ in
$\mathbb{O}$. There is, similarly to $\operatorname{Spin}(7)$, an action of
$Cl(\mathbb{R}\oplus\mathbb{O})$ on $\mathbb{C}\otimes\mathbb{O}^{2}$ given by
left multiplication. Note that in this case, $\mathbb{C}\otimes\mathbb{O}^{2}$
is the unique irreducible representation of the real Clifford algebra
$Cl(\mathbb{R}\oplus\mathbb{O})$ and the action furnishes an isomorphism
$Cl(\mathbb{R}\oplus\mathbb{O})\cong\textrm{End}_{\mathbb{C}}(\mathbb{C}\otimes\mathbb{O}^{2})$.
Restricting this to the even Clifford algebra, it splits into two isomorphic
real representations, $\mathbb{O}^{2}$ and $i\mathbb{O}^{2}$. Hence, these
both identify with $\Sigma_{9}$.

We can compute directly, as in the case of $\operatorname{Spin}(7)$, that
$\operatorname{Spin}(9)$ acts transitively on $S^{15}$. Take the vector
$(1,0)\in\mathbb{O}^{2}$. Since the orbit
$\operatorname{Spin}(9)\cdot(1,0)\subseteq S^{15}$ is closed, to show it is
all of the sphere it suffices to prove that the tangent space
$\mathfrak{so}(9)\cdot(1,0)$ has dimension at least $15$. Now,
$\mathfrak{so}(9)$ is the span in $Cl(\mathbb{R}\oplus\mathbb{O})$ of
commutators of generators. We compute that

$\Biggl{[}i\begin{pmatrix}r&L_{u}\\\
L_{\bar{u}}&-r\end{pmatrix},i\begin{pmatrix}r^{\prime}&L_{v}\\\
L_{\bar{v}}&-r^{\prime}\end{pmatrix}\Biggr{]}=-\begin{pmatrix}L_{u}L_{\bar{v}}-L_{v}L_{\bar{u}}&2(rL_{v}-r^{\prime}L_{u})\\\
2(r^{\prime}L_{\bar{u}}-rL_{\bar{v}})&L_{\bar{u}}L_{v}-L_{\bar{v}}L_{u}\end{pmatrix}.$

Hence the tangent space is the span of the vectors

$((r,u)\wedge(r^{\prime},v))\cdot(1,0)=-(u\bar{v}-v\bar{u},2(r^{\prime}\bar{u}-r\bar{v})),r,r^{\prime}\in\mathbb{R},u,v\in\mathbb{O}.$
(1)

Taking $r=1$ and $r^{\prime}=0,u=0$, we obtain $(0,-2\overline{v})$: this
spans $(0,\mathbb{O})$. Taking $u=1_{\mathbb{O}},r=r^{\prime}=0$, we obtain
$(\bar{v}-v,0)$: this spans $(\text{Im}(\mathbb{O}),0)$. So the overall span
is at least $\text{Im}(\mathbb{O})\oplus\mathbb{O}$, which is the tangent
space to $S^{15}$ and we are done.

Note that we can explicitly compute the isotropy Lie algebra, call it
$\mathfrak{h}$, of $(1,0)$ in $\mathfrak{so}(9)$. We claim that this is:

$\text{span}\bigl{(}T_{u,v}:=u\wedge v+\frac{1}{2}([u,v]\wedge
1_{\mathbb{O}})\mid
u,v\in\text{Im}(\mathbb{O})\bigr{)}\subseteq\wedge^{2}\mathbb{O}=\mathfrak{so}(8)\subseteq\wedge^{2}(\mathbb{R}\oplus\mathbb{O})=\mathfrak{so}(9).$
(2)

Indeed, plugging $T_{u,v}$ into (1) with $r=r^{\prime}=0$ we get zero, since
$(uv-vu)-\frac{1}{2}([u,v]+[u,v])=0$. So $T_{u,v}\in\mathfrak{h}$. On the
other hand, these elements span a $21=(36-15)$-dimensional subspace, which is
the correct dimension. More explicitly, a complementary subspace to (2) is
$\mathbb{R}\wedge\mathbb{O}\oplus\mathrm{Re}(\mathbb{O})\wedge\mathrm{Im}(\mathbb{O})$,
which we already showed acts injectively on $(1,0)$, with image the tangent
space to $S^{15}$.

