Normalization of Polynomials in Algebraic Invariants of Three-Dimensional Orthogonal Geometry
aa r X i v : . [ c s . S C ] F e b Normalization of Polynomials in Algebraic Invariants ofThree-Dimensional Orthogonal Geometry
Hongbo Li
KLMM, AMSS, Chinese Academy of Sciences, Beijing 100190, China [email protected]
ABSTRACT
In classical invariant theory, the Gr¨obner base of the idealof syzygies and the normal forms of polynomials of invari-ants are two core contents. To improve the performance ofinvariant theory in symbolic computing of classical geome-try, advanced invariants are introduced via Clifford product[5]. This paper addresses and solves the two key problems inadvanced invariant theory: the Gr¨obner base of the ideal ofsyzygies among advanced invariants, and the normal formsof polynomials of advanced invariants. These results beauti-fully extend the straightening of Young tableaux to advancedinvariants.
Keywords
Invariant theory; Clifford algebra; bracket algebra; non-commutative Gr¨obner base; straightening of Young tableaux.
1. INTRODUCTION
In traditional analytic approach to classical geometry, coor-dinates are introduced to represent points in the geometricspace, and equations of the coordinates are used to defineconstraints among the points, forming a representation ofhigher dimensional objects such as curves, surfaces, etc. Ba-sic manipulations of coordinates include addition and multi-plication, resulting in polynomials in the coordinates. Sincethe coordinates of generic points are independent, and themultiplication of coordinate variables are commutative, nor-malization of the polynomials in the coordinates is very easy.The normal forms of the polynomials are required in manymanipulations, e.g. , division among polynomials.Another analytic approach to classical geometry, dating backto Euclid, is to use geometric invariants such as lengths, an-gles, areas, etc . A typically algebraic system of geometricinvariants is a polynomial ring generated by basic invariants.In such a system, a vector variable in a linear space is usedto represent a point or direction in classical geometry, theinner product of a vector with itself represents the squared length of the vector, the inner product of two unit vectorsrepresents the cosine of the angle between them, etc . Suchoperators among vectors generate a set of basic invariants ,and the polynomials in these basic invariants are advancedinvariants .Although the multiplication of invariants are commutative,the basic invariants generated by generic vector variables ofthe linear space are not independent, and there are poly-nomial relations among them, called syzygy relations. Thedependency is largely caused by the dimension constraintof the linear space upon vectors. While the dimension con-straint can be easily reflected by the number of coordinatesintroduced to represent a point and the independency amongthe coordinates, for basic invariants generated by the points,fully representing the dimension constraint is by no meanstrivial. Classical invariant theory studies the generators ofinvariants, the syzygy relations among the basic invariants,and the normal forms of advanced invariants as polynomialsin the basic ones [11], [12].In symbolic geometric computing, both the coordinate ap-proach and the basic invariant approach encounter the dif-ficulty of very big polynomial size, in particular in the mid-dle of symbolic manipulations. In [5], a recipe to allevi-ate the difficulty is proposed, called long geometric product,BREEFS, and Clifford factorization, among which the longgeometric product (or Clifford product) is the foundation.The idea is to convert polynomials of basic invariants intoadvanced invariants, converse to the approach of classicalinvariant theory, by means of an associative and multilin-ear product among the vector variables representing points.The associativity of the product and the symmetries withina long bracket provide powerful manipulations that cannotbe done with basic invariants, nor with coordinates. This isa top-down approach to advanced invariants [6], while theclassical invariant theory is a bottom-up approach.Dealing with the syzygy relations among advanced invari-ants and finding the normal forms of polynomials in ad-vanced invariants are two fundamental tasks in such “ad-vanced invariant theory”. The 2D case is easy, while higherdimensional cases are difficult. Little advance has beenachieved in six years since the publication of [5] in 1997.In this paper, the two fundamental problems are solved forthe advanced invariant theory of 3D orthogonal geometry:the Gr¨obner base of the syzygy ideal of “long brackets”, andhe normal forms of Clifford bracket polynomials. It turnsout that the normal forms of such bracket polynomials aresurprisingly “beautiful”. The description is the following.In classical invariant theory for ( n − n , or in coordinateform, the n × n determinants formed by the homogeneouscoordinates of n vector variables. A bracket polynomial is innormal form if when each term is up to coefficient written inYoung tableau form, the entries in each row are increasing,while the entries in each column are non-decreasing [14]. Forexample for vector variables v ≺ v ≺ . . . ≺ v m , a bracketmonomial [ v i v i · · · v i c ][ v i v i · · · v i c ] · · · [ v i r v i r · · · v i rc ], where the v i jk are repetitive selections of the m vectorvariables, is normal if and only if in v i v i · · · v i c v i v i · · · v i c ... ... . .. ... v i r v i r · · · v i rc , (1.1) v i j ≺ v i j ≺ · · · ≺ v i jc , while v i k (cid:22) v i k (cid:22) · · · (cid:22) v i rk .In the advanced invariant theory for 3D orthogonal geom-etry, each “elementary” advanced invariant is a bracket oflength >
1, whose entries are vector variables representing3D points. In a bracket monomial, different brackets manyhave different lengths, and a bracket monomial is in normalform if and only if not only the entries in each row are in-creasing, the entries in each column are non-decreasing, butall the entries in the tableau after removing the first column,are non-decreasing. For example if (1.1) is normal in this set-ting, then the sequence ˇ v i v i · · · v i c ˇ v i v i · · · v i c · · · ˇ v i r v i r · · · v i rc is non-decreasing, where ˇ v i k denotes that v i k does not occur in the sequence.This paper is organized as follows. Section 2 introducesorthogonal geometric invariants by quaternions, Clifford al-gebra and bracket algebra. Section 3 introduces the mainresults in [7] on vector-variable polynomials and basics of“advanced bracket algebra”. Section 4 provides the Gr¨obnerbase and normal forms of long brackets in the multilinearcase. Section 6 extends the results to general case, by meansof the square-free vector-variable polynomials introduced inSection 5. Section 7 proposes a normalization algorithm forbracket polynomials.
2. QUATERNIONS, CLIFFORD ALGEBRA,AND BRACKET ALGEBRA
In the vector algebra over R , there are three multilinearproducts among vectors: (i) the inner product of two vec-tors, (ii) the cross product of two vectors, (iii) the hybridproduct of three vectors. None of them can be extended toinclude more vectors while preserving the associativity.The quaternionic product, on the other hand, is associativewhile still multilinear. Let ¯ q represent the quaternionic con-jugate of quaternion q . Among quaternions, a vector v refersto a pure imaginary quaternion, i.e. , ¯ v = − v , and a scalar s refers to a real quaternion, i.e. , ¯ s = s . All vectors span a3D real inner-product space with metric diag( − , − , − R − . We always use juxtaposition of elements to denote theirquaternionic product. The inner product of two vectors v i , v j is defined by[ v i v j ] := ( v i v j + v j v i ) / . (2.1)The cross product of two vectors v i , v j is defined by v i × v j := ( v i v j − v j v i ) / . (2.2)The result is a vector, so its inner product with a thirdvector v k is a scalar. Define the hybrid product of threevectors v i , v j , v k by[ v i v j v k ] := ( v i v j v k − v k v j v i ) / . (2.3)Then [ v i v j v k ] = [( v i × v j ) v k ].The vector algebra over R − is included in the quaternions.The latter is equipped with a powerful product, the quater-nionic product, making it possible to use quaternions to rep-resent 3D orthogonal transformations [1].The magnitude of a quaternion q is √ qq . A quaternion q is said to be unit if qq = 1. Let q be a unit quaternion,and v be a vector. The conjugate adjoint action of q on v is defined by Ad q ( v ) := qvq . (2.4)Since [ Ad q ( v ) Ad q ( v )] = [ v v ] for any two vectors v , v , Ad q realizes an orthogonal transformation in R − . A classi-cal result states that in fact all orthogonal transformationsin R − are realized in this way, and two different unit quater-nions realize the same orthogonal transformation if and onlyif they differ by sign.For a quaternion Q , the bracket [ Q ] is its scalar part:[ Q ] := ( Q + ¯ Q ) / . (2.5)The axis of Q is the vector part of Q : A ( Q ) := ( Q − ¯ Q ) / . (2.6)In particular, A ( v v ) = v × v . We interpret them ingeometrical terms below.For a unit vector v , Ad v realizes the reflection with re-spect to the plane normal to v . In general, for unit vectors v , v , . . . , v k +1 , Ad v v ··· v k +1 realizes the reflection withrespect to the plane normal to axis A ( v v · · · v k +1 ), if thelatter is nonzero.For two unit vectors v , v that are linearly independent, Ad v v realizes the rotation about the axis v × v : in theplane spanned by v , v , the rotation is from v to the re-flection of v with respect to v , i.e. , the angle of rotation is θ = 2 ∠ ( v , v ). Furthermore, [ v v ] = cos( θ/ v v ”, we mean the one induced by Ad v v .In general, for unit vectors v , v , . . . , v k , Ad v v ··· v k real-izes the rotation about the axis A ( v v · · · v k ), if the latteris nonzero. The rotation is the composition of k rotations v v , v v , . . . , v k − v k . If the angle of rotation is θ , then[ v · · · v k ] = cos( θ/ , | A ( v · · · v k ) | = | sin( θ/ | . (2.7)et A ( v v · · · v k ) = 0. By [ v v · · · v k +1 ] = [ A ( v v · · · v k ) v k +1 ], we get[ v v · · · v k +1 ] = cos ∠ ( A ( v v · · · v k ) , v k +1 ) sin( θ/ , (2.8)where θ is the angle of rotation v v · · · v k . In particularwhen k = 1, for linearly independent unit vectors v , v ,sin( θ/
