﻿ Tight Monomials with t-Value ≤ 9 for Quantum Group of Type D<sub>4</sub>

Advances in Linear Algebra & Matrix Theory
Vol.07 No.04(2017), Article ID:81010,24 pages
10.4236/alamt.2017.74009

Tight Monomials with t-Value ≤ 9 for Quantum Group of Type D4

Yuwang Hu1*, Jifang Hu2, Qiongru Wu3

1School of Mathematics and Statistics, Xinyang Normal University, Xinyang, China

2School of Sciences, Chang’an University, Xi’an, China

3School of Mathematics and Information, Xinyang University, Xinyang, China    Received: August 31, 2017; Accepted: December 9, 2017; Published: December 12, 2017

ABSTRACT

All monomials with t-value ≤9 in Canonical basis of quantum group for type D4 are determined in this paper.

Keywords:

Quantum Group, Canonical Basis, Tight Monomial 1. Introduction

Quantum group, also called quantized enveloping algebra, was introduced independently by (Drinfel’d, V. G., 1985)  and (Jimbo, M., 1985)  . It plays an important role in the study of Lie groups, Lie algebras, algebraic groups, Hopf algebras, etc. The positive part of a quantum group has a kind of important basis, i.e., Canonical basis $B$ introduced by (Lusztig, G., 1990)  , which plays an important role in the theory of quantum groups and their representations. Some efforts on $B$ have been done. (Lusztig, G., 1990)  introduced algebraic definition of $B$ for the quantum groups in the simply laced cases, and gave explicitly the longest monomials in $B$ for type ${A}_{1},{A}_{2}$ . Afterwards, (Lusztig, G., 1992)  extended algebraic definition of $B$ to the non-simply laced cases and gave 2 longest monomials in $B$ for type ${B}_{2}$ . Then, (Lusztig, G., 1993)  associated a quadratic form to every monomial, and proved that, given certain linear conditions, the monomial is tight (respectively, semi-tight) provided that this quadratic form satisfies a certain positivity condition (respectively, nonnegativity condition). He showed that the positivity condition always holds in type ${A}_{3}$ and computed 8 longest tight monomials for type ${A}_{3}$ , and he asked when we have (semi-) tightness in type ${A}_{n}$ . Based on Lusztig’s work, (Xi, N. H., 1999  ; Xi, N. H., 1999  ) found explicitly all 14 elements in $B$ for type ${A}_{3}$ and all 6 elements $B$ for type ${B}_{2}$ . For type ${A}_{4}$ , (Hu, Y. W., Ye, J. C., Yue, X. Q., 2003  ; Hu, Y. W., Ye, J. C., 2005  ; Li, X. C., Hu, Y. W., 2012  ) determined all 62 longest monomials, all 144 polynomials with one-dimensional support, 112 polynomials with two-dimensional support in $B$ . (Marsh, R., 1998)  carried out thorough investigation for type ${A}_{n}$ . He showed that the positivity condition is always satisfied in type ${A}_{4}$ for a certain orientation of the Dynkin diagram, presented a semi-tight longest monomial for type ${A}_{5}$ , and exhibited a special longest monomial for type ${A}_{r}$ (for any $r\ge 6$ ) with a quadratic form that does not even satisfy the conditions for semi-tightness, for any orientation of the Dynkin diagram (although it may turn out that the corresponding monomial is still tight). (Bedard, R., 2004)  proved that all longest monomials of type ${D}_{4}$ are semi-tight. (Reineke, M., 2001)  associated a new quadratic form to every monomial, and gave a sufficient and necessary condition for the monomial to be tight for the simply laced cases. (Deng, B. M., Du, J., 2010)  proved that the Reineke’s criterion works also for any quantized enveloping algebra associated with a symmetrizable Cartan matrix, and they gave all monomials in $B$ for type ${B}_{2}$ , in which 2 longest monomials are the same as Lusztig and Xi’s results. By use of this criterion, (Wang, X. M., 2010)  listed all tight monomials for type ${A}_{3}$ and ${G}_{2}$ , in which 8 longest monomials for type ${A}_{3}$ are same as Lusztig and Xi’s results. (Hu,Y. W., Li, G. W., Wang, J., 2015)  determined all monomials with t-value ≤ 6 in $B$ for type ${A}_{5}$ , and (Hu, Y. W., Geng, Y. J., 2015)  determined all monomials t- value ≤ 6 in $B$ for type ${B}_{3}$ .

