Hilbert series of mixed braid monoid $MB_{2, 2}$

1.   Introduction

Hilbert series helps us in expressing the growth of dimension of homogenous components of a graded algebra. Presently, these ideas are extended to graded algebras, filtered algebras, and graded or filtered modules over these algebras, monoid in commutative algebra and coherent sheaves over projective schemes in algebraic geometry. The Hilbert series is a particular case of the Hilbert-Poincaré series of a graded vector space. It is also of a great significance in computational algebraic geometry due to the easiest way for computing the dimension and the degree of an algebraic variety defined by explicit polynomial equations. The Hilbert series is also helpful in providing us the important invariants of the algebraic varieties. This particular article focuses on the presentation and Hilbert series of the mixed braid monoid $M\!B_{2, 2}$.

The classical braid group $B_{m+1}$, given by Artin ^[3], possesses the following presentation:

The braid monoid $M\!B_{m+1}$ has the same presentation as the braid group $B_{m+1}$. The braid group $B_{m+1}$ admits another presentation known as the band presentation given by Birman et al. in ^[5].

Definition 1. ^[7] Let $U$ be a set. A square matrix $G = (g_{uv})_{u, v\in U}$ is calledCoxeter matrix over $U$ such that $m_{uu} = 1$ and $m_{uv} = m_{vu}\in\{2, 3, ..., \infty\}$ for all $u, v \in U, u\neq v.$

Definition 2. ^[7] A Coxeter graph $\Gamma$ is a labeled graph. It can be defined as

i. A set of vertices of $\Gamma$ is denoted by $U$.

ii. Two vertices $u, v \in U, u\neq v$ are connected by an edge if $m_{uv}\geq 3$. The labeling of edge is $m_{uv}$ if $m_{uv}\geq 4.$

Definition 3. ^[7] Let $M\! = \!(m_{st})_{s, t\in S}$ be the Coxeter matrix of the Coxetergraph $\Gamma$. Then the group defined by

is called Coxeter group(of type $\Gamma$).

Definition 4. ^[12] In mixed braid group $B_{m, n}$ the first index $m$ denotes trivialstrings in the braid group and the next $n$ strings shows thebraiding by itself and with the $m$ strings. The mixed braid group $B_{m, n}$ generated by $m+n$ strands is defined by

If we remove the last $n$ strands of Artin group $B_{m+n}$ then wehave only identity braid with $m$ strands. The collection of allsuch elements of Artin group $B_{m+n}$ will be denoted by $B_{m, n}$.

Definition 5. ^[12] The mixed braid monoid $M\!B_{m, n}$ has the following presentation:

In the below diagram of mixed braid monoid $M\!B_{m, n}$, the single bonds shows relation of degree 3, the double bonds shows relation of degree 4 and two generators commute if they are not joined by a bond (see Figure 1).

Conveniently $\alpha_{i}$ and $\sigma_{j}$ are denoted by a single generator $a_{k}, 1\leq k \leq i+j.$ Note that the mixed braid monoid $M\!B_{1, 2}$, which is isomorphic to the Artin monoid of type $B_{2}$, is presented by

The Coxeter diagram for $M\!B_{1, 2}$ takes the form in Figure 2.

The complete presentation and Hilbert series for $M\!B_{1, 2}$ are computed in ^[13]. This motivated us to compute the Hilbert series of $MB_{2, 2}$.

Definition 6. The classical presentation of $M\!B_{2, 2}$ is given by

Let us denote the relations by

and

The Coxeter diagram of $M\!B_{2, 2}$ is given as Figure 3.

An example of a braid in $B_{2, 2}$ (see Figure 4).

