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On alphabetic presentations of Clifford algebras and their

possible applications

Francesco Toppan1,a兲and Piet W. Verbeek2,b兲

1

CBPF, Rua Dr. Xavier Sigaud 150, CEP 22290-180 Rio de Janeiro, Rio de Janeiro, Brazil

2

Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands

共Received 9 March 2009; accepted 13 November 2009; published online 28 December 2009兲 In this paper, we address the problem of constructing a class of representations of Clifford algebras that can be named “alphabetic 共re兲presentations.” The Clifford algebra generators are expressed as m-letter words written with a three-character or a four-character alphabet. We formulate the problem of the alphabetic presenta-tions, deriving the main properties and some general results. At the end, we briefly discuss the motivations of this work and outline some possible applications. © 2009 American Institute of Physics.关doi:10.1063/1.3272001兴

I. INTRODUCTION

The irreducible representations of Clifford algebras have been classified in Ref.1. Convenient reformulations of this result can be found, e.g., in Refs.2and3, where some topics, such as the connection with division algebras, are also discussed.

The Cl共p,q兲 Clifford algebra over the real is the enveloping algebra generated by the␥ireal matrices共i=1, ... ,p+q兲 and quotiented by the relation

ij+␥ji= 2␩ij1, 共1兲 where ␩ij is a diagonal matrix with p positive entries +1 and q negative entries ⫺1. In the following, a basis of p + q gamma matricesi satisfying共1兲will be called a gamma basis.

The real irreducible representations are, up to similarity transformations, unique for p − q ⫽1,5 mod 8, while for p−q=1,5 mod 8, there are two inequivalent irreducible representations, which can be recovered by flipping the sign共␥i哫−␥i兲 of all gamma basis generators. The size n of an n⫻n real matrix irreducible representation is specified in terms of p and q.

Both in Refs. 2 and4, the given gamma basis representatives of a Cl共p,q兲 real irreducible representation were explicitly constructed共for any p,q pair兲 up to an overall sign flipping in terms of tensor products of four basic 2⫻2 real matrices. In Ref.4, the four matrices were named␴1,

␴2,␴A, 1, and are defined as follows:

␴1=

0 1 1 0

, ␴2=

1 0 0 − 1

, 12=

1 0 0 1

, ␴A=

0 1 − 1 0

. 共2兲

Without loss of generality, e.g., the three irreducible gamma generators of, let us say, Cl共3,0兲, can be explicitly given by

a兲Electronic mail: toppan@cbpf.br. b兲Electronic mail: p.w.verbeek@tudelft.nl.

50, 123523-1

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␥1= 1丢␴1, ␥2= 12丢␴2, ␥3=␴A丢␴A 共3兲 关any different presentation for the Cl共3,0兲 gamma basis is equivalent by similarity兴.

Extending this result, the p + q generators of a given real irreducible Cl共p,q兲 Clifford algebra can be expressed as strings of tensor products of the above four matrices, taken m times共if n is the size of the irreducible representation, therefore n = 2m; in the previous p = 3, q = 0 example, n = 4 and m = 2兲.

In the above type of gamma basis presentations, a few points should be noticed. At first, the introduction of the tensor product symbol “丢” is redundant. Once we understood that we are dealing with tensor products, we do not need to write it explicitly. For the same reason, the four matrices given in 共2兲 can be expressed with four characters of some given alphabet. For our purposes here, we choose the four characters being given by I , X , Z , A; we associate them with the above gamma matrices according to

12⬅ I,1⬅ X,2⬅ Z,A⬅ A 共4兲

共A stands for antisymmetric sinceA is the only antisymmetric matrix in the above set兲. In the above example, the three gamma matrices␥ican be more compactly expressed through the positions

␥1⬅ IX, ␥2⬅ IZ, ␥3⬅ AA. 共5兲

With the above identifications, for any共p,q兲 pair 共with the exception of the trivial p=1, q=0 case兲 and up to an overall sign factor, we can always write down the p + q generators of a gamma basis as m-letter words共the value m is common to all words of the basis兲, written with the four I,X,Z,A characters. For obvious reasons, we call this type of gamma matrix presentations “alphabetic presentations” or “alphabetic representations,” according to the context.

