Wolfgang Bertram Graded Sets

GRADED SETS, GRADED GROUPS Clifford Algebras 2021 Hal-03332012

Clifford Algebras

Wolfgang Bertram

Keywords: 17A70 18D25 18M15 Clifford algebra, quaternion group, graded tensor product, graded group, central extension, cocycle, super algebra, braiding, monoidal category

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Introduction

Let (Γ, +) be an abelian group (our main emphasis will be on the case Γ = Z/2Z). Generally, Γ-graded structures play an important rôle in mathematics. In this work we study a class of Γ-gradings, which, when combined with what we call a central Γ-action, has very interesting and pleasant properties. 0.1. Additively and multiplicatively graded sets. A first point in the present work is to distinguish between additively, and multiplicatively graded structures: Definition 0.1. Let X and M be sets.

(1) An additive Γ-grading on M is a disjoint union structure, written

M = ⊔ γ∈Γ M γ , or M = γ∈Γ M γ ,
i.e., M is the coproduct of a family of sets (M γ ) γ∈Γ . We allow some of the M γ to be empty. Thus, an additive grading is equivalent to a degree-map,

d : M → Γ, such that M γ = d -1 (γ).
(2) A multiplicative Γ-grading on X is a direct product structure of the form

X = × γ∈Γ X γ , or X = γ∈Γ X γ .
If M is additively graded, then the space X = K M of functions from M to a set K is multiplicatively graded (a function f : M → K corresponds to the family (f | Mγ ) γ∈Γ ); in particular, taking K = {0, 1}, we see that the power set P(M) of M is multiplicatively graded: multiplicative gradings often arise as "exponentials" of additive ones. This furnishes important links between both types of gradings. 0.2. Graded groups, and Clifford groups. In this work, we focus on additively graded sets, and in particular on (additively) graded groups or monoids: Definition 0.2. A Γ-graded monoid is a monoid G which is additively Γ-graded, such that the grading map d : G → (Γ, +) is a morphism of monoids. It is called centrally graded given a morphism Z : (Γ, +) → G, x → Z(x) =: Z x such that:

(1) Z takes values in the center of G: for all x ∈ Γ and g ∈ G : Z x g = gZ x , (2) d • Z = 0, i.e., ∀x ∈ Γ : d(Z x ) = 0.

When Γ = Z/2Z = {0, 1}, then the element Z = Z(1) is called the grading element, and the power Z x is the same as Z(x). A prototype of graded group G then is what in certain texts (e.g., [L]) is called the (discrete) Clifford group, and elsewhere Salingaros Vee-groups, (cf. [S84, S82, A]): the canonical basis elements e 1 , . . . , e n of the Clifford algebra Cl p,q (K) (see Appendix A for notation) generate a group G = Q p,q of invertible elements in the Clifford algebra. This group is Γgraded in the sense defined above: the grading morphism d is the unique morphism d : Q p,q → Z/2Z such that d(e i ) = 1 (odd) for all i, and the grading element is Z = -1 (since e i , e j = 0 for i = j, we have e i e j = -e j e i , so the group G contains the element -1, which clearly is central in G). The underlying philosophy is to replace the central element -1 from the Clifford case by an "abstract central element". Thus the approach presented here will give an abstract, group-theoretic presentation of the discrete Clifford group. Such a group-theoretic approach has already been advocated by the above quoted authors, but here we will go much further by putting it into a general categorical approach, as follows. 0.3. The braided monoidal category of graded sets. A key feature in the theory of Clifford algebras is the relation (see, e.g., [BtD], Prop. I.6.6) (0.1) Cl(U ⊕ W ) = Cl(U) ⊗ Cl(W )

expressing the Clifford algebra of a quadratic space U ⊕ W by the graded tensor product of the corresponding Clifford algebras. When U, W carry the zero quadratic form, we get the corresponding rules for Grassmann algebras. Our approach is designed to furnish an analog of this for general (additvely) centrally Γ-graded sets:

Definition 0.3. A Γ-graded set, with grading map d : M → Γ, is called centrally Γ-graded if it is equipped with a (left) Γ-action

z : Γ × M → M, (x, m) → z(x, m) = x.m
preserving the grading, i.e., such that ∀x ∈ Γ, ∀m ∈ M : d(x.m) = d(m).

A centrally graded monoid G is a centrally graded set: just let z(x, m) = Z x m. First of all, we define the graded product of sets: Theorem 0.4. Assume d i : M i → Γ, i = 1, 2, are two centrally Γ-graded sets. Let M be the quotient of M 1 × M 2 for the equivalence relation defined by ∀x ∈ Γ : (x.m 1 , m 2 ) ∼ (m 1 , x.m 2 ). Then M, together with grading map and Γ-action

d([m 1 , m 2 ]) = d(m 1 ) + d(m 2 ), x.[m 1 , m 2 ] = [x.m 1 , m 2 ] = [m 1 , x.m 2 ],
Theorem 0.6. For any generalized ring, the braiding maps are isomorphisms of centrally graded sets and define a structure of (strict) braided monoidal category of centrally Γ-graded sets. If the generalized ring product is skew, meaning that ab + ba = 0 for all a, b ∈ Γ, then this structure is symmetric, i.e., we always have

β -1 M 1 ,M 2 = β M 2 ,M 1 .
In the symmetric case, there is a simple formula, in terms of permutation inversions, for the action of the symmetric group on iterated products (Theorem 2.1). Next, braiding maps are used to define graded products of graded groups: Theorem 0.7. We fix a generalized ring structure on Γ.

(1) For every centrally Γ-graded group G there is a braided dual group (b-dual) given by G ∨ = G, with same grading and central action as G, and product x) xy.

x • y = Z d(y)d(
(2) There is a group structure, denoted by G 1 × Z G 2 , on the product G 1 × z G 2 of two centrally Γ-graded groups G i , i = 1, 2, given by the group law

[g 1 , g 2 ] • [h 1 , h 2 ] = Z d(h 1 )d(g 2 ) [g 1 h 1 , g 2 h 2 ].
(3) Iteration of the preceding construction is associative: the natural map

(G 1 × Z G 2 ) × Z G 3 ∼ = G 1 × Z (G 2 × Z G 3 ) is a group isomorphism. (4) When Γ is skew, then the braiding map β G 1 ,G 2 is an isomorphism from G 1 × Z G 2 onto G 2 × Z G 1 .
Dually to the result mentioned above, there is also a simple formula for the product of n elements in G 1 × Z G 2 (Remark 2.4) and in G ∨ (Remark 2.2). 0.4. The discrete Clifford category. Turning to concrete examples, we have already mentioned that (discrete) Clifford groups provide non-trivial examples illustrating the preceding facts. More abstractly, given a generalized ring Γ (which for simplicity we assume to be skew), we define a sequence of groups: let Q 0 := (Γ, +),

Q 1,0 := Γ 2 = Γ × Γ, Q 0,1 := (Q 1,0 ) ∨ , . . . Q p,q := × p z Q 1,0 × z ( × q z Q 0,1 ),
where Γ 2 is an abelian group, but with non-trivial grading and action (see Example 2.1). When Γ = Z/2Z, then this sequence is exactly the sequence of discrete Clifford groups. In this case, the first few groups are

Q 0 = (Γ, +) = C 2 = Z/2Z, Q 1,0 = V = C 2 × C 2 (the Klein four-group), Q 0,1 = C = C 4 (the cyclic group of order 4), Q 2,0 = Q 1,0 × Z Q 1,0 = D = D 4 (the dihedral group of symmetries of the square), Q 0,2 = Q 0,1 × Z Q 0,1 = Q (the quaternion group of order 8)., Q 1,1 ∼ = D 4
(but with a different grading than above).

We investigate this discrete Clifford category in some detail in Section 3. Most facts are, essentially, known from the theory of Clifford algebras: Salingaros denotes these groups by G p,q and calls them Vee-groups (see [S84, S82], see also references given in [A, AM]). In the present work, we push the group theoretic approch given by these authors much further, by putting it in the large framework of graded monoidal categories, allowing much more general rings than Z/2Z. For instance, when Γ = R 2 with its canonical skew-symmetric product x • y = (x 1 y 2x 2 y 1 , 0), we get a sequence of groups of Heisenberg-type, called the Heisenberg category. 0.5. Group algebra, and Clifford algebra. The group algebra K[G] of the groups G = Q p,q is a super-algebra which decomposes as a direct sum of two ideals K[G] + (which in turn is a group algebra of an abelian group), and K[G] -, which is precisely the Clifford algebra (Theorem 4.2 and 4.1). More generally, when

G = G 1 × Z G 2 , then K[G] + is the ordinary (ungraded) tensor product K[G 1 ] ⊗K[G 2 ]
of group algebas, and K[G] -is the graded tensor product (Theorem 4.1). 0.6. Concluding remarks. It seems as if the category of centrally Z/2Z-graded sets supports a good deal of what can be done on ordinary, ungraded sets. One may ask if this carries as far as differential calculus and analysis -indeed, the joint paper with J. Haut, [BeH] proposes a setting of "categorical differential calculus" which seems to be suitable for an adaptation to a "graded", or "super", framework. The important challenge is to understand if, and how, such a calculus were related to other known categorical approaches ( [Mol, Sa, Sch]). We hope to investigate this question in future work.

Another aspect, already present in, e.g., [A,[START_REF] Albuquerque | Clifford algebras obtained by twisting of group algebras[END_REF][START_REF] Lundholm | Clifford algebra, geometric algebra, and applications[END_REF][START_REF] Salingaros | The relationship between finite groups and Clifford algebras[END_REF], becomes clearified in the present setting, namely, the relation with the theory of central extensions of groups: indeed, all our groups can be viewed as central extensions, and as such, can be described by certain cocycles (Section 3.3). It becomes apparent that these cocycles are all related to algebra and combinatorics of totally ordered (finite) index sets, in particular, via inversions (Fehlstand, in German). * * * Je souhaite dédier ce travail à la mémoire de mon collègue Lionel Bérard-Bergery (1945-2019). Les discussions que nous avions, autour de sujets de géométrie, où les graduations étaient toujours présentes, m'ont été une source inépuisable d'inspiration dans ce domaine. Bien que Lionel n'ait pas eu le temps de formaliser et de finaliser ces idées, j'espère qu'elles verront le jour, tôt ou tard, et que ce travail y contribuera.

Γ-graded sets and monoids

In this section, (Γ, +) is an abelian group.

1.1. Additively and multiplicatively graded sets, and grading automorphism. The definition of (additively) Γ-graded sets and monoids has been given above (Def. 0.1 and 0.2). As explained in the introduction, we'll focus on additively graded structures, and simply call them "graded". Let's turn them into a category. We have the choice between a "strong" or "weak" definition of morphisms (preserving, or permuting, the grading):

Definition 1.1. Let X, X ′ be sets and (Γ, +) an abelian group. A morphism of additively Γ-graded sets (X, d),

(X ′ , d ′ ), of degree d(f ) ∈ Γ, is a map f : X → X ′ such that ∀x ∈ X : d(f (x)) = d(f ) + d(x). When d(f ) = 0, we say that f is a grading-preserving morphism.
A morphism of centrally Γ-graded sets is a grading-preserving morphism commuting with the Γ-action (f (γ.x) = γ.f (x), for all x ∈ M and γ ∈ Γ).