The isotropy representation is the tangent space
$\text{Im}(\mathbb{O})\oplus\mathbb{O}$. The action of the isotropy Lie
algebra decomposes into two subrepresentations,
$(\text{Im}(\mathbb{O}),0)\oplus(0,\mathbb{O})$. The action on the first
factor induces a homomorphism $\varphi:\mathfrak{h}\to\mathfrak{gl}(7)$.
Explicitly, this action is given on $\text{Im}(\mathbb{O})$ as follows, for
$u,v,w\in\text{Im}(\mathbb{O})$:

$T_{u,v}\cdot w=u\cdot(v\cdot w)-v\cdot(u\cdot w)-([u,v])\cdot w,$ (3)

with multiplication taken in the octonions. This can be interpreted as “the
failure of left multiplication of $\mathrm{Im}(\mathbb{O})$ on itself to
define a Lie algebra representation” (although note that
$\mathrm{Im}(\mathbb{O})$ is not a Lie algebra under commutators).

We claim that the image of this homomorphism lies in $\mathfrak{so}(7)$.
Suppose $u=i_{1},v=i_{2},w=i_{j}$. Then $u(vw)=\pm(uv)w=\mp(vu)w=-v(uw)$, with
the sign $\pm$ positive if $j\in\\{1,2,4\\}$, and negative otherwise. It
follows that the action of $T_{u,v}$ is given by $i_{3}\mapsto-i_{6}\mapsto
i_{3}$ and $i_{5}\mapsto i_{7}\mapsto-i_{5}$. In other words,
$\varphi(T_{u,v})$ is zero on the three-dimensional space spanned by
$i_{1},i_{2},i_{4}$ (a line in the octonion Fano plane), and acts by
$90^{\circ}$ rotation on the two two-dimensional spaces spanned by
$i_{3},i_{6}$ and by $i_{5},i_{7}$ (which with $i_{4}$ form the other three
lines through $i_{4}$). In particular, $\varphi(T_{u,v})\in\mathfrak{so}(7)$
(it is skew-symmetric). Through automorphisms of the Fano plane, this
describes $\varphi(T_{i_{k},i_{\ell}})$ for all indices $k,\ell$. Indeed, each
transformation is zero on the span of $i_{k},i_{\ell}$, and $i_{k}i_{\ell}$
(which represent a line in the octonionic Fano plane), and is a $90^{\circ}$
rotation on each of the complementary two-dimensional spaces of the form
$\text{Span}(i_{m},(i_{k}i_{\ell})i_{m})$ (representing the two other lines in
the octonionic Fano plane containing $\pm i_{k}i_{\ell}$).

We claim furthermore that this gives an isomorphism
$\mathfrak{h}\to\mathfrak{so}(7)$. Since the source and target have the same
dimension, it suffices to show that the $\varphi(T_{u,v})$ are linearly
independent: since $T_{i_{k},i_{\ell}}$ acts on some of the $i_{j}$ as
multiplication by $i_{k}i_{\ell}$ and by zero on others, it suffices to check
that the three choices of indices $(k,\ell)$ with the same product
$i_{k}i_{\ell}$ (i.e., the representing the three lines in the octonionic Fano
plane through $i_{k}i_{\ell}$) give linearly independent elements
$\varphi(T_{i_{k},i_{\ell}})$. That is,
$\varphi(T_{i_{1},i_{2}}),\varphi(T_{i_{6},i_{3}})$, and
$\varphi(T_{i_{5},i_{7}})$ should be linearly independent (and cyclic
permutations of $(1,2,3,4,5,6,7)$). This is true: if
$\lambda_{1}T_{i_{1},i_{2}}+\lambda_{2}T_{i_{6},i_{3}}+\lambda_{3}T_{i_{5},i_{7}}$
acts by zero on $i_{1},i_{2}$, then $\lambda_{2}=-\lambda_{3}$; similarly to
act by zero on $i_{6},i_{3}$ we need $\lambda_{3}=-\lambda_{1}$, and finally
to act by zero on $i_{5},i_{7}$ we need $\lambda_{3}=-\lambda_{1}$. So
$\lambda_{1}=\lambda_{2}=\lambda_{3}=0$. Similarly, applying cyclic
permutations of the indices $(1,2,3,4,5,6,7)$ we get that a linear combination
of the $T_{i_{k},i_{\ell}}$ which acts by zero is the zero combination.