Not all representations are alphabetic according to the previous definition. The Cl共2,0兲 Clif-ford algebra admits X and Z as gamma bases. An equivalent gamma basis can be expressed, e.g., through the “entangled” matrices X˜ =1/

2共X+Z兲, Z˜=1/

2共X−Z兲.

In any case, due to the results in Refs. 2 and 4, it is always possible to produce a four-character alphabetic presentation of an irreducible gamma basis with words of a given length m. In the Euclidean case共q=0兲, for instance, m is explicitly given by the formula

m = log2G共k + 1兲 + 4r + 1, 共6兲

where pⱖ2 is parametrized according to

p = 8r + k + 2, 共7兲

with r = 1 , 2 , . . . and k = 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7, while G共k+1兲 is given by the Radon–Hurwitz function5

n 1 2 3 4 5 6 7 8

G共n兲 1 2 4 4 8 8 8 8 共8兲

The mod 8 property is a consequence of the famous Bott’s periodicity.

We can therefore concentrate on the subclass of the alphabetic presentations, as previously defined. Several questions can now be addressed. How many inequivalent alphabetic presentations can be defined? The notion of the equivalence group should not be based, of course, on the class of similarity transformations connecting real-valued Clifford algebras, instead the notion of a finite equivalence group of suitably defined moves transforming characters and words of an alphabeti-cally presented gamma basis into a new, equivalent, alphabetialphabeti-cally presented gamma basis should be given. Further questions can be addressed. Given the fact that A is the only character whose square is negative关A2= −I, with the position 共4兲兴, any alphabetically presented Euclidean gamma basis共for q=0兲 admits words with even numbers of A’s only 关in case共5兲above␥3contains two A’s, while1,␥2contain no A’s兲. Is it possible to define, for any p, Euclidean alphabetic

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presen-tations containing only the three characters I , X , Z共namely, limiting ourselves to a three-character alphabet兲? Furthermore, which is the minimal length m˜ of the three-character Euclidean words? Under which conditions m˜ coincides with m given in共6兲? This is just a partial list of the questions that we are addressing共and partially solve兲 in this paper. To our knowledge, this type of program has never been investigated in literature. Due to the recognized importance of Clifford algebras in several areas of mathematics and physics共for our purposes here, it is sufficient to mention the applications to higher-dimensional unification theories such as supergravities or superstrings,6or the applications to robotics7兲, we feel that it deserves being duly investigated. At the end of this paper, we provide a very rough and preliminary list of possible topics that could benefit from it. The main core of this paper is devoted to the formulation of the problem and the presentation of general results and partial answers. The scheme of this paper is as follows. In Sec. II we prove the Euclidean completeness of the three-character alphabetic presentations, introduce the equivalence group and a set of invariant functions. In Sec. III we furnish a few algorithmic constructions to induce inequivalent three-character alphabetic presentations, compute the admissible invariants, and present the results of an extensive computer search 共for three-, four-, five-, and six-letter words兲. A table with the minimal lengths for three-character alphabetic presentations of Cl共p,0兲 is also given. Further issues and an outline of four-character alphabetic presentations will be dis-cussed in Sec. IV. We will also mention there some topics that could benefit from the present investigation program.