This paper computed all monomials with t-value ≤ 9 in $B$ for type ${D}_{4}$ .

2. Preliminaries

Let $C={\left({c}_{ij}\right)}_{i,j\in {\Gamma }_{0}}$ be a Cartan matrix of finite type such that ${c}_{ii}=2,{c}_{ij}\le 0$ for any $i\ne j$ , $D=\text{diag}{\left({d}_{i}\right)}_{i\in {\Gamma }_{0}}$ be a diagonal matrix with integer entries making the matrix DC symmetric. Let $\mathfrak{g}=\mathfrak{g}\left(C\right)$ be the complex semisimple Lie algebra associated with C, and $U={U}_{v}\left(\mathfrak{g}\right)$ (here v is an indeterminate) be the corresponding quantized enveloping algebra, whose positive part ${U}^{+}=〈{E}_{i}|i\in {\Gamma }_{0}〉$ is the $ℚ\left(v\right)$ -subalgebra of $U$ , subject to the relations

$\underset{r+s=1-{c}_{ij}}{\sum }{\left(-1\right)}^{s}{E}_{i}^{\left(s\right)}{E}_{j}{E}_{i}^{\left(r\right)}=0,\forall i,j\in {\Gamma }_{0}$ , where ${E}_{i}^{\left(s\right)}={E}_{i}^{s}/{\left[s\right]}_{i}^{!}$ , ${\left[s\right]}_{i}^{!}={\left[1\right]}_{i}\cdots {\left[s\right]}_{i}$ ,

${\left[a\right]}_{i}=\left({v}^{a{d}_{i}}-{v}^{-a{d}_{i}}\right)/\left({v}^{{d}_{i}}-{v}^{-{d}_{i}}\right)$ . Let $\mathcal{A}=ℤ\left[v,{v}^{-1}\right]$ , ${U}^{+}=〈{E}_{i}^{\left(s\right)}|i\in {\Gamma }_{0},s\in ℕ〉$ be the $\mathcal{A}$ -subalgebra of ${U}^{+}$ . Corresponding to every reduced expression $i=\left({i}_{1},\cdots ,{i}_{\nu }\right)$ of the longest element ${w}_{0}={s}_{{i}_{1}}\cdots {s}_{{i}_{\nu }}$ of the Weyl group $W=〈{s}_{i}〉$ of $\mathfrak{g}$ , one constructs a PBW basis ${B}_{i}$ of ${U}^{+}$ . Lusztig  proved that the $ℤ\left[{v}^{-1}\right]$ -submodule ${\mathcal{L}}_{i}=〈{B}_{i}〉$ of ${U}^{+}$ is independent of the choice of $i$ , write it $\mathcal{L}$ ; the image of ${B}_{i}$ under the canonical projection $\pi :\mathcal{L}\to \mathcal{L}/{v}^{-1}\mathcal{L}$ is independent of the choice of $i$ , write it B; for any element $b\in B$ there is a unique element $b\in \mathcal{L}$ which is fixed by the bar map of ${U}^{+}$ defined by $v\to {v}^{-1}$ and satisfies $\pi \left(b\right)=b$ . The set $B=\left\{b|b\in B\right\}$ forms a $ℤ\left[{v}^{-1}\right]$ -basis of $\mathcal{L}$ , an $\mathcal{A}$ -basis of ${U}^{+}$ and a $ℚ\left(v\right)$ -basis of ${U}^{+}$ , Lusztig calls $B$ Canonical basis of quantum group.

