Abstract
We find the Hilbert series of the right-angled affine Artin monoid . We also discuss its recurrence relation and the growth rate.
1. Introduction
Coxeter groups were introduced by Coxeter in 1934 as abstract form of reflection groups. These groups were classified into two categories in 1935 also by Coxeter: spherical and affine. In the list of spherical Coxeter groups, is the first. The Artin group associated with is the braid group. Cardinality is an invariant of graded algebraic structures. Hilbert series deals with the cardinality of elements in the graded algebraic structures. In [1] Iqbal gave a linear system for the reducible and irreducible words of the braid monoid , which leads to compute the Hilbert series of . In [2] Iqbal and Yousaf computed the Hilbert series of the braid monoid in band generators. In [3] Berceanu and Iqbal proved that the growth rate of all the spherical Artin monoids is less than 4. In [4] Iqbal et al. studied the braid monoid of the affine type of the Coxeter systems. Authors also found the recurrence relations, the growth series of , and proved that the growth rate of is unbounded (see Figure 1).
In the present paper we study the affine-type Coxeter group and find the Hilbert series (or spherical growth series) of the associated right-angled affine Artin monoid . We also discuss its recurrence relations and the growth rate. For detailed explanation of related concepts and basic ideas about Coxeter groups and its types, readers are referred to [4] and references therein.
The affine (or infinite) Coxeter groups form another important series of Coxeter groups. These well-known affine Coxeter groups are , , , , , , , , and (for details, see [5]). In [3] authors proved that the universal upper bound for all the spherical Artin monoids is less than 4.
In this work we discuss right-angled affine Artin monoids; specifically, we study the affine monoid and compute its Hilbert series. We show that the growth of series is bounded above by 4. Along with Hilbert series we also compute the recurrence relations related to . We give a conjecture about the growth rate of , and the growth rates (maximal roots of the characteristic polynomial ) are computed using the softwares Drive6 and Mathematica.
The monoid is represented by its Coxeter graph as shown in Figure 2.
Here are vertices of the graph and all the labels are . If all the labels in a Coxeter diagram are replaced by , then there is no relation between the adjacent edges. Hence we have the associated right-angled Artin groups and the associated right-angled Artin monoids denoted by and , respectively.
In a monoid the relation will be written as in the length-lexicographic order. Let and ; then the word of the form is said to be an ambiguity. If is in the length-lexicographic order, then we say that the ambiguity is solvable. Such a presentation is complete if and only if all the ambiguities are solvable. Corresponding to the relation , the changes give a rewriting system. A complete presentation is equivalent to a confluent rewriting system. In a complete presentation of a monoid, word containing will be called reducible word and a word that does not contain will be called an irreducible word or canonical word. In a presentation of a monoid we fix a total order on the generators. Hence clearly we have the following.
Lemma 1. The monoid has the following presentation:
2. Recurrence Relations of the Monoid
In this section we discuss few interesting results relating to the recurrence relations of . First we talk about the solution of the system of linear recurrences.
Consider a system [6] of linear recurrencesThis system can be written as , where
The solution (which we need in our work) of the homogenous equation is given by , where are the eigenvalues of and is an eigenvector corresponding to . The largest eigenvalue is the growth rate of the sequence (by the definition of growth rate).
Let words of length and words starting with of length . Then we have the following.
Lemma 2. The monoid satisfies the following recurrence relations:
(a) , , and
(b) are given by the recurrence
Let be a Coxeter diagram ( is similar to the standard Coxeter diagram but opposite in direction) (see Figure 3).
We use in the solution of the recurrence of . Therefore we have the following.
Lemma 3. The monoid satisfies the recurrence relations
(a) , , and
(b) are given by the recurrence
Let and denote the characteristic polynomials of the system of recurrence relations of the monoids and , respectively. Then we have the following.
Lemma 4. The polynomials satisfy the recurrence relationwith the initial values and
Proof. Let be the matrix of order of the recurrences given in the Lemma 3. Then the characteristic polynomial of isWe write , where the determinants and are obtained by splitting such that the last rows of and are and , respectively. Therefore easily we have andSubtracting last column from 2nd last column of and after few easy computations we have the recurrence relation
Here we have an explicit formula to compute .
Lemma 5. Let ; then we have the following:
(1) ,
(2) , where
Proof. Let . Then can be written as ; i.e., This characteristic equation has the roots Let and Then we have the solution of the recurrence given by , where and are constants to determine. Since and , thereforeSolving these equations we get and . HenceFor even and odd values of we have
Theorem 6. The polynomials satisfy the recurrence relationwith , , , , and being the initial values.
Proof. Let be the matrix of order of the recurrences given in Lemma 2. Then the characteristic polynomial of isWrite , where the determinants and are obtained by splitting such that the last row of is and the last row of is . Hence easily we have . Subtracting last column from 3rd last and 2nd last columns of , respectively, we have . Hence
3. The Hilbert Series of the Monoid
Now we compute the Hilbert series of . For this we need to fix some notations first. Let denote the Hilbert series of , where words of length , and denote the Hilbert series of of words starting with , where words starting with of length .
Theorem 7. The Hilbert series of is given by the following:
(1) ,
(2) ,
(3) ,
(4) ,
(5)
Proof. From Lemma 2 we have Therefore From Lemma 2 we have . HenceSimilarly we can easily prove (3), (4), and (5).
The system of equations in Theorem 7 can be written in matrix form as , where
Lemma 8. In the monoid
Proof. The result follows immediately by factoring out from each row of .
Lemma 9. In
Proof. The system given in Theorem 7 of equations in variables is already written in the form , where , , and . Here we have only two cases.
Case I. By using Cramer’s rule we havewhere is a determinant obtained by replacing th column of by column of ; i.e.,Let denote the th column of . Adding in of and simplifying it we have a determinant of order . HenceNow by adding th column of the last determinant in its th column and simplifying it we finally haveCase II. Using again Cramer’s rule, we haveAdding in and simplifying we have the result
Now we have our main result.
Theorem 10. The Hilbert series of the monoid is
Proof. From Theorem 7 we have
4. Conjecture on the Upper Bound of Growth Rate of
In this section we compute the growth rates of and show them on the graph. We see that the growth rate is bounded above by 4. Let be the growth rate or the maximal root of the polynomial . We compute few initial growth rates (using Mathematica and derive 6) for . Let denote the growth rate of ; then we have the following few initial values of : , , , , , , , , , , , , , , , . We also compute , , , , and (using Mathematica). We have Figure 4 representing the growth rate of .
We observe that the growth rate for approaching 4 as approaches . Hence at the end we have the following.
Conjecture. The growth rate of is bounded above by 4.
Data Availability
No such data has been used to prove these results.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgments
This study was supported by research funds from Dong-A University.