In 2003, Bokut ^[6] gave the non-commutative complete presentation of the braid monoid $M\!B_{n+1}$ with the length-lexicographic order induced by $a_{1} < \cdots < a_{n}$. In ^[10] we computed the Hilbert series of braid monoid $M\!B_{4}$ in band generators. In 1972, P. Deligne ^[8] proved that the Hilbert series of all the Artin monoids are rational functions. In ^[14] Saito computed the growth series of Artin monoids. In ^[13] we computed the Hilbert series of the Artin monoids $M\!(I_{2}(p))$, where $M\!(I_{2}(4))$ is isomorphic to $M\!B_{1, 2}$. In ^[7] we showed that the growth rate of all the Artin monoids is less than 4. In ^[11] we proved that the growth rates are exponential and the growth rates for $M\!B_{3}$ and $M\!B_{4}$ are approximately 1.618 and 2.0868. In this paper we construct a linear system for canonical words and then we find Hilbert series and the growth rate of the mixed braid monoid $M\!B_{2, 2}$. It is proved in ^[1] that the Hilbert series of the free G-graded F-algebra is a rational function. In ^[2] the authors proved that the Hilbert series of relatively free algebras is a rational function.

2.   Complete presentation of $M\!B_{2, 2}$

To get a canonical form of a word in an algebra the diamond lemma by G. Bergman ^[4] is very useful. To understand the concepts of ambiguities and canonical words, we start with his terminology.

Definition 7. Let $X$ be a nonempty set and $X^*$ be the free monoid on $X$. Let $w_{1}$ and $w_{2}\in X^*$, where $w_{1} = x_{1}x_{2}\cdots x_{r}$, $w_{2} = y_{1}y_{2}\cdots y_{r}$ with $x_{i}, y_{i}\in X$. Then $w_{1} < w_{2}$ length-lexicographically if there is a $k\leq r$ such that $x_{k} < y_{k}$ and $x_{i} = y_{i}$ for all $i < k$.

Definition 8. Let $\mu = \nu$ be a relation in a monoid $M$ and $\mu_{1} = sw$ and $\mu_{2} = wt$ be two words in $M$. Then the word of the form $\mu_{1}\mathop{\times}_{w}\mu_{2} = swt$ is said to be anambiguity.

If $\mu_{1}t = s\mu_{2}$ (in the length-lexicographic order) then we say that the ambiguity $swt$ is solvable. A presentation of $M$ is said to be complete if and only if all the ambiguities are solvable. Corresponding to the relation $\mu = \nu$, the changes $\alpha\mu\beta\longrightarrow\alpha\nu\beta$ give a rewriting system. A complete presentation is equivalent to a confluent rewriting system. In a complete presentation of a monoid a word containing $\mu$ will be called reducible word and a word that does not contain $\mu$ will be called an irreducible word or canonical word. For example $a_{3}a_{2}a_{3} = a_{2}a_{3}a_{2}$ is a basic relation in the braid monoid $M\!B_{4}$. A word $v = a^{2}_{3}a_{2}a_{3}$ contains $\alpha = a_{3}a_{2}a_{3}$. Hence $v$ is a reducible word. Then $a^{2}_{3}a_{2}a_{3} = a_{2}a_{3}a_{2}^{2}$ and $a_{2}a_{3}a_{2}^{2}$ is the canonical form of $v$. In a presentation of a monoid we fix a total order $a_{1} < a_{2} < \cdots < a_{n}$ on the generators.

Definition 9. ^[9] Let $G$ be a finitely generated group and $A$ be a finite set ofgenerators of $G$. The word length $l_{A}(g)$ of an element $g \in G$ is the smallest integer $n$ for which there exist $a_{1}, \dots, a_{n} \in A\cup A^{-1}$ such that $g = a_{1}\cdots a_{n}$.

Definition 10. ^[9] The Hilbert function of a monoid $M$ is given as $H(M, n) = a_{n}$, i.e., the number of elements of $M$ of word length $n$. The Hilbertseries of the monoid $M\!B_{n}$ for arbitrary variable $t$ isdenoted by $H_{M}(t)$ and is defined by $H_{M}(t) = \sum\limits_{n = 0}^{\infty}a_{n}t^n$.