As a result, the isotropy representation
$\text{Im}(\mathbb{O})\oplus\mathbb{O}$ of $\mathfrak{h}$ realises
$\mathfrak{h}\cong\mathfrak{so}(7)$ with $\mathrm{Im}(\mathbb{O})$ the
standard $7$-dimensional representation. Note that the isotropy Lie subgroup
$H$ must be simply-connected by the long exact sequence on homotopy groups for
$H\to\operatorname{Spin}(9)\to S^{15}$, hence $H\cong\operatorname{Spin}(7)$.
Therefore the other summand $(0,\mathbb{O})$ of the isotropy representation
must be isomorphic to the spin representation, otherwise the isotropy
representation would not be faithful.

This is a contradiction, because $H$ acts effectively on $S^{15}$. More
generally, if $G$ acts effectively on a compact connected manifold $M$ and $H$
is the stabilizer of $p\in M$ is compact, we claim that the isotropy
representation is faithful. Let $K<H$ be the kernel of the isotropy
representation. Then $(T_{p}M)^{K}=T_{p}(M^{K})$, since there is an
$K$-equivariant isomorphism of a neighbourhood of $p$ in $M$ and of $p$ in
$T_{p}M$ (or because we can take an $K$-invariant metric on $M$ and then $K$
preserves the exponential flow beginning at $p$, hence all geodesics on $X$
through $p$). As a result, since $K$ acts trivially on $T_{p}M$ and $M$ is
compact and connected, $M^{K}$ is an open and closed subset of $M$, hence all
of $M$. So $K$ acts trivially on $M$. As the action of $G$ is effective, $K$
is trivial, as desired.

Thus, this explicitly realizes $H\cong\operatorname{Spin}(7)$ and the isotropy
representation as a direct sum of the standard and spin representations.

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# TCL: an ANN-to-SNN Conversion with Trainable Clipping Layers

Nguyen-Dong Ho  Kyung Hee University
<EMAIL_ADDRESS>Ik-Joon Chang  Kyung Hee University
<EMAIL_ADDRESS>
###### Abstract

Spiking-neural-networks (SNNs) are promising at edge devices since the event-
driven operations of SNNs provides significantly lower power compared to
analog-neural-networks (ANNs). Although it is difficult to efficiently train
SNNs, many techniques to convert trained ANNs to SNNs have been developed.
However, after the conversion, a trade-off relation between accuracy and
latency exists in SNNs, causing considerable latency in large size datasets
such as ImageNet. We present a technique, named as TCL, to alleviate the
trade-off problem, enabling the accuracy of 73.87% (VGG-16) and 70.37%
(ResNet-34) for ImageNet with the moderate latency of 250 cycles in SNNs.

###### Index Terms:

ANN-to-SNN conversion, trainable clipping layer, Spiking Neural Network

##  I Introduction

During the last decade, analog neural networks (ANNs) have shown rapid and
extensive progresses. ANNs demonstrates their outstanding performance by
surpassing the human-level accuracy for many applications such as image
processing, voice recognition, and language translation. However, such ANN
performance can be obtained at the cost of considerable power consumption.
This makes it difficult to operate ANNs at resource-constraint edge devices.
Unlike ANNs, spiking neural networks (SNNs) have event-driven behaviors,
delivering significantly lower power dissipation. Consequently, researchers
have considered SNNs as one of alternatives to ANNs for the resource-
constraint edge devices.

Nonetheless, the deployment of SNNs is limited since it is difficult to
efficiently train SNNs. Due to the non-differential and discontinuous
properties of SNNs, back-propagation cannot be applied for the training of
SNNs. Some researchers have overcome this problem by using approximate
techniques such as spike-base back-propagation [1, 2, 3, 4] and surrogate
gradient [5, 6, 7]. However, these techniques are only applicable to the
training of small size networks for small size datasets. Further, when SNNs
are trained based on the above techniques, forward and backward propagations
need to be computed every time-step, unlike ANNs. As a result, the direct
training approaches of SNNs suffer from considerably large overhead with
respect to computational complexity and training time.

