Some Recurrence Relations and Hilbert Series of Right-Angled Affine

1Department of Mathematics, Dong-A University, Busan 49315, Republic of Korea 2Department of Mathematics, University of Gujrat, Gujrat, Pakistan 3Faculty of Information Technology, University of Central Punjab, Lahore, Pakistan 4University of Education, Township Lahore, Pakistan 5Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 52828, Republic of Korea 6Center for General Education, China Medical University, Taichung 40402, Taiwan


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  4 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 ( Ã∞  ) of the affine type Ã of the Coxeter systems.Authors also found the recurrence relations, the growth series of ( Ã∞  ), and proved that the growth rate of ( Ã∞  ) is unbounded (see Figure 1).
In the present paper we study the affine-type Coxeter group D and find the Hilbert series (or spherical growth series) of the associated right-angled affine Artin monoid ( D∞  ).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 Ã , B , C , D , Ẽ6 , Ẽ7 , Ẽ8 , G2 , and Ĩ1 (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 ( D∞  ) and compute its Hilbert series.We show that the growth of ( D∞  ) series is bounded above by 4. Along with Hilbert series we also compute the recurrence relations related to ( D∞  ).We give a conjecture about the growth rate of Figure 2 ( D∞  ), and the growth rates (maximal roots of the characteristic polynomial   ()) are computed using the softwares Drive6 and Mathematica.
The monoid ( D∞  ) is represented by its Coxeter graph as shown in Figure 2.
Here  1 ,  2 , . . .,   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 ( D∞  ) and ( D∞  ), respectively.In a monoid the relation  =  will be written as  >  in the length-lexicographic order.Let  1 =  and  2 = V; then the word of the form V is said to be an ambiguity.If  1 V =  2 is in the length-lexicographic order, then we say that the ambiguity V 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  1 <  2 < ⋅ ⋅ ⋅ <   on the generators.Hence clearly we have the following.

Recurrence Relations of the Monoid 𝑀( D∞ 𝑛 )
In this section we discuss few interesting results relating to the recurrence relations of ( D∞  ).First we talk about the solution of the system of linear recurrences.
Let   = #{canonical words of length } and  ; = #{canonical words starting with   of length }.Then we have the following.
We use ( ∞  ) in the solution of the recurrence of ( D∞  ).Therefore we have the following.
Lemma 3. The monoid ( ∞  ) satisfies the recurrence relations Let   () and   () denote the characteristic polynomials of the system of recurrence relations of the monoids ( ∞  ) and ( D∞  ), respectively.Then we have the following.
Lemma 4. The polynomials (  ()) ≥3 satisfy the recurrence relation with the initial values Proof.Let   be the matrix of order  ×  of the recurrences given in the Lemma 3. Then the characteristic polynomial Subtracting 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  = √  2 − 4; then we have the following: Solving these equations we get  = (4/( + ) For even and odd values of  we have Theorem 6.The polynomials (  ()) ≥5 satisfy the recurrence relation

The Hilbert Series of the Monoid 𝑀( D∞ 𝑛 )
Now we compute the Hilbert series of ( D∞  ).For this we need to fix some notations first.Let H ()   () = ∑ ≥0     denote the Hilbert series of ( D∞  ), where   = #{canonical words of length }, and H ()  ; () = ∑ ≥0  () ;   denote the Hilbert series of ( D∞  ) of words starting with   , where  ; = #{canonical words starting with   of length }.
Lemma 8.In the monoid ( D∞  ) Proof.The result follows immediately by factoring out  from each row of det(  ).Case I(5 ≤  ≤  − 1).By using Cramer's rule we have where   is a determinant obtained by replacing th column of   by column of ; i.e., H Now by adding ( − 1)th column of the last determinant in its th column and simplifying it we finally have Case II( = ).Using again Cramer's rule, we have Adding   in  −2 and simplifying we have the result Now we have our main result.
Conjecture.The growth rate of ( D∞  ) is bounded above by 4.