On right-angled spherical Artin monoid of type Dn

Zaffar Iqbal; Abdul Rauf Nizami; Mobeen Munir; Amlish Rabia; Shin Min Kang

doi:10.1515/phys-2018-0061

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On right-angled spherical Artin monoid of type D_n

Zaffar Iqbal , Abdul Rauf Nizami , Mobeen Munir , Amlish Rabia und Shin Min Kang

Veröffentlicht/Copyright: 13. August 2018

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Abstract

Recently Berceanu and Iqbal proved that the growth rate of all the spherical Artin monoids is bounded above by 4. In this paper we compute the Hilbert series of the right-angled spherical Artin monoid M(Dn∞) and graphically prove that growth rate is bounded by 4. We also discuss its recurrence relations and other main properties.

Keywords: Canonical words; Hilbert series; growth rate

PACS: 02.10.Ox; 02.40.Re; 02.20.Bb; 02.30.Lt

1 Introduction

Coxeter groups are named after a British born Canadian geometer Harold Scott MacDonald Coxeter (1907–2003). Coxeter groups were introduced by Coxeter in 1934 as abstract form of reflection groups and defined as the groups with the generators a_i, i ∈ I and relations ai2 = 1 and a_ia_ja_i = a_ja_ia_j, i, j ∈ I. Coxeter classified these groups into two categories, finite and infinite, in 1935. If the relation ai2 = 1 from the presentation of Coxeter group is removed then we get the presentation of Artin groups. Thus, we can say that Coxeter groups are quotient groups of the Artin groups. A Coxeter group is called finite Coxeter group if it has the presentation as a discrete, properly acting group of reflections of the sphere [1]. Therefore these groups are also called spherical Coxeter groups. In the list of spherical Coxeter groups, A_n is the first. The Artin group associated to A_n is the braid group. A Coxeter group is affine if it is infinite. It is generated by reflections in affine spaces and has the presentation as discrete, properly acting, affine reflection group [1]. Cardinality is invariant of graded algebraic structures. Hilbert series deals with the cardinality of elements in the graded algebraic structures.

In [2] Saito computed the growth series of the Artin monoids. In [3] Parry computed the growth series of the Coxeter groups. In [4] Mairesse and Mathéus gave the growth series of Artin groups of dihedral type. In [5] Iqbal gave a linear system for the reducible and irreducible words of the braid monoid MB_n, which leads to compute the Hilbert series of MB_n and in [6] computed the Hilbert series of the braid monoid MB₄ in band generators. In [7] Berceanu and Iqbal proved that the growth rate of all the spherical Artin monoids is less than 4. In the present paper we study the affine-type Coxeter group Dn∞, and find the Hilbert series (or spherical growth series) of the associated right-angled affine Artin monoid M(Dn∞). We also discuss its recurrence relations and the growth rate. In [8, 9, 10, 11, 12, 13] authors presented some new ways to compute different solutions methods.

Let S be a set. A Coxeter matrix over S is a square matrix M = (m_st)_s,t∈S such that

m_ss = 1 for all s ∈ S;
m_st = m_ts ∈ {2, 3, 4, …, ∞} for all s, t ∈ S, s ≠ t.
A Coxeter graphΓ is a labeled graph defined by the following data:
S is a set of vertices of Γ.
Two vertices s, t ∈ S, s ≠ t are joined by an edge if m_st ≥ 3. This edge is labeled by m_st if m_st ≥ 4.

Remark 1.1

A Coxeter matrix M = (m_st)_s,t∈S is usually represented by its Coxeter graph Γ(M).

Definition 1.2

Let M = (m_st)_s,t∈S be the Coxeter matrix and Γ(M) its Coxeter graph. Then the group

W=〈s∈S∣s2=1,(st)mst=1,s,t∈S,s≠t〉

is called the Coxeter group (of type Γ(M)).

In a simple way, we can write W=〈s∈S∣s2=1,sts⋯⏟mstfactors=tst⋯⏟mstfactors,s,t∈S,s≠t〉. We call Γ to be of spherical type if W is finite.

An Artin spherical monoid (or group) is given by a finite union of connected Coxeter graphs from the well known classical list of Coxeter diagrams (see [1, 14]).

Figure 1

Spherical Coxeter graphs

By convention m_ij is the label of the edge between x_i and x_j (i ≠ j); if there is no label then m_ij = 3. If there is no edge between x_i and x_j, then m_ij = 2.