Recently, some indirect training approaches of SNNs have been proposed, where
the training results of ANNs are converted to SNN. For instance, Y. Cao et al.
[8] succeeded in converting ANNs to SNNs by mapping the output of rectified
linear unit (ReLU) in ANNs to the spike rate in SNNs. Their technique shows
good performances for the datasets of MNIST and CIFAR-10. P. U. Diehl et al.
[9] developed the data-normalization technique that uses the model and the
dataset of ANNs to estimate the normalization factors, leading to more
improved conversion results. The authors of [10, 11, 12, 13] decided more
accurate normalization factors by closely analyzing the relation between the
activation of ANNs and the spike rate of SNNs. As a result, they successfully
converted even large ANNs trained with the ImageNet dataset to SNNs. However,
there is a trade-off relation between the accuracy and the latency of the
converted SNNs, more problematic at the ImageNet dataset. As a result, the
SNNs, converted by the above techniques, suffer from considerable accuracy
degradation under low latency constraints. N. Rathi et al. [14, 15] presented
a novel technique, so called hybrid training, that retrains the SNNs obtained
by ANN-to-SNN conversion, alleviating the trade-off relation. However, the
addional SNN training phase of the hybrid training technique suffers from
extremely heavy computations, as mentioned before, roughly 10 times larger
compared to that of the ANN training [14].

Compared to [8, 9, 10, 11, 12, 13, 14, 15], our work has significant
contributions as follows.

  * •

We formulate the reason why the trade-off relation between the accuracy and
the latency of the converted SNNs exists, indicating how to mitigate the
trade-off relation.

  * •

We propose a fundamental technique to ameliorate the trade-off relation
between the accuracy and the latency of SNNs, trainable clipping layer (TCL),
where the direct training of SNN is not required unlike [14, 15]. We ensure
that a clipping layer, whose clipping region is trained, follows a ReLU layer,
finding the optimal data-normalization factor to consider both accuracy and
latency in SNNs.

  * •

We further enhance the SNN accuracy under very low latency constraints by
properly controlling a L2-regularization coefficient.

From our experiments, SNNs based on our TCL technique show the following
classification accuracies of CIFAR-10, 93.33% at VGG-16 with 100-cycle latency
and 92.06% at ResNet-20 with 150 cycle latency. For the ImageNet dataset, the
accuracies of VGG-16 and ResNet-34 are 73.87% and 70.37% respectively, with
the latency of 250 cycles. At the very low latency below 40 cycles, VGG-16
provides the SNN accuracies of 92.60% for CIFAR-10 and 70.75% for ImageNet,
well validating our proposed technique.

##  II Preliminary backgrounds

###  II-A Spiking Neural Networks theory

We can consider two representative SNN models, integrate-and-fire (IF), and
leaky-integrate-and-fire (LIF) ones. It is well-known that the IF model is
easily converted from an ANN, considered as the SNN model throughout our work.
In the IF model, at time step $t$, the neuron $i$ in the $l^{th}$ layer has
the summation of weighted spike input, $z^{l}_{i}$, as follows.

$z^{l}_{i}(t)=\sum_{j}W^{l}_{ij}\theta^{l-1}_{j}(t)+b^{l}_{i}$ (1)

, where $W^{l}_{ij}$ is the synaptic weight, $b^{l}_{i}$ refers to the bias of
the neuron, and $\theta^{l-1}_{j}$ is the spike input from the neuron $j$ that
is in the previous layer. In the $l^{th}$ layer, the spike output of the
neuron, $\theta^{l}_{i}$, remains zero until the membrane potential,
$V^{l}_{i}$, reaches the threshold $V^{l}_{thr}$. At the time that $V^{l}_{i}$
becomes larger than or equal to $V^{l}_{thr}$, the spike output is fired.
Hence,

$\theta^{l}_{i}(t)=\begin{cases}1,&\text{if}\,\>V^{l}_{i}(t)\geq
V^{l}_{thr}\\\ 0,&\text{else.}\end{cases}$ (2)

After the firing, the membrane potential, $V^{l}_{i}$, is reset. There are two
approaches to reset $V^{l}_{i}$: reset-to-zero and reset-by-subtraction. Since
the reset-to-zero suffers from considerable information loss [10], the reset-
by-subtraction is employed throughout this work. Under this situation, at the
time step of $t$, $V^{l}_{i}$ can be modeled as follows.

$V^{l}_{i}(t)=V^{l}_{i}(t-1)+z^{l}_{i}(t)-V^{l}_{thr}\theta^{l}_{i}(t)$ (3)

###  II-B ANN to SNN conversion

The ReLU function is widely used as the activation function of ANNs, given by
the following one.

$a^{l}_{i}=max\left(0,\>\>\sum_{j}W^{l}_{ij}a^{l-1}_{j}+b^{l}_{i}\right)$ (4)

By comparing (4) to (1), an ANN-to-SNN converting algorithm can be obtained.
In SNNs, spike output signals are binary, only ’1’ or ’0’, implying that the
spike outputs do not have negative values. Therefore, (4) can be simply mapped
to (1) by converting the ReLU output to the spike rates of SNNs. For the
conversion, both weights and biases need to be normalized, namely data-
normalization, where weights ($W^{l}$) and biases ($b^{l}$) are normalized by
(5), and then, the threshold voltage of neurons is set to $1.0$.

