Clifford Spinors and Root System Induction: $$H_4$$ and the Grand Antiprism

Abstract

Recent work has shown that every 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of $$H_3\rightarrow H_4$$ H 3 → H 4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan–Dieudonné theorem, giving a simple construction of the $${\mathrm {Pin}}$$ Pin and $${\mathrm {Spin}}$$ Spin covers. Using this connection with $$H_3$$ H 3 via the induction theorem sheds light on geometric aspects of the $$H_4$$ H 4 root system (the 600-cell) as well as other related polytopes and their symmetries, such as the famous Grand Antiprism and the snub 24-cell. The uniform construction of root systems from 3D and the uniform procedure of splitting root systems with respect to subrootsystems into separate invariant sets allows further systematic insight into the underlying geometry. All calculations are performed in the even subalgebra of $${\mathrm {Cl}}(3)$$ Cl ( 3 ) , including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes, and are shared as supplementary computational work sheets. This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework.

Full text

Adv. Appl. Clifford Algebras (2021) 31:57
c
The Author(s) 2021
0188-7009/030001-35
published online June 28, 2021
https://doi.org/10.1007/s00006-021-01139-2
Advances in
Applied Clifford Algebras
Clifford Spinors and Root System Induction:
H4and the Grand Antiprism
Pierre-Philippe Dechant∗
Communicated by RafalAblamowicz
To John Horton Conway, Richard Kenneth Guy and Michael Guy.
Abstract. Recent work has shown that every 3D root system allows the
construction of a corresponding 4D root system via an ‘induction theo-
rem’. In this paper, we look at the icosahedral case of H3→H4in detail
and perform the calculations explicitly. Clifford algebra is used to per-
form group theoretic calculations based on the versor theorem and the
Cartan–Dieudonn´e theorem, giving a simple construction of the Pin and
Spin covers. Using this connection with H3via the induction theorem
sheds light on geometric aspects of the H4root system (the 600-cell) as
well as other related polytopes and their symmetries, such as the famous
Grand Antiprism and the snub 24-cell. The uniform construction of root
systems from 3D and the uniform procedure of splitting root systems
with respect to subrootsystems into separate invariant sets allows fur-
ther systematic insight into the underlying geometry. All calculations are
performed in the even subalgebra of Cl(3), including the construction of
the Coxeter plane, which is used for visualising the complementary pairs
of invariant polytopes, and are shared as supplementary computational
work sheets. This approach therefore constitutes a more systematic and
general way of performing calculations concerning groups, in particular
reflection groups and root systems, in a Clifford algebraic framework.
Mathematics Subject Classification. Primary 52B15; Secondary 52B11,
15A66, 20F55, 17B22, 20G41.
Keywords. Exceptional symmetries, spinors, 600-cell, grand antiprism,
Clifford algebras, Coxeter groups, root systems, Platonic solids.
Supplementary Information The online version contains supplementary material available
at https://doi.org/10.1007/s00006-021-01139-2.
∗Corresponding author.
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57 Page 2 of 35 P. Dechant Adv. Appl. Clifford Algebras
1. Introduction
Like many other mathematical concepts, symmetry groups have a rich struc-
ture including regular families and sporadic or exceptional phenomena [16].
The exceptional symmetry group in 3D is icosahedral symmetry. This fact
has profound implications for the natural world around us, because objects
consisting of identical building blocks that are ‘maximally symmetric’ dis-
play icosahedral symmetry. This includes most viruses and many fullerenes,
as well as artificial nanocages in bionanotechnology and geodesic domes in
architecture [13]. Even before any of these examples were known, icosahedral
symmetry had inspired Plato to formulate a ‘unified theory of everything’ in
his dodecahedral ‘ordering principle of the universe’. This pattern of (excep-
tional) symmetries inspiring ‘grand unified theories’ continues to this day,
with A4=SU(5) in GUTs, and E8in string theory and GUTs, as well as
D4=SO(8) and B4=SO(9) being critical in string and M theory.
Traditionally, people seek to understand symmetries ‘top-down’. For
instance E8includes A4and H4,orH4includes H3, so that people seek
to understand the smaller groups as subgroups of the larger ones. In recent
work [6,8,10,11] the author has shown that instead there is also a ‘bottom-
up’ view, by which e.g. H4and even E8can be constructed from H3. In this
sense the key to the larger exceptional symmetry groups is already contained
in the smaller exceptional group. In particular, the author proved a uniform
theorem that any 3D root system/reflection group induces a corresponding
4D root system/reflection group in a systematised way. The unusual abun-
dance of exceptional symmetry structures in 4D could thus be based on (the
accidentalness of) this construction, because it gives rise to the 4D excep-
tional objects D4(triality), F4(the largest 4D crystallographic group) and
H4(the largest non-crystallographic group altogether). There is an immediate
connection with Arnold’s Trinities, mysterious connections between different
triplets of exceptional objects throughout mathematics [1,2,12].
There is therefore a lot of additional geometric insight to be gained
from understanding this connection between 3D and 4D geometry, rather
than looking at these phenomena from a 4D perspective alone. In fact, the
link with 3D is much wider, including an ADE-type correspondence between
3D and 4D but also between 3D and ADE-type diagrams, in addition to the
famous McKay correspondence [12,27]. The connection between 3D and 4D
geometry arises because 3D reflections give rise to rotation groups via spinors.
These 3D spinors themselves behave like 4D objects and can be shown to
satisfy the root system axioms. This paper seeks to provide an example of
concrete calculations performed entirely within the 3D Clifford algebra Cl(3)
and its even subalgebra rather than in R4. There is of course also a connection
with quaternions, because of an isomorphism. A lot of previous work has
been done using quaternions and a purely algebraic description [4,22–26]
with some somewhat haphazard results. However, we would argue that a lot
of the deeper geometric insight, and its universality and systematic approach,
have been lost by following an approach that is algebraically equivalent but
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Vol. 31 (2021) Clifford Spinors and Root System Page 3 of 35 57
is really not well suited for uncovering and understanding the underlying
geometry and the generality of the situation.
In this paper we therefore give a comprehensive and pedagogical expo-
sition of how to perform group theoretic calculations for H3in a Clifford
approach, leading to the detailed construction of H3→H4. We give explicit
results of calculations in the paper, as well as making python Jupyter note-
books available as supplementary information for convenience and repro-
ducibility. These detail the calculations, results and algorithms used based
on the galgebra python package [3] as well as some visualisations also from
SageMath [30]. These notebooks are shared in the interest of open science,
collegiality and reproducibility, and may be useful to the readers for adapting
them for their own calculations. If they are useful please cite this paper along
with the original software provided by e.g. [3] and [30].
After the basic construction of the H4root system (the 600-cell) from H3
we investigate various subrootsystems that arise within H4. These groups and
root systems are of course separately invariant within H4, and the 120 vertices
of the H4root system can be split into two separately invariant sets by
taking the complement of the subrootsystem in H4. Conway and Guy found
the Grand Antiprism computationally 55 years ago this year [5]. Conway
and Michael Guy’s father, Richard Guy, himself a famous mathematician,
populariser of mathematics and collaborator of Conway, have unfortunately
passed away this year (2020). This paper is dedicated to their memory. The
construction and the symmetries of the Grand Antiprism actually benefit
from the construction from 3D as noted in [7]. In this paper we follow in
detail how the H2×H2subgroup arises naturally within H4in the induction
process. This subrootsystem is then used to split the 120 vertices of H4
into the set of 20 roots of H2⊕H2and the 100 vertices of its complement,
which is exactly the Grand Antiprism. A completely analogous construction
works in a uniform way for other subrootsystems of H4that arise via the 3D
construction, either as subrootsystems of H3or even subgroups of 2I.The
analogous cases include D4and the snub 24-cell, A4
1,A2⊕A2and A4.
We organise this paper as follows. We review some basics of Clifford
algebras, reflection groups and root systems in Sect. 2, leading to the Versor
Theorem and the Induction Theorem. In Sect. 3we build on the Versor Theo-
rem to set up a framework for explicit group theoretic calculations, including
the construction of the Pin(H3) and Spin(H3) groups, and discussion of their
conjugacy classes. Subrootsystems can arise either as even subgroups of the
spinor group (here the binary icosahedral group 2I), via subrootsystems of
H3or generated via the inversion e1e2e3, which is discussed in Sects. 4and 5,
respectively. The Coxeter plane is a convenient way to visualise root systems
and other polytopes in any dimension, and the calculations in this case can
also be entirely performed within the even subalgebra, as shown in Sect. 6.
The next sections give detailed results for splitting the H4root system with
respect to various subrootsystems, yielding complementary pairs of invariant
polytopes, including H2⊕H2and the Grand Antiprism in Sect. 7,D4and the
snub 24-cell in Sect. 8, and examples from A4
1,A2⊕A2and A4in Sect. 9.In
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57 Page 4 of 35 P. Dechant Adv. Appl. Clifford Algebras
Sect. 10 we conclude with a few words about the insights that can be gained
from this novel approach via induction from 3D.