2) = sin ∠ ( v , v ) equals the area of the parallelogramspanned by v , v , and cos ∠ ( v × v , v ) equals the heightfrom the end of unit vector v to the plane spanned by v , v , so [ v v v ] equals the volume of the parallelepipedspanned by v , v , v .In classical invariant theory, an algebraic invariant is a poly-nomial whose variables are basic invariants. In 3D orthogo-nal geometry, there are two kinds of basic invariants: [ v i v j ]and [ v i v j v k ] for all vector variables v i , v j , v k . Given n vec-tor variables v , . . . , v n , the brackets [ v j v j · · · v j m ] for ar-bitrary 1 < m < ∞ and arbitrary repetitive selection ofelements v j , v j , . . . , v j m from the n variables, form an in-finite set of advanced algebraic invariants. That each ofthem is a polynomial of the [ v i v j ] and [ v i v j v k ] is guaran-teed by the following Caianiello expansion formulas [2], [6]:let V k = v j v j · · · v j k , then[ V l ] = P li =2 ( − i [ v j v j i ][ v j v j · · · ˇ v j i · · · v j l ]; A ( V l − ) = P (2 l − , ⊢ V l − [ V l +1(1) ] V l − ; A ( V l ) = P (2 l − , ⊢ V l [ V l (1) ] A ( V l (2) );[ V l +1 ] = P (2 l − , ⊢ V l +1 [ V l +1(1) ][ V l +1(2) ] , (2.9)where (i) ( h, m − h ) ⊢ V m is a bipartition of the m elementsin the sequence V m into two subsequences V m (1) and V m (2) of length h and m − h respectively; (ii) in [ V m (1) ], the prod-uct of the h elements in the subsequence is denoted by thesame symbol V m (1) ; (iii) the summation P ( h,m − h ) ⊢ V m isover all such bipartitions of V m , and the sign of permu-tation of the new sequence V m (1) , V m (2) is assumed to becarried by the first factor [ V m (1) ] of the addend.While quaternions are sufficient for describing orthogonaltransformations in 3D, they cannot be generalized to higherdimensions directly. In quaternions, the hybrid product[ v i v j v k ] is a scalar. To make high-dimensional generaliza-tion this requirement must be removed, at the same timethe property that this element be in the center of the alge-bra needs to be preserved. If we denote the quaternions by Q , then the above revision leads to a new algebra Q ⊕ ι Q of dimension 8, where ι := [ v v v ] for three fixed vectorvariables that are linearly independent. This algebra is the Clifford algebra over R − .The formal definition of the Clifford algebra
Cl( V n ) over an n -dimensional K -linear space V n , where the characteristic of K is = 2, is the quotient of the tensor algebra N V n over theideal generated by elements of the form v ⊗ v − Q ( v ) where Q is a K -quadratic form. The product induced from thetensor product is called the Clifford product , also denotedby juxtaposition of elements [4], [8].When V n = R − , the quaternionic product of vectors is theimage of their Clifford product under the homomorphism in-duced by mapping ι to a nonzero scalar. In Clifford algebra, ι is not a scalar, but called a pseudoscalar because it not only commutes with everything, but spans a 1D real spacecontaining all hybrid products. The concept quaternionicconjugate is replaced by the Clifford conjugate , which is thelinear extension of the following operation: for any vectors v j , v j , . . . , v j k , let V k = v j v j · · · v j k , then V k := ( − k V † k , where V † k := v i k · · · v i v i is the reversion of V k . (2.10)With the Clifford conjugate, we can define the magnitude of V k , the conjugate adjoint action Ad V k , the bracket [ V k ], the axis A ( V k ), together with the concepts not involving conju-gate: the inner product [ v i v j ], the cross product v i × v j andthe hybrid product [ v i v j v k ], just the same as in the case ofquaternions. The only difference is that since [ v i v j v k ] is nowa pseudoscalar, while A ( V l − ) remains a vector, A ( V l ) isnot, but a pseudovector . Geometrically, when A ( v v ) = 0,it represents the plane spanned by vectors v , v , or equiv-alently, the invariant plane of rotation Ad v v supporting v , v .We see that unlike the quaternions where there are only twokinds of objects of different dimensions: scalars which areusually called 0-D objects, and vectors which represent 1-Ddirections and so are usually called 1-D objects, in Cliffordalgebra Cl( R − ) there are four kinds of objects of differ-ent dimensions. Besides scalars and vectors, there are pseu-dovectors which represent 2-D directions (planes), and pseu-doscalars which represent 3-D orientations (spaces). Thisis the reason why Cl( R − ) can be extended to higher di-mensions by being capable of discerning objects of differentdimensions.To represent algebraic invariants in 3D orthogonal geome-try, using the quaternionic product or the Clifford productin the brackets does not make any difference. The geomet-ric interpretations (2.7), (2.8) and the Caianiello expansionformulas are identical for both products. This justifies theuse of juxtaposition of elements to represent both products.The two kinds of basic invariants [ v i v j ] and [ v i v j v k ] form acommutative ring, called inner-product bracket algebra . For-mally, given a set of n symbols M = { v , . . . , v n } , two kindsof new symbols can be defined as following: (1) all 2-tuplesof symbols selected repetitively from M , by requiring thateach 2-tuple be symmetric with respect to its two entries;such a 2-tuple is denoted by [ v i v j ]. (2) All 3-tuples of sym-bols selected repetitively from M , by requiring that each3-tuple is anti-symmetric with respect to its three entries,and in particular, if there are identical entries in a 3-tuple,setting the 3-tuple to be zero; such a 3-tuple is denoted by[ v i v j v k ].The two kinds of symbols must satisfy the following dimension-three constraints :IGP : For any five symbols v i , . . . , v i ,[ v i v i ][ v i v i v i ] − [ v i v i ][ v i v i v i ]+[ v i v i ][ v i v i v i ] − [ v i v i ][ v i v i v i ] = 0 . (2.11)B : For any six symbols v i , . . . , v i ,[ v i v i v i ][ v i v i v i ] = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ v i v i ] [ v i v i ] [ v i v i ][ v i v i ] [ v i v i ] [ v i v i ][ v i v i ] [ v i v i ] [ v i v i ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (2.12)The inner-product bracket algebra is the commutative ringgenerated by the above two kinds of symbols, satisfyingthe symmetry requirements and the dimension-three con-straints.To include brackets of longer length, the concept quater-nionic bracket algebra or Clifford bracket algebra needs tobe introduced. As explained before, there is no need to dis-tinguish between the two concepts in the setting of 3D or-thogonal geometry, so we simply call it bracket algebra . Todistinguish from the concept of the same name arising fromGrassmann-Cayley algebra [14], we call that in [14] classicalbracket algebra .Formally, besides the above 2-tuples and 3-tuples, a hier-archy of infinitely many new symbols can be defined: forany length l >
3, there are all l -tuples of symbols selectedrepetitively from M , with the requirement that the first andthe last equalities in Caianiello expansion (2.9) are satisfied;such an l -tuple is denoted by [ v j v j · · · v j l ]. By setting[1] = 1 (0-tuple) and [ v i ] = 0 (1-tuple) for all i , we get afull hierarchy of new symbols marked by brackets, with ar-bitrary length l ≥
0. The new symbols together with theirspecific requirements, form a a commutative ring called the bracket algebra over 3D inner-product space. This is the thebottom-up approach to defining bracket algebra. The con-cept quaternionic product or Clifford product is not needed.
3. VECTOR-VARIABLE POLYNOMIALSAND BRACKET POLYNOMIALS
In this paper, we use (1) bold-faced digital numbers andbold-faced lower-case letters to denote vector variables, e.g. , v , ; (2) bold-faced upper-case letters to denote monic vector-variable monomials, e.g. , A , X ; (3) Roman-styled lower-caseletters to denote polynomials, e.g. , f, g ; (4) Greek letters todenote K -coefficients, e.g. , λ, µ .Although the background is real orthogonal geometry, thealgebraic manipulations under investigation are independentof the real field. In fact, only the following coefficients occurin computing: ± k for k ∈ Z . We set the base field K to beof characteristic = 2.Now start from quaternions. Let v , v , . . . , v n be symbols.What properties determine that the multilinear associativeproduct among the symbols is the quaternionic one, and thatthese symbols represent vectors of a 3D real inner-productspace with metric diag( − , − , − v i , v j is a scalar, so itcommutes with a third vector v k : [ v i v j ] v k = v k [ v i v j ]. Forthree vectors v i , v j , v k , since [ v i v j v k ] is a scalar, for a fourthvector v l , the commutativity [ v i v j v k ] v l = v l [ v i v j v k ] holds.The two commutativities are all that characterize the equal- ity properties of the quaternionic product, besides the mul-tilinearity and associativity. Theorem
1. [7]
Let v , v , . . . , v n be n > symbols. De-fine the product among them, denoted by juxtaposition of el-ements, as the K -tensor product modulo the two-sided idealgenerated by the following tensors: V2 : v i ⊗ v i ⊗ v j − v j ⊗ v i ⊗ v i ;V3 : ( v i ⊗ v j + v j ⊗ v i ) ⊗ v k − v k ⊗ ( v i ⊗ v j + v j ⊗ v i );V4 : ( v i ⊗ v j ⊗ v k − v k ⊗ v j ⊗ v i ) ⊗ v l − v l ⊗ ( v i ⊗ v j ⊗ v k − v k ⊗ v j ⊗ v i ) , (3.1) for any i = j = k = l in , , . . . , n . Denote by Q or Q [[ v , . . . , v n ]] the K -algebra defined by the above productand generated by the v i . Denote by I [[ v , . . . , v n ]] the aboveideal, and call it the syzygy ideal of Q .Denote K := K ( { v i ⊗ v i , v i ⊗ v j + v j ⊗ v i , | i = j } ) , K := K ( { v i ⊗ v j ⊗ v k − v k ⊗ v j ⊗ v i | i = j = k } ) . (3.2) (1) Let ι := [ v v v ] . Then ι = 0 , and K = K ( ι ) .(2) The v i and ι v i × v j of the K -algebra Q span a 3D K -space V . For any k ≥ , A ( v j · · · v j k +1 ) and ιA ( v j · · · v j k ) are both in V .(3) The defined product is the Clifford product of the K -Clifford algebra over V .(4) If K = K = R , and the inner product of real space V induced from (2.1) is definite, then the defined product is thequaternionic product.(5) Let V k = v j v j · · · v j k . Then the following identitieshold modulo I [[ v , v , . . . , v n ]] : v i [ V k ] = [ V k ] v i , [ v i V k ] = [ V k v i ] . (3.3)The requirements in (4) distinguishing the quaternionic prod-uct from the Clifford product cannot be represented by equal-ities. So for symbolic manipulations of equalities, the quater-nionic product and the Clifford product cannot be distin-guished. The K -algebra Q is called the