A morphism of centrally Γ-graded monoids or groups is a morphism of groups or monoids satisfying the preceding condition.

Definition 1.2. On every centrally Γ-graded set (M, d, z), we define the grading automorphism by

α : M → M, x → α(x) = d(x).x = z(d(x), x).
Clearly, α is a grading-preserving morphism. Its inverse is α -1 (x) = z(-d(x), x).

Definition 1.3. We say that (M, -d) is the negative grading of (M, d). g) g is an automorphism of graded groups (again called grading automorphism).

Lemma 1.4. If (G, d, Z) is a centrally Γ-graded group, then α : G → G, g → Z d(
Proof. α(gh) = Z d(gh) gh = Z d(g) Z dh gh = Z d(g) gZ d(h) h = α(g)α(h), α(e) = e.
When Z : Γ → G is injective, then the grading can be recovered from α via G γ = {x ∈ G | ∀u ∈ Γ : ux = γx} (a kind of "pre-eigenspace"). As already noticed, examples of centrally Γ-graded sets arise from centrally graded groups:

Example 1.1. Let M := P(n) the power set of n = {1, . . . , n}. It is an abelian group for the symmetric difference A∆B of sets, and from elementary set theory we have a group morphism

d : P(n) → Γ := Z/2Z, A → |A| mod 2.
So M 0 = P(n) 0 is the subset of sets having even cardinality, and M 1 = P(n) 1 the one of subsets having odd cardinality. A central action of Z/2Z on P(n) is obtained by choosing some set B ⊂ n of even cardinality and letting z(0, A) = A, z(1, A, ) = A∆B,. We have α(A) = A if A is even, and α(A) = A∆B if A is odd.

Example 1.2. All dihedral groups D n are Z/2Z-graded, via the quotient morphism D n → D n /C n = C 2 (where C n is the cyclic group of order n). But only for n even they are centrally Z/2Z-graded: only in this case the center is non-trivial, Z(D 2n ) = {±1} ∼ = C 2 , so that Z : Z/2Z → D 2n can be taken to be Z(0) = 1, Z(1) = -1. Note that the groups D 4n admit several normal subgroups of index 2, containing the element -1, so there are several different central gradings on D 4n .

Example 1.3. The quaternion group Q is centrally Z/2Z-graded: similarly to the case of D 4 , its center ±1 is isomorphic to C 2 , and it has several normal subgroups of index 2 containing the element -1. In contrast to the D 4 -case, the different normal subgroups in question are conjugate to each other under outer automorphisms.

Remark 1.1 (Central extensions). The quotient (orbit set) of the Γ-action, M 0 := M/Γ, is again Γ-graded: the map d passes to the quotient d : M 0 → Γ. If the action of Γ is free, it will sometimes be useful to choose a set-theoretic section of the projection M → M 0 , and thus to identify M with M 0 × Γ. In particular, the setting of centrally graded groups self-dual in the sense that, reversing arrows, d and Z change their roles, and we have exact sequences of groups

G 0 → G → Γ, Γ → G → G 0 := G/Z Γ ,
the second of them realizing G as central extension of G 0 by Γ. Choosing a set-map G 0 → G, section to the projection G → G 0 , the set G can be identified with G 0 × Γ, with group law expressed via a map τ : G 0 × G 0 → Γ (with G 0 written additively),

(h, u) • (h ′ , u ′ ) = (hh ′ , u + u ′ + τ (h, h ′ )).
Associativity is equivalent to saying that τ satisfies the cocycle relation

τ (h, h ′ ) + τ (hh ′ , h ′′ ) = τ (h, h ′ h ′′ ) + τ (h ′ , h ′′ ).
(cf. Section 3.3). Note that d induces a morphism d : G 0 → Γ, but the induced [Z] : Γ → G 0 would be trivial.

1.2. Monoidal structure on the category of centrally Γ-graded sets. We define the product M = M 1 × z M 2 of two centrally Γ-graded sets as explained in Theorem 0.4. By routine check, M is again centrally Γ-graded. (Note that the equivalence relation is the orbit relation of the action of the "antidiagonal subgroup" {(γ, -γ) | γ ∈ Γ} on M 1 ×M 2 , and the grading passes to the quotient.) To complete the proof of Theorem 0.4, we check associativity

(M 1 × z M 2 ) × z M 3 = M 1 × z (M 2 × z M 3 ).
Indeed, this follows from associativiy (which we consider to be "strict") of the usual Cartesian product ×, which passes to the quotient. Namely, on both sides, the grading is induced by d 1 ⊕ d 2 ⊕ d 3 (by associativity and commutativity of the group (Γ, +)), and on both sides, the quotient is taken with respect to the "generalized antidiagonal"

N := {(γ 1 , γ 2 , γ 3 ) ∈ Γ 3 | γ 1 + γ 2 + γ 3 = 0}.
Thus both sides are identified with the quotient for the equivalence relation

z γ [x 1 , x 2 , x 3 ] = [z γ 1 x 1 , x 2 , x 3 ] = [x 1 , z γ 2 x 2 , x 3 ] = [x 1 , x 2 , z γ 3 x 3 ].
Finally, the set E = Γ with trivial grading d = 0 and action on itself by z γ (x) = γ+x is neutral for the graded product (direct check). And, given morphisms

f i : M i → M ′ i , i = 1, 2, of degree u i = d(f i ), the following is a well-defined morphism from M 1 × z M 2 to M ′ 1 × z M ′ 2 of degree u 1 + u 2 : (f 1 × z f 2 )[x 1 , x 2 ] = [f 1 (x 1 ), f 2 (x 2 )].
Summing up, we get a bifunctor, turning centrally Γ-graded sets with their graded product into a (strict) monoidal category (see [CWM], Chapter VII, p. 161).

Braidings, and graded product of groups

Next, we discuss "commutativity": what is the relation between M 1 × z M 2 and M 2 × z M 1 ? A braiding on a monoidal category is the choice of a "choherent" family of isomorphisms switching the order of product (see [CWM], Chapter XI). In general, a braiding is an additional structure. The standard braiding from set theory, A × B ∼ = B × A, induces of course a braiding on centrally Γ-graded sets: we call this again the standard braiding.

2.1. Braiding of centrally graded sets. In order to define other braidings, from now on, we assume that Γ is a generalized ring, with bi-additive product map Γ 2 → Γ, (a, b) → ab, and define the braiding map

β M 1 ,M 2 : M 1 × Z M 2 → M 2 × Z M 1
as in Definition 0.5. Let us prove Theorem 0.6. Clearly, β M 1 ,M 2 preserves gradings and commutes with the Γ-action (since (Γ, +) is commutative), hence is a morphism of centrally Γ-graded sets. It is an isomorphism, since it admits an inverse

β -1 : [y 1 , y 2 ] → Z -d(y 1 )d(y 2 ) [y 2 , y 1 ].
Note that this inverse corresponds to the original braiding map when replacing d by -d and the "product" ab of Γ by its "opposite product" ba. Thus, if ab = -ba, we have the symmetry condition β -1 M 1 ,M 2 = β M 2 ,M 1 , as claimed. Saying that our strict monoidal category is braided amounts to the following two "hexagonal diagram identities" (see [CWM], Chapter XI, p. 253): first, the following two morphisms from

M 1 × Z M 2 × Z M 3 to M 2 × Z M 3 × Z M 1 coincide (for better readability, we shall write, β 1,2 instead of β (M 1 ,M 2 ) , and β 1,23 instead of β (M 1 ,M 2 ×zM 3 ) , etc.): (id 2 × Z β (1,3) ) • (β (1,2) × Z id 3 ) = β (1,23) .
We prove this by direct computation (writing

d instead of d i ) (id 2 × Z β (1,3) ) • (β (1,2) × Z id 3 )[x 1 , x 2 , x 3 ] = (id 2 × Z β (1,3) )(Z d(x 2 )d(x 1 ) [x 2 , x 1 , x 3 ]) = z d(x 2 )d(x 1 )+d(x 3 )d(x 1 ) [x 2 , x 3 , x 1 ], β (1,23) [x 1 , x 2 , x 3 ] = z (d(x 2 )+d(x 3 ))d(x 1 ) [x 2 , x 3 , x 1 ].
By distributivity of the product in Γ, both sides agree. The second "hexagonal" condition amounts to the first condition, when replacing the braiding maps by their inverses. But, as noticed above, the inverses of the braiding are of the same form (replacing d by -d and the product in Γ by its opposite), so the computation is exactly the same.

Remark 2.1. As explained in [CWM], Section XI.5 (Braided Coherence), the preceding arguments amount to proving that the morphisms s i := β (M i ,M i+1 ) satisfy the defining relations of the generators of the braid group s i s i+1 s i = s i+1 s i s i+1 , and s i s j = s j s i whenever |i -j| = 1.

Explicitly, let's show that s 2 s 1 s 2 = s 1 s 2 s 1 , by a computation essentially equivalent to the one given above. We write (23) for s 2 and ( 12) for s 1 :

(23)(12)(23)[x 1 , x 2 , x 3 ] = (23)(12)Z d 3 d 2 [x 1 , x 3 , x 2 ] = (23)Z d 3 d 2 +d 3 d 1 [x 3 , x 1 , x 2 ] = Z d 3 d 2 +d 3 d 1 +d 2 d 1 [x 3 , x 2 , x 1 ] (12)(23)(12)[x 1 , x 2 , x 3 ] = (12)(23)Z d 2 d 1 [x 2 , x 1 , x 3 ] = (12)Z d 2 d 1 +d 3 d 1 [x 2 , x 3 , x 1 ] = Z d 2 d 1 +d 3 d 1 +d 3 d 2 [x 3 , x 2 , x 1 ]
When the braiding is symmetric, the action on the braid group passes to an action of the symmetric group S n (coherence theorem in symmetric monoidal categories, see [CWM], p. 253, Thm.1). We give an explicit formula describing this action in terms of permutation inversions:

Theorem 2.1. Assume the product in Γ is skew. Then there are well-defined isomorphisms, for every σ ∈ S n ,

β (M 1 ,...,Mn) σ : M 1 × Z . . . × Z M n → M σ(1) × Z . . . × Z M σ(n) ,
coinciding for adjacent transpositions s i = (i, i + 1) with the braiding isomorphisms, and in general given by