According to Lusztig, a monomial in ${U}^{+}$ is an element of the form

${E}_{{i}_{1}}^{\left({a}_{1}\right)}\cdots {E}_{{i}_{t}}^{\left({a}_{t}\right)}$ (1)

where ${i}_{1},\cdots ,{i}_{t}\in {\Gamma }_{0},{a}_{1},\cdots ,{a}_{t}\in ℕ$ . When $t=\nu$ and ${s}_{{i}_{1}}\cdots {s}_{{i}_{\nu }}={w}_{0}$ is the longest element of Weyl group, the monomial (1) is called the longest monomial. We say that (1) is tight (or semi-tight) if it belongs to $B$ (or is a linear combination of elements in $B$ with constant coefficients).

Let $Q=\left({Q}_{0},{Q}_{1}\right)$ be a finite quiver with vertex set ${Q}_{0}$ and arrow set ${Q}_{1}$ . Write $\rho \in {Q}_{1}$ as ${t}_{\rho }\stackrel{\rho }{\to }{h}_{\rho }$ , where ${h}_{\rho }$ and ${t}_{\rho }$ denote the head and the tail of $\rho$ respectively. An automorphism $\sigma$ of Q is a permutation on the vertices of Q and on the arrows of Q such that $\sigma \left({h}_{\rho }\right)={h}_{\sigma \left(\rho \right)}$ and $\sigma \left({t}_{\rho }\right)={t}_{\sigma \left(\rho \right)}$ for any $\rho \in {Q}_{1}$ . Denote the quiver with automorphism $\sigma$ as $\left(Q,\sigma \right)$ . Attach to the pair $\left(Q,\sigma \right)$ a valued quiver $\Gamma =\Gamma \left(Q,\sigma \right)=\left({\Gamma }_{0},{\Gamma }_{1}\right)$ as follows. Its vertex set ${\Gamma }_{0}$ and arrow set ${\Gamma }_{1}$ are simply the sets of s-orbits in ${Q}_{0}$ and ${Q}_{1}$ , respectively. The valuation of $\Gamma$ is given by ${d}_{i}$ = #{vertices in the s-orbit of i}, $\forall i\in {\Gamma }_{0}$ ; ${m}_{\rho }$ = #{arrows in the σ-orbit of ρ}, $\forall \rho \in {\Gamma }_{1}$ . The Euler form of $\Gamma$ is defined to be the bilinear form $〈,〉:ℤ\left[{\Gamma }_{0}\right]×ℤ\left[{\Gamma }_{0}\right]\to ℤ$ given by

$〈X,Y〉=\underset{i\in {\Gamma }_{0}}{\sum }{d}_{i}{x}_{i}{y}_{i}-\underset{\rho \in {\Gamma }_{1}}{\sum }{m}_{\rho }{x}_{{t}_{\rho }}{y}_{{h}_{\rho }}$ , where $X=\underset{i\in {\Gamma }_{0}}{\sum }{x}_{i}i$ , $Y=\underset{i\in {\Gamma }_{0}}{\sum }{y}_{i}i\in ℤ\left[{\Gamma }_{0}\right]$ , so

$X\cdot Y=〈X,Y〉+〈Y,X〉$ is the symmetric Euler form. The valued quiver $\Gamma$

defines a Cartan matrix ${C}_{\Gamma }={C}_{Q,\sigma }={\left({c}_{ij}\right)}_{i,j\in {\Gamma }_{0}}$ , where

${c}_{ij}=\left\{\begin{array}{l}2-2\underset{\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\rho \in {\Gamma }_{1}\\ {h}_{\rho }={t}_{\rho }=i\end{array}}{\sum }\frac{{m}_{\rho }}{{d}_{i}},i=j;\\ -\underset{\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\rho \in {\Gamma }_{1}\\ \left\{{h}_{\rho },{t}_{\rho }\right\}=\left\{i,j\right\}\end{array}}{\sum }\frac{{m}_{\rho }}{{d}_{i}},i\ne j.\end{array}$