3D vector-variablepolynomial ring generated by vector variables v i , and theproduct in it is called the vector-variable product . It is nei-ther the quaternionic product nor the Clifford product, buta more basic one.All the terminologies introduced earlier on quaternions andClifford algebra are valid for Q . Besides, there are someadditional terminologies for Q . Let v ≺ v ≺ . . . ≺ v n be vector variables. A monic monomial of vector variablesrefers to the product of a repetitive permutation of some ofthe vector variables. For a monic monomial v i v i · · · v i k ,the leading variable refers to v i , and the trailing variable refers to v i k . The monomial is said to be non-descending if v i (cid:22) v i (cid:22) . . . (cid:22) v i k , and is said to be ascending if v i ≺ v i ≺ . . . ≺ v i k . The degree , or length , of the monomial is k . The lexicographic ordering among monomials is alwaysassumed. polynomial of vector variables is a K -linear combinationof monic monomials. The leading term of a polynomial f isthe term of highest order, denoted by lt( f ). The degree ofa polynomial is that of its leading term. The leading termsof all elements in a subset S of polynomials are denoted bylt( S ). When specifying the field K or K , we can get thecorresponding concepts quaternionic polynomial and Cliffordpolynomial .Fix a multiset of vector variables M composed of m ≥ v , v , . . . , v m . Let n be the number of differentelements in M , where 3 ≤ n ≤ m . In the K -tensor al-gebra N ( v , v , . . . , v n ) generated by the n symbols takenas vectors, a tensor monomial is up to coefficient the ten-sor product of finitely many such vectors. The K -tensoralgebra over M , denoted by N [ M ], is the K -subspace of N ( v , v , . . . , v n ) spanned by tensor monomials whose vec-tor variables by counting multiplicity are in M , equippedwith the tensor product that is undefined if the result is nolonger in N [ M ].When the product among the elements in M is the vector-variable product, we have the corresponding concept of M , denotedby Q [ M ]. Each element in Q [ M ] is a polynomial whosemultiset of vector variables in each term is a submultiset of M . The syzygy ideal I [ M ] of Q [ M ] is still defined by (3.1).The concepts of Gr¨obner base and normal form are definedin Q [ M ] just as in N ( v , v , . . . , v n ) [9]. For two monomi-als h , h in vector variables, h is said to be reduced withrespect to h , if h is not a factor of h , or h is not a multi-plier of h , i.e. , there do not exist monomials l, r , includingelements of K , such that h = lh r . For two polynomials f and g , f is said to be reduced with respect to g , if theleading term of f is reduced with respect to that of g . Theterm “ non-reduced ” means the opposite.Let { f , f , . . . , f k } be a set of vector-variable polynomi-als. Another set of vector-variable polynomials { g , g , . . . , g l } is said to be a reduced Gr¨obner base of the ideal I := h f , f , . . . , f k i generated by the f i in Q [ M ], if (1) h g , . . . , g l i = I , (2) the leading term of any element in I is a multiplier ofthe leading term of some g i , (3) the g i are pairwise reducedwith respect to each other.The reduction of a polynomial f with respect to a reducedGr¨obner base g , g , . . . , g l is the repetitive procedure of di-viding the highest-ordered non-reduced term L of f by some g i whose leading term is a factor of L , then updating f byreplacing L with its remainder, until all terms of f are re-duced. The result is called the normal form of f with respectto the Gr¨obner base. Two polynomials are equal if and onlyif they have identical normal forms.In [7], two theorems are established for the Gr¨obner baseand normal forms of 3D vector-variable polynomials, one forthe multilinear case where each element in multiset M hasmultiplicity 1, the other for the general case Q [[ v , . . . , v n ]]. Theorem
2. [7]
Let I [ v , . . . , v n ] be the syzygy ideal ofthe multilinear polynomial ring Q [ v , . . . , v n ] in n differentvector variables v ≺ v ≺ . . . ≺ v n . (1) [Gr¨obner base] The following are a reduced Gr¨obner baseof I [ v , . . . , v n ] : for all ≤ i < i < . . . < i j ≤ n , G3 : [ v i v i v i ] − [ v i v i v i ] , and [ v i v i v i ] − [ v i v i v i ] ; G j : [ v i v i v i v i · · · v i j v i ] − [ v i v i v i · · · v i j v i v i ] , forall j > .(2) [Normal form] In a normal form, every term is up tocoefficient of the form V Y v z V Y v z · · · V Y k v z k or V Y v z · · · V Y k v z k V Y k +1 , where(i) k ≥ ,(ii) v z v z · · · v z k is ascending,(iii) every V Y i is an ascending monomial of length ≥ ,(iv) V Y V Y · · · V Y k (or V Y V Y · · · V Y k +1 if V Y k +1 occurs)is ascending,(v) for every i ≤ k , if v t i is the trailing variable of monomial V Y i , then v z i ≺ v t i . Theorem
3. [7]
Let I [[ v , . . . , v n ]] be the syzygy ideal ofthe polynomial ring Q [[ v , . . . , v n ]] in n different vector vari-ables v ≺ v ≺ . . . ≺ v n .(1) [Gr¨obner base] The following are a reduced Gr¨obner baseof I [[ v , . . . , v n ]] : G3 , G j : for all < j < ∞ , and all ≤ i < i < i < i ≤ . . . ≤ i j − < i j ≤ n ; EG2 : for all i < i , v i v i v i − v i v i v i , v i v i v i − v i v i v i ;EG j : [ v i v i v i v i · · · v i j v i ] − [ v i v i v i · · · v i j v i v i ] , forall < j < ∞ , and all ≤ i < i < i ≤ i ≤ . . . ≤ i j − < i j ≤ n .(2) [Normal form] In a normal form, every term is up tocoefficient of the form V Y v h v z V Y v h v z · · · V Y k v h k v z k or V Y v h v z · · · V Y k v h k v z k V Y k +1 , where(i) k ≥ ,(ii) v z v z · · · v z k is non-descending,(iii) v h v h · · · v h k is non-descending,(iv) every V Y i is a non-descending monomial of length ≥ ,(v) V Y v h V Y v h · · · V Y k v h k (or V Y v h · · · V Y k v h k V Y k +1 if V Y k +1 occurs) is non-descending,(vi) for every i ≤ k , v h i ≻ v z i ,(vii) for every i ≤ k , if the length of V Y i is nonzero, let v t i be the trailing variable of V Y i , then v t i ≺ v h i . For a general multiset M in which the different vector vari-ables are v , v , . . . , v n , the Gr¨obner base of the syzygy ideal I [ M ] is the restriction of the Gr¨obner base of I [[ v , . . . , v n ]]to Q [ M ], denoted by G [ M ]. In Q [ M ], a polynomial is saidto be I -normal if its leading term is reduced with respectto the Gr¨obner base. The procedure of deriving the normalform of a polynomial is called I -reduction .Now that any vector-variable polynomial has a normal formby I -reduction, so does a bracket polynomial when everyracket is expanded into two terms by definition. The resultis complicated.Consider the following simple example: for a single bracket2[ v v · · · v m ] = v v · · · v m + ( − m v m · · · v v , the I -reduction goes as follows: for 0 ≤ j < m , if we define V m − j = v j +1 v j +2 · · · v m , then when m ≥ V m + ( − m V † m I = v V m − + V m − v − v ( V m − + ( − m − V † m − ) I = v V m − + V m − v − v ( v V m − + V m − v )+ v v ( V m − + ( − m − V † m − )= v V m − v − v V m − v + v v ( V m − + ( − m − V † m − ) . (3.4)From this recursive formula, we get for m ≥ v v · · · v m ] I = 1 + ( − m v v · · · v m + m − X i =1 ( − i +1 ( v · · · ˇ v i · · · v m ) v i . (3.5)So the normal form of the simplest bracket [ v v · · · v m ] iscomposed of up to m terms. What is worse is that onlywhen the terms are summed up can they represent a singlealgebraic invariant (the bracket), while missing a single termdestroys the invariance of the whole expression.From the appearance, the bracket symbol only hides half ofa binomial. There are much more behind this appearance.By definition, for a sequence A of a > A ] = 2 − A + ( − a − A † . Monomial A iscalled the representative of bracket [ A ]. Later on, whenwe write [ A ], we always assume that monomial A is therepresentative of the bracket.The definition of a bracket endows the symbol with the re-version symmetry (or equivalently, the conjugate symmetry)up to sign: [ v i . . . v i a ] = ( − a [ v i a · · · v i ]. By (3.3), thebracket symbol also has shift symmetry. So up to sign abracket of a vector variables has the symmetry group D a (dihedral group).The above analysis is only for a single bracket. For bracketpolynomials, there are a lot of polynomial identities, or syzy-gies , among them. These complexities justify the separationof bracket algebra from vector-variable polynomial ring insymbolic manipulations of algebraic invariants. Finding theGr¨obner base of the syzygies and then characterizing thenormal forms of bracket polynomials are the main goal ofthis paper.The following are some terminology on brackets. The repre-sentative of a bracket polynomial is the the vector-variablepolynomial whose terms are each the product of the coeffi-cient with the representatives of the bracket factors in thesame term. The representative of a bracket polynomial isallowed to contain brackets. For example, [ v v [ v v ]] istaken as bracket binomial 2 − [ v v ( v v + v v )]; its repre-sentative is the content v v [ v v ] within the outer bracket. The lexicographic ordering of bracket polynomials is thatof their representatives. The leading variable of a bracketrefers to that of its representative. The leading term of abracket polynomial is always under the lexicographic order-ing. For example, if v ≺ v , then [ v v ] ≺ [ v v ], and[ v v ][ v v ] ≺ [ v v ][ v v ].The leader (or expanded leading term ) of a bracket polyno-mial, refers to the leading term of the bracket polynomialwhen taken as a vector-variable one, i.e. , the correspondingvector-variable polynomial obtained from expanding eachbracket into two terms by definition. For example, the leaderof bracket [ A ] refers to the one of higher order between 2 − A and ( − a − A † .Among all the bracket polynomials that are equal to thesame bracket polynomial, there are two that have strongfeatures: the first is the one whose representative is the low-est, the second is the one whose leader is the lowest. Thesecond is unique but the first is not. To make the first uniquewe introduce the following concept.In Q [ M ], where the number of elements in multiset M is m ,a uni-bracket monomial refers to a single bracket of length m . A uni-bracket polynomial is a K -linear combination ofuni-bracket monomials. All uni-bracket polynomials form a K -linear space, denoted by [ Q ][ M ]. The K -linear space ofthe representatives of elements in [ Q ][ M ] is just the K -linearspace of degree- m vector-variable polynomials, denoted by Q m [ M ]. Obviously, [ Q ][ M ] is a linear subspace of Q m [ M ].When taken as a vector-variable polynomial, a uni-bracketis a binomial. In appearance, a uni-bracket is a monomial.To distinguish between the two understandings, we need adevice to get rid of the bracket symbol and extract the rep-resentative of the uni-bracket. This can be done by taking[ Q ][ M ] as the quotient of Q m [ M ] modulo the ideal J [ M ] := I [ M ] + [ I ][ M ] , (3.6)where [ I ][ M ] is composed of the vector parts of degree- m polynomials, i.e. , the K -linear span of elements of the formR : v i v i . . . v i m − ( − m v i m · · · v i v i , (3.7)for all permutations of the m elements in M .The modulo-[ I ][ M ] operation identifies a uni-bracket withits representative, or equivalently, identifies any degree- m vector-variable polynomial with the uni-bracket polynomialit serves as the representative. This operation retains thebracket symbols of all the brackets of length < m , whileremoving the bracket symbols from all brackets of length m .
4. GRÖBNER BASE AND NORMAL FORMFOR MULTILINEAR UNI-BRACKETPOLYNOMIALS
From this section on, we use bold-faced digital numbers todenote vector variables, and use bold-faced capital letters todenote monic monomials of vector variables.In this section, the multiset M is composed of m ≥ I ][ M ] operation is al-ways assumed. Then [ Q ][ M ] and Q m [ M ] are identical, and uni-bracket no longer has the outer bracket symbol. Ideal[ I ][ M ] is called the uni-bracket removal ideal in Q [ M ], andideal J [ M ] is called the syzygy ideal of [ Q ][ M ] in N [ M ].Below we compute the Gr¨obner base of J [ M ] and charac-terize the normal forms of uni-bracket polynomials.The following are elements of J [ M ]: for all monomials A to F such that A1B , , , are of length m and E has length e > : A1B − , S N : the S in which A1B is I -normal , R : 2( [ C ]) = − ( − m † , R : 2( [ D2 ]) = − ( − m † , R [ ∗ ] : 2( [ E2 [ F ]]) = [ F ] − ( − e † [ F ] . (4.1)Since the reduced Gr¨obner base of I [ M ] is G [ M ], we onlyneed to consider the elements of type R in J [ M ], as theyspan [ I ][ M ]. By A1B − ( − m B † † = ( A1B − ) + ( − ( − m † B † ) − ( − m ( B † † − † B † ) , (4.2)we get Lemma
4. R is a subset of the ideal h S , R i . For anytype- R element f but not of type R , lt( f ) ∈ lt(S ) . Lemma
5. S is a subset of h S N , R i + I [ M ] . For anytype- S element f but not of type S N , lt( f ) ∈ lt( I [ M ]) . Proof.