To a given Coxeter graph X_n we associate the monoid M(X_n) with generators (corresponds to vertices) x₁, x₂, …, x_n and relations (corresponds to labels m_ij of the graphs) xixjxixj⋯⏟mijfactors=xjxixjxi⋯⏟mijfactors, where 1 ≤ j < i ≤ n.

The corresponding group G(X_n) associated to X_n is defined by the same presentation. From now onward we will use X_n for G(X_n) for simplicity.

Definition 1.3

If W is a Coxeter group with Coxeter matrix M = (m_st)_s,t∈S, then the Artin group associated to W is defined by

A=〈s∈S|sts⋯⏟mstfactors=tst⋯⏟mstfactors〉.

If W is finite then 𝓐 is called a spherical Artin group.

Definition 1.4

In the spherical type Coxeter graphs, if all the labels m_st ≥ 3 are replaced by ∞ then the associated groups (monoids) are called right-angled Artin groups (monoids).

Definition 1.5

[15] Let G be a finitely generated group and S be a finite set of generators of G. The word lenthl_S(g) of an element g ∈ G is the smallest integer n for which there exists s₁, …, s_n ∈ S ∪ S⁻¹ such that g = s₁ ⋯ s_n.

Definition 1.6

[15] Let G be a finitely generated group and S be a finite set of generators of G. The growth function of the pair (G, S) associates to an integer k ≥ 0 the number a_k of elements g ∈ G such that l_S(g) = k. The corresponding spherical growth series (also known as the generating function) or the Hilbert series is given by HG(t)=∑k=0∞aktk.

The affine (or infinite) Coxeter groups form another important series of Coxeter groups. These well-known affine Coxeter groups are A͂_n, B͂_n, C͂_n, D͂_n, E͂₆, E͂₇, E͂₈, G͂₂ and I͂₁ (for details, see [14]). In [7] we 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 M(Dn∞) and compute its Hilbert series. We show that the growth of M(Dn∞) series is bounded above by 4. Along with Hilbert series we also compute the recurrence relations related to M(Dn∞). Based on some computations with mathematical softwares Derive 6 and Mathematica, we give a conjecture of the growth rate of M(Dn∞).

The monoid M(Dn∞) is represented by its Coxeter graph as follows:

Here x₁, x₂, …, x_n 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 G(Dn∞) and M(Dn∞), respectively.

Definition 1.7

Let X be a nonempty set and X^* be the free monoid on X. Let w₁ and w₂ ∈ X^*, where w₁ = x₁x₂ ⋯ x_r, w₂ = y₁y₂ ⋯ y_r with x_i, y_i ∈ X. Then w₁ < w₂length-lexicographically (or quasi-lexicographically) if there is a k ≤ r such that x_k < y_k and x_i = y_i for all i < k.

Let α = β be a relation in a monoid M and α₁ = uw and α₂ = wv be the words in M.

Then the word of the form α₁ ×_wα₂ = uwv is said to be an ambiguity. If α₁v = uα₂ (in the length-lexicographic order) then we say that the ambiguity uwv is solvable. A presentation of M is complete if and only if all the ambiguities are solvable. Corresponding to the relation α = β, the changes γαδ ⟶ γβδ give a rewriting system. A complete presentation is equivalent to a confluent rewriting system. In a complete presentation of a monoid a word containing α will be called reducible word and a word that does not contain α will be called an irreducible word or canonical word. For example x₂x₁x₂ = x₁x₂x₁ is a basic relation in the braid monoid MB₃. A word v=x22x1x2 contains α = x₂x₁x₂. Hence v is a reducible word. Then x22x1x2=x2x1x2x1=x1x2x12 is the canonical form of v [16].

In a presentation of a monoid we fix a total order x₁ < x₂ < ⋯ < x_n on the generators. Hence clearly we have

Lemma 1.8

The presentation of the monoidM(Dn∞)has generatorsx₁, x₂, …, x_nand has relationsx_ix_j = x_jx_i, 3 ≤ j + 2 ≤ i ≤ n − 1, andx_nx_k = x_kx_n, k ≠ n − 2.

2 Results and discussion

Here we present our main results. First we compute recurrence relations for monoid M(Dn∞). Then we formulate our results about Hilbert series of monoid M(Dn∞). At the end we formulate a conjecture about the growth rate of these monoids, graphically proving that it is bounded by 4.

2.1 Recurrence relations of the monoid M(Dn∞)

In this section we discuss few interesting results relating the recurrence relations of M(Dn∞). First we talk about the solution of the system of linear recurrences.

