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Volume 16, Issue 1

Issues

Volume 13 (2015)

On right-angled spherical Artin monoid of type Dn

Zaffar Iqbal / Abdul Rauf Nizami / Mobeen Munir / Amlish Rabia / Shin Min Kang
  • Corresponding author
  • Department of Mathematics and RINS, Gyeongsang National University, Jinju, Korea
  • Center for General Education, China Medical University, Taichung, Taiwan
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Published Online: 2018-08-13 | DOI: https://doi.org/10.1515/phys-2018-0061

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 ai, iI and relations ai2 = 1 and aiajai = ajaiaj, i, jI. 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, An is the first. The Artin group associated to An 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 MBn, which leads to compute the Hilbert series of MBn 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 = (mst)s,tS such that

  • mss = 1 for all sS;

  • mst = mts ∈ {2, 3, 4, …, ∞} for all s, tS, st.

    A Coxeter graph Γ is a labeled graph defined by the following data:

  • S is a set of vertices of Γ.

  • Two vertices s, tS, st are joined by an edge if mst ≥ 3. This edge is labeled by mst if mst ≥ 4.

Remark 1.1

A Coxeter matrix M = (mst)s,tS is usually represented by its Coxeter graph Γ(M).

Definition 1.2

Let M = (mst)s,tS be the Coxeter matrix and Γ(M) its Coxeter graph. Then the group

W=sSs2=1,(st)mst=1,s,tS,st

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

In a simple way, we can write W=sSs2=1,stsmstfactors=tstmstfactors,s,tS,st. 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]).

Spherical Coxeter graphs
Figure 1

Spherical Coxeter graphs

By convention mij is the label of the edge between xi and xj (ij); if there is no label then mij = 3. If there is no edge between xi and xj, then mij = 2.

To a given Coxeter graph Xn we associate the monoid M(Xn) with generators (corresponds to vertices) x1, x2, …, xn and relations (corresponds to labels mij of the graphs) xixjxixjmijfactors=xjxixjximijfactors, where 1 ≤ j < in.

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

Definition 1.3

If W is a Coxeter group with Coxeter matrix M = (mst)s,tS, then the Artin group associated to W is defined by

A=sS|stsmstfactors=tstmstfactors.

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 mst ≥ 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 lenth lS(g) of an element gG is the smallest integer n for which there exists s1, …, snSS−1 such that g = s1sn.

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 ak of elements gG such that lS(g) = k. The corresponding spherical growth series (also known as the generating function) or the Hilbert series is given by HG(t)=k=0aktk.

The affine (or infinite) Coxeter groups form another important series of Coxeter groups. These well-known affine Coxeter groups are n, n, n, n, 6, 7, 8, 2 and 1 (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, …, xn 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 w2X*, where w1 = x1x2xr, w2 = y1y2yr with xi, yiX. Then w1 < w2 length-lexicographically (or quasi-lexicographically) if there is a kr such that xk < yk and xi = yi 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 ×w α2 = 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 γαδγβδ 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 < ⋯ < xn on the generators. Hence clearly we have

Lemma 1.8

The presentation of the monoid M(Dn) has generators x1, x2, …, xn and has relations xixj = xjxi, 3 ≤ j + 2 ≤ in − 1, and xnxk = xkxn, kn − 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, …, λk are the eigenvalues of A(t) and ui is an eigenvector corresponding to λi. The largest eigenvalue is the growth rate of the sequence (ui(t)) (by the definition of growth rate).

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

Lemma 2.1

The monoid M(Dn) satisfies the following recurrence relations

  1. c0 = 1, c1;i = 1, ck=i=1nck;i (k ≥ 1).

  2. ck;i (k ≥ 2) is given by following recurrence relations

    ck;i={i=1nck1;i,i=1,i=j1nck1;i(j=2,3,,n1),ck1;i2+ck1;i,i=n.

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

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

Mn=[111111111111011111001111000111000101]

and its characteristic polynomial Dn(λ) is

|λ1111111λ11111011111001λ1110001λ1100010λ1|.

Lemma 2.2

[7] The polynomials 𝓐n(λ) for the monoid M(An) satisfy the recurrence

An(λ)=λAn1(λ)λAn2(λ)(n2)(2.1)

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

Theorem 2.3

The polynomials (Dn(λ))n≥4 satisfy the recurrence

Dn(λ)=(λ1)An1(λ)λ2An4(λ),n4(2.2)

with D0(λ) = 1, D1(λ) = λ − 1, D2(λ) = λ2 − 2λ and D3(λ) = λ(λ2 − 3λ + 1) as the initial values.

Proof

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

Dn(λ)=|λ1111111λ11111011111001λ1110001λ1100010λ1|.

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

Dn(λ)=|λ1111111λ11111011111001λ1110001λ1100000λ1|+|λ1111111λ11111011111001λ1110001λ11000100|.

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

Dn(λ)=(λ1)An1(λ)V(λ),

where

V(λ)=|λ1111111λ11111011111001λ1110001110000λ11|.

By subtracting last column from second last column we get

V(λ)=|λ1111101λ11110011110001λ1100001100000λ1λ|

and expanding by last row we get

=λ|λ1111111λ1111101λ11110001λ11000011|.