II. ALPHABETIC PRESENTATIONS

In Sec. I we defined the alphabetic presentations of the gamma basis generators of a Cl共p,q兲 Clifford algebra as given by p + q words of m-letters constructed with the four alphabetic charac-ters I , A , X , Z 关the alphabetic characters are in one-to-one correspondence共4兲 with the four 2⫻2 matrices共2兲兴. We also pointed out that for Euclidean Clifford algebras 共q=0兲 with pⱖ2, three-character alphabetic presentations of the p gamma basis generators could exist. Their words are constructed with the I , X , Z characters alone. It is indeed easily proved that a three-character alphabetic presentation is Euclidean complete. This means the following: for any p, it is always possible to find p words satisfying 共1兲 and written with I , X , Z alone. The completeness of the four-character alphabetic presentations is guaranteed by the results given, e.g., in Refs.2and4. If the given Cl共p,0兲 alphabetic presentation contains no A’s, a three-character presentation immedi-ately follows. If at least one word contains an A in the jth position, we can replace all jth letter characters by two characters 共in jth and j+1th positions兲 according to, for instance, I哫II, X哫XX, Z哫ZX, A哫IZ. 关This position leaves the anticommutation relation 共1兲 between two different characters unchanged. The square of A changes its sign. This, however, has no overall effect since each word of the Euclidean gamma basis contains an even number of A’s.兴 If the original words possess m letters, the transformed words possess m + 1 letters. We can repeat the procedure every time we need to get rid of all A’s. The replacement leaves the relation 共1兲 unchanged. Applying the transformations to the Cl共3,0兲 gamma basis共5兲, we obtain, for instance, the three-character presentation

␥1= IIXX, ␥2= IIZX, ␥3= IZIZ. 共9兲

It follows that a three-character presentation 共not necessarily with minimal length words兲 can always be found for any p. Translated back into the matrix language 共tensor products of 2⫻2 matrices兲, it produces representations of共1兲q = 0 generating relations in terms of matrices that are not necessarily irreducible. “Alphabetic” irreducibility should not be confused with matrix irre-ducibility.

A. The alphabetic group of equivalence

We are now in the position to introduce the finite group of equivalence acting on alphabetic presentations. It is easier to discuss at first the three-character alphabetic presentations. It is convenient to arrange the p words of m letters each of a given alphabetic Cl共p,0兲 gamma basis

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into a p⫻m rectangular matrix, whose entries are the three alphabetic characters. The equivalence group G acting on the p⫻m rectangular matrices is obtained by combining the following three types of moves:

共i兲 permutations of the rows共they correspond to irrelevant reordering of the p words兲; 共ii兲 permutations of the columns关the anticommutative property共1兲between two distinct given

words is unaffected by this operation兲; and

共iii兲 transmutation of the characters in a given column: X , Z are exchanged共X↔Z兲, while I is unchanged关as before, the anticommutative property 共1兲between two distinct given words is unaffected by this operation兴.

It should be noticed that the rectangular matrices can be simplified, without affecting the relation共1兲, by erasing the columns possessing entries with either a single character or the two characters I and X or I and Z共the columns possessing both X and Z as entries cannot be erased兲. The process of erasing columns will be referred to as “simplification of the rectangular matrix.” A simple rectangular matrix is a rectangular matrix that cannot be further simplified. It produces a simple alphabetic presentation of a gamma basis. To be explicit, the three-character presentation of the Cl共3,0兲 gamma basis共9兲is associated with a 3⫻4 rectangular matrix, which can be simplified, erasing the first and the second columns, to produce a 3⫻2 rectangular matrix according to

I I X X I I Z X I Z I Z X X Z X I Z 共10兲 The simple rectangular matrix on the right hand side corresponds to three two-letter 共length 2兲 words. This is the minimal length for an alphabetic presentation of Cl共3,0兲. It coincides with the minimal length of the presentation共5兲, which, on the other hand, requires four characters instead of just three.

Two problems will be addressed in Sec. III.

共1兲 Which is the minimal length m˜ of the words for a three-character alphabetic presentation of Cl共p,0兲?

共2兲 How many inequivalent simple presentations of length m can be found for a three-character alphabetic presentation of Cl共p,0兲?

The second problem can be investigated with the help of invariants that detect the inequivalent classes under the finite group of transformations defined above. We introduce a few invariants, a “horizontal invariant” and the “vertical invariants.”