Definition 11. For a sequence of positive numbers $\{b_{l}\}_{l\geq 1}$ the rate ofgrowth $r\in \mathbb{R}$ is given by

Lemma 1. ^[13]The following equations hold for the canonical words in $M\!I_{2}(4)$.

(1) $P^{(4)}_{1} = \frac{t}{(1-t)}+\frac{t}{(1-t)}P^{(4)}_{2}$

(2) $P^{(4)}_{2} = t+tP^{(4)}_{2}+P^{(4)}_{21}$

(3) $P^{(4)}_{21} = tP^{(4)}_{1}-t^{2}P^{(4)}_{21}-\frac{t^{3}}{1-t}P^{(4)}_{212}$

(4) $P^{(4)}_{212} = t^{2}P^{(4)}_{2}-t^{2}P^{(4)}_{21}$.

The solution of the above equations is given in ^[13] which is as follows:

Lemma 2. ^[13]The Hilbert series of $M\!I_{2}(4)$ is given by

While solving all the ambiguities we now give a complete presentation of $M\!B_{2, 2}$.

Theorem 1. The braid monoid $M\!B_{2, 2}$ has a complete presentation

where

where $n, m \geq 0.$

Proof. We denote the ambiguity formed by left sides of the relations $R_i$ and $R_j$ in $M\!B_{2, 2}$ by $R_i-R_j = swt$ (say). If in the ambiguity $z = swt$, $L(z) = (sw)t$ and $R(z) = s(wt)$ are different lexicographically, then we get a new relation in $M\!B_{2, 2}$ and if $L(z) = (sw)t$ and $R(z) = s(wt)$ are reduced to an identical word, then we say that ambiguity is solvable and no relation is formed. Here $L(z)$ denotes the canonical form of $(sw)t$ and $R(z)$ denotes the conical form of $s(wt).$ The above relations are obtained by solving the ambiguities involving the relations $R_0^{(4)}$, $R_1^{(4)}$ and $R_2^{(4)}$ and the new relations. In ^[13] we computed the relation (for $p = 4$) $R_3^{(4)}$, which is given by

For an ambiguity $R_2^{(4)}-R_2^{(4)} = a_3a_2a_3a_2a_3a_2 = w_1 (say)$, we have

Hence we have a relation $R_{w_1}\!:a_3a_2^{2}a_3a_2a_3 = a_2a_3a_2a_3^{2}a_2.$ Again by solving a new ambiguity $R_{w_1}-R_2^{(4)} = a_3a_2^{2}a_3a_2a_3a_2 = w_2$ we have

which gives another relation $R_{w_2}\!:a_3a_2^{3}a_3a_2a_3 = a_2a_3a_2a_3^{2}a_2^{2}.$ By continuing the same process we have the general relation

In the ambiguity $R_2^{(4)}-R_1^{(4)} = a_{3}a_{2}a_{3}a_{2}a_{1}a_{2}a_{1} = w_3$, we have

Hence we have a relation $R_{w_3}:a_{3}a_{2}a_{1}a_{3}a_{2}a_{1}a_{2} = a_{2}a_{3}a_{2}a_{1}a_{3}a_{2}a_{1}.$ Again by solving a new ambiguity $R_{w_3}-R_1^{(4)} = a_{3}a_{2}a_{1}a_{3}a_{2}a_{1}a_{2}a_{1} = w_4$ we have

which gives another relation $R_{w_4}:a_{3}a_{2}a_{1}^{2}a_{3}a_{2}a_{1}a_{2} = a_{2}a_{3}a_{2}a_{1}a_{3}a_{2}a_{1}^{2}.$ By continuing the same process we have the general relation

In the ambiguity $R_4^{(4)}-R_0^{(4)} = a_3a_2^{n+1}a_3a_2a_3a_{1} = w_5$, we have