Consider a general type-S element f = A1B − , where A is not empty. If A1B is I -normal, then f ∈ S N . If I -reductions are carried out to A , B , say A = A N + I and B = B N + I , where A N , B N are both I -normal, then f = A N N − N A N + I , and A N N is the new leadingterm. So by I -reductions we can assume that both A and B are I -normal.Assume that A1B is not I -normal. Further assume thatany type-S element g ≺ f is in h S N , R i + I [ M ]. We provethe conclusion for f by reduction on the order. There arethree possibilities to apply Gr¨obner base elements of I [ M ]to make reduction to A1B : (i) G3 at the end of A1 , (ii) G i for i > A1 , (iii) G3 on and two variablesfrom A , B respectively.Case (i). Let A1B = Cuv1B , where u ≻ v ≻ . Then f G3 = − Cvu1B + C1 ( uv + vu ) B − induction = − ( vu + uv ) + ( uv + vu ) BC I = 0 . Case (ii). Let
A1B = CuvD1B , where 1 ≺ v ≺ u ≺ allvariables of D . Let the length of D be d >
0. Let the lengths of B , C be b, c respectively. Then b + c = m − d − f G i = C ( vD1 + ( − d † v ) uB − ( − d Cu1D † vB − induction = { ( uBC − BCu ) vD + ( − d D † v ( uBC − BCu ) } R = ( − d † v { ( − m − d ( C † B † u − uC † B † )+( uBC − BCu ) } = ( − d † v {− (( − b + c C † B † + BC ) u + u (( − b + c C † B † + BC ) } I = 0 . Case (iii). Let
A1B = Cu1vD , where u ≻ v ≻ . Let thelengths of C , D be c, d respectively. Then c + d = m − f G3 = − C1uvD + Cv ( u1 + ) D − induction = ( − uvDC + DCvu + uDCv − vDCu ) R = {− (( − m C † D † − DC ) vu + v (( − m C † D † − DC ) u } I = 0 . ✷ Lemma
6. R is a subset of the ideal h R i + I [ M ] . Forany type- R element f but not of type R , lt( f ) ∈ lt( I [ M ]) . Proof.
Let f = 2( [ E2F ]) be a general element of typeR . Then f I = 2( [ FE2 ]), and the latter is in R . ✷ Theorem Let M = { , , . . . , m } be m > differentsymbols, where ≺ ≺ . . . ≺ m , and let Q [ M ] be thevector-variable polynomial ring over M . Let [ Q ][ M ] be thespace of uni-bracket polynomials, and let J [ M ] be its syzygyideal in N [ M ] .(1) [Gr¨obner base] The following are a reduced Gr¨obner baseof J [ M ] : G [ M ] : G i for all ≤ i ≤ m ; S N : A1B − , where A is an ascending sequence oflength a > , B is of length m − a − ≥ , and A1B is I -normal; R N [ j ] : for all ≤ j ≤ m − , [ A2 ][ Y z ][ Y z ] · · · [ Y j z j ] ,where (i) A is an ascending sequence of length a > ;(ii) when j = 1 , then R N [1] = [ A2 ] ;(iii) each Y i is a non-empty ascending sequence, andeach z i is a variable such that z i Y i is ascending;(iv) z · · · Y j z j is I -normal and length- m .(2) [Normal form] In a normal form, every term is I -normal,and is up to coefficient of one of the following forms:(I) , where C is I -normal;(II) z Y z · · · Y k − z k − Y k , where k ≥ , A andthe Y i are each a non-empty ascending sequence, and each z i ≺ t i , the latter being the trailing variable of Y i ;III) z Y z · · · Y k z k , where k ≥ , A and the Y i , z i are as in (II), and for some ≤ i ≤ k , if l i is the leadingvariable of Y i , then l i ≺ z i . Proof.
There are several steps.
Step 1 . We need to prove that R is a subset of the ideal h R N i + I [ M ]. Once this is done, then since the leader of anyelement of R N is I -normal and cannot be cancelled by theleader of any other element of R N , the G i and R N [ j ] forma reduced Gr¨obner base of h R i + I [ M ]. By this and theprevious three lemmas, we get conclusion (1) of the theorem.Once conclusion (1) holds, then any I -normal monomial oflength m with leading variable is the representative of auni-bracket in normal form if and only if it is not the leaderof an R N -typed element. Conclusion (2) follows. Step 2 . The idea of proving the statement in Step 1 is to use G [ M ] to decrease the order of the leader of every elementof type R or R [ ∗ ], at the same time keep the reductionresult to be within the K -linear space spanned by elementsof type R or R [ ∗ ]. Then ultimately all the leaders ofthese elements become I -normal.We start with the I -reduction on the leading term of a gen-eral R -typed element f = − ( − m † , where thelength of A is m −
2. If by I -reduction, A = A N + I , then f = N − ( − m N † + I . So we can assume that A is I -normal.If A2 is I -normal, then f is just R N [1]. When A2 is not I -normal, if A2 is non-reduced with respect to G3, let A2 = Buv2 , where u ≻ v ≻ , then = − + 2 { [ uv ] } . (4.3)The result consists of the leading terms of one R -typedelement and one R [ ∗ ]-typed element. The leaders of bothterms are lower than f .If A2 is non-reduced with respect to G i for some i >
3, let A2 = BuvC2 , where u ≻ v ≻ , and uC is ascending, andthe length of C is c >
0. Then G i = ( vC2 + ( − c † v ) u − ( − c † v = + (( − c C † vu + uvC ) − + ( − ( − c C † v + vC ) − I = 2 { [ uvC ] } + 2 { [ vC ] } − , (4.4)The result consists of the leading terms of two R -typedelements and one R [ ∗ ]-typed element. The leaders of thethree terms are lower than f .By (4.3) and (4.4), a monomial that is non-reduced withrespect to a G i for some i ≥ uDv , where (i) the length of D is d >
0; (ii) u ≻ v ; (iii) if l D is the leading variable of D , then u ≻ l D ;(iv) if D contains more than one variable, then l D ≻ v . (4.3)and (4.4) can be written in the following unified form: uDv G( d +2) = 2( uv [ D ]) + 2( v [ uD ]) − Duv . (4.5) It is called the fundamental I -reduction formula . Step 3 . Consider I -reductions on the leader of a generalR [ ∗ ]-typed element f = [ B ] − ( − a † [ B ], wherethe length of B is b > B ] = ( − b [ B † ], henceforth we assume that in anyR [ ∗ ]-typed element to be normalized, the leading variableof any bracket has higher order than the trailing variableof the bracket. Then the leader of the bracket is always itsrepresentative.In this step, we consider I -reduction to the representative B of [ B ]. Let [ B ] = [ CuDvE ]. Substituting (4.5) into it,we get[
CuDvE ] I = 2[ CuvE ][ D ] + 2[ CvE ][ uD ] − [ CDuvE ] . (4.6) Step 4 . Consider I -reductions on the leader of f = [ B ] − ( − a † [ B ] involving both the tail part of A and the head part of B , where the leading variable of B isassumed to be higher than the trailing variable.As is lower than any element of A , B , the only possiblereduction is by G3. Let = , where C maybe empty but D is not. Assume a ≻ b ≻ t D , where t D is the trailing variable of D . Let the length of D be d . Itis easy to prove that applying G3 to a2b in vector-variablebinomial [ bD ] is equivalent to the following absorptionof bracket : [ bD ] I = [ bD ] a2 = 2 − ( ) + ( − d − ( † ba2 ) . (4.7)Each term in the result is a leading term of an R -typedelement lower than f . Step 5 . In Step 3, we have seen that a single bracket after I -reduction, may be split into two brackets. The split cancontinue and we gradually get expressions of the formR [ j ] : [ F ][ F ] · · · [ F j ] − ( − e † [ F ][ F ] · · · [ F j ] , (4.8)where the length of E is e >
0, and the length of F · · · F j is m . R [ j ] is a K -linear combination of elements oftype R [ ∗ ] if all but one bracket are each expanded into twoterms.Consider I -reductions of R [ j ] involving more than twobracket factors, and I -reductions involving and morethan one bracket factor. Since is lower than all elements of E and the F i , G3 is the only possible Gr¨obner base elementthat may apply to and its neighbors on both sides simul-taneously. G3 can involve only [ F ] among the brackets.In [ F ][ F ] · · · [ F j ], only G i where i > i is of the form uDv where D is ascending, if the product of the leaders of threebrackets is non-reduced with respect to some G i , then themiddle bracket must be composed of a subsequence of D of length ≥
2, contradicting with the assumption that theleading variable in the middle bracket be higher than thetrailing variable.o each I -reduction of R [ j ] by a single G i where i ≥ and onebracket factor. Step 6 . Consider I -reductions on [ F ][ F ], where the leadingvariable in each bracket is higher than the trailing variable.If the leading variable l F of F is higher than the leadingvariable l F of F , then an I -reduction commuting the twobrackets reduces the order of their product. Below we alwaysassume l F ≺ l F .For G3, there are two possibilities to involve both F , F in the leader F F : two variables at the end of F and thethird at the beginning of F , or one variable at the end of F and the other two at the beginning of F . The latter isimpossible because l F ≺ l F . For G i where i >
3, thereare also two possibilities: two variables at the end of F andthe rest at the beginning of F , or one variable at the endof F and the rest at the beginning of F . The latter is alsoimpossible due to l F ≺ l F .Case G3. Let [ F ][ F ] = [ Buv ][ wCd ], where u ≻ v , and u ≻ w ≻ d . Let the length of C be c ≥
0, and let theleading variable of Bu be l . Then w ≻ l ≻ v . ApplyingG3 to uvw is equivalent to the following absorption of thesecond bracket :[ Buv ][ wCd ] I = [ B [ wCd ] uv ]= 2 − [ BwCduv ] + ( − c − [ BdC † wuv ] . (4.9)The leader of each bracket monomial in the result has lowerorder than the leader of [ F ][ F ].Case G i . Let [ F ][ F ] = [ Buv ][ aDwC ], where aD is as-cending, and a ≻ u ≻ v ≻ w . Let the lengths of B , C , D be b, c, d respectively. Let the leading variable of Bu be l , andlet the trailing variable of wC be t . Then a ≻ l ≻ v and a ≻ t . Applying G( d + 4) to uvaDw , we get4 [ Buv ][ aDwC ]= BuvaDwC + ( − c + d BuvC † wD † a +( − b vuB † aDwC + ( − b + c + d vuB † C † wD † a I = ( vaDw − ( − d wD † av ) BuC + ( − d BuwD † avC +( − d (( − c vC † w + wCv ) BuD † a − ( − d BuwCvD † a +( − b v ( aDwC + ( − c + d C † wD † a ) uB † = v { aDw ( BuC − ( − b + c C † uB † )+ ( − b aDw ( C + ( − c C † ) uB † + ( − c + d C † w ( BuD † a + ( − b + d aDuB † ) − ( − b + c C † w ( aD − ( − d D † a ) uB † } +( − d w {− D † av ( BuC − ( − b + c C † uB † ) − ( − b + c D † avC † uB † − ( − b + d CvaDuB † + Cv ( BuD † a + ( − b + d aDuB † ) } +( − d Buw ( D † avC − CvD † a ) I = 4([ vaDw ][ BuC ] + ( − d [ wCv ][ BuD † a ])+( − b v ( aDC + ( − c + d D † aC † ) wuB † − ( − b w ( CvaD + ( − c + d D † avC † ) uB † +( − d Buw ( D † avC − CvD † a ) I = 4([ vaDw ][ BuC ] + ( − d [ wCv ][ BuD † a ])+( − b { v ( aD [ C ] − ( − c [ aD ] C † ) wuB † }− Buw ][ CvaD ] +
Buw { C ( aD − ( − d D † a )+( − d D † a ( C + ( − c C † ) } v I = 4 { [ vaDw ][ BuC ] + ( − d [ wCv ][ BuD † a ] − [ Buw ][ CvaD ] + [
Buw ( C [ aD ] + ( − d D † a [ C ]) v ] } I = 4 { [ vaDw ][ BuC ] + ( − d [ wCv ][ BuD † a ] − [ Buw ][ CvaD ] + [