B. Alphabetic invariants

The horizontal invariant is defined as follows: at first, the number mIof I entries in any one of the p rows is computed. Let us suppose we obtain i different results k1, . . . , ki. We order them according to k1⬎k2⬎ ¯ ⬎kiⱖ0. Let hr be the number of rows producing the kr result 共r = 1 , 2 , . . . , i兲. Obviously, h1+ h2+¯+hi= p. The horizontal invariant hor is expressed as an ordered set of the hrvalues with kras the suffix. We write it as hor共h1k1, h2k2, . . . , hiki兲. It is easily checked that hor is invariant under the group transformations 共permutations and transmutations兲. As an example, the hor invariant of the simple rectangular matrix in the right hand side of 共10兲 is hor共11, 20兲.

The first vertical invariant ver is analogously defined; the difference is that the number nIof

I entries is computed in terms of the columns. Let us suppose we get j different results l1, . . . , lj, ordered according to l1⬎l2⬎ ¯ ⬎ljⱖ0. Let vrbe the number of columns producing the lrresult 共v1+v2+¯+vj= m兲. The vertical invariant ver is expressed as ver共v1l1,v2l2, . . . ,vjlj兲. The ver invariant of the simple rectangular matrix in the right hand side of 共10兲 is explicitly given by ver共11, 10兲.

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The second vertical invariant ver is defined as ver, but instead of counting the number nIof I’s in a given column, we compute the absolute difference nXZ=兩nX− nZ兩 between the number of X and the number of Z entries in any given column. Applied to共10兲, we obtain ver共11, 10兲. A less refined invariant under the group generated by permutations and transmutations is the total number NIof

I entries in a simple rectangular matrix.

A more refined invariant is ver, counting the number v共k

i,li兲 of columns presenting the given

pair共nI= ki, nXZ= li兲. The result is presented as ver共v共k1,l1兲,v共k2,l2兲, . . . ,v共kj,lj兲兲 共the pairs are

conve-niently ordered兲. Applied to共10兲, we obtain ver共1共1,0兲, 1共0,1兲兲.

In Sec. III it is sufficient to use the invariants hor and ver共based on the counting of I’s兲 to detect the inequivalent three-letter and four-letter alphabetic presentations.

For m = 1共single-letter word兲, we have a unique Cl共2,0兲 gamma basis given by 兵X,Z其. For m = 2, we have four equivalent 共under permutations and transmutations兲 presentations of Cl共3,0兲, given by 兵XX,ZX,IZ其, 兵XX,XZ,ZI其, 兵ZZ,XZ,IX其, 兵ZZ,ZX,XI其.

In Sec. III we discuss the construction of three-character alphabetic presentations with m-letter words for higher values of m.

III. INEQUIVALENT THREE-CHARACTER ALPHABETIC PRESENTATIONS

In Sec. II we furnished the m-letter three-character alphabetic presentations for m = 1 , 2. We discuss now the situation for mⱖ3. In order to do that, besides the already introduced notion of “simple alphabetic presentation,” we also need to define the notion of “maximally extended alphabetic presentation.” It corresponds to an m-letter gamma basis B such that no further word, anticommuting with all the words in B, can be added 共in the following, explicit examples of nonmaximally extended gamma basis will be given; they are obtained by erasing at least one word from a maximally extended gamma basis兲. It turns out that, at any given m, the classification of the inequivalent gamma basis is recovered from the classification of the simple, maximally extended, gamma basis.

In Ref.4 an algorithmic presentation was given to induce new gamma basis from the previ-ously known ones. In a very simple form共which is applied to the Euclidean case兲, it corresponds to produce an 共m+1兲-letter gamma basis for the Cl共p+1,0兲 Clifford algebra in terms of an m-letter gamma basis for Cl共p,0兲. If we denote withithe words in the Cl共p,0兲 gamma basis, it is sufficient to express the␥˜jwords共j=1,2, ... ,p+1兲 in the Cl共p+1,0兲 gamma basis as