2. Root System Induction
The setting for reflection groups and root systems stipulates the existence of
a vector space with an inner product, so without loss of generality one can
construct the corresponding Clifford algebra. The Clifford algebraic frame-
work used here is very standard, though we broadly follow [14,17]. We define
an algebra product via the geometric product xy =x·y+x∧y, where
the inner product (given by the symmetric bilinear form) is the symmetric
part x·y=1
2(xy +yx), and the wedge/exterior product the antisymmetric
part x∧y=1
2(xy −yx). These relations also mean that parallel vectors com-
mute whilst orthogonal vectors anticommute. The full 2n-dimensional algebra
is generated via this geometric product, linearity and associativity. For our
purposes we only consider the Clifford algebra of 3D Cl(3) generated by three
orthogonal unit vectors e1,e2and e3, though some of the following statements
hold under very general conditions. This yields an eight-dimensional vector
space generated by the elements
{1}

1 scalar
{e1,e
2,e
3}
 
3 vectors
{e1e2=Ie3,e
2e3=Ie1,e
3e1=Ie2}
 
3 bivectors
{I≡e1e2e3}
 
1 trivector
,
with an even subalgebra consisting of the scalar and bivectors, which is 4-
dimensional. Note that for the orthogonal unit vectors e.g. e1e2=e1∧e2.
We follow the galgebra L
A
T
EX output, which has the wedge version.
Root systems and in particular their simple roots are convenient objects
to characterise reflection and Coxeter groups. We therefore briefly introduce
the relevant terminology here:
Definition 2.1. (Root system)Aroot sy ste m is a collection Φ of non-zero
(root) vectors αthat span an n-dimensional Euclidean vector space V
endowed with a positive definite bilinear form, that satisfies the two axioms:
1. Φ only contains a root αand its negative, but no other scalar multiples:
Φ∩Rα={−α, α}∀α∈Φ.
2. Φ is invariant under all reflections corresponding to root vectors in Φ:
sαΦ=Φ∀α∈Φ. The reflection sαin the hyperplane with normal
vector αis given by
sα:x→sα(x)=x−2(x·α)
(α·α)α,
where (x·y) denotes the inner product on V.
Unlike other popular conventions here we assume unit normalisation for
all our root vectors for later convenience.
Proposition 2.2. (Reflection) In Clifford algebra, the reflection formula sim-
plifies to
sα:x→sα(x)=−αxα
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Vol. 31 (2021) Clifford Spinors and Root System Page 5 of 35 57
for normalised root vectors, so that double-sided (‘sandwich’)application of
a root vector generates the corresponding reflection.
Proof. Using the Clifford form for the inner product x·y=1
2(xy +yx)inthe
(simple) reflection formula in Definition 2.1 si:x→si(x)=x−2(x·α)
(α·α)αand
assuming unit normalisation of roots αi·αi= 1 yields the much simplified
version
si:x→si(x)=x−2·1
2(xαi+αix)αi=x−xα2
i−αixαi=−αixαi.

Proposition 2.3. (Double cover) In Clifford algebra, the reflections αand −α
doubly cover the same reflection sα.
Proof. Straightforward, since due to the double-sided application the signs
cancel out. 
A subset Δ of Φ, called simple roots α1,...,α
nis sufficient to express
every element of Φ via linear combinations with coefficients of the same sign.
For a crystallographic root system, these are Z-linear combinations, whilst for
the non-crystallographic root systems one needs to consider certain extended
integer rings. For instance for H2,H3and H4one has the extended integer
ring Z[τ]={a+τb|a, b ∈Z}, where τis the golden ratio τ=1
2(1+√5) =
2cos π
5,andσis its Galois conjugate σ=1
2(1 −√5) (the two solutions to
the quadratic equation x2=x+ 1), and linear combinations are with respect
to this Z[τ]. This integrality property of the crystallographic root systems
(types A-G) leads to an associated lattice which acts as a root lattice for Lie
algebras, which are named accordingly. In contrast, no such lattice exists for
the non-crystallographic groups (types Hand I), which accordingly do not
have associated Lie algebras, and are perhaps less familiar as a result.
The reflections corresponding to simple roots are also called simple
reflections. The geometric structure of the set of simple roots encodes the
properties of the reflection group and is summarised in the Cartan matrix and
Coxeter–Dynkin diagrams, which contain the geometrically invariant infor-
mation of the root system as follows:
Definition 2.4. (Cartan matrix and Coxeter–Dynkin diagram)TheCartan
matrix of a set of simple roots αi∈Δ is defined as the matrix
Aij =2
(αi·αj)
(αi·αi).(2.1)
A graphical representation of the geometric content is given by Coxeter–
Dynkin diagrams, in which nodes correspond to simple roots, orthogonal roots
are not connected, roots at π
3have a simple link, and other angles π
mhave a
link with a label m.
Example. The Cartan matrices for H3and H4are respectively given by
A(H3)=⎛
⎝
2−10
−12−τ
0−τ2
⎞
⎠,A(H4)=⎛
⎜
⎜
⎝
2−10 0
−12−10
0−12−τ
00−τ2
⎞
⎟
⎟
⎠.
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57 Page 6 of 35 P. Dechant Adv. Appl. Clifford Algebras
Possible choices of simple roots are e.g.
a1=e2,a
2=1
2(−τe1−e2−(τ−1)e3),a
3=e1
and
α1=1
2(τe1−e2+(τ−1)e4),α
2=e2,
α3=−1
2((τ−1)e1+e2+τe3)andα4=e3.
Remark 2.5. What is a slight drawback of the diagrammatic approach is that
it is rather unobvious which subgroups are contained e.g. in H4–atleast
exhaustively. We will see some non-obvious examples later on.
Root systems and simple roots are therefore convenient paradigms for
considering reflection groups: each root vector defines a hyperplane that it is
normal to and thereby acts as a generator of a reflection in that hyperplane. In
Clifford algebra this root vector in fact directly acts as a reflection generator
via the geometric product. Multiplying together such simple reflections si:
x→si(x)=x−2(x|αi)
(αi|αi)αi=−αixαitherefore generates a reflection group.
This is in fact a Coxeter group, since the simple reflections sisatisfy the
defining relations:
Definition 2.6. (Coxeter group)ACoxeter group is a group generated by
a set of involutory generators si,s
j∈Ssubject to relations of the form
(sisj)mij = 1 with mij =mji ≥2fori=j.
Definition 2.7. (Coxeter element and Coxeter number) The product of all the
simple reflections in some order is called a Coxeter element. All such elements
are conjugate and as such their order is well-defined and called the Coxeter
number.
These reflection groups are built up in Clifford algebra by performing
successive multiplication with the unit vectors defining the reflection hyper-
planes via ‘sandwiching’
s1...s
k:x→s1...s
k(x)=(−1)kα1...α
kxαk...α
1=: (−1)kAx ˜
A, (2.2)
where the tilde denotes the reversal of the order of the constituent vectors
in the product A=α1...α
k. In order to study the groups of reflections one
therefore only needs to consider products of root vectors in the Clifford alge-
bra, which form a multivector group under the geometric product and yield
a Pin double cover of the corresponding reflection group [29]. The inverse of
each group element is of course simply given by the reversal, because of the
assumed normalisation condition. Since αiand −αiencode the same reflec-
tion, products of unit vectors are double covers of the respective orthogonal
transformation, as Aand −Aencode the same transformation. We call even
products R, i.e. products of an even number of vectors, ‘spinors’ or ‘rotors’,
and a general product A‘versors’ or ‘pinors’. They form the Pin group and
constitute a double cover of the orthogonal group, whilst the even prod-
ucts form the double cover of the special orthogonal group, called the Spin
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Vol. 31 (2021) Clifford Spinors and Root System Page 7 of 35 57
group. Clifford algebra therefore provides a particularly natural and simple
construction of the Spin groups.
In fact, more general groups can be constructed in this way because
of the fundamental importance of reflections according to the Cartan–
Dieudonn´e Theorem [15].
Theorem 2.8. (Cartan–Dieudonn´e Theorem) Every orthogonal transforma-
tion in an n-dimensional symmetric bilinear space can be described as the
composition of at most nreflect ion s.
The above approach to group theory via multivector groups (2.2)is
therefore a much more general way of doing group theory.
In Clifford algebra, instead of using matrices to perform linear transfor-
mations one can use spinors/rotors/pinors/versors to perform linear transfor-
mations that leave the inner product invariant i.e. orthogonal transformations
[18–20]. Here the normalisation condition has been dropped as long as the
vectors are non-null since the inverse of multiplication with a non-null vector
xis simply x−1=x
|x|2since xx =x·x=|x|2(in the positive signature spaces
we will consider there are no null vectors anyway). Therefore the multivector
Athat is a product of vectors is invertible and preserves the inner product,
though the inverse is no longer just given by the reverse:
Theorem 2.9. (Versor Theorem) Every orthogonal transformation Acan be
expressed in the canonical form A:x→x=A(x)=±A−1xA where Ais a
versor and the sign is its parity.