BuwCv ][ aD ]+ ( − d [ BuwD † av ][ C ] } . (4.10)The leader of each bracket monomial in the result has lowerorder than the leader of [ F ][ F ]. Step 7 . Consider a bracket of the form h = [ a B c a B c · · · a k B k c k ], where (1) k >
1, (2) a i ≻ c i for every i , (3) a ≻ c j for all 1 ≤ j ≤ k . Let the length of B i be b i .When k = 2,[ a B c a B c ] − a B c ][ a B c ]= 2 − { ( − b + b c B † a c B † a − ( − b a B c c B † a − ( − b c B † a a B c − ( − b + b c B † a c B † a } I = − ( − b [ c B † a a B c ] . The leader in the result has lower order than h .For k > a B c · · · a k B k c k ] − a B c · · · a k − B k − c k − ][ a k B k c k ] I = − ( − b k [ c k B † k a k ( a B c · · · a k − B k − c k − )] , and for [ a B c · · · a k − B k − c k − ], the split into [ a B c · · · a k − B k − c k − ][ a k − B k − c k − ] can continue. In the end,we get[ a B c · · · a k B k c k ] I = 2 k − [ a B c ] · · · [ a k B k c k ] + g, (4.11)where g is a bracket polynomial whose leader has lower orderthan h .(4.11) can be used to split a long bracket whose representa-tive is I -normal. It can also be used in the converse direc-tion, to concatenate short brackets into a long one. Step 8 . So far we have proved that for any R -typed element [ E2 ] or any R [ j ]-typed element [ A2 ][ F ][ F ] · · · [ F j ], aslong as the leader is not I -normal, I -reductions can alwaysbe carried out to change or [ F ][ F ] · · · [ F j ] intothe following form: T = X α λ α α + X β µ β β [ F β ][ F β ] · · · [ F β j ]+ X γ τ γ [ D γ ][ D γ ] · · · [ D γ k ] , (4.12)where the leading variable in each bracket is higher than thetrailing variable, the α and β β F β · · · F β j are all I -normal.Since any I -normal form is of type Y z . . . Y k z k or Y z . . . Y k z k Y k +1 , it must be thati) α = Y z , i.e. , E α = · · · m .(ii) β β F β · · · F β j = Y z . . . Y j z j , and β = Y z , F β = Y z ,. . . . . . F β j = Y j z j . (i) is obvious. In (ii), the trailing variable of each bracketmust be some z i . If an F β i is Y h z h . . . Y h + p z h + p for some p >
0, for h ≤ s ≤ h + p , let the leading variable of Y s be l s , then l h ≻ z h + p ≻ z h + p − ≻ . . . ≻ z h . By (4.11), [ F β i ] issplit into 2 p [ Y h z h ] . . . [ Y h + p z h + p ] plus some bracket mono-mials of lower leader. Then I -reductions continue to theterms involving such bracket monomials. Ultimately eachbracket is of the form [ Y i z i ].In (4.12), α = ( − m ( E α ) † by R N [1], β [ F β ] · · · [ F β j ] = ( − m ( A β [ F β ] · · · [ F β j ]) † by R N [ j ], and [ D γ ] · · · [ D γ k ] = ( − m ( [ D γ ] · · · [ D γ k ]) † by I [ M ]. So T − ( − m T † is reduced to zero by I [ M ] and R N . ✷
5. SQUARE-FREE VECTOR-VARIABLEPOLYNOMIAL RING
When M is a general multiset of vector variables, a square in Q [ M ] refers to the product of a vector with itself. Denote v i := v i v i . It commutes with everything in Q [ M ]. Proposition In Q [ M ] , let f = g v i h be a multiplier of v i . If f is not I -normal, then by doing I -reduction to g, h ,together with rearranging the position of v i in each term, f can become I -normal. Proof.
Suppose g, h are I -normal. There are three casesfor f to be non-reduced with respect to the Gr¨obner base G [ M ]:(1) If f contains as a factor the leader of G k for k >
3, orEG j for j > v i , then v i is preserved bythe reduction with the Gr¨obner base element.(2) If g v i is non-reduced, then switch the element of G [ M ]with respect to which g v i is non-reduced:Case EG2: Let g v i = Auv i v i or Auuv i , where u ≻ v i .Then Auv i v i h = Av i uv i h or Auuv i h = Av i uu h .Case G3: Let g v i = Auwv i where u ≻ w and u ≻ v i . Then Auwv i h = Av i uw h .Case G k for k > j for j >
2: Let g v i = CuwDv i where u ≻ w ≻ v i . Then CuwDv i h = Cv i uwD h .(3) If v i h is non-reduced, then switch the element of G [ M ]with respect to which v i h is non-reduced:Case EG2: Let v i h = v i v i zB or v i zzB , where v i ≻ z .Then g v i v i zB = g v i zv i B or g v i zzB = g zzv i B .Case G3: Let v i h = v i yzB where v i ≻ y and v i ≻ z . Then g v i yzB = g yzv i B .Case G k for k > j for j >
2: Let v i h = v i yCzD where v i ≻ y ≻ z . Then g v i yCzD = g yv i CzD . In all the cases, the order of f is decreased while preserving v i . By induction on the order we get the conclusion. ✷ Proposition 8 suggests a“square-free normalization” of vector-variable polynomials, by moving all squares to a set free ofany reduction operation, and maintaining the set of squaresin a normal form.In a vector-variable monomial, let the set of squares be sep-arated from the remainder of the monomial by a symbol“ (cid:3) ”, such that all elements on the right side of the symbolare squares. Two things need to be established before sucha symbol can be used in algebraic manipulations: (1) alge-braic structure of the new symbolic system, (2) connectionwith the canonical system based on V2, V3, V4.Let S be a commutative monoid. All elements in S span a K -vector space whose dimension equals the number of elementsin S . The product in the vector space is the multilinearextension of the product in S . The vector space equippedwith this product forms a commutative K -algebra, called the K -algebra extension of monoid S , denoted by K S .For a K -algebra A , when S is a subset of the center of A ,then A is not only a module over the ring K S , but a multi-linear algebra over K S , called a K S -algebra .For K -tensor algebra N [ M ], let N (cid:3) [ M ] := N [ M ] / h V2 i . (5.1)It is easy to see that when setting S to be generated byelements of the form v i := v i ⊗ v i , for all v i ∈ M , then N (cid:3) [ M ] is a K S -algebra, called the K S -tensor algebra overmultiset M , or the square-free tensor algebra over M . Theproduct in N (cid:3) [ M ] is induced from the tensor product. Forbrevity we still denote the product by “ ⊗ ”, but denote thecommutative product in S by juxtaposition of elements.In N (cid:3) [ M ], for all q ∈ N [ M ] and s ∈ S , we introduce thenotations q (cid:3) s := q ⊗ s ∈ N [ M ] ,q (cid:3) := q ∈ N [ M ] , (cid:3) s := s ∈ S . (5.2)Then ( q (cid:3) s ) ⊗ ( q (cid:3) s ) := q ⊗ q (cid:3) s s . (5.3)Formally, an element q ∈ Q [ M ] is taken as q (cid:3)
1, and anelement s ∈ K S is taken as 1 (cid:3) s . In other words, factor (cid:3) (cid:3) ) in q (cid:3) q (cid:3) ) is usually omitted. So q (cid:3) s = q ⊗ ( (cid:3) s ) = ( (cid:3) s ) ⊗ q, (5.4)and (cid:3) st = ( (cid:3) s ) ⊗ ( (cid:3) t ) = ( (cid:3) t ) ⊗ ( (cid:3) s ) . (5.5)That S is generated by squares can be succinctly expressedby the following identity: v i ⊗ v i = (cid:3) v i . (5.6)making left multiplication with f and right multiplicationwith g on both sides of the identity, we get f ⊗ ( v i ⊗ v i ) ⊗ g = ⊗ g (cid:3) v i . It includes V2 as a special case.The degree , or length , of a monomial in N (cid:3) [ M ] is the degreeof the monomial when taken as an element in N [ M ]. The left degree or left length of a monomial refers to the degreeof the monomial on the left side of the square symbol. Fora monomial f ∈ N (cid:3) [ M ], its canonical form in N [ M ] isdefined to be the monomial of lowest lexicographic orderamong all monomials equal to f modulo V2. The order of f is that of its canonical form. This ordering is still calledthe lexicographic ordering .The canonical form of f = v i ⊗ v i ⊗ · · · ⊗ v i k (cid:3) v r j v r j · · · v r l j l , where v j ≺ v j ≺ . . . ≺ v j l , can be obtained asfollows:1. Set g = v i ⊗ v i ⊗ · · · ⊗ v i k .2. For p from 1 to l , let v i t be the first variable in thesequence of g such that v i t ≻ v j p (cid:23) v i t − . Insert v j p ⊗ v j p ⊗ · · · ⊗ v j p | {z } r p to the position before v i t in g , and update g .3. Output g .The vector-variable polynomial ring Q [ M ] when taken asthe quotient of N (cid:3) [ M ] modulo the two-sided ideal I (cid:3) [ M ]generated by V3, V4, is a K S -algebra, called the square-free polynomial ring , denoted by Q (cid:3) [ M ]. The product in Q (cid:3) [ M ] is still denoted by juxtaposition of elements. I (cid:3) [ M ]is called the syzygy ideal of Q (cid:3) [ M ].Theorem 3 has the following square-free version for multiset M : Theorem Let M be a multiset of m symbols, amongwhich n are different ones: v ≺ v ≺ . . . ≺ v n , and let I (cid:3) [ M ] be the syzygy ideal of the square-free polynomial ring Q (cid:3) [ M ] .(1) [Gr¨obner base] The following are a reduced Gr¨obner baseof I (cid:3) [ M ] : G3 , and G j : for all < j < n + 1 , and i < i < . . . < i j , [ v i v i v i v i · · · v i j v i ] − [ v i v i v i · · · v i j v i v i ] , EG k : for all ≤ k ≤ n + 1 , and i < i < . . . < i k , [ v i v i v i v i · · · v i k v i ] − [ v i v i v i · · · v i k v i v i ] . The above Gr¨obner base is denoted by G (cid:3) [ M ] .(2) [Normal form] In a normal form, every term is up tocoefficient of the form V Y v z V Y v z · · · V Y k v z k (cid:3) s or V Y v z · · · V Y k v z k V Y k +1 (cid:3) s , where(i) k ≥ ,(ii) v z v z · · · v z k is non-descending,(iii) every V Y i is an ascending monomial of length > ,(iv) V Y V Y · · · V Y k (or V Y · · · V Y k V Y k +1 if V Y k +1 occurs)is non-descending,(v) for every i ≤ k , let v t i be the trailing variable of V Y i , then v t i ≻ v z i ,(vi) s is either 1 or the product of several squares. In Q (cid:3) [ M ], a polynomial is said to be I (cid:3) -normal if its leadingterm is reduced with respect to the Gr¨obner base G (cid:3) [ M ].A monic square-free uni-bracket monomial is of the form[ A ] (cid:3) s , where A is either 1 or a monomial of length > s is a product of squares, and the length of A s is m . A square-free uni-bracket polynomial is a K -linear combinationof square-free uni-bracket monomial. The space of square-free uni-bracket polynomials is denoted by [ Q ] (cid:3) [ M ].The K -linear space of degree- m square-free polynomials isdenoted by Q (cid:3) m [ M ]. The space [ Q ] (cid:3) [ M ] can be taken asthe quotient of Q (cid:3) m [ M ] modulo the ideal J (cid:3) [ M ] := I (cid:3) [ M ] + [ I ] (cid:3) [ M ] , (5.7)where [ I ] (cid:3) [ M ] is composed of the vector parts of degree- m square-free polynomials, i.e. , the K -linear span of elementsof the form R (cid:3) : A (cid:3) s − ( − a A † (cid:3) s, (5.8)where A is a monomial of length a > s is a product of squares, and the length of A s is m .The modulo-[ I ] (cid:3) [ M ] operation identifies a square-free uni-bracket with its representative. It removes the outer bracketsymbol on the left side of the “ (cid:3) ” symbol from every square-free uni-bracket, disregarding the length of the bracket. Ideal[ I ] (cid:3) [ M ] is called the uni-bracket removal ideal in Q (cid:3) [ M ],and ideal is called the syzygy ideal of [ Q ] (cid:3) [ M ] in N (cid:3) [ M ].