β (M 1 ,...,Mn) σ [x 1 , . . . , x n ] = Z u [x σ(1) , . . . , x σ(n) ],
with u given in terms of permutation inversions for σ:

u = (i,j): i<j,σ(i)>σ(j) d(x j )d(x i ).
Proof. Existence of the action follows from the coherence theorem in symmetric monoidal categories (loc cit.) All that remains to be proved is the explicit formula for u in terms of inversions. Indeed, for n = 2 and n = 3, the explicit formulae given above prove the claim. For general n, the claim is given by induction. To this end, one may decompose any permutation into a product of adjacent transpositions, each of which adding an inversion term (which may cancel out with another term obtained precedingly, and so really counts the inversions: this is where skewness of the product in Γ becomes crucial), and performing a computation which is essentially the same as the one given above for the permutation (13) = (12)(23)(12). 2.2. Braided dual of a centrally graded group. We continue to assume that Γ is a generalized ring. The braiding maps show up in constructions on graded groups (or monoids) from Theorem 0.7. We prove Item (1) of this theorem: the product of G ∨ is associative:

g 1 • ∨ (g 2 • ∨ g 3 ) = Z d(g 1 ) d(g 2 g 3 ) (g 1 • (g 2 • ∨ g 3 )) = Z d(g 1 )d(g 2 )+d(g 1 )d(g 3 )+d(g 2 )d(g 3 ) g 1 g 2 g 3 = Z d(g 1 g 2 ) d(g 3 ) (g 1 • ∨ g 2 ) • g 3 = (g 1 • ∨ g 2 ) • ∨ g 3 .
The neutral element is e, and the inverse of

g is Z (d(g) 2 ) g -1 , since d(g -1 ) = -d(g), g • ∨ Z (d(g) 2 ) g -1 = Z d(g)•d(g -1 ) Z (d(g)) 2 gg -1 = Z -(d(g)) 2 +(d(g)) 2 e = e = Z (d(g) 2 ) g -1 • ∨ g.
Since Γ is commutative and Z(Γ) central, the maps d and Z define on this group the structure of a Γ-graded group. Since the product in (G

∨ ) ∨ is (g 1 , g 2 ) → Z d(g 2 )d(g 1 )+d(g 1 )d(g 2 ) g 1 g 2 , it follows that (G ∨ ) ∨ = G if Γ is skew.
Remark 2.2. By induction, the product of n elements in G ∨ is given by

g 1 • ∨ . . . • ∨ g n = Z u g 1 • • • g n , where u = (i,j):i<j d(g i )d(g j ).
Note that there are also other "duals": the opposite group G opp of a group G, and the braided duals belonging to the products -ab and to ba, and any combination of these operations. We are mainly interested in the case Γ = Z/2Z, where most of them coincide, and will not investigate systematically the family of "duals" thus obtained.

Example 2.1 (Central extensions of Γ). Let G = Γ 2 with group law here written (x 0 , x 1 ) • (y 0 , y 1 ) = (x 0 + y 0 , x 1 + y 1 ), and d(x 0 , x 1 ) = x 1 and Z(x) = (x, 0). This clearly defines a (commutative) centrally Γ-graded group, denoted by Γ 1,0 . Its b-dual, which we denote by Γ 0,1 = Γ ∨ 1,0 , is given by the same d and Z, and product

(x 0 , x 1 ) • ∨ (y 0 , y 1 ) = Z(x 1 y 1 ) • (x 0 + y 0 , x 1 + y 1 ) = (x 0 + y 0 + x 1 y 1 , x 1 + y 1 ).
Indeed, Γ 1,0 is the central extension of Γ by Γ via the trivial cocycle 0, whereas Γ 0,1 is the extension via the (in general) non-trivial cocycle (x, y) → xy.

For later use, let us have a closer look at the case Γ = Z/2Z = {0, 1} with its usual (field) product. Let 1 := e 0 := (0, 0), Z := (1, 0), e 1 := (0, 1), Ze 1 := (1, 1).

In the following, we'll write the group law multiplicatively (although at this stage it will still be commutative). So let t ∈ {1, Z}. When t = 1, the following gives the group law of Γ 1,0 , and when t = Z, it gives the group law of Γ 0,1 :

• 1 Z e 1 Ze 1 1 1 Z e 1 Ze 1 Z Z 1 Ze 1 e 1 e 1 e 1 Ze 1 t tZ Ze 1 Ze 1 e 1 tZ t
The elements 1 and Z are even, and e 1 and Ze 1 are odd. For t = 1, we see that Γ 1,0 ∼ = C 2 × C 2 as group (usual direct product). For t = Z, we have Γ 0,1 ∼ = C 4 , because e 1 becomes here an element of order four.

2.3. Graded product of centrally graded groups. We define the graded product of two centrally Γ-graded groups G i , i = 1, 2, as in Item 5)2) of Theorem 0.7. Let us show that the new product is associative:

(f 1 , f 2 ) • (g 1 , g 2 ) • (h 1 , h 2 ) = (f 1 , f 2 ) • (Z d(g 2 ) d(h 1 ) 1 g 1 h 1 , g 1 h 2 ) = (Z d(f 2 )d(g 1 h 1 ) 1 Z d(g 2 )d(h 1 ) 1 f 1 g 1 h 1 , f 2 g 2 h 2 ) = (Z d(f 2 )d(g 1 )+d(f 2 )d(h 1 )+d(g 2 )d(h 1 ) 1 f 1 g 1 h 1 , f 2 g 2 h 2 ) = (Z d(f 2 )d(g 1 )+d(f 2 g 2 )d(h 1 ) 1 f 1 g 1 h 1 , f 2 g 2 h 2 ) = (Z d(f 2 )d(g 1 ) 1 f 1 g 1 , f 2 g 2 ) • (h 1 , h 2 ) = (f 1 , f 2 ) • (g 1 , g 2 ) • (h 1 , h 2 ).
The inverse element of (g 1 , g 2 ) is, as is directly checked, (Z

d(g 2 )d(g 1 ) 1 g -1 1 , g -1 2 ), and so (G 1 × G 2 , •) is a group. Moreover, d(g 1 , g 2 ) := d(g 1 ) + d(g 2 ) clearly defines a morphism from this group to (Γ, +). The elements (Z k 1 , Z -k 2 )
belong to the center of this group, and their degree is zero, hence the quotient G/N is again a graded group. The equivalence relation generated by

N comes from (g 1 , g 2 ) ∼ (Z k 1 g 1 , Z -k 2 g 2 ), that is, (g 1 , Z k 2 g 2 ) ∼ (Z k 1 g 1 , g
2 ) for all k, and hence the quotient group G 1 × G 2 /N fulfills the conditions from the theorem. Moreover, if the product in Γ is skew, the braiding map is a group isomorphism from

G 1 × Z G 2 onto G 2 × Z G 1 : β([g 1 , g 2 ] • [h 1 , h 2 ]) = β([Z d(g 2 )d(h 1 ) 1 g 1 h 1 , g 2 h 2 ]) = Z d(g 2 )d(h 1 )-d(g 2 h 2 )d(g 1 h 1 ) [g 2 h 2 , g 1 h 1 ] = Z d(g 2 )d(h 1 )-(d(g 2 )+d(h 2 ))(d(g 1 )+d(h 1 )) [g 2 h 2 , g 1 h 1 ] = Z -d(h 2 )d(g 1 )-d(h 2 )d(h 1 )-d(g 2 )d(g 1 ) [g 2 h 2 , g 1 h 1 ] = Z d(g 1 )d(h 2 ) [Z d(g 1 )d(g 2 )+d(h 1 )d(h 2 ) 2 g 2 h 2 , g 1 h 1 ] = [Z d(g 1 )d(g 2 ) 2 g 2 , g 1 ] • [Z d(h 1 )d(h 2 ) 2 h 2 , h 1 ] = β[g 1 , g 2 ] • β[h 1 , h 2 ]
Remark 2.3 (Product of n elements). With different notation, the formula from the proof for the product of three elements reads, with

d ij := d(g ij ), (g 11 , g 12 ) • (g 21 , g 22 ) • (g 31 , g 32 ) = Z d 12 d 21 +d 12 d 31 +d 22 d 31 1 (g 11 g 21 g 31 , g 12 g 22 g 32 ).
By induction, the product of n elements in G is given by

[g 11 , g 12 ] • • • [g n1 , g n2 ] = Z (i,j)∈n 2 : i<j d i2 d j1 g 11 • • • g n1 , g 12 • • • g n2 .
Note that the proof of associativity and the proof of the braiding property are essentially equivalent.

To finish the proof of Theorem 0.7, we prove Item (3): the set-theoretic equality

(G 1 ×G 2 ) ×G 3 = G 1 ×(G 2 ×G 3 ) is an isomorphism of groups: on the one hand, (g 1 , g 2 ), g 3 • (h 1 , h 2 ), h 3 = Z d(g 3 ) d(h 1 ,h 2 ) (g 1 , g 2 ) • (h 1 , h 2 ), g 3 h 3 = Z d(g 3 )(d(h 1 )+d(h 2 )) 1 Z d(g 2 )d(h 1 ) 1 g 1 h 1 , g 2 h 2 , g 3 h 3 = Z d(g 3 )d(h 1 )+d(g 3 )d(h 2 )+d(g 2 )d(h 1 ) 1 g 1 h 1 , g 2 h 2 , g 3 h 3 .
On the other hand, using distributivity in the ring Γ, we get the same result:

g 1 , (g 2 , g 3 ) • h 1 , (h 2 , h 3 ) = Z d(g 2 ,g 3 ) d(h 1 ) 1 g 1 h 1 , (g 2 , g 3 ) • (h 2 , h 3 ) = Z (d(g 2 )+d(g 3 )) d(h 1 ) 1 Z d(g 3 )d(h 2 ) 1 g 1 h 1 , g 2 h 2 , g 3 h 3 = Z d(g 2 )d(h 1 )+d(g 3 )d(h 1 )+d(g 3 )d(h 2 ) 1 g 1 h 1 , g 2 h 2 , g 3 h 2
Remark 2.4. The preceding computation shows that the group product in

G 1 × Z G 2 × Z G 3 is given by, with d ij := d (i,j) := d(g ij ), [g 11 , g 12 , g 13 ] • [g 21 , g 22 , g 23 ] = Z d 12 d 21 +d 13 d 21 +d 13 d 22 [g 11 g 21 , g 12 g 22 , g 13 g 23 ].
By induction, we get the product of two elements in

G = G 1 × Z G 2 × Z . . . × Z G p , [g 11 , . . . , g 1p ] • [g 21 , . . . , g 2p ] = Z i>j d 1i d 2j [g 11 g 21 , . . . , g 1p g 2p ].
Note the "duality" with the formula from Remark 2.3! Applying it twice, we get