For $t\in ℕ$ , let $i=\left({i}_{1},\cdots ,{i}_{t}\right)\in {\Gamma }_{0}^{t}$ , $a=\left({a}_{1},\cdots ,{a}_{t}\right)\in {ℕ}^{t}$ , and write ${E}_{i}^{\left(a\right)}={E}_{{i}_{1}}^{\left({a}_{1}\right)}\cdots {E}_{{i}_{t}}^{\left({a}_{t}\right)}\in {U}^{+}$ . Define ${\mathcal{M}}_{i,a}=\left\{A={\left({a}_{rm}\right)}_{tt}|{a}_{rm}\in ℕ,\text{ro}\left(A\right)=\text{co}\left(A\right)=a,{a}_{rm}=0,\forall {i}_{r}\ne {i}_{m}\right\}$ , where

$\text{ro}\left(A\right)=\left(\underset{m=1}{\overset{t}{\sum }}\text{ }{a}_{1m},\cdots ,\underset{m=1}{\overset{t}{\sum }}\text{ }{a}_{tm}\right)$ , $\text{co}\left(A\right)=\left(\underset{r=1}{\overset{t}{\sum }}\text{ }{a}_{r1},\cdots ,\underset{r=1}{\overset{t}{\sum }}\text{ }{a}_{rt}\right)$ . Obviously, ${D}_{a}=\text{diag}\left({a}_{1},\cdots ,{a}_{t}\right)\in {\mathcal{M}}_{i,a}$ .

Lusztig gave the following criterion for a monomial to be tight or semi-tight.

Theorem 2.1 ([Lusztig, 1993, §6 Theorem)  . Let $U$ be the quantum group of type ${A}_{n},{D}_{n},{E}_{n}$ , $i\in {\Gamma }_{0}^{t},a\in {ℕ}^{t}$ as above. If the following quadratic form takes only values < 0 on ${\mathcal{M}}_{i,a}\\left\{{D}_{a}\right\}$ (respectively, ≤ 0 on ${\mathcal{M}}_{i,a}$ ), then monomial ${E}_{i}^{\left(a\right)}$ is tight (respectively, semi-tight).

${{q}^{\prime }}_{i,a}\left(A\right)=\underset{\begin{array}{c}1\le m\le t\\ 1\le p

It should be noticed that the above theorem is sufficient but not necessary, M. Reineke gave a sufficient and necessary condition by symmetrizing Lusztig’s quadratic form.

Theorem 2.2 ([Reineke, 2001, Theorem 3.2])  . Let $U$ be the quantum group of type ${A}_{n},{D}_{n},{E}_{n}$ , $i\in {\Gamma }_{0}^{t},a\in {ℕ}^{t}$ as above, the monomial ${E}_{i}^{\left(a\right)}$ is tight if and only if the following quadratic form takes only values <0 on ${\mathcal{M}}_{i,a}\\left\{{D}_{a}\right\}$

${q}_{i,a}\left(A\right)=\underset{\begin{array}{c}1\le m\le t\\ 1\le p

In fact, ${q}_{i,a}\left(A\right)={{q}^{\prime }}_{i,a}\left(A\right)+{{q}^{\prime }}_{i,a}\left({A}^{T}\right)$ (see [Reineke, 2001  , Lemma 3.3]).

Deng and Du generalized the tight monomial criterion given by Reineke to any quantum group associated with symmetrizable matrices.

Theorem 2.3 ([Deng, Du, 2010, Theorem 2.5])  . Let $U$ be the quantum group associated with any symmetrizable matrices, $i\in {\Gamma }_{0}^{t},a\in {ℕ}^{t}$ as above, the monomial ${E}_{i}^{\left(a\right)}$ is tight if and only if the following quadratic form takes only values < 0 on ${\mathcal{M}}_{i,a}\\left\{{D}_{a}\right\}$

${q}_{i,a}\left(A\right)=\underset{\begin{array}{c}1\le m\le t\\ 1\le p

By Theorem 2.3, we have the following Corollaries.