Hence we have a relation $R_{w_5}:a_3a_2^{n+1}a_3a_2a_1a_{3} = a_{2}a_{3}a_{2}a_{3}^{2}a_{2}^{n}a_{1}.$ Again by solving a new ambiguity $R_{w_5}-R_0^{(4)} = a_3a_2^{n+1}a_3a_2a_3a_{1}a_3a_{1} = w_6$ we have

which gives another relation $R_{w_6}:a_3a_2^{n+1}a_3a_2a_1^{2}a_{3} = a_{2}a_{3}a_{2}a_{3}^{2}a_{2}^{n}a_{1}^{2}.$ By continuing the same process we have the general relation

All other ambiguities all solvable. Hence we have the complete set of relations.

3.   Hilbert series of $M\!B_{2, 2}$

For our convenience we use these notations for canonical and reducible words in $M\!B_{2, 2}$. In general $B^{(4)}_{*}$ denotes the set of reducible words and $A^{(4)}_{*}$ denotes the set of canonical words in $M\!B_{2, 2}$. Particularly, $B^{(4)}_{i}$ denotes the reducible word of $R^{(4)}_{i}$ in $M\!B_{2, 2}$ and $A_{j(j-1)\ldots k}^{(n+m)}$ denotes collection of all canonical words in $M\!B_{m, n}$ starting with $a_j a_{j-1}\ldots a_k$. These sets are graded by length-lexicographic order, so we can compute the Hilbert series of these sets. The words $Xa_{3}\times_{3} a_{3}Y$ and $Xa_{3}a_{2}\times_{32} a_{3}a_{2}Y$ are equivalent to products $Xa_{3}Y$ and $Xa_{3}a_{2}Y$, respectively. Let $Q^{(4)}_{*}$ denote the Hilbert series for $B^{(4)}_{*}$ and $P^{(4)}_{*}$ denote the Hilbert series for $A^{(4)}_{*}$. Let $H^{(m+n)}_{M}(t)$ denote the Hilbert series for the set

(for some $k \in \mathbb{Z}$) for a monoid $M$. Then we have

Theorem 2. The following relations hold for the Hilbert series of reducible words in $M\!B_{2, 2}$.

Proof. Let $\alpha (i, \ldots, j)$ be a wordin $a_{i}, a_{i+1}, \ldots, a_{j}$ in $M\!B_{2, 2}$, then $\Sigma\alpha(i, \ldots, j)$ be a word in $a_{i+1}, a_{i+2}, \ldots, a_{j+1}$ in $M\!B_{2, 2}$. $B^{(4)}_{*}$ denotes the reducible wordsin $M\!B_{2, 2}$ corresponding to relation $R^{(4)}_{*}$. $Q^{(4)}_{*}$ denotes the Hilbert series of $B^{(4)}_{*}$. Hilbertseries of $A^{(2)}_{1} = \{a_{1}, a_{1}^{2}, \ldots\}$ is $P^{(2)}_{1} = t+t^{2}+\cdots = \frac{t}{1-t}.$ Therefore we have

a. $B^{(4)}_{1} = \{a_{2}a_{1}a_{2}a_{1}\}.$ This implies $Q^{(4)}_{1} = t^{4}.$

b. Since $B^{(4)}_{2} = \{a_{2}a^{n+1}_{1}a_{2}a_{1}a_{2}\} = \{a_{2}a_{1}\}\times A^{(2)}_{1}\times \{a_{2}a_{1}a_{2}\}.$ Hence $Q^{(4)}_{2} = \frac{t^{6}}{1-t}.$

c. Similarly $B^{(4)}_{3} = \{a_{3}a_{2}a_{3}a_{2}\}$ implies $Q^{(4)}_{3} = t^{4}$.