˜i=␥iX,

˜p+1= I共m兲Z, I共m兲⬅ II, ... ,I 共taken m times兲. 共11兲 It is easily shown that the above position, in general, does not exhaust the class of inequivalent共in the sense specified in Sec. II兲 共m+1兲-letter alphabetic presentations of Cl共p+1,0兲. A general algorithm can be presented through the following construction. LetA, B1,B2be the three sets of m-letter words共whose respective cardinalities are nA, nB1, nB2兲 satisfying the following properties: bothC1=A艛B1 andC2=A艛B2 are a gamma basis and, furthermore, the words inB1 commute with all the words in B2. Under these conditions, an 共m+1兲-letter presentation B˜ of a Cl共nA + nB1+ nB2, 0兲 gamma basis can be produced by setting, symbolically,

B˜ ⬅ 兵AI,B1X,B2Z其. 共12兲

One should notice thatA could be the empty set, while both B1,B2must necessarily be nonempty in order forB˜ to be a simple gamma basis.

We applied this algorithm to induce, for m = 3 , 4, the whole set of inequivalent, simple, maximally extended, gamma basis. In parallel, we produce a systematic computer search of the inequivalent, simple, maximally extended gamma basis for m = 3 , 4 , 5 , 6. The results are reported below.

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A. Three-letter alphabetic presentations

For m = 3, there are only three inequivalent, simple, maximally extended gamma bases 共two for p = 4, one for p = 5兲. The representatives in each given class and their associated invariants are explicitly given by 4共p = 4兲: X X X X X Z X Z I Z I I 关hor共12,11,20兲; ver共12,11,10兲; NI= 3兴, 4共p = 4兲: X X I X Z I Z I X Z I Z 关hor共41兲; ver共22,10兲; NI= 4兴, 5គ共p = 5兲: X X X X I Z I Z X Z X I Z Z Z 关hor共31,20兲; ver共31兲; NI= 3兴. 共13兲

One should notice that two inequivalent p = 4 nonmaximally extended gamma bases are obtained by erasing one word from 5គ; if the word to be erased is XXX, we obtain a gamma basis with horizontal invariant hor共31, 10兲, while if the word to be erased is XIZ, we obtain a gamma basis 共兵XXX,IZX,ZXI,ZZZ其兲 with horizontal invariant hor共21, 20兲. The NIinvariant of the first case共the 兵XIZ,IZX,ZXI,ZZZ其 gamma basis兲 is NI= 3, which means that it is not sufficiently refined to detect a difference between this nonmaximally extended representation and the maximally ex-tended 4gamma basis. Erasing from both cases above an extra, conveniently chosen, word, we produce two inequivalent p = 3 simple nonmaximally extended gamma bases. They are given by 兵XIZ,ZXI,IZX其 with horizontal invariant hor共31兲 and 兵XXX,XIZ,ZZZ其 with hor共11, 20兲.

On the other hand, erasing a word from either 4 or 4 produces, in both cases, a p = 3 nonsimple gamma basis.

It is quite illustrative to show how 4, 4, and 5គ in共13兲can be algorithmically computed in terms of共12兲. We get

A = 兵IZ其 B1=兵ZX,XX其 B2=兵IX其

⇒ 兵IZI,ZXX,XXX,IXZ其 苸 4␣,

A = 兵XX,XZ其 B1=B2=兵IZ其

⇒ 兵XXI,ZXI,IZX,IZZ其 苸 4␤,

A = 兵XX其 B1=兵ZX,IZ其 B2=兵XZ,ZI其

⇒ 兵XXI,ZXX,IZX,XZZ,ZIZ其 苸 5គ. 共14兲

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B. Four-letter alphabetic presentations

Starting from mⱖ4, a new feature arises. Simple, maximally extended gamma basis with nonminimal length words are produced. Indeed, four inequivalent such representations for p = 5 can be found. On the other hand, as we have seen, a p = 5 gamma basis is already encountered for m = 3. Translated back into matrix representations, the four p = 5, m = 4 gamma bases produce reducible共in matrix, not alphabetic, sense兲 16⫻16 gamma matrices, whose size is twice the 8 ⫻8 irreducible representation obtained from 5គ in共13兲. The representatives of the four inequivalent p = 5, m = 4 gamma bases and their associated invariants are explicitly given by