A concept familiar from abstract group theory via generators and rela-
tions is that group elements can be written as words in the generators. It
is noteworthy here that in contrast to for instance the Coxeter group ele-
ments as words in the generators sα, in the Clifford algebra approach the
root vectors are directly generators for the Pin double cover under multipli-
cation with the geometric product. To stress this slight distinction we call
these ‘generator paths’ for each versor v, with ±veach versor corresponding
to one Coxeter group word.
Therefore Clifford algebras and root systems are frameworks that per-
fectly complement each other since performing reflections in Clifford algebras
is so simple and only assumes the structure of a vector space with an inner
product that is already given in the root system definition. Therefore Clif-
ford algebras are perhaps the most natural framework for studying reflection
groups and root systems, and through the above arguments also more general
groups [9]. In the next section we perform a detailed computation of the Pin
and Spin covers of the icosahedral groups to illustrate the principles. Fol-
lowing the above two frameworks, given (simple) roots in a root system one
can start multiplying these together using the geometric product. General
products will be in Pin whilst even products are in Spin. For now we will
concentrate on the Spin and the even subalgebra.
Proposition 2.10. (O(4)-structure of spinors) The space of Cl(3)-spinors has
a 4D Euclidean structure.
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57 Page 8 of 35 P. Dechant Adv. Appl. Clifford Algebras
Proof. For a spinor R=a0+a1e2e3+a2e3e1+a3e1e2, the norm is given by
R˜
R=a2
0+a2
1+a2
2+a2
3, and the inner product between two spinors R1and
R2is (R1,R
2)=1
2(R1˜
R2+R2˜
R1). 
Remark 2.11. For rotors, the inner product 1
2(R1˜
R2+R2˜
R1)isofcourse
invariant under Ri→RRi˜
Rbut also just under Ri→RRi.
From the double cover property of Proposition 2.3 we have the following
corollary:
Corollary 2.12. (Discrete spinor groups) Discrete spinor groups are of even
order since if a spinor Ris contained in the group then so is −Rsince it
encodes the same orthogonal transformation.
Proposition 2.13. (Spin group closure properties) Spin groups are closed
under:
•multiplication using the geometric product
•reversal
•multiplication by −1
Proof. Straightforward:
•by definition of multivector groups via the geometric product
•by the inverse element group axiom since reversal is equivalent to the
inverse
•by Corollary 2.12 both Rand −Rare contained in the group

Following the formula for fundamental reflections from Definition 2.1
one can likewise define reflections on this spinor space with respect to the
inner product between spinors.
Proposition 2.14. (Spin reflections) Reflections between spinors using the
spinor inner product are given by
R2→R
2=−R1˜
R2R1.
Proof. In analogy to Proposition 2.2, for normalised spinors R1and R2
and using the definition of the spinor inner product from Proposition 2.10
this amounts to R2→R
2=R2−2(R1,R
2)/(R1,R
1)R1=R2−(R1˜
R2+
R2˜
R1)R1=R2−R1˜
R2R1−R2˜
R1R1=−R1˜
R2R1.
Proposition 2.15. (3D spinor–4D vector correspondence) Spinor reflections
in the spinor R=a0+a1e2∧e3−a2e1∧e3+a3e1∧e2are equivalent to 4D
reflections in the 4D vector (a0,a
3,−a2,a
1).
Proof. By direct calculation or see supplementary material. 
In practice, the exact mapping of the components is a matter of con-
vention and often irrelevant, since most root systems contain roots up to ±
and (cyclic) permutations anyway.
Theorem 2.16. (Induction Theorem) Any rank-3 root system induces a root
system of rank 4.
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Vol. 31 (2021) Clifford Spinors and Root System Page 9 of 35 57
Table 1. List of the rank 3 root systems and their induced
root systems in four dimensions, as well as the binary poly-
hedral groups that act as the spinor group intermediaries
Start root system Induced root system Binary polyhedral group
A3
1A4
1Q
A1⊕I2(n)I2(n)⊕I2(n)2,2,n
A3D42T
B3F42O
H3H42I
Proof. A root system in three dimensions Φ(3) gives rise to a group of spinors
by taking even products of the root vectors. From Corollary 2.12, this group
contains −Rif it contains R, and therefore satisfies the first root system
axiom from Definition 2.1. The set of spinors has a 4D Euclidean structure
by Proposition 2.10 and can thus be treated as a collection of 4D vectors Φ(4)
with the inner product as given in the Proposition. It remains to show that
this collection of vectors Φ(4) is invariant under reflections (axiom 2), which
is satisfied by Proposition 2.3. 
Closure of the root system is thus ensured by closure of the spinor group.
This also has very interesting consequences for the automorphism group of
these spinorial root systems, which contains two factors of the spinor group
acting from the left and the right [7] (in this sense, the above closure under
reflections amounts to a certain twisted conjugation).
There is a limited number of cases which we can just enumerate. The
3D root systems are listed in Table 1along with the 4D root systems that
they induce as well as the intermediate spinor groups (the binary polyhedral
groups). In this article, we focus on the case H3→H4.
Definition 2.17. (Subrootsystem) By a subrootsystem Φ1of a root system Φ2
we mean a subset Φ1of the collection of vectors Φ2that itself satisfies the
root system axioms.
From the Induction Theorem 2.16 we immediately get:
Corollary 2.18. (Induced subrootsystems) A subrootsystem Φ(3)
1of a root sys-
tem Φ(3)
2induces a subrootsystem Φ(4)
1of the induced root system Φ(4)
2.
Any subrootsystem of H3therefore induces a subrootsystem of H4.For
instance, the A3
1inside H3induces the rather boring A4
1in H4. Similarly, A2
and H2are contained, if rather boring as 2D root systems. If H3contained
A1⊕A2and A1⊕H2subrootsytems (which it doesn’t), then this would lead
to the doubling A2⊕A2and H2⊕H2inside the H4. This is not quite the
case, but nearly so, which we will return to later. Similarly, it follows from
the Induction Theorem that any even subgroup of a spinor group will also
yield a subrootsystem. We will explore these points in the Sects. 4and 5
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57 Page 10 of 35 P. Dechant Adv. Appl. Clifford Algebras
which follow the next section, where we will discuss the multivector group
calculation framework.
3. Pin Group and Spin Group
In this section we pick up on the practical implications of the versor Theorem
2.9 and the Cartan–Dieudonn´e theorem in order to do explicit calculations in
group theory in the Clifford algebra approach. We use the concrete example of
the H3reflection/Coxeter group. The icosahedral rotation group (in SO(3))
is the alternating group A5of order 60, also known simply as I(which we will
avoid due to the pseudoscalar often being denoted by that too). This group
is of course doubly covered in Spin(3) by its spin double cover, with its nice
Clifford algebra construction via the reflection formula in Proposition 2.2.We
might denote this group by Spin(H3) but it is also commonly known as the
binary icosahedral group 2I. From the induction theorem 2.16 of the previous
section the elements of this group of course give the 120 roots of the H4root
system. The rotational group is of course also doubly covered in O(3) by its
double cover H3=A5×Z2. Both double covers are of course of order 120.
H3however is itself also doubly covered in Pin(3) by a group of order 240,
which doesn’t have a common name since it is simply H3×Z2=A5×Z2×Z2
but which we might for consistency call Pin(H3).
For the reader’s convenience and for reproducibility in Tables 2,3,4,5
we list the different group elements of Pin(H3) in our Clifford approach explic-
itly. The python Jupyter notebooks in the supplementary material contain
the algorithms used, which are based on the galgebra software package [3].
For convenience we group the elements as whole conjugacy classes but we
give the order in which the elements are generated by repeated application
of the generating simple roots. This gives a number for each group element
as a reference for ease of access, as well as the word in the generators (the
‘generator path’) that generates this particular group element in terms of the
H3simple roots/generators a1,a
2,a
3. For the simple roots of H3we pick
a1=e2,a
2=1
2(−τe1−e2−(τ−1)e3),a
3=e1.
Note that the wedge could be omitted since we have picked the orthogo-
nal unit vectors e1,e
2,e
3so the wedge product is synonymous with the full
geometric product. We multiply all group elements by 2 to save clutter.
The spin group Spin(3) = 2Iis given in Tables 2and 3. Its nine conju-
gacy classes lead to irreducible representations of dimensions 1,3,3,4,5which
are shared by A5, as well as the spinorial ones of dimensions 2,2,4,6. (Inter-
esting connections with the binary polyhedral groups and the McKay corre-
spondence [27] are explored elsewhere [12]). Since Pin(H3) = Spin(H3)×Z2,
the remaining Tables 4and 5list the remaining 9 conjugacy classes achieved
by multiplying those of Spin(H3) with the inversion e1e2e3.
Tables 2and 3: The only normal subgroup of 2Iconsists of the first two
conjugacy classes i.e. ±1. We note that the conjugacy class of order 4 consists
of thirty pure bivectors, and that they give rise to the 2-fold rotations of the
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Vol. 31 (2021) Clifford Spinors and Ro ot System Page 11 of 35 57
Table 2. The first set of the conjugacy classes of Spin(H3),
the ones with orders of ‘crystallographic type’ 1,2,3,4,6.