[g 11 , g 12 , g 13 ] • [g 21 , g 22 , g 23 ] • [g 31 , g 32 , g 33 ] = Z u [g 11 g 21 g 31 , g 12 g 22 g 32 , g 13 g 23 g 33 ],
with the degree term given by The 9 terms are structured by the graph of the usual total order on {1, 2, 3} (the three pairs (1 < 2), (1 < 3), (2 < 3)). By induction (the details of which we omit), we get for the product of n elements g i = (g i1 , . . . , g ip ), for i = 1, . . . , n:

g 1 • • • g n = Z u (g 11 • • • g n1 , . . . , g 1p • • • g np )
with degree term given by a kind of "matrix product"

u = (i,j,k,ℓ)∈p 2 ×n 2 : i>j,k<ℓ d ki d ℓj . Remark 2.5. The group inverse in G 1 × Z G 2 × Z G 3 is given by [g 1 , g 2 , g 3 ] -1 = Z d 2 d 1 +d 3 d 1 +d 3 d 2 [g -1 1 , g -1 2 , g -1 3 ]. By induction, [g 1 , . . . , g n ] -1 = Z i<j d j d i [g -1 1 , . . . , g -1 n ].
Remark 2.6 (Choices and conventions). The graded product of graded groups could have been defined by other choices and conventions. To explain this, note that we could have defined the group law on G 1 × z G 2 in a different way: every matrix X ∈ M(2, 2; Z) gives rise to a bi-additive product on Γ ⊕ Γ, via

a, b = (a 1 , a 2 )X(b 1 , b 2 ) t = a 1 X 11 b 1 + a 1 X 12 b 2 + a 2 X 21 b 1 + a 2 X 22 b 2
(where products in Γ are given by the fixed bi-additive product, and coefficients from Z associate with everything). Thus we can define a braided dual group structure on G 1 × G 2 , as above, with respect to the bi-additive product on Γ ⊕ Γ, and then mod out the normal subgroup N as before. The result is a new group structure on G 1 × z G 2 , depending on the matrix X. When Γ = F 2 = Z/2Z, then all 16 matrices from M(2, 2; F 2 ) give rise to a group structure. For instance, the matrix X = E 11 gives rise to the group G ∨ 1 × z G 2 , and

X = I = E 11 + E 22 gives rise to the group G ∨ 1 × Z G ∨ 2 .
However, for these choices, Item (3) of Theorem 0.7 would fail to hold: for this we need that X is strict (upper or lower) triangular. For X = 0, this gives the ungraded product; our choice is X = E 12 , but X = E 21 would give an isomorphic theory.

Remark 2.7 (Internal graded product). With notation as in Theorem 0.7, G has two normal subgroups

{[g 1 , 1] | g 1 ∈ G 1 } ∼ = G 1 , {[1, g 2 ] | g 2 ∈ G 2 } ∼ = G 2 , such that G 1 ∩ G 2 = Z Γ = im(Z). Commutators of elements of G 1 with elements of G 2 belong to Z Γ . Indeed, both groups normalize each other: [1, g 2 ] • [g 1 , 1] • [1, g 2 ] -1 = Z d(g 2 )d(g 1 ) [g 1 , g 2 ] • [1, g -1 2 ] = [Z d(g 2 )d(g 1 ) 1 g 1 , 1] = Z d(g 2 )d(g 1 ) [g 1 , 1].
These facts imply that G is a certain amalgamated semi-direct product of G 1 with G 2 . Conversely, such data can be used to recover the graded product (so we may speak of an "internal graded product" of subgroups of a given group, corresponding to the "external graded product", given by the theorem). Thus we may write g 1 g 2 instead of [g 1 , g 2 ], and compute products by using the "commutation relation"

(2.1)

g 2 g 1 = Z d(g 2 )d(g 1 ) g 1 g 2 .
Example 2.2 (D 4 and Q revisited). To prepare the grounds for the following section, let us compute explicitly the graded product of two groups of type Γ 1,0 or Γ 0,1 , as defined in Example 2.1.

Let G i = {1, Z i , e i , Z i e i }, given by the table from Example 2.1, with t i ∈ {1, Z i }, where Z i corresponds to Z. The cardinality of G = G 1 × Z G 2 is 4×4 2 = 8
, and its elements are

1 = [(1, 1)], Z = [(1, Z 2 )] = [(Z 1 , 1)], e 1 = [(e 1 , 1)], e 2 = [(1, e 2 )],
Ze 1 , Ze 2 , e 12 := e 1 e 2 = [(e 1 , e 2 )], Ze 12 .

These elements are multiplied by using the rules:

e 1 e 2 = Ze 2 e 1 , e 2 i = t i , Z 2 i = 1, Z is central.
G 0 ∼ = C 4 .
In the other cases, it contains exactly 2 elements of order 4, and hence is isomorphic to D 4 . The even subgroup then is isomorphic to

C 4 if t 1 = 1 = t 2 , and to C 2 × C 2 if t 1 = Z, t 2 = 1, or if t 1 = 1, t 2 = Z.
Example 2.3. Let m ≥ 2, and Γ = Z/mZ. We have three groups, of cardinal m 3 ,

C 2 m × Z C 2 m , C m 2 × Z C 2 m , C m 2 × Z C m 2 .
It follows from Remark 2.7 that these groups satsify [G, G] ⊂ Z , hence are 2-step nilpotent. E.g., for m = 3, this gives the two non-abelian groups of order 27 (see [D] p. 565 and p. 566 for detailed information on these groups).

The discrete Clifford category

3.1. The Γ-category. Assume the product in Γ is skew, and recall from Example 2.1 the graded groups

Q(1) := Γ 1,0 = Γ 2 , Q(Z) := Γ 0,1 = Γ ∨ 1,0 .
The symmetric monoidal category of Γ-graded groups generated by Q(1) and Q(Z) will be called the Γ-category. Explicitly, these groups are of the form:

Definition 3.1. For t = (t 1 , . . . , t n ) ∈ {1, Z} n , if t i = 1, let Q(t i ) be a copy of Q(1) = Γ 2 , if t i = Z, let Q(t i ) be a copy of Q(Z) = Q(1) ∨ , and define
Q(t) := Q(t 1 ) × Z . . . × Z Q(t n ).
We let also, Q 0 := Γ with trivial grading (as we have seen, this is the neutral element for the graded product), and for p + q = n,

Q p,q := Q(1, . . . , 1 p× , Z, . . . , Z q× ), Q n := Q n,0 = Q(1, . . . , 1), Q ∨ n = Q 0,n .
By Item (3) of Theorem 0.7, we can recast this definition:

Definition 3.2. Let K be a set. For (t, s) ∈ K n × K m , we denote by t ⊕ s := (t 1 , . . . , t n , s 1 , . . . , s m ) ∈ K n+m their "juxtaposition". Note that this operation is associative, but not commutative (in fact, it is the composition law of the free monoid over K ).

Lemma 3.3. For t ∈ {1, Z} n , s ∈ {1, Z} m , we have

Q(t ⊕ s) = Q(t) × Z Q(s).
In particular, Q p,q = Q p × Z Q 0,q .

Remark 3.1. Fom Item (4) of Theorem 0.7, we get an isomorphism φ (t,s) : Q(t⊕s) → Q(s ⊕ t). For instance, re-arranging the order of factors, we get an isomorphism

Q p,p ∼ = Q 1,1 × Z . . . × Z Q 1,1 .
Example 3.1. Let Γ = R k with k ≥ 2 and skew-symmetric product (x 1 , x 2 ) = (0, . . . , 0, x 1 x 2x 2 x 1 ). Then Q 1,0 is the abelian group R 2k , whereas Q 0,1 , and all higher Q p,q are are 2-step nilpotent Lie groups, isomorphic to a Heisenberg group.

We propose to call the Γ-category, in this case, the Heisenberg category. A systematic study of this category is certainly an interesting topic for subsequent work.

3.2. The discrete Clifford category. From now on, we assume that Γ = Z/2Z with its usual ring (field) structure. In this case, the Γ-category will be called the (discrete) Clifford category, since the groups Q p,q then are (discrete) Clifford groups, as we shall see. The specific feature of this case is that Γ is both skew and admits a unit element 1 (which is not the case in the Heisenberg category). To give an explicit description of all groups by generators and relations, we follow the notation from Example 2.2: for m ∈ N, let Z m be a copy of the element Z (copy called "of m-th generation"). For each t m ∈ {1, Z m }, we define a graded group of four elements,

Q(t m ) = {e 0 , e m , Z m , Z m e m },
just as in Example 2.2: 1 = e 0 is neutral, Z 2 m = e 0 , and e 2 m = t m e 0 . The elements e i , Z i e i are odd, and e 0 , Z i are even. Thus, as we have seen in Exemple 2.1, (3.1)

Q 1 = Q(1) = C 2 2 , Q 0,1 = Q(Z) = C 4 .
From Example 2.2, we now get (3.2)

Q 2 = Q(1, 1) = D 4 , Q 0,2 = Q(Z, Z) = Q, Q 1,1 = Q(1, Z) = D 4 .
As already said, Q(t) will be seen to be isomorphic to the discrete Clifford group of the Clifford algebra Cl(t) (Appendix A). In the following, let us prove some basic structure results on these groups, without using the theory of Clifford algebras. Recall from Example 1.1 the abelian group (P(n), ∆), isomorphic to (Z/2Z)) n . Theorem 3.4. Fix t ∈ {1, Z} n . Then:

(1) The group G := Q(t) is of cardinality 2 n+1 .

(2) The quotient group G 0 = G/{1, Z} is abelian, isomorphic to P(n) = (Z/2Z) n , i.e., the following is an exact sequence of groups (central extension)

{1, Z} → Q(t) → (P(n), ∆).
Elements of Q(t) are of order either 1, 2 or 4.

(3) The group G is generated by the elements e i ∈ Q(t i ) (which are identified with the corresonding element in G). For A ⊂ n with k = |A|, whose elements are ordered, a 1 < . . . < a k , we let

e A := e a 1 • • • e a k .
Then G is a disjoint union, as follows:

G = {e A | A ∈ P(n)} ⊔ {Ze A | A ∈ P(n)}.
We often identify the first set with P(n) (i.e., we fix this set theoretic splitting of the exact sequence from Item (2)). There is a function τ : P(n) × P(n) → Z/2Z such that, for all A, B ∈ P(n), (5) Relations (a) -(d) are contained in the preceding items. Conversely, given a group having generators satisfying these relations, it is seen by induction that it is isomorphic to Q(t): for n = 1 and t 1 = 1, we have the defining relations of C 2 × C 2 , and for n = 1 and t 1 = Z, we have the defining relations of C 4 ; likewise, for n = 2, we have the defining relations of D 4 , resp., of Q. For the induction step, assume the claim holds at order n, and let G n+1 be a group having generators and relations of the given form at order n + 1. Then H := {1, e n+1 , Z, Ze n+1 } is a subgroup, isomorphic to C 4 if t n+1 = Z and to C 2 2 if t n+1 = 1, and the given relations show that G is a homomorphic image of G n × Z H. For reasons of cardinality, G is actually equal to this group, i.e., given by (3.3).

Remark 3.2. Following the pattern of "classification of real Clifford algebras", we can now "classify" the groups Q p,q . In our opinion, this gives a transparent and conceptual version of that part of Clifford theory. Since, in the main text, we are rather interested in classification-free theory, we relegate the presentation of this issue to Appendix B.

Remark 3.3. The function τ is the cocycle of a central extension. See Section 3.3 for more on this; cf. also [L] p.12-14.

