Growth Series of the Braid Monoid in Band Generators
Growth series is an important invariant associated with group or monoid which classifies all the words of group or monoid. Therefore, the growth series of braid monoids and Hecke algebras in Artin’s generators is presented in many scholarly published articles. The growth series of braid monoids and in band generators is known. In this work, we compute the complete presentation of braid monoid in band generators by solving all the ambiguities of . The words on the left-hand of each relation are reducible words, and the words on the right-hand side are canonical words. We partially find the growth series of reducible words. Then, we construct a linear system for canonical words of in band presentation and compute the corresponding growth series. We also find the growth rate of growth series of in band generators.
The growth series also known as Hilbert series is an important invariant in the study of modern geometry. In physics, growth series have recently become a power full tool in high energy theory, appearing, for example in the study of Bogomol’nyi-Prasad-Sommerfield operators of supersymmetric gauge theories [1, 2]; supersymmetric quantum chromodynamics [3, 4], and instanton moduli space [5, 6]. In , Hilbert series was used to construct an operator basis in 1/m expansion of a theory with a nonrelativistic heavy fermion in an electromagnetic (NRQED) or color gauge field (NRQCD/HQET).
The braid group admits the presentation given by Artin .
The braid group admits other presentations such as Sergiescu graph-presentation and Birman-Ko-Lee presentation or band presentation. The last presentation is given by
Growth series of braid monoid and is computed in . In , growth series of the finite dimensional Hecke algebra is presented. Growth series for graded S-module was computed by Haider in . The growth series of binomial edge ideals was computed by Kumar and Sarkar in . In , growth series of the graded algebra of real regular functions on the symplectic quotient associated to an -module is computed and given an explicit expression for the first nonzero coefficient of the Laurent expansion of the growth series at Growth series and the coefficient of Laurent expansion of growth series of special linear group of matrices are computed in . In , Saito proved that the growth functions associated with Artin monoids of finite type are rational functions whose nominator is 1 and the denominator is polynomial having distinct roots. Growth series of symplectic quotients by 2-torus is computed in . In , the Laurent coefficients of the growth series of a Gorenstein algebra is presented. In the growth series of (for ) for Artin generators, the degrees of the polynomials are 3, 4, 10, 15, respectively (for detail see ). Universal upper bound for the growth of Artin monoid is computed in . Growth series of braid monoid and in band generators is computed in . The degrees of polynomial in case of band generators are 2 and 3 (for and 4). In this paper, we compute the growth series of braid monoid using band presentation, and we see that the degree of the polynomial is 4. We note that growth rate of in band generators is much slower than that of Artin generators.
2. Materials and Methods
In , we fix a total order on the generators. In the monoid, the relation will be written as in the length-lexicographic order. Let and ; then the word of the form is said to be ambiguity (for detail see ). If as a relation as well as in the length-lexicographic order, then, we say that the ambiguity is solvable. A presentation is complete if and only if all the ambiguities are solvable (for detail see [21, 22]). Corresponding to the relations the changes give a rewriting system. A presentation will be called a complete presentation if and only if all the ambiguities are solvable.
In a complete presentation (or in the general presentation) of if a word contains, then is called a reducible word and we denote it by in general. If word does not contain, then is called canonical word or canonical form. Let and be nonempty words; then, the word will be denoted as .
Definition 1 (see ). Let be a finitely generated group and be a finite set of generators of . Then, the word length of an element is the smallest integer for which there exist such that
Definition 2 (see ). Let be a finitely generated group and be a finite set of generators of . Then, the growth function of the pair associates to an integer the number of the element such that , and the corresponding growth series is given by .
In 2008, Bokut  gave the Grbner-shirshov basis (GSB) of in band generators. The notion of this basis is in [14, 20, 24–27] under different names: complete presentation, presentations with solvable ambiguities, Grbner-shirshov basis, rewriting system, and so on. In , we proved the subset of GSB of given by Bokut  is a GSB of Using the notation (used in ) for generator and or for the words in such that , we have.
Theorem 3 (see ). A GSB of braid monoid consist of following relations: where
Proposition 4 (see ). Solution of linear system for canonical words of braid monoid is given by And the growth series of in band generators is given as
3. Results and Discussion
In this paper, we compute the growth series of (in band presentation). From Equation (2), we have the following band-presentation of : where are given basic relations.
For the braid monoid , we have given another form of Theorem 3 that is directly used to compute the growth series of . This form is obtained by solving all the ambiguities in the band presentation of .
Proposition 5. A complete presentation of for band presentation is given by where the new relations … are given as follows where and are positive integers, a canonical word in starting with and (i): is a canonical word in starting with (ii): and (as mentioned in Theorem 3)
Proof. We denote the ambiguity formed by left sides of the relation and in by - (say). If in the ambiguity and are different lexicographically, then we get a new relation in the complete presentation of , and if and are reduced to an identical word, then we say the ambiguity is solvable and no new relation is formed. The above relations are formed by solving the ambiguities involving basic relations and new relations.
For an ambiguity (say), we have Hence, we have a new relation . Again, by solving new ambiguity (say), we have Hence, we have another relation Now by solving ambiguity (say), we have Hence, we have a relation . By continuing the same process, we have the general relation For an ambiguity (say), we have Hence, we have a new relation . Again, by solving new ambiguity (say), we have which give another relation Now by solving ambiguity (say), we have Hence, . By continuing the same process, we have the general relation For an ambiguity (say), we have where are reduced to identical word, so ambiguity is solvable and no new relation is formed. Using a similar procedure, we obtained all above new relations in complete presentation of Hence the proof is omitted.
As defined above, denotes the set of canonical words and the set of reducible words in in general. In particular, , where denote the set of reducible words starting with and ending on In this notation is a generator in : , denotes the canonical word (possibly empty) in starting with , denotes the canonical word (possibly empty) in starting with and denotes the canonical word (possibly empty) in starting with We will denote the empty word by We denote the set all reducible words starting with and ending on by . Hence, is a subset of
We are using other notions as follows: (i)We denote the set by (ii) denotes the set of canonical words starting with in (iii) denotes the set of all the word in such that the index of each generator is increased by . Hence, , i.e., the cardinality remains unchanged. In particular for the set , we have (iv) denotes the set of canonical words starting with in (v)The growth series of , and is denoted by and , respectively
Note that growth series of is
Proposition 6. The following equalities hold for the reducible words in :
Proof. Using simple decomposition of words and denotes the disjoint union of sets then, we have (1)Since = , hence , (2) == implies (3) implies Using similar procedure, we obtained all above of Hence, the proof is omitted.
Next, we construct linear system for canonical form in
Proposition 7. The following equalities hold for the canonical words in :
Proof. We compute the growth series inductively. By using decomposition of words, we have (1) gives (2) implies (3)