These conjugacy classes all contain their own reverses i.e.
inverses. The only normal subgroup consists of the first two
conjugacy classes and is ±1, and the order 4 conjugacy class
consists of pure bivectors
Order Number Element ×2 Generator path
14 2 11
226 −2 1313
35 −1+τe1∧e2+σe2∧e312
37 −1−τe1∧e2−σe2∧e321
336 −1−τe1∧e2+σe2∧e33123
338 −1+τe1∧e2−σe2∧e33213
382 −1+e1∧e2+e1∧e3−e2∧e3231232
383 −1−e1∧e2−e1∧e3+e2∧e3232132
3 124 −1−e1∧e2+e1∧e3+e2∧e312312321
3 125 −1+e1∧e2−e1∧e3−e2∧e312321321
3 128 −1−e1∧e2+e1∧e3−e2∧e312323123
3 131 −1−e1∧e2−e1∧e3−e2∧e313213232
3 134 −1−σe1∧e2+τe1∧e321321323
3 137 −1+e1∧e2+e1∧e3+e2∧e323213213
3 141 −1+σe1∧e2−τe1∧e332132132
3 143 −1+e1∧e2−e1∧e3+e2∧e332132321
3 170 −1+σe1∧e2+τe1∧e31213213231
3 172 −1−σe1∧e3−τe2∧e31213231232
3 174 −1+σe1∧e3−τe2∧e31232132312
3 177 −1−σe1∧e2−τe1∧e31321321321
3 184 −1−σe1∧e3+τe2∧e32132312321
3 187 −1+σe1∧e3+τe2∧e32321323121
46 −2e1∧e213
49 2e1∧e231
430 τe1∧e2−σe1∧e3−e2∧e32132
431 −τe1∧e2+σe1∧e3+e2∧e32312
466 −τe1∧e2−σe1∧e3+e2∧e3121321
468 τe1∧e2+σe1∧e3−e2∧e3123121
489 −τe1∧e2+σe1∧e3−e2∧e3321323
491 τe1∧e2−σe1∧e3+e2∧e3323123
4 129 −τe1∧e2−σe1∧e3−e2∧e313213213
4 130 τe1∧e2+σe1∧e3+e2∧e313213231
4 139 e1∧e2+τe1∧e3+σe2∧e323213232
4 140 −e1∧e2−τe1∧e3−σe2∧e323231232
4 175 −e1∧e2+τe1∧e3−σe2∧e31232132321
4 176 e1∧e2−τe1∧e3+σe2∧e31232312321
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57 Page 12 of 35 P. Dechant Adv. Appl. Clifford Algebras
Table 2. continued
Order Number Element ×2 Generator path
4 181 −σe1∧e2+e1∧e3−τe2∧e32132132132
4 182 σe1∧e2−e1∧e3+τe2∧e32132132312
4 186 −e1∧e2−τe1∧e3+σe2∧e32321321323
4 188 e1∧e2+τe1∧e3−σe2∧e32321323123
4 209 σe1∧e2+e1∧e3+τe2∧e3121321321321
4 211 −σe1∧e2−e1∧e3−τe2∧e3121321323121
4 213 −e1∧e2+τe1∧e3+σe2∧e3123213213213
4 214 e1∧e2−τe1∧e3−σe2∧e3123213213231
4 221 σe1∧e2−e1∧e3−τe2∧e3321321321323
4 222 −σe1∧e2+e1∧e3+τe2∧e3321321323123
4 233 −2e2∧e312132132132132
4 234 2e2∧e312132132132312
4 235 −σe1∧e2−e1∧e3+τe2∧e312321321321323
4 236 σe1∧e2+e1∧e3−τe2∧e312321321323123
4 237 2e1∧e321321321321323
4 238 −2e1∧e321321321323123
628 1+τe1∧e2−σe2∧e31323
632 1+τe1∧e2+σe2∧e32313
637 1−τe1∧e2−σe2∧e33132
639 1−τe1∧e2+σe2∧e33231
681 1−e1∧e2−e1∧e3+e2∧e3213232
684 1+e1∧e2+e1∧e3−e2∧e3232312
6 123 1 + e1∧e2−e1∧e3−e2∧e312132321
6 126 1 + e1∧e2−e1∧e3+e2∧e312321323
6 127 1 −e1∧e2+e1∧e3+e2∧e312323121
6 132 1 + e1∧e2+e1∧e3+e2∧e313231232
6 136 1 + σe1∧e2−τe1∧e321323123
6 138 1 −e1∧e2−e1∧e3−e2∧e323213231
6 142 1 −σe1∧e2+τe1∧e332132312
6 144 1 −e1∧e2+e1∧e3−e2∧e332312321
6 169 1 −σe1∧e2−τe1∧e31213213213
6 171 1 + σe1∧e3+τe2∧e31213213232
6 173 1 −σe1∧e3+τe2∧e31232132132
6 179 1 + σe1∧e2+τe1∧e31321323121
6 183 1 + σe1∧e3−τe2∧e32132132321
6 185 1 −σe1∧e3−τe2∧e32321321321
icosahedron around its 30 edges. The 20 3-fold rotations around the 20 trian-
gular faces are split into two conjugacy classes of orders 3 and 6, which are
related by multiplication by −1, and each contain their own reverse/inverse.
Similarly, the two sets of twelve 5-fold rotations are doubly covered by 4
conjugacy classes which are related by multiplication by −1. The two classes
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Vol. 31 (2021) Clifford Spinors and Ro ot System Page 13 of 35 57
Table 3. The second set of conjugacy classes of Spin(H3),
the ones with orders of ‘non-crystallographic type’ i.e. the
ones related to 5-fold symmetry. These conjugacy classes
also all contain their inverses. The first column denotes the
order of the elements in each conjugacy class. The second
column is the position in the order in which our algorithm
generates this element, for convenience (c.f. the supplemen-
tary material). The final column denotes the order in which
the generators with the corresponding labels are applied to
generate this group element, i.e. is effectively the ‘word in
the generators’ that yields this element. To avoid confusion,
pairs of such words doubly cover the rotations of the icosa-
hedral group A5, which are also often considered in terms
of words in the A5generators. The ones meant here are the
root vectors multiplied by using the geometric product
Order Number Element ×2 Generator path
58 −τ+e1∧e2−σe1∧e323
510 −τ−e1∧e2+σe1∧e332
523 −τ−e1∧e2−σe1∧e31231
527 −τ+e1∧e2+σe1∧e31321
565 −τ−σe1∧e2−e2∧e3121312
567 −τ+e1∧e3+σe2∧e3121323
572 −τ+σe1∧e2−e2∧e3123232
575 −τ−e1∧e3+σe2∧e3132312
577 −τ+σe1∧e2+e2∧e3213121
580 −τ+e1∧e3−σe2∧e3213231
585 −τ−σe1∧e2+e2∧e3232321
590 −τ−e1∧e3−σe2∧e3323121
534 −σ−τe1∧e2−e1∧e32323
540 −σ+τe1∧e2+e1∧e33232
571 −σ+τe1∧e2−e1∧e3123231
576 −σ−τe1∧e2+e1∧e3132321
5 122 −σ−e1∧e2+τe2∧e312132312
5 135 −σ+e1∧e2−τe2∧e321323121
5 178 −σ+e1∧e2+τe2∧e31321321323
5 190 −σ−e1∧e2−τe2∧e33213213231
5 212 −σ−τe1∧e3+e2∧e3121321323123
5 215 −σ+τe1∧e3+e2∧e3132132132132
5 217 −σ−τe1∧e3−e2∧e3213213213213
5 220 −σ+τe1∧e3−e2∧e3232132132312
10 22 τ+e1∧e2+σe1∧e31213
10 24 τ+σe1∧e2+e2∧e31232
10 25 τ+e1∧e2−σe1∧e31312
10 29 τ−e1∧e2+σe1∧e32131
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57 Page 14 of 35 P. Dechant Adv. Appl. Clifford Algebras
Table 3. continued
Order Number Element ×2 Generator path
10 33 τ−σe1∧e2−e2∧e32321
10 35 τ−e1∧e2−σe1∧e33121
10 69 τ−e1∧e3−σe2∧e3123123
10 74 τ+e1∧e3−σe2∧e3132132
10 79 τ−e1∧e3+σe2∧e3213213
10 87 τ−σe1∧e2+e2∧e3312323
10 88 τ+e1∧e3+σe2∧e3321321
10 92 τ+σe1∧e2−e2∧e3323213
10 70 σ−τe1∧e2+e1∧e3123213
10 73 σ−τe1∧e2−e1∧e3131232
10 78 σ+τe1∧e2+e1∧e3213123
10 86 σ+τe1∧e2−e1∧e3312321
10 121 σ+e1∧e2−τe2∧e312132132
10 133 σ−e1∧e2+τe2∧e321321321
10 180 σ−e1∧e2−τe2∧e31321323123
10 189 σ+e1∧e2+τe2∧e33213213213
10 210 σ+τe1∧e3−e2∧e3121321321323
10 216 σ−τe1∧e3−e2∧e3132132132312
10 218 σ+τe1∧e3+e2∧e3213213213231
10 219 σ−τe1∧e3+e2∧e3232132132132
describe rotations by ±2π/5and±4π/5, respectively, around the 5-fold axes
of symmetry, the icosahedral vertices.