Corollary 2.4. When ${i}_{1},\cdots ,{i}_{t}$ are mutually different, monomial ${E}_{{i}_{1}}^{\left({a}_{1}\right)}\cdots {E}_{{i}_{t}}^{\left({a}_{t}\right)}$ is tight.

Proof: In fact, ${i}_{1},\cdots ,{i}_{t}$ are mutually different, so ${\mathcal{M}}_{i,a}=\left\{{D}_{a}\right\}$ .

Corollary 2.5. If ${E}_{{i}_{p+1}}^{\left({a}_{p+1}\right)}\cdots {E}_{{i}_{p+q}}^{\left({a}_{p+q}\right)}$ is tight, then for any mutually different ${i}_{1},\cdots ,{i}_{p}\notin \left\{{i}_{p+1},\cdots ,{i}_{p+q}\right\}$ and any mutually different ${i}_{p+q+1},\cdots ,{i}_{t}\notin \left\{{i}_{1},\cdots ,{i}_{p};{i}_{p+1},\cdots ,{i}_{p+q}\right\}$ , $t\le l\left({w}_{0}\right)$ , ${E}_{i}^{\left(a\right)}={E}_{{i}_{1}}^{\left({a}_{1}\right)}\cdots {E}_{{i}_{t}}^{\left({a}_{t}\right)}$ is also tight.

Proof: Write $j=\left({i}_{p+1},\cdots ,{i}_{p+q}\right)$ , $b=\left({a}_{p+1},\cdots ,{a}_{p+q}\right)$ , $i=\left({i}_{1},\cdots ,{i}_{t}\right)$ , $a=\left({a}_{1},\cdots ,{a}_{t}\right)$ , then

${E}_{i}^{\left(a\right)}={E}_{{i}_{1}}^{\left({a}_{1}\right)}\cdots {E}_{{i}_{t}}^{\left({a}_{t}\right)},\text{\hspace{0.17em}}{E}_{j}^{\left(b\right)}={E}_{{i}_{p+1}}^{\left({a}_{p+1}\right)}\cdots {E}_{{i}_{p+q}}^{\left({a}_{p+q}\right)}.$

For any $\stackrel{˜}{A}\in {\mathcal{M}}_{i,a}$ , we have

$\stackrel{˜}{A}=\left(\begin{array}{ccc}{D}_{{a}^{\prime }}& 0& 0\\ 0& A& 0\\ 0& 0& {D}_{{a}^{″}}\end{array}\right),$

where ${a}^{\prime }=\left({a}_{1},\cdots ,{a}_{p}\right)$ , ${a}^{″}=\left({a}_{p+q+1},\cdots ,{a}_{t}\right)$ . It is easy to see that $A\in {\mathcal{M}}_{j,b}$ and $q\left(\stackrel{˜}{A}\right)=q\left(A\right)$ . Moreover, $\stackrel{˜}{A}={D}_{a}$ if and only if $A={D}_{b}$ . Since ${E}_{j}^{\left(b\right)}$ is tight, we get by Theorem 2.3 that $q\left(\stackrel{˜}{A}\right)=q\left(A\right)<0$ for all $\stackrel{˜}{A}\in {\mathcal{M}}_{i,a}\\left\{{D}_{a}\right\}$ , applying Theorem 2.3 again, we conclude that ${E}_{i}^{\left(a\right)}$ is tight.

The following two theorems are very useful in determining tight monomials.