5共p = 5兲: X X X X X X I Z I Z X I Z I Z Z Z Z Z X 关hor共12,21,20兲; ver共41兲; NI= 4兴, 5共p = 5兲: X X X X X X X Z X X Z I X Z I I Z I I I 关hor共13,12,11,20兲; ver共13,12,11,10兲; NI= 6兴, 5共p = 5兲: X X X I X X Z I X Z I X X Z I Z Z I I I 关hor共13,41兲; ver共23,11,10兲; NI= 7兴, 5共p = 5兲: X X X I X X Z I X Z I I Z I I X Z I I Z 关hor共32,21兲; ver共23,12,10兲; NI= 8兴. 共15兲

Here, m = 4 is the minimal length for an alphabetic presentation of the Euclidean Clifford algebra with p = 6 , 7 , 8. The complete list共representatives and their associated invariants兲 of inequivalent, simple, maximally extended gamma basis for p = 6 , 7 , 8, and m = 4 is explicitly given by

6共p = 6兲: X X X X Z I X X X Z I X I X Z X Z Z Z X I I I Z 关hor共13,31,20兲; ver共32,10兲; NI= 6兴,

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6共p = 6兲: X X X X Z X X X I Z I X I X Z I I I X Z I Z Z Z 关hor共32,11,20兲; ver共14,31兲; NI= 7兴, 6共p = 6兲: I X X X I Z X X X I Z X Z I Z I Z I X Z X I I Z 关hor共22,41兲; ver共14,12,21兲; NI= 8兴, 7គ共p = 7兲: X X X X Z I X X X Z I X I X Z I Z Z Z I I I X Z X Z Z Z 关hor共22,31,20兲; ver共32,11兲; NI= 7兴, 8គ共p = 8兲: I X X X X Z I X Z I Z X Z Z X I X X Z I X I X Z Z X I Z I Z Z Z 关hor共81兲; ver共42兲; NI= 8兴. 共16兲

All the gamma bases entering共15兲and共16兲can be algorithmically produced with the construction 共12兲. For simplicity, we limit ourselves to present the algorithmic construction of the largest of such representations, the gamma basis 8គ in共16兲which generates Cl共8,0兲. The sets A, B1,B2are given by

A = 兵XXX,ZZZ其, B1=兵ZXI,XIZ,IZX其 B2=兵XZI,ZIX,IXZ其

⇒兵XXXI,ZZZI,ZXIX,XIZX,IZXX,XZIZ,ZIXZ,IXZZ其 苸 8គ. 共17兲

This is the first example of the subclass of “cyclic” algorithmic constructions that will be dis-cussed later.

We made an exhaustive computer search and listed all inequivalent, simple, maximally ex-tended, three-character alphabetic presentations for m = 5 and m = 6. To save space, we just limit ourselves to mention that five-letter words can produce a Euclidean gamma basis for at most p = 9, while six-letter words can produce a gamma basis for at most p = 10.

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C. The minimal lengths

We are now in the position to present a table with the minimal length m˜ required to produce a three-character alphabetic presentation of Cl共p,0兲 at a given p. We compare m˜ with m, the minimal length for four-character alphabetic presentations, given by共6兲. We get

p 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 . . .

m 1 2 3 3 4 4 4 4 5 6 7 7 8 8 8 8 9 . . .

m

˜ 1 2 3 3 4 4 4 5 6 7 7 8? 8 8 8 9? 10? . . .

共18兲 The m˜ values for p = 13, 17, 18 are conjectured since a formal proof is lacking.

The above table is the result of an explicit computer search for m˜ⱕ6, combined with algo-rithmic constructions for m˜⬎6.

D. The cyclic prescription and another algorithm

There is a class of gamma basis共let us call them cyclic兲, obtained by a specific choice of A, B1,B2 entering共12兲.

For integral values n = 1 , 2 , 3 , . . ., we constructB1 as a set of 2n + 1 words of共2n+1兲-length obtained by cyclically permuting IZXZX¯ZX⬅I共ZX兲共n兲, whileB2 is the set of 2n + 1 words of 共2n+1兲-length obtained by cyclically permuting IXZXZ¯XZ⬅I共XZ兲共n兲,

B1=兵I共XZ兲共n兲 and its cylic permutations其,

B2=兵I共ZX兲共n兲 and its cyclic permutations其. 共19兲

Clearly, the words inB1 commute with the words inB2. Two subcases are now considered.