$W^{l}=W^{l}\frac{\lambda^{l-1}}{\lambda^{l}}\text{\> and
\>}b^{l}=\frac{b^{l}}{\lambda^{l}}$ (5)

, where $\lambda^{l}$ is the normalization factor of the current layer, so
called norm-factor, $\lambda^{l-1}$ is the norm-factor of the previous layer.
The decision of norm-factors is more discussed in Section III-A.

It is well-known that it is difficult to model max-pooling and batch-
normalization in SNNs [10]. The max-pooling can be replaced with the other
pooling techniques such as average-pooling, well-modeled in SNNs. The authors
of [10] remove the batch-normalization, of (6) after the training of ANNs.

$BN(a)=\frac{\gamma}{\sigma}\left(a-\mu\right)+\beta$ (6)

, where $a$ is the input, $\mu$ and $\sigma$ are mean and variance of mini-
batch, and $\gamma$ and $\beta$ are two learned parameters of batch-
normalization. To prevent the accuracy loss of ANNs due to the removal of the
batch-normalization, they scale weights and biases of corresponding
convolution layers by using the following equation:

$W^{l}_{ij}=\frac{\gamma^{l}_{i}}{\sigma^{l}_{i}}W^{l}_{ij}\text{\> and
\>}b^{l}_{i}=\frac{\gamma^{l}_{i}}{\sigma^{l}_{i}}(b^{l}_{i}-\mu^{l}_{i})+\beta^{l}_{i}.$
(7)

Our ANN-to-SNN conversion is based on the above discussion. We apply the data-
normalization based on (5) and remove batch-normalization by using (7). We
replace max-pooling layers with average-pooling ones. Instead of using a soft-
max layer, not modeled in SNNs, we simply count the number of spiking signals
and take the maximum to classify outputs. At the first SNN layer, we feed
input signals with analog values, so called real coding, same as the technique
which is used in [10, 13, 15].

##  III Our contribution

In this section, we discuss how our TCL technique alleviates the trade-off
relation latency and accuracy of SNNs. Before this, we explain the reason why
the trade-off relation exists after ANN-to-SNN conversion.

###  III-A The trade-off between accuracy and latency

Figure 1: (A) Distribution of ANN activations (plotted in log-scale), (B)
Conversion with small $\lambda$, (C) Conversion with large $\lambda$.

In ANN-to-SNN conversion, an activation of ANNs, $a_{i}$, is mapped to a spike
rate of SNNs, $f_{i}$, as mentioned above. The spike rate, $f_{i}$, can be
estimated by following equation.

$f_{i}\approx\begin{cases}\frac{\left\lfloor\frac{a_{i}}{\Delta}\right\rfloor}{T},&\text{if}\,\>a_{i}\leq
a_{limit}\,\>(\Delta=\frac{a_{limit}}{T})\\\
1.0,&\text{if}\,\>a_{i}>a_{limit}\end{cases}$ (8)

, where $a_{limit}$ is the maximum value of activations mapped to the maximum
spike rate ($f_{max}=1$), and $\Delta$ is the quantization resolution. In ANN-
to-SNN conversion, the activations of ANNs are quantized and then, the
quantized activations are mapped to their corresponding spike rate, where some
quantization error is unavoidable. To reduce the quantization error due to
ANN-to-SNN conversion, we need to make smaller $\Delta$, which can be obtained
by increasing the latency, $T$, or decreasing $a_{limit}$. This is the reason
why after ANN-to-SNN conversion, latency and accuracy have trade-off relation.
In the ANN-to-SNN conversion with data-normalization, $\lambda$ is
proportional to $a_{limit}$, where smaller $\lambda$ with the fixed $T$ tends
to reduce $a_{limit}$. This results in the reduction of the quantization
error. However, from a certain point, $a_{limit}$ is lower than $a_{max}$ and
then, we have the conversion loss due to the clipping of activations, shown in
Fig. 1.B, degrading the accuracy of converted SNNs as well. On the other hand,
larger $\lambda$ accompanies the increment of the quantization error, as shown
in Fig. 1.C.

