206Chapter 7
pointthereadertotheworkofWeilerandCesa(2019b).Furthermore,while
our text remainsfocused onstandarddomains andavoids thebroader topicof
Liegroups,readersinterestedinextendingequivariancetoarbitrarycontinu-
ous groupsoperating onirregular data,such aspoint clouds,will finda critical
milestone in the LieConv architecture proposed by Finzi et al. (2020).
Exercises
1.ConsiderahomogeneousdomainΩ,andasymmetrygroupGactingonit.Given
anyelementu∈Ω,we canform thestabiliser subgroupG
u
={g| g.u=u},consisting
of those elements of Gthat leave u invariant.
(i)Show that the stabiliser G
u
is indeed a subgroup of G.
(ii)LetG=O(3)bethegroupoforthogonal3×ばつ3matrices(AA
⊤
=I,i.e.,thegroup
ofcontinuous3Drotationsandreflections)actingonthesphereΩ=S
2
.Whatisthe
stabiliser of an arbitrary point u∈Ω (e.g., the north pole)?
(iii)Let G=S
n
be thegroupofpermutationsofnelements,Ω={1,2,...,n}.Whatisthe
stabiliser of an arbitrary point u∈Ω?
(iv)Letu,v∈Ωbetwoelementsonthedomain.ByhomogeneityofΩ,wecanfinda
g∈Gsuch that g.u=v. Show that, if h∈G
v
, then g
–1
hg∈G
u
.
(v)Fromthepreviouspart,itfollowsthatthereexistsamappingf:G
v
→G
u
definedby
f(h)=g
–1
hg. Provide an inverse of f, i.e., a map f
–1
:G
u
→G
v
.
(vi)Showthatf(hh
′
)=f(h)f(h
′
)forallh,h
′
∈G
v
;i.e.,thatfisagrouphomomor-
phism.This,coupledwiththeexistenceoff
–1
,impliesthatanytwopointsu,vofa
homogeneous space Ωhave isomorphic stabiliser subgroups.
2.Let G=S
n
bethe groupof permutationsacting ona setofn elementsΩ={1,...,n}.
LetS
n–1
bethesubgroupofS
n
thatleavesinvarianttheelement1,i.e.g.1=1for
allg∈S
n–1
.WewillstudythecosetsgS
n–1
={gh|h∈S
n–1
}(whereg∈S
n
)andthe
quotient S
n
/S
n–1
={gS
n–1
| g∈S
n
} (the set of cosets).
(i)What is the size, N, of S
n
/S
n–1
, i.e., how many cosets are there?
(ii)CharacterisethecosetsgS
n–1
,i.e.,findapropertythatallelementsofagivencoset
have,andthatnoelementoutsideofthecosethas.Inthefollowingparts,wewill
leverage this to identify S
n
/S
n–1
with the set {1,...,N} in the natural way.
(iii)Considerthemapπ:S
n
→S
n
/S
n–1
definedbyπ(g)=gS
n–1
.Whatarethefibers,F
u
=
π
–1
(u), where u∈S
n
/S
n–1
is a coset? (Note: in this context, πis a fiber bundle.)
(iv)Considertheright actionofS
n–1
onS
n
,definedasg.h=gh,for g∈S
n
,h∈S
n–1
.Show
that this action preserves fibers, i.e., π(g.h)=π(g) for all g∈S
n
,h∈S
n–1
.
(v)Additionally,show thatthisrightactionistransitiveon thefibers,i.e.,forany g,g
′
∈
π
–1
(u), there exists h∈S
n–1
such that g.h=g
′
.
(vi)Additionally, show that thisright action is fixed-pointfree. That is, if g.h=g for some
g∈S
n
and h∈S
n–1
, then h=e (the identity).