Tables 4and 5: The first two conjugacy classes are the inversion and
its negative. The conjugacy class consisting of pure vectors of course corre-
sponds to the 30 roots of H3which generate the reflections, and which are
of course related to the 30 2-fold rotations since the inversion is contained in
the group (so one can dualise a (root) vector to a pure bivector). The con-
jugacy classes of order 12 are the two inversion-related versions of the 3-fold
rotations, and are rotoreflections. The four conjugacy classes of order 20 are
both related to the 5-fold rotations, as well as serving as the versor analogues
of the Coxeter elements e.g. w=a1a2a3. These are in one conjugacy class in
the reflection/Coxeter group framework where their order gives the Coxeter
number. But in this Clifford double cover setup these ‘Coxeter versors’ are
given in 4 conjugacy classes that are related by reversal and multiplication
by e1e2e3.
Remark 3.1. It has been noted that a1a2and a2a3generate the quaternionic
root system multiplicatively e.g. for H4. This is pretty obvious when thought
of in terms of the 3D simple roots and the Induction Theorem, as they of
course generate Spin(H3), which gives rise to the H4root system.
This example illustrates how one can perform practical computations in
group theory via versors in this Clifford algebra framework, and in galgebra
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Vol. 31 (2021) Clifford Spinors and Ro ot System Page 15 of 35 57
in particular. We will discuss group and representation theoretic aspects in
more detail elsewhere.
4. Subgroups
The Induction Theorem 2.16 from Sect. 2showed that every 3D root system
determines a 4D root system. This proceeded essentially via using the 3D
roots to construct a group of spinors (via multiplication with the geometric
product), which satisfies the properties of a root system. Our main example
is of course H3which induces H4in four dimensions via the binary icosa-
hedral group of order 120 as the spinor group intermediary. It is therefore a
straightforward corollary of the Induction Theorem that each even subgroup
of 2Ialso yields a root system.
Corollary 4.1. (Subgroups of 2I)Each even subgroup Gof the binary icosa-
hedral group 2Idetermines a corresponding root system Φthat is a subset of
the H4root system, the 600-cell.
Theorem 4.2. (Induced subrootsystems of H4)The binary icosahedral group
2Ihas the following subgroups that determines the corresponding root sys-
tems:
•The normal subgroup ±1which gives A1.
•The quaternion group Qconsisting of ±1,±e1e2,±e2e3and ±e3e1,
which gives A1×A1×A1×A1.
•The binary dihedral groups of orders 6and 10, which yield A2and H2.
•The binary tetrahedral group, which yields D4.
Remark 4.3. Note that although the A3root system is not contained in H3,
2Tis a subgroup of 2Iand therefore D4is contained in H4. We will revisit
these examples in later sections and in the next section investigate this deli-
cate relationship between subgroups and other subrootsystems further.
Proposition 4.4. (Simple roots of induced subrootsystems) A2and H2are
generated straightforwardly from the H3generators a1,a
2,a
3e.g. via the
‘spinorial simple roots’ a1a1=1and a1a2for A2and a1a1=1and a2a3
for H2.
One possible choice of simple roots for D4contained in H4is given by
(a1a1,a
1a2,a
1a2a3a2a3a1a2a3,a
3a2a1a3a2a1a3a2),
but it is of course not unique.
Explicit versions of these simple roots can be looked up in the earlier
tables via the ‘generator path’, which can be used to explicitly verify the
correct Cartan matrix and closure of the root system (see e.g. supplementary
information).
5. Subrootsystems
There is a subtlety at play here since there are subrootsystems of H4that
are neither induced by 3D subrootsystems nor by even spinor subgroups. We
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57 Page 16 of 35 P. Dechant Adv. Appl. Clifford Algebras
Table 4. The first set of the remaining conjugacy classes of
Pin(H3). Note that the first two conjugacy classes show that
the inversion e1e2e3is contained in the group. The class of
order 2 consists of pure vectors and are thus their own reverse
i.e. inverse. The other pair are each other’s reverses/inverses
Order Number Element ×2 Generator path
4 240 2e1∧e2∧e3121321321323123
4 239 −2e1∧e2∧e3121321321321323
21 2e21
22 −τe1−e2+σe32
23 2e13
211 τe1−e2−σe3121
213 −2e1131
217 −σe1+τe2+e3232
219 −2e2313
221 −τe1+e2−σe3323
244 −τe1+e2+σe312313
245 σe1+τe2−e312321
248 τe1−e2+σe313123
249 τe1+e2−σe313132
250 −τe1−e2−σe313213
251 τe1+e2+σe313231
253 σe1−τe2−e321312
260 −σe1−τe2−e323232
293 −σe1−τe2+e31213121
2 101 σe1−τe2+e31232321
2 103 σe1+τe2+e31312323
2 108 −σe1+τe2−e31323213
2 109 −e1+σe2−τe32132132
2 110 e1−σe2+τe32132312
2 145 e1+σe2+τe3121321321
2 147 −e1−σe2−τe3121323121
2 166 −e1−σe2+τe3321321323
2 168 e1+σe2−τe3321323123
2 199 −e1+σe2+τe313213213213
2 200 e1−σe2−τe313213213231
2 229 −2e32132132132132
2 230 2e32132132132312
12 52 −τe1−σe2−e1∧e2∧e313232
12 55 −τe2+σe3−e1∧e2∧e321323
12 58 τe1+σe2−e1∧e2∧e323213
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Vol. 31 (2021) Clifford Spinors and Ro ot System Page 17 of 35 57
Table 4. continued
Order Number Element ×2 Generator path
12 62 τe2−σe3−e1∧e2∧e332132
12 64 τe1−σe2−e1∧e2∧e332321
12 94 −τe1+σe2−e1∧e2∧e31213123
12 96 −τe2−σe3−e1∧e2∧e31213231
12 98 −e1−e2−e3−e1∧e2∧e31231232
12 100 −e1+e2−e3−e1∧e2∧e31232312
12 104 τe2+σe3−e1∧e2∧e31321321
12 112 e1−e2+e3−e1∧e2∧e32312321
12 115 e1+e2+e3−e1∧e2∧e32323121
12 116 e1−e2−e3−e1∧e2∧e32323123
12 120 −e1+e2+e3−e1∧e2∧e33231232
12 149 −e1−e2+e3−e1∧e2∧e3123213213
12 156 e1+e2−e3−e1∧e2∧e3132312321
12 191 σe1−τe3−e1∧e2∧e312132132132
12 201 −σe1+τe3−e1∧e2∧e321321321321
12 228 σe1+τe3−e1∧e2∧e31321321323123
12 232 −σe1−τe3−e1∧e2∧e32321321323123
12 46 τe1−σe2+e1∧e2∧e312323
12 57 τe2−σe3+e1∧e2∧e323123
12 59 −τe1−σe2+e1∧e2∧e323231
12 61 τe1+σe2+e1∧e2∧e331232
12 63 −τe2+σe3+e1∧e2∧e332312
12 95 τe2+σe3+e1∧e2∧e31213213
12 97 e1+e2+e3+e1∧e2∧e31213232
12 99 e1−e2+e3+e1∧e2∧e31232132
12 102 −τe1+σe2+e1∧e2∧e31312321
12 106 −τe2−σe3+e1∧e2∧e31323121
12 111 −e1+e2−e3+e1∧e2∧e32132321
12 113 −e1−e2−e3+e1∧e2∧e32321321
12 114 −e1+e2+e3+e1∧e2∧e32321323
12 119 e1−e2−e3+e1∧e2∧e33213232
12 150 e1+e2−e3+e1∧e2∧e3123213231
12 155 −e1−e2+e3+e1∧e2∧e3132132321
12 192 −σe1+τe3+e1∧e2∧e312132132312
12 203 σe1−τe3+e1∧e2∧e321321323121
12 227 −σe1−τe3+e1∧e2∧e31321321321323
12 231 σe1+τe3+e1∧e2∧e32321321321323
have observed in the spinor Induction Theorem that A1⊕I2(n) root systems
experience a doubling to I2(n)⊕I2(n) in the induction process. Indeed, such
A2⊕A2and H2⊕H2within H4are induced, but not because they have an
orthogonal A1. Instead, it is because the group H3contains the inversion,
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57 Page 18 of 35 P. Dechant Adv. Appl. Clifford Algebras
Table 5. The four conjugacy classes of order 20 are related
to the 5-fold rotations in Spin(H3) via multiplication by
e1e2e3. They are also the spinor version of the Coxeter ele-
ments e.g. w=a1a2a3. They are related by reversal and
multiplication by e1e2e3
Order Number Element ×2 Generator path
20 14 e1−τe2−σe1∧e2∧e3132
20 15 −e1+τe2−σe1∧e2∧e3213
20 20 −e1−τe2−σe1∧e2∧e3321
20 41 e1+τe2−σe1∧e2∧e312131
20 43 τe1+e3−σe1∧e2∧e312312
20 56 −τe1−e3−σe1∧e2∧e323121
20 105 τe1−e3−σe1∧e2∧e31321323
20 118 −τe1+e3−σe1∧e2∧e33213231
20 159 e2+τe3−σe1∧e2∧e3213213232
20 162 −e2−τe3−σe1∧e2∧e3232132312
20 193 e2−τe3−σe1∧e2∧e312132132321