6. GRÖBNER BASE AND NORMAL FORMFOR UNI-BRACKET POLYNOMIALS
In this section, we extend Theorem 7 to the case of generalmultiset M with m ≥ I ][ M ] operation is always assumed, i.e. , [ Q ] (cid:3) [ M ]and Q (cid:3) m [ M ] are identical, and a uni-bracket does not havethe outer bracket symbol on the left side of the “ (cid:3) ” symbol.The following are elements of J (cid:3) [ M ]:R (cid:3) ( k ) : ( K − ( − k K † ) (cid:3) s, S ( k ) : ( iAb B − b BiA ) (cid:3) s, for i = b , S ( k ) : ( iCb − b iC ) (cid:3) s, for i = b , S ( k ) : ( b Ab B − b Bb A ) (cid:3) s, Sq ( k ) : ( b Cb − C (cid:3) b ) (cid:3) s, Sq [ ∗ ]( k ) : ( b Eb − E (cid:3) b )[ F ] (cid:3) s, R ( k ) : b [ D ] (cid:3) s, R ( k ) : b [ Cb ] (cid:3) s, R [ ∗ ]( k ) : b [ Eb ][ F ] (cid:3) s, R ( k ) : b [ Cb ] (cid:3) s, R [ ∗ ]( k ) : b [ Eb ][ F ] (cid:3) s, (6.1)where(a) s has length m − k , and the left length of each expressionis k > b , b are respectively the variables of the lowest orderand the second lowest order on the left side of the squaresymbol;c) in S , either A or B can be empty, while in S , both A and B are non-empty;(d) in R , D is either empty ( i.e. , D = 1), or of length > and R , C is non-empty;(f) in Sq [ ∗ ]( k ) and R [ ∗ ], E , F are non-empty, and F doesnot contain b ;(g) in R , C is non-empty and does not contain b ;(h) in R [ ∗ ], E , F are non-empty and do not contain b ,and F does not contain b .In bracket [ ], we have [ ] I (cid:3) = [ A ] (cid:3) . That the lead-ing variable has higher order than the trailing variable isalways possible. This is taken as a postulate for all thebrackets in (6.1).Consider a general element f = ( K − ( − k K † ) (cid:3) s of typeR (cid:3) ( k ):1. If b occurs in K both as the leading variable and trailingvariable, then f ∈ h Sq ( k ) , R (cid:3) ( k − i .2. If b occurs in K at only one end, then f ∈ h R ( k ) , S ( k ) i .3. If b occurs at the interior of K , set K = Ab B , where A , B are both non-empty, and b does not occur at any endof A or B . By (4.2), f ∈ h R ( k ) , S ( k ) i .By induction on k , we get Lemma (cid:3) ( k ) ⊆ X h ≤ k ( h S ( h ) i + h Sq ( h ) i + h R ( h ) i ) . In R ( k ), when D contains b , let D = Ab B , where thelengths of A , B are respectively a, k − a −
2, then b [ Ab B ] (cid:3) s I = b [ BAb ] (cid:3) s = ( b BAb − ( − k A † B † ) (cid:3) b s = ( b BAb − BA (cid:3) b ) (cid:3) s +( BA − ( − k A † B † ) (cid:3) b s ∈ h Sq ( k ) , R (cid:3) ( k − i . (6.2)By induction on k , we get that both b [ Ab B ] (cid:3) s and b [ BAb ] (cid:3) s ∈ R ( k ) are equivalent to Sq ( k ): ( b BAb − BA (cid:3) b ) (cid:3) s in the sense that their difference is in the ideal P h ≤ k − ( h S ( h ) i + h R ( h ) i + h R ( h ) i ) + h R (1) i + I (cid:3) [ M ]. Lemma
11. R (cid:3) ( k ) is a subset of the ideal X h ≤ k ( h S ( h ) i + h R ( h ) i + h R ( h ) i ) + h R (1) i + I (cid:3) [ M ] . Theorem
Let M be a multiset of m > symbols,among which n ≥ are different ones: ≺ ≺ . . . ≺ n ,and let Q (cid:3) [ M ] be the square-free vector-variable polynomialring over M . Let [ Q ] (cid:3) [ M ] be the space of square-free uni-bracket polynomials, and let J (cid:3) [ M ] be its syzygy ideal in N (cid:3) [ M ] .(1) [Gr¨obner base] The following are a reduced Gr¨obner baseof J (cid:3) [ M ] , denoted by BG [ M ] : G (cid:3) [ M ] : G i for all ≤ i ≤ n ; EG j for all ≤ j ≤ n + 1 . S (cid:3) : ( Ab B − b BA ) (cid:3) s , where the length of each term is m , b is the variable of the lowest order on the leftside of the square symbol, A is a non-empty ascendingsequence not containing b , and Ab B is I (cid:3) -normal; R (cid:3) [ j, l ] : b [ Y b ][ Y b ] · · · [ Y j b ][ Y j +1 z j +1 ][ Y j +2 z j +2 ] · · · [ Y j + l z j + l ] (cid:3) s, (6.3) where (i) j, l ≥ , and the length of each term is m ,(ii) b is the variable of the lowest order on the left side ofthe square symbol,(iii) each Y i is a non-empty ascending sequence not con-taining b ,(iv) for all j + 1 ≤ i ≤ j + l , z i = b , and z i Y i is ascending;(v) b Y b · · · Y j b Y j +1 z j +1 · · · Y j + l z j + l is I (cid:3) -normal.(2) [Normal form] In a normal form, every term is I (cid:3) -normal, and is up to coefficient of one of the following forms,where statements (ii), (iii) on b and the Y i are still valid:(I) b Y b · · · Y j b Y j +1 z j +1 · · · Y j + l z j + l Y j + l +1 (cid:3) s , where j, l ≥ , and for all j + 1 ≤ i ≤ j + l , z i ≺ t i , the latter beingthe trailing variable of Y i ;(II) b Y b · · · Y j b Y j +1 z j +1 · · · Y j + l z j + l (cid:3) s , where j, k ≥ but j + k > , each z i ≺ t i , but for some ≤ h ≤ l , if l j + h is the leading variable of Y j + h , then l j + h (cid:22) z j + h .Remark . The set of R (cid:3) can be replaced by the followingthree sets of degree- m polynomials: for all j > l ≥ (cid:3) [0 ,
0] : b (cid:3) s, Sq (cid:3) [ j, l ] : ( b Y b − Y (cid:3) b )[ Y b ] · · · [ Y j b ][ Y j +1 z j +1 ] · · · [ Y j + l z j + l ] (cid:3) s, R (cid:3) [ j, l ] : b [ Y b ][ Y b ] · · · [ Y j b ][ Y j +1 z j +1 ][ Y j +2 z j +2 ] · · · [ Y j + l z j + l ] (cid:3) s. (6.4)In R (cid:3) [ j, l ], Y j +1 , . . . , Y j + l do not contain b , and the z i ≻ b . The replacement has no effect upon the normal forms. Proof.
There are several steps.