The work in [9] determined the norm-factor of each layer by taking the maximum
value among the activation parameters of the layer. However, in order to
reduce quantization error, this approach requires large latency in SNNs. The
authors of [10] and [13] decreased $\lambda$ by the following technique. In
ANNs, most activations, roughly 99.0% to 99.99%, are placed in the range of of
$[0,\;max/3]$. From this observation, they decide $\lambda$ by selecting the
value of 99.9%. The authors of [11] and [12] chose $\lambda$ based on the
operation of SNN. They sequentially run SNN layers and take the maximum
weighted spike output for $\lambda$, then scaled with the factor of $0.7\sim
0.9$. Although the smaller $\lambda$ due to these techniques reduces
quantization error, the conversion loss due to clipping error still causes
significant accuracy degradation in SNNs.

In this work, we propose a novel technique to alleviate the trade off between
quantization and clipping errors, where we decrease $a_{max}$ as small as
possible while maintaining the accuracy of ANNs accuracy. Such an approach
provides both low latency and high accuracy in SNNs, discussed in Section
III-B.

###  III-B Trainable Clipping Layer

Figure 2: Clipping Layer for ANN activation

To decide the norm-factor of a certain layer, instead of analyzing activations
of the corresponding ANN layer, we add a clipping layer after the ReLUs of the
ANN layer, shown in Fig. 2. The forward function of the clipping layer is
described in (9). Please, note that the clipping layer has a trainable
parameter, $\lambda$, which becomes the norm-factor for the data-
normalization. When backward computations of ANNs are processed, the gradients
of $\lambda$ are formulated as (10). $\lambda$ is trained based on the
learning rule of (11), where L2-regularization is used to optimize $\lambda$.

$\bar{a}=clip(a,\,\lambda)=\begin{cases}\lambda,&\text{if }a\geq\lambda\\\
a,&\text{else.}\end{cases}$ (9)

$\frac{\partial\bar{a}}{\partial\lambda}=\begin{cases}1,&\text{if
}a\geq\lambda\\\ 0,&\text{else.}\end{cases}$ (10)

$\lambda\rightarrow\lambda-\eta\alpha\lambda-\eta\frac{\partial
L}{\partial\lambda}=\lambda-\eta\alpha\lambda-\eta\frac{\partial\bar{a}}{\partial\lambda}\frac{\partial
L}{\partial\bar{a}}$ (11)

, where $\eta$ is the learning rate, $\alpha$ is the L2-regularization
coefficient, and $L$ is the ANN loss. We visualize the training process of
norm-factors, shown in Fig. 3. The results are obtained from the training of
VGG-16 for two datasets, CIFAR-10 and ImageNet. In (11), the L2-regularization
term ($-\eta\alpha\lambda$) tends to decrease $\lambda$, the norm-fact of the
current layer, as already mentioned. From a certain point, the decrement of
$\lambda$ causes training loss. Under this situation, the optimization term
($-\eta\frac{\partial\bar{a}}{\partial\lambda}\frac{\partial
L}{\partial\bar{a}}$) increases $\lambda$ to complement the training loss.
After some epochs, $\lambda$ starts to be stabilized to the optimal value with
respect to the training loss. Fig. 4 shows training and test losses of each
training epoch. We compare training and test losses of the original ANN
training, where our TCL is not used, to those of our TCL-based ANN training.
Further, we observe validation accuracies of each epoch, shown in Fig. 5,
clearly demonstrating that our TCL hardly affect the training results of
VGG-16.

Figure 3: Variation of each norm-factor during training VGG-16 Figure 4:
Training and test losses during training VGG-16 Figure 5: Training and test
accuracies during training VGG-16 Figure 6: Accuracy loss due to ANN-SNN
conversion for the dataset of CIFAR-10

To further prove the efficacy of our TCL, we investigate how our TCL affect
the loss of ANN-to-SNN conversion by comparing the conversion loss due to our
TCL to those due to various norm-factors, 99.99%, 99.9%, 99.8%, 99.7%, and
99.5%, shown in Fig. 6. The conversion loss due to our TCL is significantly
lower compared to the others, well showing that our TCL improves the accuracy
of SNNs compared to [10, 11, 12, 13].