Subcase (i): For odd values n,A is given by the two word sets

A = 兵Z共ZZ兲共n兲, X共XX兲共n兲其. 共20兲

Subcase (ii): For even values n, A is the empty set

A =  . 共21兲

The prescription共12兲gives us, in both cases, a共2n+2兲-letter gamma basis such that subcase 共i兲 for odd values n, p = 2共2n+1兲+2=4共n+1兲 and subcase 共ii兲 for even values n, p=2共2n+1兲=4n+2.

For n = 1 we recover the construction of the 8គ gamma basis given in共17兲.

As a result, we obtain a relation, for cyclic three-character representations, between p and the length m¯ of their words, given by

p 8 10 16 18 . . . m

¯ 4 6 8 10 . . . 共22兲

We know that in subcase共i兲, for p=8k 共k=1,2,...兲, m¯ = 4k is a minimal length because it coincides with the known minimal length for four-character presentations. On the other hand, we explicitly checked that in subcase共ii兲, for p=10, m¯ = 6 corresponds to a minimal length, while subcase共ii兲 provides an upper bound for the minimal length for p = 18.

Another algorithmic construction, different from the cyclic prescription and generalizing the algorithm共11兲, allows us to prove that m˜ = 7 in共18兲is indeed the three-character minimal length for p = 12. Let C1, C2be two gamma basis for, respectively, Cl共p1, 0兲, Cl共p2, 0兲 with m1, m2length of their words. Let␥be a word of C1and Cthe complement set of 兵1其 in C1. A new gamma basis C for Cl共p1+ p2− 1 , 0兲, with words of length m=m1+ m2, is symbolically given by

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C =兵␥C2,CI1 共m2兲其. 共23兲 By taking, e.g., 5គ in共13兲as C1and 8គ in共16兲as C2, we obtain a three-character gamma basis with p = 5 + 8 − 1 = 12 and m = 3 + 4 = 7.

IV. CONCLUSIONS AND OUTLOOK

In this work, we investigated the alphabetic representations of the Cl共p,q兲 Clifford algebra gamma basis. The gamma basis generators are expressed as words written in up to four character alphabets. The four characters, I , X , Z , A, are associated with four 2⫻2 matrices according to共4兲 共I corresponds to the identity matrix, A to the antisymmetric matrix, etc.兲 and satisfy the anticom-mutation relation 共1兲. The words of an alphabetic representation are in correspondence with the matrix tensor products共in the correspondence, the tensor product symbol is omitted兲.

The interesting alphabets to consider are the whole four-character alphabet or a three-character alphabet. A two-three-character alphabet given by, e.g., X and Z, is too poor; indeed, it can only produce a Euclidean gamma basis for p = 1 , 2. On the other hand, the three-character alphabet given by I , X , Z is Euclidean complete. It produces Cl共p,0兲 Euclidean gamma basis for any value of p. For this alphabet, we introduced the notion of the alphabetic group of equivalence, con-structed invariants, and derived general and partial results共concerning, e.g., the minimal length of the words which produce a gamma basis for a given p兲. The alphabetic group of equivalence G can be extended to the whole four-character alphabet or to a three-character alphabet containing A 共namely, the character associated with the antisymmetric matrix兲. G is based on three types of moves, the permutations 共of rows and columns兲 and the transmutations of characters. In the extended case, the transmutations have to be suitably restricted since an A↔X 共or an A↔Z兲 transmutation maps a Cl共p,q兲 gamma basis into a Cl共p

, q

兲 gamma basis 共the constraint p

+ q

= p + q is satisfied; in the general case, on the other hand, p

differs from p兲. A viable restriction in the definition of the alphabetic group of equivalence consists in disregarding the transmutations involving the A character. Besides the invariants discussed in Sec. II B, extra horizontal and vertical invariants, counting the number of the A’s character, have to be introduced. The analysis of the four-character case 共invariants, inequivalent alphabetic presentations, etc.兲 is left for the forthcoming publications. It is worth pointing out that the introduction of a fourth character greatly increases the time needed for computer search of the inequivalent alphabetic presentations.