Step 1 . We need to prove by induction on k that S ( k ),R ( k ) , R ( k ) are all in the ideal X left length ≤ k h S (cid:3) , R (cid:3) [ ∗ ] i + I (cid:3) [ M ] , (6.5)where the asterisk stands for the ( j, l ).Once this is done, then since the leader of any element oftype S (cid:3) or R (cid:3) [ ∗ ] is I (cid:3) -normal and cannot be cancelled bythe leader of any other element of type S (cid:3) or R (cid:3) [ ∗ ], theS (cid:3) , R (cid:3) [ ∗ ] and G (cid:3) [ M ] must be a reduced Gr¨obner base of h S , R , R , R (1) i + I (cid:3) [ M ]. By Lemma 11, this ideal isjust J (cid:3) [ M ]. This proves conclusion (1), and conclusion (2)follows. tep 2 . S ( k ) requires k >
1; R ( k ) and R ( k ) both require k >
2. When k = 2, S (2) = ( b b − b b ) (cid:3) s is in S (cid:3) .When k = 3, R (3) = ( b b b − b (cid:3) b ) (cid:3) s = R (cid:3) [1 , (3) = b [ b b ] (cid:3) s = R (cid:3) [0 , b ≻ b .Consider S (3). There are 3 elements led by variable b :( b b b − b (cid:3) b ) (cid:3) s , ( b b b − b b b ) (cid:3) s , and ( b b b − b b b ) (cid:3) s . They all belong to S (cid:3) . There are two otherelements in S (3): ( b b b − b b b ) (cid:3) s and ( b b b − b b b ) (cid:3) s . By b b b − b b b = b b b − b b b ; b b b − b b b = ( b b b − b b b )+( b b b − b b b ) , both are in h S (cid:3) i + I (cid:3) [ M ].So the statement in Step 1 holds for k ≤
3. Assume thatit holds for all k < h . When k = h , we need to make I (cid:3) -reduction to the leaders of the elements of any of the typesS ( h ) , R ( h ) , R ( h ) , R [ ∗ ]( h ) , R [ ∗ ]( h ) , (6.6)at the same time keep the reduction result to be within the K -linear space spanned by elements of the types listed in(6.6) but where the left length h is replaced by all i ≤ h .Then ultimately all the leaders of these elements become I (cid:3) -normal. Step 3 . Consider types R ( h ) , R [ ∗ ]( h ) , R ( h ) , R [ ∗ ]( h ).Let there be an R ( h )-typed element f = 2( b [ Ab ] (cid:3) s ),and an R ( h )-typed element g = 2( b [ Bb ] (cid:3) s ), where B does not contain b . In the following we omit the factor (cid:3) s .Do I (cid:3) -reductions to A , B , and assume that the results are A I (cid:3) = C A b + b D A + b E A b + A N , B I (cid:3) = C B b + b D B + b E B b + B N , (6.7)where (i) none of the terms in C A , D A , E A , A N has b atany end;(ii) none of the terms in C B , D B , E B , B N has b at any end;(iii) any of the four terms in each result may not occur;(iv) the component on the right side of the square symbolin each term, together with the symbol itself, are omitted,as they do not affect the analysis below;(v) in the extreme case, A N or B N may be in K , if allvector variables in the term form squares and are moved tothe right side of the square symbol.Substituting the reduction results into f, g , we get f I (cid:3) = ( D A b − ( − h D † A b ) (cid:3) b ∈ h S ( h − , R ( h − i +( b C A − ( − h b C † A ) (cid:3) b ∈ R ( h − E A (cid:3) b − ( − h b E † A b ) (cid:3) b ∈ R ( h − b [ A N b ]) , ∈ R ( h ) g I (cid:3) = 2( b [ C B ]) (cid:3) b ∈ R ( h − b [ D B ]) (cid:3) b ∈ R ( h − b [ E B b ]) (cid:3) b ∈ R ( h − b [ B N b ]) . ∈ R ( h )Notice that the left lengths indicated on the right column arethe maximal possible ones for the corresponding types. So by induction hypothesis, we can assume that in f, g , mono-mials A = A N , B = B N and both are I (cid:3) -normal.The I (cid:3) -reduction to the leaders of f, g are much the samewith the procedure in the proof of Theorem 7 starting fromStep 3 there to Step 8, with negligible revisions. Formula(4.11) can also be used to split the leader of factor b [ Y b · · · Y k b ] in a type-R [ ∗ ] element, and the leader of factor b [ Y b · · · Y k b ] in a type-R [ ∗ ] element.By induction on the order of the leader, we get that R ( h ),R ( h ) , Sq ( h ) , Sq [ ∗ ]( h ) , R [ ∗ ]( h ), R [ ∗ ]( h ) are all in (6.5)where k = h . Step 4 . Consider a general type-S ( h ) element f = ( Ab − b A ) (cid:3) s . Let the I (cid:3) -reduction result of A be as in (6.7).Then if omitting “ (cid:3) s ”, f I (cid:3) = C A (cid:3) b − b C A b ∈ Sq ( h )+ b D A b − D A (cid:3) b ∈ Sq ( h )+( b E A − E A b ) (cid:3) b ∈ S ( h − A N b − b A N . ∈ S ( h )So we can assume that in f = ( Ab − b A ) (cid:3) s , monomial A = A N and is I (cid:3) -normal.The I (cid:3) -reduction to the leading term of f is much the samewith the procedure in the proof of Lemma 5 starting fromCase (i) there to Case (ii). By induction on the order of theleading term, we get that S ( h ) is in (6.5) where k = h . Step 5 . Consider a general type-S ( h ) element g = ( Ab B − b BA ) (cid:3) s , where A , B are both non-empty. Let the I (cid:3) -reduction results of A , B be as in (6.7), where every b isreplaced by b . Then if omitting “ (cid:3) s ”, g I (cid:3) = C A C B b (cid:3) b − b C B b C A b +( C A b D B − D B C A b ) (cid:3) b +( C A b E B b − E B b C A b ) (cid:3) b + C A B N (cid:3) b − b B N C A b + b D A b C B b − b C B D A (cid:3) b +( b D A D B − D B b D A ) (cid:3) b + b D A E B b (cid:3) b − E B D A (cid:3) b + b D A b B N − b B N b D A +( b E A C B b − b C B E A b ) (cid:3) b +( b E A b D B − D B b E A b ) (cid:3) b + b E A b E B b (cid:3) b − E B E A b (cid:3) b + b E A B N b − b B N b E A b + A N b C B b − b C B b A N +( A N D B − D B A N ) (cid:3) b + A N b E B b − b E B b A N + A N b B N − b B N A N . In the above result, the lines that do not belong to the ideal h Sq ( h ) , S ( h − , Sq ( h − , S ( h − i are b D A b B N − b B N b D A ∈ h S ( h ) , Sq ( h ) i + A N b C B b − b C B b A N ∈ h S ( h ) , S ( h ) i + A N b E B b − b E B b A N ∈ h S ( h ) , S ( h ) i + A N b B N − b B N A N . ∈ S ( h ) (6.8)Some remarks on (6.8) are necessary. The first line of (6.8), ifonzero, is S ( h ) when B N / ∈ K , and Sq ( h ) otherwise. By A N b C B b − b C B b A N = ( A N b C B b − b A N b C B )+( b A N b C B − b C B b A N ), the second line of (6.8) is a K -linear combination of an element of type S ( h ) and anotherelement of type S ( h ).Consider a general type-S ( h ) element p = ( b Ab B − b Bb A ) (cid:3) s , where the length of A is a >
0. When omitting (cid:3) s , p I (cid:3) = b B ( b A − ( − a A † b )+( − a A † B (cid:3) b − b Bb A = ( − a ( A † B (cid:3) b − b BA † b ) Sq ( h ) = ( − a ( A † B − BA † ) (cid:3) b ∈ h S ( h − i . So S ( h ) is in (6.5) where k = h .By (6.8), we can assume that in g = ( Ab B − b BA ) (cid:3) s ,monomials A = A N , B = B N and both are I (cid:3) -normal. The I (cid:3) -reduction to the leading term of g is much the same withthat in the proof of Lemma 5 starting from Case (i) thereto Case (iii). By induction on the order of the leading term,we get that S ( h ) is in (6.5) where k = h . ✷ Consider a bracket polynomial f whose multiset of variablesis M . In any term of f , when all the brackets but one areexpanded into two terms by definition, f is changed into auni-bracket polynomial g . Using the Gr¨obner base B G [ M ]to make reduction to g results in a uni-bracket polynomial h ,where each term is I (cid:3) -normal. h must have the lowest orderlexicographically among all uni-bracket polynomials equal to f . It is called the lowest-representative normal form of g , orthe uni-bracket normal form of f . Remark . In the above definition of normal forms, we onlyconsidered square-free ones. Of course any normal form canbe converted to a canonical one, where all representativesare I -normal instead of I (cid:3) -normal. Later on, we consideronly square-free ones.Let the size of M be m . Given any partition ( i , . . . , i k )of integer m , where each i j >
1, there is a
Caianiello ex-pansion [6] of uni-bracket polynomials into bracket polyno-mials where each term is composed of k brackets of length i , . . . , i k respectively. Each expansion produces a normalform. It is not clear if such a normal form is of any value.
7. NORMALIZATION OF BRACKETPOLYNOMIALS
In this section, we do NOT remove bracket symbols fromuni-brackets in multiset of variables M .Consider attaching an additional vector variable v / ∈ M to M to form a bigger multiset ˜ M . Let v ≺ all variables in M . In the procedure of obtaining R (cid:3) by the I (cid:3) -reduction of b [ A ] in the proof of Theorem 12, or in more details, in theproof of Theorem 7, if the input is v [ A ] where A ∈ Q m [ M ],then among the Gr¨obner base of I (cid:3) [ ˜ M ], only those elementsin G (cid:3) [ M ] are needed in I (cid:3) -reduction. The reduction of v [ A ] is a procedure of recursively do-ing I (cid:3) -reductions to the leader of the bracket polynomialobtained from previous I (cid:3) -reductions to [ A ]. At any in-stance, the representative of a bracket monomial in reduc-tion is the leader of the bracket monomial. The reductionresults in a K -linear combination of monomials of the form v [ Y z ] · · · [ Y k z k ], where z i Y i is ascending for every i . Af-ter removing v from the result, we get another normal formof uni-bracket [ A ] in M . Theorem
Let f be a square-free bracket polynomialin multiset of variables M . Do the following to f :1. Always select the leader of a bracket as its representative.2. Rearrange the order of the brackets in the same term, sothat the leading variables of the brackets are non-descending.3. Use (4.6) to normalize the interior of a bracket; it alsosplits a bracket into two.4. Use (4.9) to absorb a bracket into the one ahead of it.5. Use (4.10) to decrease the order of the product of twobrackets by lifting a lower-order variable from the secondbracket to the first.6. Use (4.11) to segment a long bracket of type Y z Y z · · · Y k z k into short ones.7. Once the representative of the leading term of f is I (cid:3) -normal, output the leading term, and continue the above I (cid:3) -normalization to the remainder of f .The output, called the lowest-leader normal form, or leader-normal form is a bracket polynomial where the representativeof each term is up to coefficient of the form [ Y z ][ Y z ] · · · [ Y k z k ] (cid:3) s , where each z i Y i is ascending, Y Y · · · Y k and z z · · · z k are both non-descending, and the leading variable l i of each Y i satisfies l i ≻ z i . Two bracket polynomials areequal if and only if their leader-normal forms are identical. Proof.