##  IV Experimental results

We implemented our TCL on a Pytorch framework [16]. We trained ANNs by using a
stochastic gradient descent algorithm. We ensured that the total training
epochs are 200 for CIFAR-10, and 100 for ImageNet. We initialize learning
rates to $0.1$ besides VGG-16 and CNET with CIFAR-10, whose initial learning
rates are $0.01$. We scaled the learning rates by $0.1$ at the training epochs
of [100, 150] for CIFAR-10 and [30, 60, 80] for ImageNet. The initial value of
$\lambda$ is set to $4.0$ for all networks with both CIFAR-10 and ImageNet
datasets.

###  IV-A The Effect of a L2-regularization Coefficient

In our TCL, the L2-regularization coefficient of (11), $\alpha$, affects the
accuracy of SNNs with the following two styles. Firstly, as shown in Fig. 7,
larger $\alpha$ tends to provide smaller norm-factors, resulting in enhanced
SNN accuracy under very small latency constraints below 50 cycles. Secondly,
in our TCL-based training the accuracy of ANNs shows the trend to become lower
as $\alpha$ increases, potentially lowering corresponding SNN accuracies with
moderate latency conditions above 200 cycles. The above two trends clearly
appear in our experiment results, shown in Table I. Since the above two trends
oppositely influence the accuracy of SNNs, we need to find the optimal
$\alpha$, the largest value not to affect the accuracy of ANNs. However, the
algorithm to find the optimal values of hyper-parameters such as the
L2-regularization coefficient is not well-established in previous researches
yet, not explored in this work. From some trial and errors, we properly
selected the values of $\alpha$ in this work.

Figure 7: The norm-factors of VGG-16 with two alpha values for CIFAR-10 TABLE I: SNN accuracy at low latency Network | $\alpha$ |  ANN accuracy | SNN accuracy
---|---|---|---
T=20 | T=25 | T=30 | T=35
CIFAR-10 |
VGG-16 | 5e-4 | 93.49 | 88.33 | 92.02 | 92.48 | 92.66
| 1e-3 | 93.25 | 92.60 | 93.03 | 93.07 | 93.12
ResNet-20 | 1e-4 | 92.20 | 85.55 | 87.88 | 89.30 | 89.88
| 5e-4 | 91.58 | 89.63 | 90.50 | 90.98 | 91.22
MobileNet | 1e-4 | 91.68 | - | - | 10.02 | 20.53
| 5e-4 | 91.17 | - | - | 64.29 | 87.50
ImageNet |
VGG-16 | 1e-4 | 73.91 | - | 57.42 | 64.20 | 67.59
| 1e-3 | 73.22 | - | 65.72 | 69.36 | 70.75