Subtracting last column from second last column we have

V(λ)=λ|λ111011λ110101101001λ100001|.

Simplifying last determinant, we finally have

V(λ)=λ2|λ111111λ111101λ1110001λ1|.

Therefore we have the result

Dn(λ)=(λ1)An1(λ)λ2An4(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)=k0cktk denote the Hilbert series of M(Dn), where ck = #{canonical words of length k}. Let HM;i(n)(t)=k0ck;i(n)tk denote the Hilbert series of M(Dn) of words starting with xi, where ck;i = #{canonical words starting with xi of length k}.

Theorem 2.4

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

  1. HM(n)(t)=1+i=1nHM;i(n)(t)

  2. HM;1(n)(t)=HM;2(n)(t)

  3. HM;j(n)(t)=t+ti=j1nHM;i(n)(t)   (j = 2, 3, …, n − 1)

  4. HM;n(n)(t)=t+tHM;n2(n)(t)+tHM;n(n)(t)

Proof

  1. From Lemma 2.1 we have ck=i=1nck;i, (k ≥ 1). Therefore HM(n)(t)=k0cktk=c0+k1cktk = 1 + k1i=1nck;itk=1+i=1nk1ck;itk=1+i=1nHM;i(n)(t).

  2. is obvious as ck;1 = ck;2.

  3. From Lemma 2.1 we have, ck;j=i=j1nck1;i (j = 2, 3, …, n − 1). Hence HM;j(n)(t)=k1nck;j(t)tk = c1;j(t)t+k2nck;j(t)tk=t+k2ni=j1nck1;i(t)tk = t + ti=j1nk2nck1;i(t)tk1=t+ti=j1nHM;i(n)(t).

    Similarly we can easily prove (4). □

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

Wn=[1ttttttt1ttttt0ttttt00t1ttt000t1tt000t01t],X=[HM;1(n)(t)HM;2(n)(t)HM;3(n)(t)HM;n(n)(t)],andB=[tttt].

Lemma 2.5

In the monoid M(Dn)

det(Wn)=tnDn(1t).

Proof

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

Lemma 2.6

In M(Dn)

HH;m(n)(t)={tm1Am2(1t)tnDn(1t),2mn1tm2(1t)Am3(1t)tnDn(1t),m=n

Proof

The system explained in Theorem 2.4 of n equations in n variables HM;i(n)(t),1in is already written in the form WnX = 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:

  • Case I

    2 ≤ mn − 1. By using Cramer’s rule we have

    HM;m(n)(t)=Tmdet(Wn),

    where Tm is a determinant obtained by replacing mth column of Wn by column of B. That is, [HM;m(n)(t) is now becomes

    1tnDn(1t)|1ttttttt1ttttt0ttttt00tttt000t1tt000t01t|.

    Let Ci denote the ith column of Tm. Adding Cm in Cm+1, Cm+2, …, Cn of Tm and simplifying we have a determinant of order m. Hence

    HM;m(n)(t)=1tnDn(1t)|1tttt00t1ttt000ttt0000tt00000t10000t01|.

    Now simplifying it we finally have

    HM;m(n)(t)=tm1Am2(1t)tnDn(1t).

  • Case II

    m = n.

    Again using the Cramer’s rule, HM;n(n)(t) becomes

    1tnDn(1t)|1ttttttt1ttttt0ttttt00t1ttt000t1tt000t0t|.

    Let Ci denote the ith column of Dn. Adding Cn in Cn−1 and Cn−2, we get

    HM;n(n)(t)=1tnDn(1t)|1ttt00tt1tt00t0tt00t00t10t00001t00000t|.

    Therefore

    HM;n(n)(t)=tm2(1t)Am3(1t)tnDn(1t),m=n

 □

Now we have our main result.

Theorem 2.7

The Hilbert series of the monoid M(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;n1[n)(t)+HM;n(n)(t)=1tnDn(1t)(tnDn(1t)+2t+t2A1(1t)++tn3An4(1t)+2tn2An3(1t)+tn1An3(1t))=1tnDn(1t)(tn1An1(1t)tnAn1(1t)tn2An4(1t)+2t+t2A1(1t)++tn3An4(1t)+2tn2An3(1t)+tn1An3(1t))=1tnDn(1t)(tn2An2(1t)tn2An3(1t)tn1An2(1t)+tn1An3(1t)tn2An4(1t)+2t+t2A1(1t)+t3A2(1t)++tn3An4(1t)+2tn2An3(1t)tn1An3(1t))=1tnDn(1t)(2t+t2A1(1t)+t3A2(1t)++tn4An5(1t)+tn3An3(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 rn be the growth rate (or the maximal root of the polynomial Dn(λ)).

We compute few initial growth rates (using Mathematica and Derive 6) for M(Dn). We have the following few initial values of rn: 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.

Acknowledgement

Authors are extremely thankful for reviewer’s comments.

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About the article

Received: 2018-03-09

Accepted: 2018-06-11

Published Online: 2018-08-13


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.


Citation Information: Open Physics, Volume 16, Issue 1, Pages 441–447, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0061.

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© 2018 Zaffar Iqbal et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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