After operations 1 and 2, for any bracket mono-mial in the reduction procedure, its representative is alsoits leader, so that the representative of the leading term ofbracket polynomial f is the leader of f . Once the leader of f is I (cid:3) -normal, it is the leading term of the normal formof vector-variable polynomial f with respect to the Gr¨ob-ner base G (cid:3) [ M ]. By induction on the order of the outputterms from the highest down, we get the uniqueness of theleader-normal form for f .For two equal bracket polynomials, they have identical uni-bracket normal forms, and so have identical leader-normalforms. ✷ From the above proof, we see that the leader-normal formof a bracket polynomial f has the following properties: (1)the representative of any term is the leader of the term; (2)the representative of the leading term is the leading term ofthe I (cid:3) -normal form of vector-variable polynomial f .By Theorem 12, the Gr¨obner base B G [ ˜ M ] of J (cid:3) [ ˜ M ] is com-posed of G (cid:3) [ M ] and the R (cid:3) [ ∗ ] : v g (cid:3) s for all monomials g in the leader-normal forms of bracket polynomials in M ,such that gs has length m . This phenomenon is easy tounderstand: the K -linear subspace of [ I ] (cid:3) [ ˜ M ] composed ofpolynomials whose terms are led by variable v , is the spacef degree-( m + 1) polynomials of the form v f (cid:3) s , for allbracket polynomials f in M such that fs has length m .The leader-normal forms are a basis of the K -linear space oflength- m square-free bracket polynomials in M .In a leader-normal form, if we commute Y i and z i in [ Y i z i ],we get another normal form whose terms are up to coefficientof the form [ z Y ] · · · [ z k Y k ] (cid:3) s . If we write the left sideof the square symbol in the following tableau form, where Y i = y i y i · · · y it i , we get z y . . . . . . y t z y . . . y t ... ... . . . z k y k . . . . . . . . . y kt k , where (1) each row does not need to have equal length, andit is not required that the length be non-increasing as inYoung tableau;(2) each row is an ascending sequence of variables;(3) each column is a non-descending sequence of variables;(4) y it i (cid:22) y ( i +1)1 for 1 ≤ i < k .Such a normal form is called the straight form . Feature (4)above makes this definition stronger than the straight form (or standard form ) of Young tableau. In comparison, inclassical bracket algebra a bracket monomial is in straightform if and only if the entries are ascending along each row,and non-descending along each column.The procedure of deriving the straight form of a bracketpolynomial is called straightening . Among the formulas usedin Theorem 13 for straightening, (4.10) is highly nontrivialand requires further investigation.Set Bu , aD in (4.10) to be new A , B respectively, and letthe lengths of B , C be b, c . Then (4.10) can be writtensuccinctly as follows:[ Av ][ BwC ] = [
AwCv ][ B ] − ( − b [ AwB † v ][ C ] − ( − b [ wCv ][ AB † ] + ( − b [ wB † v ][ AC ] − [ Aw ][ CvB ] . (7.1)It is called the shuffle formula for bracket normalization . Proposition
For any two monomials A , B of length a, b respectively, AB + BA AB ] + ( − b ( A [ B † ] − [ A ] B † ) , AB − ( − a + b A † B † − a ([ A † ] B − A † [ B ]) . (7.2) Proof. AB + BA = 2[ AB ] − ( − a + b B † A † + 2[ B ] A − ( − b B † A = 2([ AB ] + [ B ] A − ( − b B † [ A ]) . ✷ The fundamental I -reduction formula (4.5) is a direct con-sequence of the first identity in (7.2) for AB = uD . The shuffle formula (7.1) is a consequence of the following iden-tity by making left multiplication with A and then applyingthe bracket operator to both sides of the identity: v [ BwC ] + w [ CvB ] = ( − b ( wCv [ B † ] − [ wCv ] B † − wB † v [ C ] + [ wB † v ] C ) . (7.3)The identity can be obtained as follows: by the second iden-tity of (7.2) from right to left, the right side of (7.3) equals2 − ( BwCv + vBwC ) − ( − b + c − ( B † vC † w + wB † vC † ) , which by the first equality of (7.2), equals[ vBwC ] − ( − b + c v [ C † wB † ] − ( − b + c [ wB † vC † ] + w [ CvB ]= v [ BwC ] + w [ CvB ] . The shuffle formula can be further generalized. In monomial[
AvD ][ BwC ], let the leading variable and trailing variableof any sequence F be l F and t F respectively. Assume l A (cid:22) l B , and l A ≻ t D , and l B ≻ t C . Further assume l A ≻ v ≻ w . Then I (cid:3) -reductions can be made to [ AvD ][ BwC ] todecrease its leader, leading to the following result:
Proposition
Let the lengths of monomials A , B , C , D be a, b, c, d respectively. Then [ AvD ][ BwC ] = [ vDBw ][ AC ] − ( − b + c [ vDC † w ][ AB † ] − ( − d [ Aw ][ D † vBC ] − [ vD ][ AwBC ] − ( − b [ AwB † vD ][ C ] + [ AwCvD ][ B ] . (7.4) Proof.
AvD ][ BwC ]= AvDBwC − ( − b + c AvDC † wB † − ( − a + d D † vA † BwC + ( − a + b + c + d D † vA † C † wB †I (cid:3) = ( vDBw + ( − b + d wB † D † v ) AC − ( − b (( − c vDC † w + ( − d wCD † v ) AB † − ( − b + d AwB † D † vC + ( − b + d AwCD † vB † − ( − a + d D † v ( BwC − ( − b + c C † wB † ) A †I (cid:3) = 4([ vDBw ][ AC ] − ( − b + c [ vDC † w ][ AB † ])+( − a + d ( BCD † v − D † vCB ) wA † +( − b + d Aw ( CD † vB † − B † D † vC ) . (7.5)By (7.2), ( − d ( BCD † v − D † vCB )= ( − d BCD † v − ( − b + c vDC † B † +( − b + c ( vD − ( − d D † v ) C † B † +( − c + d { D † v ( − [ C † ] B + C † [ B ]) } , we get from (7.5) the following:4 [ AvD ][ BwC ] I (cid:3) = 4([ vDBw ][ AC ] − ( − b + c [ vDC † w ][ AB † ])+4 [ { ( − d [ BCD † v ] + ( − b + c [ vD ] C † B † − ( − c + d D † vB [ C † ] + ( − c + d D † vC † [ B ] } ( − a wA † ]+( − b Aw {− ( − c vDC † B † + ( − c B † C † vD +( − d CD † vB † − ( − d B † D † vC } (cid:3) = 4([ vDBw ][ AC ] − ( − b + c [ vDC † w ][ AB † ]) − { ( − d [ Aw ][ BCD † v ] + [ AwBC ][ vD ]+( − b [ AwB † vD ][ C ] − [ AwCvD ][ B ] } . ✷ Clearly each term in the result of (7.4) has lower leader thanthe input. To better understand this reduction formula, wewrite it in tableau form: (cid:20)
AvDBwC (cid:21) = (cid:20) AwCvDB (cid:21) + (cid:20) A (( − b +1 Bw ) † vDC (cid:21) + (cid:20) vD (( − b B † ) w (( − a A ) † (( − c C † ) (cid:21) + (cid:20) wC (( − d +1 vD ) † B (( − a A ) † (cid:21) + (cid:20) (( − b +1 Bw ) † (( − d +1 vD ) † AC (cid:21) + (cid:20) Aw (( − b B † ) vD (( − c C † ) (cid:21) . (7.6)As l A ≻ v ≻ w and l A ≻ t D , to decrease the order of theleader, in the first line of (7.6), a subsequence or reversedsubsequence of the second row of the input tableau is movedup between A , v of the first row, with the requirement thatthe subsequence be led by w . In the second line of (7.6), A is moved down behind w of the second row.In the third and fourth lines of (7.6), the leading subsequence A of the first row is commuted with a subsequence or re-versed subsequence of the second row led by w . In the lastline of (7.6), the trailing subsequence vD of the first row iscommuted with variable w of the second row.We make comparison with the shuffle formula for straighten-ing in classical bracket algebra [13]. In the classical bracketalgebra over 3D vector space, where the exterior product isalso denoted by juxtaposition of elements, suppose that (cid:20) a v db w c (cid:21) is not straight: avd is ascending, so is bwc ; ab is non-descending, but v ≻ w . The shuffle formula is obtainedas follows. For any four vectors v , d , b , w of the 3D vectorspace, their exterior product equals zero. By0 = a ∨ ( vdbw ) ∨ c = [ avd ][ bwc ] − [ avb ][ dwc ] + [ avw ][ dbc ]+[ adb ][ vwc ] − [ adw ][ vbc ] + [ abw ][ vdc ] , (7.7)where “ ∨ ” is the dual of the exterior product called the meetproduct [14], we get the following shuffle formula, also called van der Waerden relation : (cid:20) a v db w c (cid:21) = (cid:20) a v bd w c (cid:21) + (cid:20) a v wb d c (cid:21) + (cid:20) a b dv w c (cid:21) + (cid:20) a w db v c (cid:21) − (cid:20) a b wv d c (cid:21) . (7.8)(1) The first line commutes d of the first row and one of thefirst two vectors of the second row;(2) the second line commutes v of the first row and one ofthe first two vectors of the second row;(3) the last line commutes vd of the first row and the firsttwo vectors of the second row.For straightening in classical bracket algebra, there are otheroperations besides (7.8). When a ≻ b , we only need tocommute the two brackets. When d ≻ c while a (cid:22) b and v (cid:22) w , we have another formula for straightening. By0 = ( av ) ∨ ( dbwc )= [ avd ][ bwc ] − [ avb ][ dwc ]+[ avw ][ dbc ] − [ avc ][ dbw ] , we get the following Grassmann-Pl¨ucker relation : (cid:20) a v db w c (cid:21) = (cid:20) a v bd w c (cid:21) + (cid:20) a v wb d c (cid:21) + (cid:20) a v cb w d (cid:21) . The last vector d of the first row commutes in turn withevery vector of the second row.
8. CONCLUSION
In the bottom-up approach to manipulating brackets, longbrackets are expanded into basic ones by Caianiello expan-sion, in the end only brackets of length 2 and 3 are leftfor further algebraic manipulations. This approach provesto be inefficient in practice, despite the fact that there arestraightening algorithms for polynomials of basic invariants[3], [10].Uni-bracket polynomials provide a top-down approach tomanipulating brackets. Given a bracket polynomial, by “un-grading”, each bracket but one in every term is expandedinto a vector-variable binomial, and the bracket polynomialis changed into a uni-bracket one. Algebraic manipulationsof uni-bracket polynomials can take full advantage of theassociativity of the vector-variable product and the symme-tries within a uni-bracket. The Gr¨obner base G (cid:3) [ M ] pro-vided by this paper further fulfills the arsenal of symbolicmanipulations on uni-bracket polynomials.The last section of this paper suggests a third approach tomanipulating brackets by algebraic manipulations directlyupon the input brackets. To establish this approach thereare many research topics ahead: division among bracketpolynomials, properties of principal ideals, bracket polyno-mial factorization, and simplification by reducing the num-ber of terms, etc . This seems to be a promising approach.This paper is supported partially by NSFC 10871195,60821002/F02, and NCMIS of CAS. The Gr¨obner bases for m = 5 , . REFERENCES [1] Altmann, S.L. Rotations, Quaternions, and DoubleGroups . Oxford University Press, Oxford, 1986.[2] Bravi, P. and Brini, A. Remarks on invariant geometriccalculus, Cayley-Grassmann algebras and geometricClifford algebras. In: Crapo, H. and Senato, D. (eds.),
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