###  IV-B Experiment results and discussion

TABLE II: Comparison of the results Model | Architecture | ANN accuracy | ANN accuracy | SNN | Latency | Conversion
---|---|---|---|---|---|---
(without TCL) | (with TCL) | accuracy | loss
CIFAR-10
Cao et al. (2015)[8] | CNN (3Conv, 2Linear) | 79.12 | - | 77.43 | 400 | 1.69
Rueckaur et al. (2017) [10] | CNN (4Conv, 2Linear) | 91.91 | - | 90.85 | 400 | 1.06
Sengupta et al. (2019) [11] | VGG-16 | 91.70 | - | 91.55 | 2500 | 0.15
| ResNet-20 | 89.10 | - | 87.46 | 2500 | 1.64
Han et al. (2020) [12] | VGG-16 | 93.63 | - | 93.51 | 1024 | 0.12
| ResNet-20 | 91.47 | - | 91.36 | 2048 | 0.11
Rathi et al. (2020) [14] | VGG-16 | 92.81 | - | 92.02 | 200 | 0.79
| ResNet-20 | 93.15 | - | 92.22 | 250 | 0.93
Rathi & Roy (2020) [15] | VGG-16 | 93.72 | - | 92.64 | 20 | 1.08
| ResNet-20 | 92.79 | - | 92.14 | 25 | 0.65
Ours | CNET (4Conv, 2Linear) | 89.96 | 90.05 ($\alpha=5\mathrm{e}{-4}$) | 90.04 | 100 | 0.01
| VGG-16 | 93.28 | 93.49 ($\alpha=5\mathrm{e}{-4}$) | 93.33 | 100 | 0.16
|  |  | 93.25 ($\alpha=1\mathrm{e}{-3}$) | 92.60 | 20 | 0.65
| ResNet-20 | 92.26 | 92.20 ($\alpha=1\mathrm{e}{-4}$) | 91.78 | 100 | 0.42
|  |  |  | 92.06 | 150 | 0.14
|  |  | 91.58 ($\alpha=5\mathrm{e}{-4}$) | 90.50 | 25 | 1.08
| MobileNet | 91.81 | 91.68 ($\alpha=1\mathrm{e}{-4}$) | 91.26 | 150 | 0.42
|  |  |  | 91.48 | 200 | 0.20
|  |  | 91.17 ($\alpha=5\mathrm{e}{-4}$) | 87.50 | 35 | 3.67
ImageNet
Rueckaur et al. (2017) [10] | VGG-16 | 63.89 | - | 49.61 | 400 | 14.28
| Inception-V3 | 76.12 | - | 74.60 | 550 | 1.52
Sengupta et al. (2019) [11] | VGG-16 | 70.52 | - | 69.96 | 2500 | 0.56
| ResNet-34 | 70.69 | - | 65.47 | 2500 | 5.22
Lu & Sengupta (2020) [13] | VGG-15 | 69.05 | - | 66.56 | 64 | 2.49
Han et al. (2020) [12] | VGG-16 | 73.49 | - | 73.09 | 4096 | 0.40
| ResNet-34 | 70.64 | - | 69.89 | 4096 | 0.75
Rathi et al. (2020) [14] | VGG-16 | 69.35 | - | 65.19 | 250 | 4.16
| ResNet-34 | 70.02 | - | 61.48 | 250 | 8.54
Rathi & Roy (2020) [15] | VGG-16 | 70.08 | - | 66.52 | 25 | 3.56
Ours | VGG-16 | 73.85 | 73.91 ($\alpha=1\mathrm{e}{-4}$) | 73.79 | 150 | 0.12
|  |  |  | 73.87 | 250 | 0.04
|  |  | 73.22 ($\alpha=1\mathrm{e}{-3}$) | 70.75 | 35 | 2.47
| ResNet-34 | 70.87 | 70.85 ($\alpha=1\mathrm{e}{-4}$) | 70.37 | 250 | 0.48
|  |  |  | 70.66 | 300 | 0.19
| ResNet-50 | 75.42 | 75.33 ($\alpha=1\mathrm{e}{-4}$) | 74.59 | 350 | 0.74
| MobileNet | 70.54 | 66.82 ($\alpha=1\mathrm{e}{-4}$) | 66.57 | 350 | 0.25

Our experiment results are summarized and compared to those of state-of-the-
arts (SOTAs) related to ANN-to-SNN conversion in Table II. For CIFAR-10, [8],
[10], [11], and [12] achieve good SNN accuracies after the ANN-to-SNN
conversion. However, large latencies are required for those techniques.
Although the authors of [14, 15] alleviate the large latency problem by using
their proposed hybrid training technique, where they reduce their latency to
the cycles of 20$\sim$250, the accuracy loss due to the ANN-to-SNN conversion
is still significant, larger than $0.5\%$. Our TCL technique makes the
following two significant accomplishments compared to SOTAs.

  * •

In spite of limiting the range of activations, our TCL technique hardly affect
the accuray of ANNs.

  * •

After ANN-to-SNN conversion, SNNs show accuracies comparable to their ANN
counterparts with moderate latency conditions.

In the dataset of CIFAR-10, with the latency of 100$\sim$150 cycles, the
accuracy loss to the ANN-to-SNN conversion is almost negligible, less than
$0.5\%$. The results of ImageNet further validate our contributions, where the
training results of ANNs based on our TCL are almost same to their original
accuracies. In addition, with the moderate latency of 250 cycles, we obtain
good SNN accuracies, almost comparable to their ANN counterparts.

For very low latencies below 50 cycles , we train ANNs with a little larger
$\alpha$. The results show almost comparable accuracy to the hybrid training
of [15], optimized for small latency constraints. Unlike [15], where the
direct SNN training phase in addition to the training one of ANN requires very
large computational overhead, our TCL delivers good SNN accuracy with the only
ANN training phase, our significant contribution as well.

##  V Conclusion

Many researches have shown that ANN-to-SNN conversion can become a realistic
alternative to the direction training of SNNs. However, SNNs suffer from large
latency, more problematic for large size dataset such as ImageNet, limiting
the possibility of SNNs. In this work, we present a trainable clipping layer
technique based on the ANN-to-SNN conversion, namely TCL, alleviating the
trade-off relation between accuracy and latency of SNNs. Our experiment
results shows that our TCL-based SNNs obtain almost comparable accuracy to
their ANN counterpart for ImageNet even with the small latency of 250 clock
cycles, well validating the efficacy of our TCL technique.

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