d. The decomposition $B^{(4)}_{4} = \{a_{3}a^{n+1}_{2}a_{2}a_{2}a_{3}\} = \{a_{3}a_{2}\}\times\Sigma A^{(2)}_{1}\times \{a_{3}a_{2}a_{3}\}$ gives $Q^{(4)}_{4} = \frac{t^{6}}{1-t}.$

e. $B^{(4)}_{5} = \{a_{3}a_{2}a^{m}_{1}a_{3}a_{2}a_{1}a_{2}\} = \{a_{3}a_{2}\}\times A^{(2)}_{1}\times \{a_{3}a_{2}a_{1}a_{2}\}$ gives $Q^{(4)}_{5} = \frac{t^{7}}{1-t}$.

f. $B^{(4)}_{6} = \{a_{3}a^{n+1}_{2}a_{2}a_{2}a^{m}_{1}a_{3}\} = \{a_{3}a_{2}\}\times\Sigma A^{(2)}_{1}\times \{a_{3}a_{2}\}\times A^{(2)}_{1}\times a_{3}$ gives $Q^{(4)}_{6} = \frac{t^{7}}{(1-t)^{2}}.$

Next we construct a linear system for canonical forms in $M\!B_{2, 2}$.

Theorem 3. The following relations hold for the Hilbert series of canonicalwords in $M\!B_{2, 2}$.

(1) $P^{(4)}_{1} = P^{(3)}_{1}+P^{(3)}_{1}P^{(4)}_{3}.$

(2) $P^{(4)}_{2} = P^{(3)}_{2}+P^{(3)}_{2}P^{(4)}_{3}.$

(3) $P^{(4)}_{3} = t+tP^{(4)}_{3}+P^{(4)}_{32}.$

(4) $P^{(4)}_{21} = P^{(3)}_{21}+P^{(3)}_{21}P^{(4)}_{3}.$

(5) $P^{(4)}_{32} = tP^{(4)}_{2}-t^{2}P^{(4)}_{32}-\frac{t^{3}}{1-t}P^{(4)}_{323}-\frac{t^{3}}{1-t}P^{(4)}_{3212}-\frac{t^{6}}{(1-t)^{2}}P^{(4)}_{3}.$

(6) $P^{(4)}_{212} = P^{(3)}_{212}+P^{(3)}_{212}P^{(4)}_{3}$

(7) $P^{(4)}_{323} = t^{2}P^{(4)}_{3}-t^{2}P^{(4)}_{32}.$

(8) $P^{(4)}_{3212} = tP^{(4)}_{212}.$

Proof. As we have already defined $A^{(4)}_{*}$ denotes the set of canonical words in $M\!B_{2, 2}$ and $B^{(4)}_{*}$ denotes the set of reducible words in $M\!B_{2, 2}$. Now we define the corresponding Hilbert series. Let $Q^{(4)}_{*}$ denotes the Hilbert series for $B^{(4)}_{*}$ and $P^{(4)}_{*}$ denotes the Hilbert series for $A^{(4)}_{*}$. Then using the set decomposition we have

(1) The decomposition of $A^{(4)}_{1}$ can be given as $A^{(4)}_{1} = A^{(3)}_{1} \cup (A^{(3)}_{1}\times A^{(4)}_{3})$. This implies Relation 1.

(2) It follows directly from $A^{(4)}_{2} = A^{(3)}_{2}\cup (A^{(3)}_{2}\times A^{(4)}_{3})$.

(3) The set $A^{(4)}_{3}$ can be written as $A^{(4)}_{3} = \{a_{3}\} \cup ({a_{3}\times A^{(4)}_{3}}) \cup A^{(4)}_{32}$. This results the Relation 2.

(4) It follows from $A^{(4)}_{21} = A^{(3)}_{21}\cup (A^{(3)}_{21}\times A^{(4)}_{3})$.

(5) The set $A^{(4)}_{32}$ is decomposed as

which gives the required equation.

(6) The decomposition of $A^{(4)}_{212}$ can be written as $A^{(4)}_{212} = A^{(3)}_{212} \cup (A^{(3)}_{212}\times A^{(4)}_{3})$. This implies Relation 6.