Consider a system [17] of linear recurrences

u1(t+1)=a11(t)u1(t)+⋯+a1n(t)un(t)+f1(t)u2(t+1)=a11(t)u1(t)+⋯+a1n(t)un(t)+f2(t)⋮un(t+1)=a11(t)u1(t)+⋯+a1n(t)un(t)+fn(t).

This system can be written as u(t + 1) = A(t)u(t) + f(t), where

u(t)=[u1(t)⋮un(t)],A(t)=[a11(t)⋯a1n(t)⋮⋱⋮an1(t)⋯ann(t)]andf(t)=[f1(t)⋮fn(t)].

The solution (which we need in our work) of the homogenous equation u(t + 1) = A(t)u(t) is given by u(t) = c1λ1tu1+⋯+ckλktuk, where λ₁, …, λ_k are the eigenvalues of A(t) and uⁱ is an eigenvector corresponding to λ_i. The largest eigenvalue is the growth rate of the sequence (u_i(t)) (by the definition of growth rate).

Let c_k = #{canonical words of length k} and c_k;i = #{canonical words starting with x_i of length k}. Then by Lemma 1.8 we have the following

Lemma 2.1

The monoidM(Dn∞)satisfies the following recurrence relations

c₀ = 1, c_1;i = 1, ck=∑i=1nck;i (k ≥ 1).
c_k;i (k ≥ 2) is given by following recurrence relations
ck;i={∑i=1nck−1;i,i=1,∑i=j−1nck−1;i(j=2,3,…,n−1),ck−1;i−2+ck−1;i,i=n.

From the above equations it is clear that c_k;1 = c_k;2.

Let M_n be the matrix of order n × n of the system of linear recurrences given in Lemma 2.1. Then

Mn=[11⋯111111⋯111101⋯1111⋮⋮⋮⋮⋮⋮00⋯111100⋯011100⋯0101]

and its characteristic polynomial D_n(λ) is

|λ−1−1⋯−1−1−1−1−1λ−1⋯−1−1−1−10−1⋯−1−1−1−1⋮⋮⋮⋮⋮⋮00⋯−1λ−1−1−100⋯0−1λ−1−100⋯0−10λ−1|.

Lemma 2.2

[7] The polynomials 𝓐_n(λ) for the monoidM(An∞)satisfy the recurrence

An(λ)=λAn−1(λ)−λAn−2(λ)(n≥2)(2.1)

with 𝓐₀(λ) = 1 and 𝓐₁(λ) = λ − 1 the initial values.

Theorem 2.3

The polynomials (D_n(λ))_n≥4satisfy the recurrence

Dn(λ)=(λ−1)An−1(λ)−λ2An−4(λ),n≥4(2.2)

withD₀(λ) = 1, D₁(λ) = λ − 1, D₂(λ) = λ² − 2λandD₃(λ) = λ(λ² − 3λ + 1) as the initial values.

Proof

The characteristic polynomial of M_n is given (as above) by

Dn(λ)=|λ−1−1⋯−1−1−1−1−1λ−1⋯−1−1−1−10−1⋯−1−1−1−1⋮⋮⋮⋮⋮⋮00⋯−1λ−1−1−100⋯0−1λ−1−100⋯0−10λ−1|.

Now by decomposing D_n(λ) by last row as sum of two determinants, we have

Dn(λ)=|λ−1−1⋯−1−1−1−1−1λ−1⋯−1−1−1−10−1⋯−1−1−1−1⋮⋮⋮⋮⋮⋮00⋯−1λ−1−1−100⋯0−1λ−1−100⋯000λ−1|+|λ−1−1⋯−1−1−1−1−1λ−1⋯−1−1−1−10−1⋯−1−1−1−1⋮⋮⋮⋮⋮⋮00⋯−1λ−1−1−100⋯0−1λ−1−100⋯0−100|.

Expanding both the determinants by last rows, respectively, we get

Dn(λ)=(λ−1)An−1(λ)−V(λ),

where

V(λ)=|λ−1−1⋯−1−1−1−1−1λ−1⋯−1−1−1−10−1⋯−1−1−1−1⋮⋮⋮⋮⋮⋮00⋯−1λ−1−1−100⋯0−1−1−100⋯00λ−1−1|.

By subtracting last column from second last column we get

V(λ)=|λ−1−1⋯−1−1−10−1λ−1⋯−1−1−100−1⋯−1−1−10⋮⋮⋮⋮⋮00⋯−1λ−1−1000⋯0−1−1000⋯00λ−1−λ|

and expanding by last row we get

=−λ|λ−1−1−1⋯−1−1−1−1λ−1−1⋯−1−1−10−1λ−1⋯−1−1−1⋮⋮⋮⋮⋮⋮000⋯−1λ−1−1000⋯0−1−1|.