To our knowledge, this investigation program has not been addressed in literature. We have proven here that it is based on a well-posed mathematical problem admitting interesting and quite nontrivial solutions.

We have postponed so far discussing its possible applications. In the light of this, we should mention that the whole idea of constructing and analyzing the alphabetic presentations was deeply rooted in the investigations in our respective fields. Clifford algebras共in their alphabetic presen-tations兲 are the basis to construct8

representations of the N-extended supersymmetric quantum mechanics. These representations are nicely encoded in a graphical interpretation共see Refs.9and 10兲 in terms of colored, oriented, graphs. The equivalence group of transformations acting on graphs is related to the alphabetic group of transformations of the associated Clifford algebra.

The applications of Clifford algebras to robotics have been detailed, e.g., in Ref. 7. An interesting possibility is offered by the construction of cellular automata, which manipulate words in an alphabetic presentation of Clifford algebras. The three-character alphabet, here investigated in detail, is the simplest of such settings which allows the necessary complexity 共Euclidean completeness, inequivalent alphabetic representations, etc.兲.

At the end, let us just mention a seemingly far-fetched possibility, which, nevertheless, we believe deserves being duly investigated. The DNA codon problem concerns the yet to be ex-plained degeneracies found in associating amino acids with the triplets of the DNA nucleotides, cytosine共C兲, adenine 共A兲, thymine 共T兲, guanine 共G兲 for DNA or their respective G, U 共for uracil兲, A, C complements for mRNA. In the vertebral mitochondrial code, for instance, the 43= 64 nucleotides triples are associated with 20 amino acids and a stop signal according to a decompo-sition assigning 2, 4, or 6 different words to each amino acid and the stop signal: 64= 2⫻6+7

(11)

⫻4+12⫻2. One can consult Ref. 11 for an updated discussion of the codon problem and the attempted solutions共based on p-adic distance, deformed superalgebras, etc.兲. It is quite tempting to reformulate this problem in terms of alphabetic presentations of Clifford algebras共identifying each nucleotide with one of the four characters I, X, Z, and A兲 and check whether the alphabetic invariants could play a role in the association with the amino acids.

ACKNOWLEDGMENTS

F.T. is grateful to Roldão da Rocha for useful comments. This work has been supported by Edital Universal CNPq under Processo No. 472903/2008-0.

1M. F. Atiyah, R. Bott, and A. Shapiro,Topology3, 3共1964兲. 2S. Okubo,J. Math. Phys.32, 1657共1991兲; 32, 1669 共1991兲.

3I. R. Porteous, Clifford Algebras and the Classical Groups共Cambridge University Press, Cambridge, 1995兲. 4H. L. Carrion, M. Rojas, and F. Toppan,J. High Energy Phys.0304, 040共2003兲, e-print arXiv:hep-th/0302113v1. 5A. Pashnev and F. Toppan,J. Math. Phys.42, 5257共2001兲, e-print arXiv:hep-th/0010135.

6M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory共Cambridge University Press, Cambridge, 1987兲. 7G. Sommer, Geometric Computing with Clifford Algebras共Springer, London, 2001兲.

8Z. Kuznetsova, M. Rojas, and F. Toppan,J. High Energy Phys.3, 98共2006兲, e-print arXiv:hep-th/0511274. 9M. Faux and S. J. Gates, Jr.,Phys. Rev. D71, 065002共2005兲, e-print arXiv:hep-th/0408004.

10Z. Kuznetsova and F. Toppan,Mod. Phys. Lett. A23, 37共2008兲, e-print arXiv:hep-th/0701225. 11B. Dragovich and A. Dragovich, e-print arXiv:q-bio.OT/0707.3043.

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