20 197 −e2+τe3−σe1∧e2∧e312321323121
20 12 −e1−τe2+σe1∧e2∧e3123
20 16 e1−τe2+σe1∧e2∧e3231
20 18 −e1+τe2+σe1∧e2∧e3312
20 42 −τe1−e3+σe1∧e2∧e312132
20 47 e1+τe2+σe1∧e2∧e313121
20 54 τe1+e3+σe1∧e2∧e321321
20 107 −τe1+e3+σe1∧e2∧e31323123
20 117 τe1−e3+σe1∧e2∧e33213213
20 160 −e2−τe3+σe1∧e2∧e3213231232
20 161 e2+τe3+σe1∧e2∧e3232132132
20 194 −e2+τe3+σe1∧e2∧e312132312321
20 195 e2−τe3+σe1∧e2∧e312321321321
20 146 σe1−e2−τe1∧e2∧e3121321323
20 151 −e1+σe3−τe1∧e2∧e3123213232
20 154 σe1+e2−τe1∧e2∧e3132132312
20 158 −σe1−e2−τe1∧e2∧e3213213231
20 163 e1−σe3−τe1∧e2∧e3232132321
20 167 −σe1+e2−τe1∧e2∧e3321323121
20 198 −e1−σe3−τe1∧e2∧e312321323123
20 202 σe2−e3−τe1∧e2∧e321321321323
20 205 e1+σe3−τe1∧e2∧e323213213213
20 207 −σe2+e3−τe1∧e2∧e332132132132
20 224 σe2+e3−τe1∧e2∧e31213213213231
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Vol. 31 (2021) Clifford Spinors and Ro ot System Page 19 of 35 57
Table 5. continued
Order Number Element ×2 Generator path
20 226 −σe2−e3−τe1∧e2∧e31232132132312
20 148 −σe1+e2+τe1∧e2∧e3121323123
20 152 e1−σe3+τe1∧e2∧e3123231232
20 153 −σe1−e2+τe1∧e2∧e3132132132
20 157 σe1+e2+τe1∧e2∧e3213213213
20 164 −e1+σe3+τe1∧e2∧e3232312321
20 165 σe1−e2+τe1∧e2∧e3321321321
20 196 e1+σe3+τe1∧e2∧e312321321323
20 204 −σe2+e3+τe1∧e2∧e321321323123
20 206 −e1−σe3+τe1∧e2∧e323213213231
20 208 σe2−e3+τe1∧e2∧e332132132312
20 223 −σe2−e3+τe1∧e2∧e31213213213213
20 225 σe2+e3+τe1∧e2∧e31232132132132
which is manifested at the level of the pin group by virtue of containing the
pseudoscalar e1e2e3from Table 4. These are all contained in H3, and have
the effect of creating a second orthogonal I2(n) in the even subalgebra. We
illustrate the idea and its complexities with some examples.
Example. Take as a first example A3
1with simple roots e1,e
2,e
3but think of
it as I2(2) ⊕A1. Now this gets doubled to I2(2) ⊕I2(2) = A4
1via the spinor
group ±1, ±e1e2,±e2e3and ±e3e1.
Example. Now consider the following twist: take A2
1with simple roots e1,e
2
but instead of having e3available as another orthogonal simple root, we just
have the inversion e1e2e3available. So we get ±1and±e1e2from multiplying
the simple roots. But by operating in the whole pin group we can multiply e1
by the pseudoscalar e1e2e3, which yields e2e3,whichis in the spin part. So
similarly we get ±e2e3and ±e3e1, i.e. we get the same spinor group as in the
previous example, without actually having the third simple root e3available.
This therefore induces the same A4
1root system.
Proposition 5.1. (Doubling—even case) For even nthe root system I2(n)
together with the inversion e1e2e3yields the doubling I2(n)⊕I2(n).
Proof. Without loss of generality take e1as the first simple root. Since n
is even, the number of roots is a multiple of 4 and therefore e2is also a
root. Therefore having e1,e2and e1e2e3available is equivalent to having
e3available as well, which via the Induction Theorem leads to a doubling
I2(n)⊕I2(n). 
A convenient choice of simple roots for I2(n)isα1=e1and α2=
−cos π
ne1+sinπ
ne2.
Proposition 5.2. (Doubling—general case) The root system I2(n)together
with the inversion e1e2e3yields the doubling I2(n)⊕I2(n). A possible choice
of simple roots for I2(n)⊕I2(n)is given by α1α1,α
1α2,α
1e1e2e3,α
2e1e2e3).
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57 Page 20 of 35 P. Dechant Adv. Appl. Clifford Algebras
Proof. Direct calculation confirms that these constitute two orthogonal I2(n)
root systems with respect to the spinor inner product, and the simple roots
give the correct Cartan matrix
A(I2(n)⊕I2(n)) = ⎛
⎜
⎜
⎝
2−2cos π
n00
−2cos π
n200
002−2cos π
n
00−2cos π
n2
⎞
⎟
⎟
⎠.
C.f. also the computational proof in the supplementary information. 
Remark 5.3. The inclusion of the A2⊕A2or H2⊕H2subrootsystems there-
fore arises in the more oblique way for odd nroot systems as n=3and
n= 5. The Cartan matrices are given by
A(A2⊕A2)=⎛
⎜
⎜
⎝
2−10 0
−12 0 0
002−1
00−12
⎞
⎟
⎟
⎠,A(H2⊕H2)=⎛
⎜
⎜
⎝
2−τ00
−τ20 0
002−τ
00−τ2
⎞
⎟
⎟
⎠
Proposition 5.4. (Simple roots of other subrootsystems) A2⊕A2and H2⊕H2
are generated straightforwardly from the H3generators a1,a
2,a
3e.g. via the
‘spinorial simple roots’
a1a1=1,a
1a2,a
1e1e2e3,a
2e1e2e3
for A2and
a1a1=1,a
2a3,a
2e1e2e3,a
3e1e2e3
for H2.
One possible choice of simple roots for A4within H4is given by
(a1a1,a
1a2,a
1a3a2a1a3a2a1a3,a
3a2a1a3a2a1a3a2a3a1a2a3),
but other choices are of course possible.
Remark 5.5. The fact that H3can’t have A1⊕A2or A1⊕H2subrootsystems
is clear from the following: the H3root system is the icosidodecahedron with
vertices at the 2-fold axes. It contains decagonal/hexagonal grand circles
which are valid A2or H2subrootsystems. However, a root normal to those
can’t exist because they would be the vertices of the icosahedron (5-fold
axes) or dodecahedron (3-fold axes), which is of course inconsistent with the
vertices being the 2-fold axes of the icosidodecahedron.
Remark 5.6. For o d d nthe 4D root systems induced by A1⊕I2(n) and via
I2(n) in combination with the inversion e1e2e3are subtly different (related
via e1↔e2). However, for even nthey of course coincide.
Remark 5.7. The inversion e1e2e3is often contained in Coxeter groups, but
is famously not contained in the Anfamily for odd n. As such A3, the tetrahe-
dral group, is a prime example of where this isn’t the case; it is even obvious
from the tetrahedron itself that it is not inversion invariant. The existence of
the inversion in a group means that one can use this pseudoscalar to dualise
root vectors to pure bivectors. In work on quaternions it was often regarded
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Vol. 31 (2021) Clifford Spinors and Ro ot System Page 21 of 35 57
as deeply meaningful that the pure quaternion roots e.g. of H4are exactly
the H3root system, and analogously for other cases (e.g. [28]). However, a
more useful way of viewing this is that this is pretty obvious from the Induc-
tion Theorem as long as the inversion is contained in the group. And that
rather than it being proof that the ‘top-down’ approach is somehow deeply
significant it is rather a sign of the opposite: that the ‘bottom-up’ approach
constructs H4from H3whilst one can dualise the 30 roots of H3directly
to pure bivectors/quaternions using the pseudoscalar/inversion. This is not
possible for A3where the inversion is missing and no pure quaternion repre-
sentation of A3within D4exists; however, the Induction Theorem still holds
and yields A3→D4[6].
Having shown the existence and nature of various H4subrootsystems
we now briefly discuss a nice way of visualising 4D polytopes in the Coxeter
plane, before using the subrootsystems of H4in order to construct pairs of
invariant polytopes which we then visualise in the Coxeter plane.
6. The Coxeter Plane
The Coxeter plane is a convenient way of visualising any root system in
any dimension. The exposition is not necessary for the following sections
but helps with the visualisation. We will briefly summarise the construction
of this plane that is invariant under a corresponding Coxeter element. Its
existence relies on the bipartite nature of the corresponding graphs (a two-
colouring) [21], which means that the simple roots can be partitioned into two
mutually orthogonal sets (e.g. black and white), as can the reciprocal basis,
the basis of fundamental weights. The properties of the Cartan matrix further
mean that a Perron–Frobenius eigenvector with all positive entries exists.