Next, we compute an explicit formula for the inner automorphism given by e A : Lemma 3.5. For all A, B ∈ P(n),

e A • e B • (e A ) -1 = Z |A|•|B|-|A∩B| e B .
Proof. When |A| = 1 = |B|, then the claim is in keeping with the defining relations of Q(t), namely e i e j (e i ) -1 = ue j , with u = 1 if i = j, and u = Z else. Now let |A| = 1, say, A = {i}, and B arbitrary. When i / ∈ B, then we have to exchange the position of e i and each e j , j ∈ B, which gives a factor Z |B| , and when i ∈ B, we get a factor Z |B|-1 (since we exchange for all i ∈ B with i = j), which again is in keeping with the claim. For general A and B, conjugation by e A = e a 1 • • • e a k is composition of conjugation by the e a i . We count the total number of exchanges of elements e i and e j arising in the whole procedure: whenever (i, j) ∈ A × B with i = j, we get an additional factor Z, so in the end we get a factor Z p with e n e B (e n ) -1 = Z (n-1)|B| e B , which is equal to e B if n is odd, and equal to Z |B| e B if n is even.

p = |{(i, j) ∈ A × B | i = j}| = |A × B \ diag| = |A| • |B| -|A ∩ B|,
Theorem 3.6 (Center, inner automorphisms). Let t ∈ {1, Z} n and G := Q(t). Assume n is even. Then:

(1) the center of G is the subgroup {1, Z}, (2) there are 2 n + 1 conjugacy classes: two of them are singletons, {1} and {Z}, and the other are of cardinal two, {e A , Ze A }, for A = ∅, (3) the inner automorphism group of Q(t) is isomorphic to P(n) ∼ = (Z/2Z) n , (4) the grading automorphism α(e A ) = Z |A| e A is an inner automorphism.

Assume n is odd. Then:

(1) the center of G is the subgroup {1, Z, e n , Ze n } (when t = (1, . . . , 1), it is isomorphic to C 4 if n ≡ 1 mod 4, and to C 2 2 if n ≡ 3 mod 4), (2) there are 2 n + 2 conjugacy classes, four of them singletons (elements of the center), the other 2 n -2 classes of the form {e A , Ze A }, A = ∅, A = n, (3) the inner automorphism group is isomorphic to P(n -1) ∼ = (Z/2Z) n-1 , (4) the grading automorphism α is an outer automorphism. In both cases, α, together with the inner automorphisms, generates a subgroup of automorphisms isomorphic to P(n) ∼ = (C 2 ) n .

Proof. From Lemma 3.5, e A commutes with all e i iff |A| -|A ∩ {i} is even, for all i, that is, for all i, the set A\{i} has an even number of elements. The only possibility to realize this case is A = n = {1, . . . , n}, where n is odd. Since Z is always central, and e 2 n = Z n(n-1) 2

, this proves the statements about the center. Concerning conjugation classes, Lemma 3.5 shows that that each class has at most 2 elements, say e B and Ze B . It has 1 element iff the element is central, and 2 elements else. Thus the claims follow from those about the center. Likewise, those on the inner automorphism group G/Z(G) follow from those on the center.

The grading automorphism from Lemma 1.4 satisfies the condition α(e k ) = Z d(e k ) e k = Ze k , whence α(e A ) = Z |A| e A . As seen in Example 3.2, when n is even, it is inner. When n is odd, then there is no set A ∈ P(n) such that e A e k (e A ) -1 = Ze k for all k = 1, . . . , n (since A = n does not satisfy the condition, and when A = n, the value of e A e k (e A ) -1 depends whether k ∈ A or k / ∈ A). Thus the grading automorphism then is outer.

In general, the group of all automorphisms of Q(t) is not a suitable object, for our purposes. We are interested in the case that all t i are equal, because then permutations of the e i induce automorphisms. But in this case, the group of all automorphisms may be too big, as is illustrated by the following examples: (2) When t 1 = Z = t 2 , then G = Q and Aut(G) ∼ = S 4 . Namely, the inner automorphisms form a Klein 4-group as above, and permutations of i, j, k (that is, of e 1 , e 2 , e 12 ), form a complementary S 3 -subgroup. However, exchange of e 1 and e 12 does not preserve the grading (since e 1 is odd and e 12 even).

Example 3.3. Let n = 1, G = Q(t 1 ) = {e 0 , e 1 , Z, Ze 1 }, e 2 1 = t 1 . (1) When t 1 = Z, so G ∼ = C 4 , then Aut(G) ∼ = C 2 (
The subgroup of grading-preserving automorphisms is again a D 4 -subgroup of S 4 , having same types of elements as in (1).

Theorem 3.7 (Hyperoctahedral automorphism group). The symmetric group S n acts by automorphisms on Q n : for every σ ∈ S n , there is a unique automorphism

Φ σ : Q n → Q n such that ∀k = 1, . . . , n : Φ σ (e k ) = e σ(k) .
This group of automorphisms normalizes the group (C 2 ) n described in Theorem 3.6 , and together they form a group B n of grading-preserving automorphisms of cardinality n!2 n . The grading automorphism α belongs to the center of this group. The same statements hold for the group Q 0,n .

Proof. By induction: for n = 1 the statement is uninteresting; for n = 2, the braiding automorphism the unique automorphism

τ : Q 2 = Q 1 × Z Q 1 → Q 2 , [g 1 , g 2 ] → Z d(g 1 )d(g 2 ) [g 2 , g 1 ]
such that τ (e 1 ) = e 2 , τ (e 2 ) = e 1 , τ (Z) = Z, τ (e 12 ) = Ze 12 (cf. also Example 3.4). Assuming that S n acts by automorphisms on Q n , we write

Q n+1 = Q n ×Q 1 = Q n-1 ×Q 2 .
Then S n = S n × id Q 1 acts on Q n+1 , and another copy of S 2 = id Q n-1 acts on Q n+1 (transposition (n n + 1)). Together, these two actions generate the action of S n+1 .

Similarly for

Q 0,n = Q 0,1 × . . . ×Q 0,1 .
It is clear that the action of S n normalizes the inner automorphisms as well as the grading automorphism α, hence the group (Z/2Z) n is a normal subgroup, intersecting trivially the S n -subgroup. Thus B n is a semidirect product, of cardinality n!2 n . Clearly, α commutes with the S n -action, hence belongs to the center of B n .

Remark 3.4. Recall that, by definition, the hyperoctahedral group is the group of signed permutation matrices (of size n × n). It is Coxeter group of type B n , generated by the usual permutation matrices (subgroup S n ) and the diagonal ±1matrices (subgroup C n 2 ). It is isomorphic to the automorphism group defined in the preceding theorem. Its center is equal to {id, α}. Definition 3.8. We call the group B n of automorphisms defined in the preceding theorem the hyperoctahedral automorphism group of Q n , resp. of Q 0,n .

Remark 3.5. The whole group of grading preserving automorphisms is in general bigger than the hyperoctahedral automorphism group.

Remark 3.6. For Q p,q , the symmetry group S n must be replaced by S p × S q , permuting the e i 's with t i = 1, resp. those with t i = Z.

Remark 3.7 (Inversions). In the Clifford category, all algebraic operations can be described by explicit formulae. From an algebraic point of view, this amounts to describe structures (central extensions) by cocycles, and from a combinatorial point of view, the main tool is given by inversions. For instance, to compute a formula for the automorphism Φ σ from Theorem 3.7, we start by writing A = {a 1 , . . . , a k } with a 1 < . . . < a k , and

Φ σ (e A ) = Φ σ (e a 1 • • • e a k ) = e σ(a 1 ) • • • e σ(a k ) .
We have to re-order these terms so that the indices are in increasing order. The number of terms Z appearing is u := |(i, j) ∈ A × A | i < j, σ(i) > σ(j)|, so we end up with the formula of "inversion type" Φ σ (e A ) = Z (u mod 2) e σ(A) .

Cocycles. Recall that the central extension of a group H by an abelian group

A can be described by a cocyle C : H × H → A. In our case, by Theorem 3.4 we have the central extension A = {1, Z} → Q(t) → H = P(n), with cocycle denoted by τ , defined by the rule e A e B = τ (A, B)e A∆B .

Expanding both sides of (e A e B )e C = e A (e B e C ), we get the cocycle relations:

(3.4) τ (A, B)τ (A∆B, C) = τ (A, B∆C)τ (B, C).
Since our cocyle τ depends on t we call it a "relative cocycle". We shall separate it into two parts, the first of which contains the explicit dependence on t, and the second being independent of t ("absolute").

Definition 3.9. Let K be a commutative monoid, t ∈ K n and A ∈ P(n). We let

t ∅ := 1, t A := a∈A t a ∈ K.
The map A → t A behaves like a "multiplicative measure": when A ∩ B = ∅, then t A∪B = t A • t B , and, for general A, B,

t A t B = t A∩B t A∪B = t 2 A∩B t A\B t B\A = t 2 A∩B t A∆B . If k 2 = 1 for all k ∈ K (which is the case for K = {1, Z}), then t A∆B = t A • t B .
Lemma 3.10. Let t ∈ {1, Z} n . The map (A, B) → t A∩B defines a cocyle on P(n).

Proof. t A∩B t (A∆B)∩C = t A∩B t A∩C t B∩C = t A∩(B∆C) t B∩C .
We call the cocycle from the preceding lemma "relative". It will be combined with an "absolute" cocyle γ:

Definition 3.11. Let L := {(a, b) ∈ n 2 |
a < b} be the usual total order relation on n, and Theorem 3.12 (The absolute cocycle). Let t ∈ {1, Z} n and A, B ∈ P(n). Then the product e A e B in Q(t) is given by

L -1 = {(a, b) ∈ n 2 | a > b} its opposite order. For a subset R ⊂ (n × n), call Inv(R) := R ∩ L -1 = {(a, b) ∈ R | a > b}
e A e B = t A∩B • γ(A, B) • e A∆B .
The map γ : P(n) 2 → {1, Z} is the cocylce defining Q n as central extension of P(n).

Proof. As in the proof of Theorem 3.4, we re-order the product e

A e B = e a 1 • • • e a k e b 1 • • • e b l .
After re-ordering, indices i ∈ A ∩ B give rise to a square e 2 i = t i , whence the factor t A∩B . While re-ordering, a relation of type e i e j = Ze j e i is applied when i > j, first for j = b 1 , then for j = b 2 , and so on, so the number of times we apply it is

b∈B |{a ∈ A | a > b}| = |{(a, b) ∈ A × B | a > b}| = |Inv(A, B)|.
When t = (1, . . . , 1) (case of Q n ), the factor t A∩B is always 1, so γ(A, B) is the cocyle of Q n (and in particular satisfies again the cocycle relation).

Besides satisfying the cocycle relation (3.4), the cocyle γ has several other properties, which in turn can be used to develop Clifford theory based on the discrete Clifford group (see [L] for a systematic exposition of this viewpoint).