(7) It follows from $A^{(4)}_{323} = \{a_{3}a_{2}\}\times A^{(4)}_{3}\setminus (B^{(4)}_{3}\times_{32}A^{(4)}_{32})$.

(8) The decomposition of $A^{(4)}_{3212} = \{a_{3}\}\times A^{(4)}_{212}$ immediately gives us Relation 8.

Theorem 4. Hilbert Series for the canonical words in $M\!B_{2, 2}$ is given as

Proof. As we have already computed the Hilbert series of canonical words and reducible words in Theorem 3 and in Theorem 2, respectively. Now we construct a linear system forcanonical words in $M\!B_{2, 2}.$ For this we put the values of $P^{(3)}_{1}, P^{(3)}_{2}, P^{(3)}_{21}$ and $P^{(3)}_{212}$ form Lemma 1 in Theorem 3, we get following system of equations:

(1) $P^{(4)}_{1} = \frac{t}{(1-t)(1-t-t^{2}-t^{3})}+\frac{t}{(1-t)(1-t-t^{2}-t^{3})}P^{(4)}_{3}$

(2) $P^{(4)}_{2} = \frac{t(1+t+t^{2})}{(1-t-t^{2}-t^{3})}+\frac{t(1+t+t^{2})}{(1-t-t^{2}-t^{3})}P^{(4)}_{3}$

(3) $P^{(4)}_{3} = t+tP^{(4)}_{3}+P^{(4)}_{32}$

(4) $P^{(4)}_{21} = \frac{t^{2}(1+t)}{(1-t-t^{2}-t^{3})}+\frac{t^{2}(1+t)}{(1-t-t^{2}-t^{3})}P^{(4)}_{3}$

(5) $P^{(4)}_{32} = tP^{(4)}_{2}-t^{2}P^{(4)}_{32}-\frac{t^{3}}{1-t}P^{(4)}_{323}-\frac{t^{3}}{1-t}P^{(4)}_{3212}-\frac {t^{6}}{(1-t)^{2}}P^{(4)}_{3}$

(6) $P^{(4)}_{212} = \frac{t^{3}}{1-t-t^{2}-t^{3}}+\frac{t^{3}}{1-t-t^{2}-t^{3}}P^{(4)}_{3}$

(7) $P^{(4)}_{323} = t^{2}P^{(4)}_{3}-t^{2}P^{(4)}_{32}$

(8) $P^{(4)}_{3212} = tP^{(4)}_{212}.$

Now we use Matrix Inversion Method to solve the system of these equations. Writing above equations in matrix form $AX = B$ we have

By solving the above system of equations, we get

Hence the Hilbert series of $M\!B_{2, 2}$ is computed as

4.   Growth rate for $M\!B_{2, 2}$

In the following remark we find the growth rate of $M\!B_{2, 2}$. Using Maple we convert the Hilbert series into partial fraction and by expanding its each term.

Remark 1. The partial fraction of above Hilbert series is calculated as

As the last term

gives us the approximation of the series and the rest of the termshave negligible effect on it. Hence $a^{(4)}_{l}\approx1.48827(2.45917)^{l}.$ Thus, in this case the growth function $a^{(4)}_{l}$ of $M\!B_{2, 2}$ is also exponential and growth rate isapproximately equal to $2.45917$.

5.   Conclusions

The mixed braid group $B_{m, n}$ is the subgroup of the Artinian braid group $B_{m+n}$. In this article we compute the canonical forms(ambiguity free presentation) of the words in the braid monoid and the corresponding Hilbert series. This work can also be utilized to compute Hilbert series of braid monoid $MB_{m, n}$.

Acknowledgements

The authors thank the National Key Research and Development Program under Grant 2018YFB0904205, Science and Technology Bureau of Chengdu (2020-YF09-00005-SN), Sichuan Science and Technology program (2021YFH0107), and Erasmus+SHYFTE Project (598649-EPP-1-2018-1-FR-EPPKA2-CBHE-JP).

Conflict of interest

Authors declare that there are no conflicts of intereset.