The components of this eigenvector corresponding to the black, respectively
white, roots are used in a linear combination of the black, respectively white,
fundamental weights. This gives a pair of (black and white) vectors which
together determine a plane, which can be shown to be invariant under the
Coxeter element. Since of course several such Coxeter elements exist (that are
conjugate to one another), there are likewise several such planes. However,
they give an equivalent description.
The Clifford view of the Coxeter plane more generally has been inves-
tigated in [11]. Here, we instead perform all calculations in the 3D even
subalgebra. The 4D simple roots can be chosen as follows in terms of the 3D
H3simple roots:
α1=a1a1=1
α2=a1a2=−1
2+1
2τe1∧e2+1
2σe2∧e3
α3=e1e2a2e3=1
2σe1∧e2−1
2e1∧e3+1
2τe2∧e3
α4=a2e1e2e3=1
2σe1∧e2+1
2e1∧e3−1
2τe2∧e3
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57 Page 22 of 35 P. Dechant Adv. Appl. Clifford Algebras
The reciprocal basis in this spinorial setup (with respect to the spinor
inner product) is given by the following basis of fundamental weights:
ω1=1−τe2∧e3−(τ+1)e1∧e3
ω2=−2τe2∧e3−2(τ+1)e1∧e3
ω3=−(2τ+1)e2∧e3−3(τ+1)e1∧e3−τe1∧e2
ω4=−(2τ+1)e2∧e3−(3τ+1)e1∧e3−τe1∧e2
The Perron–Frobenius eigenvector of the H4Cartan matrix is given by
v=
⎛
⎜
⎜
⎜
⎜
⎜
⎝
4+4
√5
21+√5√5+6√5+30+7
6√5+30+8+4
√5+√56√5+30
−1+√5+6√5+30
√5+6√5+30+7
⎞
⎟
⎟
⎟
⎟
⎟
⎠
whilst the eigenvalue is
λ=2−1
27+√5+6√5+30.
Figure 1shows the projection of the 120 vertices and 720 edges of the
H4root system (aka the 600-cell) into its Coxeter plane. The projection is
performed via the respective inner product with the ‘black and white spinors’.
The computations are shown in the supplementary information.
We will use this Coxeter plane as a means to visualise the H4substruc-
tures of the following sections, including the Grand Antiprism and the snub
24-cell with their H2⊕H2and D4(aka 2T) complements, as well as anal-
ogous constructions with A1⊕A1⊕A1⊕A1,A2⊕A2and A4and their
complements in the 600-cell.
7. The Grand Antiprism and H2×H2
It is of course simple to show that H3contains an H2root system (generated
by the a2and a3simple roots), which leads to a corresponding root system H2
in the 4D space of spinors. However, since the inversion e1e2e3is contained in
the group H3this gets doubled to two orthogonal copies H2⊕H2sitting inside
the H4root system, as seen above. A possible set is shown below, which is the
one multiplicatively generated by a2and a3in combination with e1e2e3via
the geometric product. Of course the H3root system, the icosidodecahedron,
contains many such decagonal circles, but for this set of simple roots this set
could be considered preferred:
Of course they are invariant under their own automorphism group
Aut(H2⊕H2) of order 20×20 = 400. In previous work the author has already
argued that the automorphism group of a spinorial/quaternionic root system
is just the group acting on itself by left and right multiplication, leading to
two factors of the same group [7]. This spinorial group multiplication is to
be distinguished from the above ‘spin reflections’, which one could consider
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Vol. 31 (2021) Clifford Spinors and Ro ot System Page 23 of 35 57
Figure 1. Projection of the 120 roots of H4into the Coxeter
plane. This is a familiar view—however, all the calculations
have been done in the even subalgebra of 3D
Element ×2 Element ×2
22e2∧e3
−2−2e2∧e3
−τ−e1∧e2+σe1∧e3σe1∧e2+e1∧e3−τe2∧e3
σ−τe1∧e2−e1∧e3−e1∧e2+τe1∧e3+σe2∧e3
τ−e1∧e2+σe1∧e3σe1∧e2+e1∧e3+τe2∧e3
τ+e1∧e2−σe1∧e3−σe1∧e2−e1∧e3+τe2∧e3
−σ+τe1∧e2+e1∧e3e1∧e2−τe1∧e3−σe2∧e3
−σ−τe1∧e2−e1∧e3−e1∧e2+τe1∧e3−σe2∧e3
−τ+e1∧e2−σe1∧e3−σe1∧e2−e1∧e3−τe2∧e3
σ+τe1∧e2+e1∧e3e1∧e2−τe1∧e3+σe2∧e3
a different type of multiplication, generating the reflection/Coxeter groups,
which was termed ‘conjugal’ in the above paper. The H2⊕H2root system
is invariant under this conjugal group multiplication (i.e. 4D reflections) by
virtue of being a spinor group by Proposition 2.13.
But since this H2×H2is a subgroup of H4, its complement in the 600-
cell is separately left invariant. By this we mean the collection of vertices
derived from the 600-cell by subtracting from the 120 vertices the 20 vertices
from the H2⊕H2root system. This orbit / collection of points is separately
invariant and therefore has the same automorphism group of order 400. This
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57 Page 24 of 35 P. Dechant Adv. Appl. Clifford Algebras
Figure 2. The projection of the 20 roots of the H2⊕H2sit-
ting inside the 600-cell/H4root system into the H4Coxeter
plane
Figure 3. Removal of those 20 roots of the H2⊕H2from
the 600-cell/H4root system leads to the Grand Antiprism
with 100 vertices. Their projection into the H4Coxeter plane
is shown above
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Vol. 31 (2021) Clifford Spinors and Ro ot System Page 25 of 35 57
can also be verified straightforwardly by direct computation (see supplemen-
tary material). This complement of the H2⊕H2root system is a 4D polytope
with 100 vertices and 500 edges which was found in 1965 in [5] by computa-
tional means and is called the ‘Grand Antiprism’. This is also discussed from
a quaternionic perspective in [22–25]. The construction from 3D seems both
simpler in terms of deriving the vertex set and in shedding light on its sym-
metry group, as well as the conceptual and uniform construction that carries
over in the other cases below. The projection of the two orthogonal H2sinto
the H4Coxeter plane is shown in Fig. 2. The corresponding projection of the
Grand Antiprism is shown in Fig. 3. For ease of visualisation the edges are
also computed and projected, and plotted in 3D in a SageMath visualisation
that can be further explored [30] (supplementary information).
Proposition 7.1. (H2×H2split of H4vertices) The sets H2⊕H2and its
complement the Grand Antiprism are separately invariant under H2×H2.
Proof. Straightforward or by straightforward explicit calculation (supplemen-
tary material). 
8. Snub 24-Cell
From a 3D perspective it is obvious that although A3is not a subrootsystem
of H3, the binary tetrahedral group 2Tis contained in the binary icosahedral
group 2I(this can also easily been seen since the tetrahedral group is the
alternating group A4whilst the icosahedral group is A5and thus the former
is contained in the latter, and this also holds for their spin double covers) and
the root system D4is thus contained in H4. As a spin/quaternionic group,
D4is of course just 2Tand its automorphism group is just 2T×2Tof order
242= 576. Removing the 24 vertices from the 600-cell leads to a set of 96
points that is separately invariant under D4, in analogy to the construction
of the Grand Antiprism above. This collection of 96 vertices connected by
432 edges is known as the ‘snub 24-cell’. Again we believe the construction
from 3D to be conceptually clearer, more systematic and more economical.
The projection of the 24 2Tspinors (listed below) aka the D4root system
along with its 96 edges into the H4Coxeter plane is shown in Fig. 4.The
corresponding projection of the snub 24-cell and its edges is shown in Fig. 5.
Note that because of the spin double cover property they come at least
in pairs. But if they are not their own reverse, then they even come in quadru-
plets related via reversal and multiplication by −1 (c.f. Proposition 2.13).
Proposition 8.1. (D4split of H4vertices) The sets D4and its complement
the snub 24-cell are separately invariant under D4.
Proof. Straightforward or by straightforward explicit calculation (supplemen-
tary material). 
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57 Page 26 of 35 P. Dechant Adv. Appl. Clifford Algebras
Figure 4. The projection of the 24 roots of the D4root
system sitting inside the 600-cell/H4root system into the
H4Coxeter plane. These vertices of course come from the
binary tetrahedral group 2Tinside the 2I
Element ×2 Element ×2
2−2
e1∧e2−τe1∧e3+σe2∧e3−e1∧e2+τe1∧e3−σe2∧e3
σe1∧e2−e1∧e3+τe2∧e3−σe1∧e2+e1∧e3−τe2∧e3
τe1∧e2−σe1∧e3+e2∧e3−τe1∧e2+σe1∧e3−e2∧e3
1+e1∧e2−e1∧e3+e2∧e3−1−e1∧e2+e1∧e3−e2∧e3
1−e1∧e2+e1∧e3−e2∧e3−1+e1∧e2−e1∧e3+e2∧e3
1+τe1∧e2+σe2∧e3−1−τe1∧e2−σe2∧e3
1−τe1∧e2−σe2∧e3−1+τe1∧e2+σe2∧e3
1+σe1∧e2−τe1∧e3−1−σe1∧e2+τe1∧e3
1−σe1∧e2+τe1∧e3−1+σe1∧e2−τe1∧e3
1+σe1∧e3−τe2∧e3−1−σe1∧e3+τe2∧e3
1−σe1∧e3+τe2∧e3−1+σe1∧e3−τe2∧e3
9. A4
1,A2×A2and A4
We follow the above uniform construction of splitting the H4root system
with respect to its subrootsystems, using the remaining examples of A1⊕
A1⊕A1⊕A1,A2⊕A2and A4.