Group algebra, and Clifford algebras

In the following, we start to develop some "harmonic analysis" on graded groups -far from complete, with the main aim to clarify the rôle of the Clifford algebra in case of the discrete Clifford category. Remarkably, this shows some analogy with the harmonic analysis of the Heisenberg group. This deserves to be developed elsewhere, and in the following we assume that all groups are discrete, or even finite. 4.1. Group (super) algebra. From now on we fix a commutative unital base ring K. We assume that the scalar 2 is invertible in K. For a set M, we denote by K[M] the free K-module with basis denoted (δ m ) m∈M , so elements of K[M] are finite linear combinations m∈M λ m δ m ; if M = G is a semigroup, then K[G] carries an associative product * defined by δ x * δ y = δ xy (semigroup algebra; cf. [FH]); if G is a monoid with unit e, the basis element δ e becomes a neutral element of

K[G], and if G is a group, then inversion G → G induces an anti-automorphism t : K[G] → K[G]
, sometimes called the canonical anti-automorphism or antipode.

In the following, let G be a centrally Z/2Z-graded group, and assume the element Z( 1) is non-trivial (we denote it again by Z, so that the image of Z : Z/2Z → G is the group {e, Z}). Let π : G → G 0 = G/{e, Z} be the canonical projection, and A := K[G] be the group algebra. There are two maps of order two on G, which induce two linear maps of order two on A:

Z : G → G, g → Zg = gZ, α : G → G, g → Z d(g) g, (4.1) Z * : A → A, f → Z * f, α * : A → A, f → α * f. (4.2)
Since 2 is invertible in K, both linear maps can be diagonalized; since α • Z = Z • α, they can be jointly diagonalized. We denote the eigenspace decompositions by for Z * :

A = A + ⊕ A -, A ± = {f ∈ A | Z * f = ±f }, (4.3) for α * : A = A 0 ⊕ A 1 , A k = {f ∈ A | α * f = (-1) k f }. (4.4)
Since Z is central, the map Z * commutes with elements of the group algebra, and hence the eigenspaces A ± are ideals of the group algebra. Since α is a group automorphism, by functoriality of the group algebra construction, it follows that the induced linear map is an algebra automorphism of order two, and hence the eigenspaces would behave multiplicatively if we indexed them by the eigenvalues ±1. However, following the usual convention for super-algebras, we take 0 and 1 as indices, so the eigenspaces obey the "super algebra rules"

A i * A j ⊂ A i+j mod 2 .
Each of the two ideals A ± is, in turn, again a superalgebra.

* : K[G] → K[G 0 ] having kernel K[G] -and image isomorphic to K[G] + , so K[G] + ∼ = K[G 0 ] as algebra.
(2) With respect to α * , the group algebra and the ideals from the preceding item are super-algebras. The disjoint union G = G 0 ⊔G 1 induces a decomposition

K[G] = K[G 0 ] ⊕ K[G 1 ] such that K[G 0 ] ⊂ A 0 , A 1 ⊂ K[G 1 ].
Proof.

(1) Let P ± be the two projectors onto the eigenspaces. To see that both eigenspaces are isomorphic, choose a set H of representatives for the {e, Z}-cosets in G (i.e., a section of π), so that G = H ⊔ZH as set, whence

K[G] = K[H]⊕Z * K[H]
as K-module. Then K[G] + represents the diagonal, and K[G] -the antidiagonal in this decomposition, and the restrictions and corestrictions

P + : K[H] → K[G] + , P -: K[H] → K[G] - are linear isomorphisms, so both K[G] ± are isomorphic to K[H] as K-modules. (When G is finite, both spaces have dimension |H| = |G|/2).
Since π(Z) = π(1), it follows that π * (1 -Z) = 0. Thus the ideal generated by 1 -Z, which is

P -(K[G]) = K[G] -,
belongs to the kernel of π * . On the other hand, the basis (h + Zh) h∈H , is sent to the basis π

* (h + Zh) = 2π * (h) of K[H], so the restriction of π * to K[G] + defines a bijection K[G] + → K[H], whence K[G] -is the kernel of π * , and π * induces an isomorphism of K[G] + onto K[G 0 ].
(2) We have already noticed that the decompositions define superalgebras. To prove the remaining statements, we decompose further H = H 0 ⊔ H 1 into even and odd elements, whence

G = H 0 ⊔ ZH 0 ⊔ H 1 ⊔ ZH 1 . Since α is the identity on G 0 , we get K[G 0 ] ⊂ A 0 . Also, P + K[H 1 ] ⊂ A 0 , since, if g ∈ H 1 , then α * (δ g + Z * δ g ) = δ Zg + δ g = δ g + Z * δ g . It follows that A 1 = P -K[H 1 ] ⊂ K[G 1 ].
4.2. Graded and ungraded tensor product. If A, B are sets, the free module with basis

A ⊔ B is K[A ⊔ B] = K[A] ⊕ K[B]
, and the one with basis

A × B is K[A × B] = K[A] ⊗ K[B]
. The tensor product of two associative algebras is again an associative algebra, for the "usual" product a ⊗ b • a ′ ⊗ b ′ = aa ′ ⊗ bb ′ . If A, B are Z/2Z-graded, then their graded tensor product is the tensor product, as module, together with bilinear product

(4.5) a ⊗ b • a ′ ⊗ b ′ = (-1) |b| |a ′ | aa ′ ⊗ bb ′ ,
where a, b, a ′ , b ′ are assumed to be homogeneous elements. If A and B are associative, then this is again an associative algebra (cf. [BtD], p. 56).

Theorem 4.2. Let (G i , d i , Z i ), i = 1, 2, two centrally Z/2Z-graded groups, and

G = G 1 × Z G 2 .
Then the algebras K[G] ± are given in terms of ungraded, resp. graded tensor products of algebras by those of G i , i = 1, 2, as

K[G] + = K[G 1 ] + ⊗ K[G 2 ] + , K[G] -= K[G 1 ] -⊗ K[G 2 ] -.
In particular, it follows that for G = Q(t), the ideal A -is a Clifford algebra: for t ∈ {1, Z} n , define Cl(t) to be the Clifford algebra of K n for the diagonal quadratic form with coefficient

1 if t i = 1 and -1 if t i = Z. Then K[Q(t)] + ∼ = K[P(n)], K[Q(t)] -∼ = Cl(t).
In particular, K[Q p,q ] -∼ = Cl p,q (K).

Proof. Choose H in G, and H i in G i , as in the preceding proofs. As above, elements

P ± (δ h ) = 1 2 (δ h ± Z * δ h i ), h ∈ H, form a basis in K[G] ± ,

and likewise such elements with h

i ∈ H i form a basis in K[G i ] ± . We compute, in K[G], δ h 1 ⊗ δ h 2 • δ h ′ 1 ⊗ δ h ′ 2 = δ [h 1 ,h 2 ] • δ [h ′ 1 ,h ′ 2 ] = Z d(h 2 )d(h ′ 1 ) * δ [h 1 h ′ 1 ,h 2 h ′ 2 ] = Z d(h 2 )d(h ′ 1 ) * δ h 1 h ′ 1 ⊗ δ h 2 h ′ 2
, and project with P -onto K[G] -. Since Z projects to -1, this gives

P -(δ h 1 ) ⊗ P -(δ h 2 ) • P -(δ h ′ 1 ) ⊗ P -(δ h ′ 2 ) = (-1) d(h 2 )d(h ′ 1 ) P -(δ h 1 )P -(δ h ′ 1 ) ⊗ P -(δ h 2 )P -(δ h ′ 2 )
, so the algebra product is given by the graded tensor product. On K[G] + , since Z projects to 1, there is no sign change, and we get the usual tensor product.

The statement on the Clifford agebras follows from the preceding statement by induction, using Relation (0.1) for Clifford algebras. (Cf. the arguments given in loc.cit., [BtD], where this is used to prove that the dimension of the Clifford algeba Cl(K n ) is 2 n . On the other hand, knowing the dimension of the Clifford algebra, the proof could also be given by noticing that the generators of A -satisfy exactly the same relations as the generators of the Clifford algebra Cl(t).)

Example 4.1 (Group algebra of Q and of D 4 ). The preceding arguments can be used to show, in a transparent way, that the Clifford algebra Cl 2,0 is isomorphic to M 2 (K), and that Cl 0,2 is the usual quaternion algebra. Namely, by the preceding results, the group Q p,q is (isomorphic to) the group generated by the basis elements e i in the Clifford algebra Cl p,q (K) ∼ = K[Q p,q ] -. For instance, the quaternion group G = Q if often defined as group of matrices {±1, ±i, ±i, ±k} ⊂ H * ⊂ Gl(2, C). Now, modulo -1, these matrices furnish a basis of H, the elements multiply as they should in the group, and the matrix Z = -1 acts as it should on K[G] -(by the scalar -1): so we can conclude that K[G] -∼ = H as algebra, whence Cl 0,2 (R) ∼ = H. The same argument works for the group G = D 4 , which we realize as matrix group {±1, ±I, ±J, ±K} ⊂ Gl(2, Z). Modulo -1, the four matrices give a basis of M 2 (K), and Z = -1 acts by -1, whence K[G] -∼ = M 2 (K).

Remark 4.1. The fact that Clifford algebras are parts of group algebras, or images of them, is known (cf. [A,[START_REF] Albuquerque | Clifford algebras obtained by twisting of group algebras[END_REF][START_REF] Salingaros | The relationship between finite groups and Clifford algebras[END_REF]), and we hope the framework of graded groups proposed here clarifies those approaches. For instance, the abstract ingredients of the classification of Clifford algebras are quite neatly featured in the abstract approach (see Appendix B.3, cf. also [S82] and [L], p.53, for this issue). 4.3. Harmonic analysis of the group algebra. The group G = Q(t) acts on K[G] in the usual way (left-regular representation). This representation decomposes into sub-representations (ideals) K

[G] = K[G] + ⊕K[G] -.
We'll decompose it further. Since K is just a ring and not a field, we do not speak about decomposition into "irreducible" modules, but rather determine a basis of the space of class functions (which in case of K = C correspond to the irreducible characters). Recall that a class function is just an element f ∈ K[G] commuting with every g ∈ G, that is, an element of the center of K [G].

see that all groups Q(t) can be decomposed as ungraded product ×

C 2 of groups of type C 2 , C 2 × C 2 , C 4 , D = D 4 , Q, corresponding to the decomposition of Clifford algebras into tensor products involv- ing R, R ⊕ R, C, M 2 (R), H.
The key lemma for classifying discrete Clifford groups is: Lemma B.1. For all n ≥ 3 and t ∈ {1, Z} n , there is a group isomorphism

Q(t) ∼ = Q(t 1 , t 2 ) × Z Q(Zt 1 t 2 t 3 , . . . , Zt 1 t 2 t n ).
Proof. Elements of Q(t) are either of the form e A or Ze A , with A ∈ P(n). When A ⊂ {1, 2}, we get the 8 elements of the first factor Q(t 1 , t 2 ) = {e 0 , e 1 , e 2 , e 12 , Z, Ze 1 , Ze 2 , Ze 12 }. Next, let H ⊂ Q(t) be the subgroup generated by the elements hence Q(t 1 , t 2 ) commutes with the generators of H, hence with H. Moreover, H and Q(t 1 , t 2 ) together generate the group Q(t), and therefore H contains at least 2 n-1 elements. We compute, for 3 ≤ k < ℓ (using These are the defining relations of the group Q(Zt 1 t 2 t 3 , . . . , Zt 1 t 2 t n ), hence, by Theorem 3.4, there is a surjective morphism from Q(Zt 1 t 2 t 3 , . . . , Zt 1 t 2 t n ) to H, which must be an isomorphism for reasons of cardinality.