Subtracting last column from second last column we have

V(λ)=−λ|λ−1−1⋯−10−1−1λ−1⋯−10−10−1⋯−10−1⋮⋮⋮⋮⋮00⋯−1λ−100⋯00−1|.

Simplifying last determinant, we finally have

V(λ)=λ2|λ−1−1−1⋯−1−1−1λ−1−1⋯−1−10−1λ−1⋯−1−1⋮⋮⋮⋮⋮000⋯−1λ−1|.

Therefore we have the result

Dn(λ)=(λ−1)An−1(λ)−λ2An−4(λ).

□

2.2 Hilbert series of M(Dn∞)

Now we compute the Hilbert series of M(Dn∞). For this we need to fix some notations first. Let HM(n)(t)=∑k≥0cktk denote the Hilbert series of M(Dn∞), where c_k = #{canonical words of length k}. Let HM;i(n)(t)=∑k≥0ck;i(n)tk denote the Hilbert series of M(Dn∞) of words starting with x_i, where c_k;i = #{canonical words starting with x_i of length k}.

Theorem 2.4

The Hilbert series ofM(Dn∞)is represented by the following system of equations:

HM(n)(t)=1+∑i=1nHM;i(n)(t)
HM;1(n)(t)=HM;2(n)(t)
HM;j(n)(t)=t+t∑i=j−1nHM;i(n)(t) (j = 2, 3, …, n − 1)
HM;n(n)(t)=t+tHM;n−2(n)(t)+tHM;n(n)(t)

Proof

From Lemma 2.1 we have ck=∑i=1nck;i, (k ≥ 1). Therefore HM(n)(t)=∑k≥0cktk=c0+∑k≥1cktk = 1 + ∑k≥1∑i=1nck;itk=1+∑i=1n∑k≥1ck;itk=1+∑i=1nHM;i(n)(t).
is obvious as c_k;1 = c_k;2.
From Lemma 2.1 we have, ck;j=∑i=j−1nck−1;i (j = 2, 3, …, n − 1). Hence HM;j(n)(t)=∑k≥1nck;j(t)tk = c1;j(t)t+∑k≥2nck;j(t)tk=t+∑k≥2n∑i=j−1nck−1;i(t)tk = t + t∑i=j−1n∑k≥2nck−1;i(t)tk−1=t+t∑i=j−1nHM;i(n)(t).
Similarly we can easily prove (4). □

The system of equations in Theorem 2.4 can be written in matrix form as W_nX = B, where

Wn=[1−t−t⋯−t−t−t−t−t1−t⋯−t−t−t−t0−t⋯−t−t−t−t⋮⋮⋮⋮⋮⋮00⋯−t1−t−t−t00⋯0−t1−t−t00⋯0−t01−t],X=[HM;1(n)(t)HM;2(n)(t)HM;3(n)(t)⋮HM;n(n)(t)],andB=[ttt⋮t].

Lemma 2.5

In the monoidM(Dn∞)

det(Wn)=tnDn(1t).

Proof

The result follows immediately by factoring out t from each row of det(W_n). □

Lemma 2.6

InM(Dn∞)

HH;m(n)(t)={tm−1Am−2(1t)tnDn(1t),2≤m≤n−1tm−2(1−t)Am−3(1t)tnDn(1t),m=n

Proof

The system explained in Theorem 2.4 of n equations in n variables HM;i(n)(t),1 ≤ i ≤ n is already written in the form W_nX = B, where X=[HM;1(n)(t),…,HM;n(n)(t)]t,det(Wn)=tnDn(1t) and B = [t, …, t]^t. Here we have two cases:

2 ≤ m ≤ n − 1. By using Cramer’s rule we have
HM;m(n)(t)=Tmdet(Wn),
where T_m is a determinant obtained by replacing mth column of W_n by column of B. That is, [HM;m(n)(t) is now becomes
1tnDn(1t)|1−t−t⋯−tt−t−t−t1−t⋯−tt−t−t0−t⋯−tt−t−t⋮⋮⋮⋮⋮⋮00⋯−tt−t−t00⋯0t1−t−t00⋯0t01−t|.
Let C_i denote the ith column of T_m. Adding C_m in C_m+1, C_m+2, …, C_n of T_m and simplifying we have a determinant of order m. Hence
HM;m(n)(t)=1tnDn(1t)|1−t−t⋯−tt00−t1−t⋯−tt000−t⋯−tt00⋮⋮⋮⋮⋮⋮00⋯−tt0000⋯0t1000⋯0t01|.
Now simplifying it we finally have
HM;m(n)(t)=tm−1Am−2(1t)tnDn(1t).
m = n.
Again using the Cramer’s rule, HM;n(n)(t) becomes
1tnDn(1t)|1−t−t⋯−t−t−tt−t1−t⋯−t−t−tt0−t⋯−t−t−tt⋮⋮⋮⋮⋮⋮00⋯−t1−t−tt00⋯0−t1−tt00⋯0−t0t|.
Let C_i denote the ith column of D_n. Adding C_n in C_n−1 and C_n−2, we get
HM;n(n)(t)=1tnDn(1t)|1−t−t⋯−t00t−t1−t⋯−t00t0−t⋯−t00t⋮⋮⋮⋮⋮⋮00⋯−t10t00⋯001t00⋯000t|.
Therefore
HM;n(n)(t)=tm−2(1−t)Am−3(1t)tnDn(1t),m=n

□

Now we have our main result.

Theorem 2.7

The Hilbert series of the monoidM(Dn∞)is

HM(n)(t)=1tnDn(1t).

Proof

From Theorem 2.4 we have

HM(n)(t)=1+∑i=1nHM;i(n)(t)=1+HM;1(n)(t)+HM;2(n)(t)+⋯+HM;n−1[n)(t)+HM;n(n)(t)=1tnDn(1t)(tnDn(1t)+2t+t2A1(1t)+⋯+tn−3An−4(1t)+2tn−2An−3(1t)+tn−1An−3(1t))=1tnDn(1t)(tn−1An−1(1t)−tnAn−1(1t)−tn−2An−4(1t)+2t+t2A1(1t)+⋯+tn−3An−4(1t)+2tn−2An−3(1t)+tn−1An−3(1t))=1tnDn(1t)(tn−2An−2(1t)−tn−2An−3(1t)−tn−1An−2(1t)+tn−1An−3(1t)−tn−2An−4(1t)+2t+t2A1(1t)+t3A2(1t)+⋯+tn−3An−4(1t)+2tn−2An−3(1t)−tn−1An−3(1t))=1tnDn(1t)(2t+t2A1(1t)+t3A2(1t)+⋯+tn−4An−5(1t)+tn−3An−3(1t))=1tnLn(1t)(2t+t2A2(1t))=1tnDn(1t)(2t+tA1(1t)−tA0(1t))=1tnDn(1t)

□

2.3 Conjecture on the upper bound of growth rate of M(Dn∞)

Now we compute the growth rates of M(Dn∞) and construct the graph using these growth rates. We see that the growth rate of M(Dn∞) is bounded above by 4. Let r_n be the growth rate (or the maximal root of the polynomial D_n(λ)).

We compute few initial growth rates (using Mathematica and Derive 6) for M(Dn∞). We have the following few initial values of r_n: r₃ = 2.6180, r₄ = 3.1478, r₅ = 3.4142, r₆ = 3.5618, r₇ = 3.6657, r₈ = 3.7320, r₉ = 3.7793, r₁₀ = 3.8144, r₁₁ = 3.8413, r₁₂ = 3.8624, r₁₃ = 3.8793, r₁₄ = 3.8932, r₁₅ = 3.9046, r₁₆ = 3.9142, r₁₇ = 3.9224, r₁₈ = 3.9294, r₁₉ = 3.9354, r₂₀ = 3.9407.

We also compute r₄₀ = 3.9817, r₆₀ = 3.9911, r₈₀ = 3.9947, r₁₀₀ = 3.9965, and r₁₂₀ = 3.9975 using Mathematica. We have the following graph representing the growth rate of M(Dn∞):

We observe that the growth rate for M(Dn∞) approaching 4 as n approaches ∞. Hence at the end we have

Conjecture: The growth rate of M(Dn∞) is bounded above by 4.

3 Conclusions

We computed the Hilbert series of the right-angled affine Artin monoid M(Dn∞) for the first time. We also formulated new recurrence relations for this monoid. In the end we graphically proved that growth rate is bounded by 4 as it is the case in [7] with all spherical Artin monoids. We also posed it as an open problem for its rigorous proof.

Competing interests: The authors declare that they have no competing interests.
Authors’ contributions: All the authors jointly worked on deriving the results and approved the final manuscript.

Acknowledgement

Authors are extremely thankful for reviewer’s comments.

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Received: 2018-03-09

Accepted: 2018-06-11

Published Online: 2018-08-13

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