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Vol. 31 (2021) Clifford Spinors and Ro ot System Page 27 of 35 57
Figure 5. The projection of the snub 24-cell into the H4
Coxeter plane. The 96 vertices can be derived by removing
the 24 vertices of D4/the binary tetrahedral group from the
600-cell
The basic unit vectors e1,e2,e3of course generate the quaternion group
consisting of ±1, ±e1e2,±e2e3and ±e3e1. The projection of this 16-cell with
its 8 vertices and 24 edges into the H4Coxeter plane is shown in Fig. 6.Of
course its complement is also invariant under A1×A1×A1×A1:
Proposition 9.1. (A1×A1×A1×A1split of H4vertices) The sets A1⊕A1⊕
A1⊕A1and its complement are separately invariant under A1×A1×A1×A1.
Proof. Straightforward or by straightforward explicit calculation (supplemen-
tary material). 
The projection of this complement of this 16-cell with its 112 vertices
and 624 edges into the H4Coxeter plane is shown in Fig. 7.
Similarly to the H2case above, it is simple to show that H3contains
an A2root system (generated via reflections in the a1and a2simple roots),
which leads to a corresponding root system A2in the 4D space of spinors.
However, since the inversion e1e2e3is contained in the group H3this A2
again gets doubled to two orthogonal copies A2⊕A2sitting inside the H4
root system, as seen above. A possible set is shown below, which is the one
multiplicatively generated by a1and a2simple roots of H3in combination
with the inversion e1e2e3. But other choices would be possible, since of course
the H3root system, the icosidodecahedron, contains many such hexagonal
circles.
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57 Page 28 of 35 P. Dechant Adv. Appl. Clifford Algebras
Figure 6. The projection of the 8 roots of the A4
1sitting
inside the 600-cell/H4root system into the H4Coxeter plane
Element ×2 Element ×2
2−2e1∧e3
−22e1∧e3
−1+τe1∧e2+σe2∧e3σe1∧e2+1e1∧e3−τe2∧e3
1+τe1∧e2+σe2∧e3σe1∧e2−1e1∧e3−τe2∧e3
1−τe1∧e2−σe2∧e3−σe1∧e2−1e1∧e3+τe2∧e3
−1−τe1∧e2−σe2∧e3−σe1∧e2+1e1∧e3+τe2∧e3
The projection of these 12 points into the H4Coxeter plane as before
is shown in Fig. 8. Its complement consisting of 108 vertices and 576 edges
is shown in Fig. 9.
Proposition 9.2. (A2×A2split of H4vertices) The sets A2⊕A2and its
complement are separately invariant under A2×A2.
Proof. Straightforward or by straightforward explicit calculation (supplemen-
tary material). 
Our final example is the A4root system contained in H4as we saw
above. Again the A4root system and its complement are both invariant.
Proposition 9.3. (A4split of H4vertices) The sets A4and its complement
are separately invariant under A4.
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Vol. 31 (2021) Clifford Spinors and Ro ot System Page 29 of 35 57
Figure 7. The projection of the 112 vertices into the H4
Coxeter plane that are left from the 600-cell by removing
the above 8 vertices constituting the A4
1root system, along
with its 624 edges
Proof. Straightforward or by straightforward explicit calculation (supplemen-
tary material). 
The root system consists of 20 vertices and 60 edges (shown in Fig. 10)
whilst its complement consists of 100 vertices and 480 edges and is shown in
Fig. 11.
A possible choice of A4roots is shown here:
Element ×2 Element ×2
2−2
τe1∧e2+σe1∧e3+e2∧e3−τe1∧e2−σe1∧e3−e2∧e3
e1∧e2−τe1∧e3+σe2∧e3−e1∧e2+τe1∧e3−σe2∧e3
σe1∧e2−e1∧e3−τe2∧e3−σe1∧e2+e1∧e3+τe2∧e3
1+τe1∧e2+σe2∧e3−1−τe1∧e2−σe2∧e3
1−τe1∧e2−σe2∧e3−1+τe1∧e2+σe2∧e3
1+σe1∧e3+τe2∧e3−1−σe1∧e3−τe2∧e3
1−σe1∧e3−τe2∧e3−1+σe1∧e3+τe2∧e3
1+σe1∧e2−τe1∧e3−1−σe1∧e2+τe1∧e3
1−σe1∧e2+τe1∧e3−1+σe1∧e2−τe1∧e3
Note that because of the spin double cover property these elements come
at least in pairs (which satisfies the corresponding root system axiom). But
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57 Page 30 of 35 P. Dechant Adv. Appl. Clifford Algebras
Figure 8. The projection of the 12 roots of the A2⊕A2sit-
ting inside the 600-cell/H4root system into the H4Coxeter
plane
Figure 9. The projection of the 108 vertices into the H4
Coxeter plane that are left from the 600-cell by removing
the above 12 vertices from the A2⊕A2root system
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Vol. 31 (2021) Clifford Spinors and Ro ot System Page 31 of 35 57
Figure 10. The projection of the 20 roots of the A4root
system sitting inside the 600-cell/H4root system into the
H4Coxeter plane
Figure 11. The projection of the complement of the above
into the H4Coxeter plane. The 100 vertices can be derived
by removing the 20 vertices of A4from the 600-cell
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57 Page 32 of 35 P. Dechant Adv. Appl. Clifford Algebras
if they are not their own reverse, then they even come in quadruplets related
via reversal and multiplication by −1 (c.f. Proposition 2.13).
This concludes our listed examples, which illustrate the uniform
approach of constructing subrootsystems via the connection with 3D and
splitting the vertices of the 600-cell into two separately invariant sets. The
H4Coxeter plane provides a nice visualisation for each complementary pair
based on H4subrootsystems.
10. Conclusions
The intention for this article was to firstly shed light on 4D geometry, root
systems and polytopes through the connection with 3D spinors via the uni-
form Induction Theorem. This gives additional insight into 4D root systems
and polytopes along with their symmetries via another uniform construc-
tion that splits the H4root system into a complementary pair of separately
invariant polytopes, consisting of a subrootsystem and its complement in
H4. These can also be consistently visualised via a projection into the Cox-
eter plane. As we have shown in previous work there are many connections
across exceptional objects throughout mathematics, including Trinities and
ADE correspondences, which this work relates to. In particular, the ‘bottom-
up’ view of exceptional objects has led to a very fruitful and insightful line
of research, and is perhaps mirrored by some constructions in finite group
theory for some of the sporadic groups. We continue to advocate the use of
geometric insight in addition to the purely algebraic manipulation in terms of
quaternions, by viewing quaternions as arising in a geometric guise as spinors
in 3 dimensions with a much clearer and consistent geometric interpretation.
Secondly, we continue to advocate that Clifford algebras and root sys-
tems/reflection groups are natural and complementary frameworks - in a
setting with a vector space with an inner product, and a powerful reflection
formula - and can therefore be synthesised into one powerful and coherent
framework. Thirdly, this provides a general arena to do group theoretical cal-
culations in as demonstrated with some detailed examples in this paper, and
we will investigate group and representation theoretic topics in more detail
in future work. In particular, in the spirit of open science and reproducibility
I hope that the computational work sheets provided as supplementary mate-
rial are useful and can help further research in this area by adapting my code
and using other free software provided by the community such as galgebra
and Sage. Finally, at the end of 2020, we wish to commemorate and honour
some of the giants in this field, Conway and the Guys.
Acknowledgements
This paper is dedicated to the memory of the late John Horton Conway
and Richard Kenneth Guy, as well as Michael Guy. I would like to thank
Yang-Hui He, Yvan Saint-Aubin, Jiˇri Patera, Peter Cameron, John McKay,
Robert Wilson, Alexander Konovalov, Jean-Jacques Dupas, Tony Sudbery,
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Reidun Twarock, David Hestenes, Anthony Lasenby, Joan Lasenby and Eck-
hard Hitzer for interesting discussions over the years, and Hugo Hadfield and
Eric Wieser for help with setting up the galgebra software package [3].
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Vol. 31 (2021) Clifford Spinors and Ro ot System Page 35 of 35 57
Pierre-Philippe Dechant
School of Science, Technology and Health
York St John University
York YO31 7EX
UK
and
Department of Mathematics
University of York
Heslington YO10 5GE
UK
and
York Cross-disciplinary Centre for Systems Analysis
University of York
Heslington YO10 5GE
UK
e-mail: ppd22@cantab.net
Received: December 29, 2020.
Accepted: April 8, 2021.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

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