e 2 i = t i e 0 ) b 2 k = e 1 e 2 e k e 1 e 2 e k = Z 3 e 2 1 e 2 2 e 2 k = Zt 1 t 2 t 3 e 1 e 2 e k = Zt 1 t 2 t 3 b k , b k b ℓ =
In particular, the lemma implies for (t 1 , t 2 ) = (1, 1), or (1, Z), or (Z, Z):

Q(1, 1, t 3 , . . . , t n ) ∼ = Q 2,0 × Z Q(Zt 3 , . . . , Zt n ) ∼ = D 4 × Z Q(Zt 3 , . . . , Zt n ), Q(1, Z, t 3 , . . . , t n ) ∼ = Q 1,1 × Z Q(t 3 , . . . , t n ) ∼ = D 4 × Z Q(t 3 , . . . , t n ), Q(Z, Z, t 3 , . . . , t n ) ∼ = Q 2,0 × Z Q(Zt 3 , . . . , Zt n ) ∼ = Q × Z Q(Zt 3 , . . . Zt n ).
From this, the first statements of the following corollary immediately follow:

Corollary B.2. For all p, q ∈ N, Q p+2,q ∼ = Q 2,0 × Z Q q,p ∼ = D 4 × Z Q q,p , Q p+1,q+1 ∼ = Q 1,1 × Z Q p,q ∼ = D 4 × Z Q p,q , Q p,q+2 ∼ = Q 0,2 × Z Q q,p ∼ = Q × Z Q q,p .
Applying this several times, using notation

G 3 Z = G × Z G × Z G, etc., Q p+4,q ∼ = Q 2,0 × Z Q 0,2 × Z Q p,q ∼ = D 4 × Z Q × Z Q p,q ∼ = Q p,q+4 , Q p+2,q+2 ∼ = D 4 × Z D 4 × Z Q p,q ∼ = Q × Z Q × Z Q p,q , Q p+8,q ∼ = Q p,q+8 ∼ = Q p+4,q+4 ∼ = (D 4 ) 4 Z × Z Q p,q ∼ = Q 4 Z × Z Q p,q .
Moreover, for any Z/2Z-graded group G,

G × Z Q 1,0 = G × C 2 (C 2 × C 2 ) ∼ = G × C 2 . Proof. The last statement is obvious from (C 2 × C 2 )/C 2 = C 2 .
Remark B.1. In particular, the corollary gives the group isomorphisms

Q 2,2 ∼ = D 4 × Z D 4 ∼ = Q × Z Q, Q 1,2 ∼ = C 4 × Z D ∼ = C 4 × Z Q.
On the level of real Clifford algebras, this corresponds to the algebra isomorphisms

H ⊗ H ∼ = M(2, R) ⊗ M(2, R) (in turn, this algebra is isomorphic to M(4, R)) and H ⊗ C ∼ = M(2, C).
Q n+4 ∼ = DQ Q n , Q n+8 ∼ = D 4 Q n , Q 0,n+4 ∼ = DQ Q 0,n , Q 0,n+8 ∼ = D 4 Q 0,n .
Proof. Repeated application of Corollary B.2.

For sake of completeness, we give in Section B.3 the full "periodic table" of the groups Q p,q , together with the corresponding classification of Clifford algebras. Due to the relation Q p+1,q+1 = Q p,q × C D 4 = D Q p,q , this table is best presented as a triangle (like Pascal's triangle). This triangle is not symmetric with respect to the "main diagonal" Q n,n , but it is symmetric with respect to the "secondary diagonal"

Q n+1,n = C 2 × (D 4 ) n Z = D n C 2 2
, by the following lemma: Lemma B.4. We have an isomorphism of (ungraded) groups:

Q(t 1 , . . . , t n ) ∼ = Q(t 1 , Zt 1 t 2 , . . . , Zt 1 t n )
In particular, there is a group isomorphism Q 1+p,q ∼ = Q 1+q,p . so the group generated by b 1 , . . . , b n , Z is a quotient of Q(t 1 , Zt 1 t 2 , . . . , Zt 1 t n ). On the other hand, since b 1 b 2 = t 1 e 2 , etc., these elements generate Q(t), so for reasons of cardinality, Q(t) ∼ = Q(t 1 , Zt 1 t 2 , . . . , Zt 1 t n ). (Another proof is by starting from the isomorphism Q 2 ∼ = Q 1,1 , and to note that it induces the claimed symmetry.) B.2. The even part. Recall that the even part G 0 of a Z/2Z-graded group G = G 0 ⊔ G 1 is a (normal) subgroup. If G is finite and the grading non-trivial, then |G 0 | = 1 2 |G|. The element Z belongs to G 0 , but the grading of G 0 is a priori trivial. Theorem B.5. Let n ≥ 2 and t ∈ {1, Z} n . The even part of Q(t) is isomorphic to

Q(t) 0 ∼ = Q(Zt 1 t 2 , . . . , Zt 1 t n ).
It follows that

(Q n ) 0 ∼ = Q 0,n-1 ∼ = (Q 0,n ) 0 , (Q 1+p,q ) 0 ∼ = Q q,p , (Q p,1+q ) 0 ∼ = Q p,q .
Explicitly, the even part of Q n , resp. Q 0,n , is given by n 0 1 2 3 4 5 6 7 8

Q n C 2 C 2 2 D DC 4 DQ DQC 2 2 D 2 Q D 3 C 4 D 4 Q 0,n C 2 C 4 Q QC 2 2 DQ D 2 C 4 D 3 D 3 C 2 2 D 4 (Q n ) 0 ∼ = (Q 0,n ) 0 C 2 C 2 C 4 Q QC 2 2 DQ D 2 C 4 D 3 D 3 C 2 2
Proof. For k = 2, . . . , n, define as above the even element b k := e 1 e k . As noted above, (b k ) 2 = Zt 1 t k , and, for i = j, b i b j = e 1 e i e 1 e j = Z 2 e 2 1 e j e i = Z 3 e 1 e j e 1 e i = Zb j b i , so we get the desired relations for the generators b 2 , . . . , b n of Q(t) 0 . For reasons of cardinality we conclude that Q(t) 0 ∼ = Q(Zt 1 t 2 , . . . , Zt 1 t n ). Using Theorem B.3, we get the table for the even parts.

Remark B.2. As said above, a priori, the even part is ungraded; but since it is again a Clifford group, it "remembers" the grading of the preceding extensions, and so does ((Q n ) 0 ) 0 , until we reach C 0 , which is ungraded: the table from the theorem gives us a chain of even parts C 2 = (C 4 ) 0 = ((Q) 0 ) 0 = (((QC 2 2 ) 0 ) 0 ) 0 = . . . = ((((((D 4 ) 0 ) . . .) 0 . B.3. The periodic tables of discrete Clifford groups and of Clifford algebras. Finally, here are the tables classifying discrete Clifford groups and Clifford algebras. We present them in a form akin to Pascal's triangle: for the groups, Cl 0,1 (K) Cl 2,0 (K) Cl 1,1 (K) Cl 0,2 (K) . . . For the following tables, cf. also [S82, S84, L].

Q 0 Q 1,0 Q 0,1 Q 2,0 Q 1,1 Q 0,2 . . .
Table of groups Q p,q (p + q runs vertically and pq runs horizontally, notation V = C 2 × C 2 , C = C 4 , D = D 4 , see above). Note that the column labelled 1 is a symmetry axis of both tables (cf. Lemma B.4 ).

D 2 Q D 2 Q D 3 D 3 D 2 Q D 2 Q D 3 128 7 D 3 C D 2 QV D 3 C D 3 V D 3 C D 2 QV D 3 C D 3 V 256 8 D 4 D 3 Q D 3 Q D 4 D 4 D 3 Q D 3 Q D 4 D 4 512
Remark. For very detailed information on the four groups of cardinality 16 listed in line 3 of the table, see [D], page 529 for the group DV , page 530 for DC, and page 531 for QV . (You can recognize them in loc.cit. by looking at the structure of the center, containing the derived group -always of type C 2 -and the quotient with respect to this C 2 -always of type C 3 2 ). E.g., Aut(DC) is exactly the hyperoctahedral group of order 48, while Aut((QV ) is bigger, of cardinality 192, and Aut(DV ) is smaller, of cardinality 32. For the groups D 2 = D × Z D and DQ = D × Z Q (line 4), cf. loc. cit., p. 626, 627.

Table of classification of Clifford algebras

Cl p,q (K) (notation C K = K ⊕ iK, H K = K ⊕ iK ⊕ jK ⊕ kK, see above): ß

8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 0 K 1 K 2 C K

u

  = d 12 d 21 + d 13 d 21 + d 13 d 22 + (d 12 + d 22 )d 31 + (d 13 + d 23 )d 31 + (d 13 + d 23 )d 32 = (d 12 d 21 + d 13 d 21 + d 13 d 22 ) + (d 12 d 31 + d 13 d 31 + d 13 d 32 ) + (d 22 d 31 + d 23 d 31 + d 23 d 32 ).

e

  A e B = Z τ (A,B) e A∆B . (4) The group G is 2-step nilpotent: the commutator subgroup [G, G] belongs to the center of G, whence [G, [G, G]] = {1}. (5) Defining relations between the generators e 1 , . . . , e n , Z are: (a) e 0 is neutral, (b) (for all i = 1, . . . , n) e 2 i = t i e 0 , (c) (for k < ℓ) e k e ℓ = Ze ℓ e k , and (d) Z is central, even, and of order 2. Proof. (1), (2), (3) are essentially contained in Theorem 0.7 and Remark 2.4. More formally, these items are proved by induction: for n = 1 and n = 2, this has been noticed above. Assume the claim already proved at level n ∈ N, and write (3.3) Q(t 1 , . . . , t n+1 ) = Q(t) ×Q(t n+1 ), and use "internal notation" (2.1) together with induction hypothesis to write elements of this group as claimed. Concerning the product of two elements, e A e B = e a 1 • • • e a k e b 1 • • • e b ℓ , re-order elements in ascending order of indices; when two indices coincide, the square yields either 1 or Z; only terms with indices from the symmetric difference survive, together with a term Z u with u = 0 or u = 1. It follows that (e A ) 4 = 1 and (Ze A ) 4 = 1, for all A ∈ P(n). (4) The morphism from item (2) sends the commutator group [G, G] to [P(n), P(n)] = 0 since P(n) is abelian. Therefore [G, G] ⊂ {1, Z}, and hence [G, G] belongs to the center of G. (See Lemma 3.5 for an explicit formula describing commutators.)

  Example 3.2. For the pseudoscalar (case A = n, A ∩ B = B), e n = e 1 • • • e n , we get:

  the unique non-trivial automorphism exchanges the two elements of order 4, namely e 1 and Ze 1 ).(2) When t 1 = 1, so G ∼ = C 2 2 , then Aut(G) ∼ = S 3 is the permutation group of the three elements of order 2, namely of e 1 , Z, Ze 1 .Example 3.4. Let n = 2, G = Q(t 1 , t 2 ) = {e 0 , e 1 , e 2 , e 12 , Z, Ze 1 , Ze 2 , Ze 12 }, e 2 i = t i . (1) When t 1 = 1 = t 2 , then Aut(G) ∼ = D 4 ∼ = G. Conjugation by elements e 1 , e 2 , e 12 defines three non-trivial inner automorphisms of order 2, and exchange of e 1 and e 2 yields an outer automorphism of order 2. Conjugation by e 12 commutes with all automorphisms.

  the set of inversions in R, and for (A, B) ∈ P(n) 2 , let γ(A, B) := Z |Inv(A×B)| = Z |{(a,b)∈A×B| a>b}| .

  Theorem 4.1 (Eigenspaces of Z * ; superalgebras). With notation as above:(1) The algebraK[G] is a direct sum of ideals, K[G] = K[G] + ⊕ K[G] -,and both eigenspaces are isomorphic as K-modules. The projection π : G → G 0 = G/{e, Z} induces an algbra morphism π

  b 3 := e 1 e 2 e 3 , b 4 := e 1 e 2 e 4 , . . . , b n := e 1 e 2 e n Elements of Q(t 1 , t 2 ) and H commute: indeed, for all k = 3, . . . , n, e 1 b k = e 1 e 1 e 2 e k = Z 2 e 1 e 2 e k e 1 = b k e 1 , e 2 b k = e 2 e 1 e 2 e k = Z 2 e 1 e 2 e k e 2 = b k e 2 ,

  e 1 e 2 e k e 1 e 2 e ℓ = Ze 1 e 2 e ℓ e 1 e 2 e k = Z b ℓ b k .

Notation.

  To abbreviate, in the following, we are going to writeD := D 4 , C := C 4 , V := C 2 2 := C 2 × C 2 (the Klein four-group), and since all direct products are of the form × Z , we omit this sign, and just write it as juxtaposition, and we writeD 2 = DD, etc. For instance, DQ = D × Z Q, D 2 = D × Z D. The isomorphisms from Remark B.1 thus read D 2 ∼ = Q 2 and DC 4 ∼ = QC 4 . Note also that ×C 2 = × Z C 2 2 , so, e.g., DC 2 2 = D × Z C 2 2 = D × C 2 .Using these isomorphisms and notation, every group Q p,q is isomorphic to one in "normal form"C 2 , D k C 2 2 , D k C 4 , D k Q, QC 2 2 , D k QC 2 2 . Theorem B.3. The discrete Clifford groups Q n , resp. Q 0,n , are given by 4 DQ DQC 2 2 D 2 Q D 3 C 4 D 4 Q 0,n C 2 C 4 Q QC 2 2 DQ D 2 C 4 D 3 D 3 C 2 2 D 4and we have the periodicity relations:

Proof.

  The proof is similar to the one of Lemma B.1: define b 1 := e 1 , b 2 := e 1 e 2 = e 12 , . . . b n := e 1n . These elements satisfy the relations b 2 1 = t 1 , and for ℓ > k ≥ 2, b 2 k = e 1 e k e 1 e k = Zt 1 t k , b ℓ b k = e 1 e ℓ e 1 e k = Z 3 e 1 e k e 1 e ℓ = Zb k b ℓ ,

  8

  The elements 1, Z, e 12 , Ze 12 are even, and e 1 , e 2 , Ze 1 , Ze 2 are odd. For instance e 2 12 = e 1 e 2 e 1 e 2 = Ze 2 1 e 2 2 = Zt 1 t 2 . The group table of G is as follows: Ze 12 Zt 1 t 2 t 1 t 2 Zt 1 e 2 t 1 e 2 t 2 e 1 Zt 2 e 1 Ze 12 Ze 12 e 12 t 1 t 2 Zt 1 t 2 t 1 e 2 Zt 1 e 2 Zt 2 e 1 t 2 e 1 e 1 e 1 Ze 1 t 1 e 1 Zt 1 e 1 Since G is non-commutative and of cardinality 8, it is either isomorphic to the dihedral group D 4 or to the quaternion group Q. When t 1 = Z = t 2 , then G contains 6 elements of order 4, and hence is isomorphic to Q. Its even subgroup is

	•	1	Z	e 12	Ze 12	e 1	Ze 1	e 2	Ze 2
	1	1	Z	e 12	Ze 12	e 1	Ze 1	e 2	Ze 2
	Z	Z	1	Ze 12	e 12	Ze 1	e 1	Ze 2	e 2
	e 12	e 12 t 1	Zt 1	e 12	Ze 12
	Ze 1 Ze 1	e 1 Zt 1 e 1 t 1 e 1	Zt 1	t 1	Ze 12	e 12
	e 2	e 2	Ze 2 Zt 2 e 1 t 2 e 1 Ze 12	e 12	t 2	Zt 2
	Ze 2 Ze 2	e 2	t 2 e 1 Zt 2 e 1 e 12	Ze 12	Zt 2	t 2

Since K[G] + ∼ = K[P(n)], and the group is abelian and each element is of order at most two, a complete decomposition of this algebra is easy: Proposition 4.3. The group characters of P(n) are the maps, for A ∈ P(n), is symmetric, it follows that

A is a group morphism from P(n) to its dual group. Its kernel is trivial: χ A (X) = 1 for all X means that |A ∩ X| is even, for all X, and taking for X the singletons, if follows that A contains no element, so A = ∅. Thus the morphism A → χ A is injective. Since characters are always linearly independent, the cardinality of H cannot exceed the one of H, and thus the morphism is a bijection of P(n) onto its dual group.

Remark 4.2. Parametrizing the characters as in the proposition, the character table

A specific feature of these matrices is that they are symmetric:

Explicitly, for n = 1, P(1) = {∅, 1}, resp. n = 2, P(2) = {∅, {1}, {2}, 2}, we get

Then

The element e - 0 is a basis of the center of the Clifford algebra K[G] -.

(2) Assume n is odd. Then a basis of K[G] G is given by the elements (E + A ) A∈P(n) , along with e - 0 := (1 -Z)e 0 and e - n := (1 -Z)e - n (the pseudoscalar). The elements (e - 0 , e - n ) form a basis of the center of the Clifford algebra

If K is a field, then each basis element corresponds to an irreducible representation of G, and each irreducible G-representation arises in this way.

Proof. The dimension of K[G]

G is equal to the number of conjugacy classes of G, since the characteristic functions of conjugacy classes form a basis. By Theorem 3.6, we know the number of conjugacy classes. We have already exhibited 2 n linearly independent central functions; thus one (resp. two) additional independent elements suffice for defining a basis. Since K[G] G is Z-invariant, it decomposes into Zeigenspaces, so we can choose these additional elements in K[G] -, i.e., in the Clifford algebra. These elements being central, and since K[G] + * K[G] -= 0, it is necessary and sufficient that these elements belong to the center of K[G] -. Now, it is obvious that e - 0 , resp. e - n (for n odd), satisfy these conditions, and for reasons of dimension, there are no other independent elements satisfying them. This proves (1) and ( 2). If K is a field, then by the general theory of representations of finite groups (see [FH]) every irreducible G-representation arises in K[G]. The elements E + A define one-dimensional (hence irreducible) representations; the other elements generate representations that may decompose into direct sums of several irreducibles (cf. the following remark).

Remark 4.3. To get more information, in case n is odd, a further distinction should be made, according to the square of the pseudoscalar being 1 or Z. When all t j = 1, this corresponds to the cases n ≡ 1, 5, resp. n ≡ 3, 7 modulo 8. This determines whether the two submodules of dimension bigger than one are isomorphic to each other, or not (cf. [BtD], p. 288, see also [FH], Exercise 3.9, p. 30).

Appendix A. Clifford algebras

A.1. General definitions. Let K be a commutative base ring in which 2 is invertible. To every quadratic space (V, q) is associated, in a functorial way, a unital associative algebra Cl(V, q), its Clifford algebra, together with a linear map j : V → Cl(V, q), such that ∀v ∈ V : (j(v)) 2 = q(v)1, and which is universal for this property, see e.g., [BtD, L]. We shall be interested in the case V = K n , together with a quadratic form q = q t , where t = (t 1 , . . . , t n ) ∈ K n ,

that is, the canonical basis e 1 , . . . , e n is orthogonal, and q(e i ) = t i . The corresponding Clifford algebra is then denoted by

When t = 0 = (0, . . . , 0), this is the exterior algebra: Cl(V, q 0 ) = ∧(K n ). When t = (1, . . . , 1, -1, . . . , -1), with p terms 1 and q terms -1, we write Cl p,q (K) := Cl(t) = Cl(1, . . . , 1, -1, . . . , -1).

Among these algebras, Cl(1, . . . , 1) = Cl n,0 (K) plays a special role: it is anchored.

Lemma A.1. For every t ∈ K n , the linear map

The linear map is a morphism of quadratic spaces from (K(t 2 1 , . . . , t 2 n )) to (K, (1, . . . , 1)), and induces, by the universal property, an algebra morphism.

We call anchor the morphism defined in the lemma, and say that the Clifford algebra Cl K (t 2 1 , . . . , t 2 n ) is anchored. Of course, when t i ∈ {±1}, then the only anchored Clifford algebra is Cl K (1, . . . , 1). The terminology is motivated by [BeH]: indeed, the anchored Clifford algebras should be the correct "super-analog" of the anchored tangent algebras defined in loc.cit., and "differential calculus" then is the theory describing the contraction for t → 0 (which in the graded case still remains to be worked out).

A.2. Clifford basis. The standard basis e 1 , . . . , e n of K n defines n independent elements in the Clifford algebra Cl K (t). For a subset A = {a 1 , . . . , a k } ⊂ n, with a 1 < . . . < a n , we define the element Proof. The dimension of the Clifford algebra is 2 n (cf. [BtD]), and the elements e A generate it, hence form a basis. To compute the product e A e B , using the Clifford rules e Cf. page 12 -14 of [L] for a similar presentation of the preceding result.

Appendix B. Classification of Clifford groups and algebras B.1. Graded and ungraded product. Both the classification of real Clifford algebras and the one of discrete Clifford groups proceeds by decomposing them into ungraded tensor products of elementary algebras, resp. products of groups. On the level of algebras, we have, e.g., Cl R (1, -1, -1) ∼ = M 4 (C) ∼ = C ⊗ M 2 (R) ⊗ M 2 (R). On the level of groups, recall the ungraded product of groups G 1 × Z G 2 (quotient of G 1 ×G 2 under (x 1 , Zx 2 ) ∼ (Zx 1 , x 2 ), written also G 1 × C 2 G 2 when Γ = Z/2Z). We'll