Theorem 2.6 ([Deng & Du, 2010, Corollary 2.6, Theorem 6.2)  . Let $i=\left({i}_{1},\cdots ,{i}_{t}\right)\in {\Gamma }_{0}^{t}$ and $a=\left({a}_{1},\cdots ,{a}_{t}\right)\in {ℕ}^{t}$ . If ${E}_{i}^{\left(a\right)}$ is tight, then

(a) For $\forall 1\le r\le s\le t$ , monomial ${E}_{{i}_{r}}^{\left({a}_{r}\right)}\cdots {E}_{{i}_{s}}^{\left({a}_{s}\right)}$ is also tight;

(b) For $\forall 1\le r , ${i}_{r}\ne {i}_{r+1}$ .

Theorem 2.7 ([Lusztig, 1990, Proposition 3.3 and Lusztig, 1993, §13])  . Let $\Phi$ be the non-trivial automorphism of ${U}^{+}$ induced by Dynkin diagram automorphism $\phi$ of $\mathfrak{g}$ , and $\Psi :{U}^{+}\to {\left({U}^{+}\right)}^{\text{o}pp}$ be the unique $ℚ\left(v\right)$ -algebra isomorphism such that ${E}_{j}\to {E}_{j}$ . If ${E}_{i}^{\left(a\right)}$ is tight, then $\Phi \left({E}_{i}^{\left(a\right)}\right)$ and $\Psi \left({E}_{i}^{\left(a\right)}\right)$ are all tight.

A quadratic form $f:{ℤ}^{n}\to ℤ$ denoted by

$f\left({x}_{1},\cdots ,{x}_{n}\right)=\underset{i=1}{\overset{n}{\sum }}\text{ }\text{ }{x}_{i}^{2}+\underset{i

with ${a}_{ij}\in ℤ$ is called a unit form.

The symmetric matrix ${A}_{f}=\left({a}_{ij}\right)$ (when $i>j$ , set ${a}_{ij}={a}_{ji}$ ) with ${a}_{ii}=2$ , defines a bilinear form $f\left(X,Y\right)=X{A}_{f}{Y}^{\text{T}}$ , where

$X=\left({x}_{1},\cdots ,{x}_{n}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Y=\left({y}_{1},\cdots ,{y}_{n}\right).$

In particular, we have

$f\left(X\right)=\frac{1}{2}f\left(X,X\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(X,Y\right)=f\left(X+Y\right)-f\left(X\right)-f\left(Y\right).$

For a vector $w\in {ℚ}^{n}$ with non-negative coordinates, we write $w\ge 0$ . The vector in ${ℤ}^{n}$ which has a 1 in the ith coordinate ( $1\le i\le n$ ) and 0’s elsewhere is denoted by ${r}_{i}$ .

Let $f$ be a unit form. We define the set of positive roots of $f$ as $\left\{w\in {ℤ}^{n}:0\le w\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}f\left(w\right)=1\right\}$ . The linear transformation ${\sigma }_{i}:{ℤ}^{n}\to {ℤ}^{n}$ defined by ${\sigma }_{i}\left(w\right)=w-f\left(w,{r}_{i}\right){r}_{i}$ is called the reflection with respect to ${r}_{i}$ . The transformation ${\sigma }_{i}$ has the property that ${\sigma }_{i}^{2}=\text{id}$ and $f\left({\sigma }_{i}\left(w\right)\right)=f\left(w\right)$ for every $w\in {ℤ}^{n}$ .

Let $f$ be a quadratic form, if $0 for every $0\ne X\in {ℕ}^{n}$ , then we call $f$ weakly positive.

The following algorithm and theorem are taken from (Blouin, Dean, Denver, Pershall, 1995)  :

Let $f:{ℤ}^{n}\to ℤ\left(n\ge 3\right)$ be a unit form. First of all, we define

${R}_{1}=\left\{{r}_{i}:1\le i\le n\right\}.$

Next we want to construct ${R}_{j}$ recursively. Assume that we have defined a set of positive roots of $f$ as

${R}_{s}=\left\{{z}_{1},\cdots ,{z}_{m}\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}s\ge 1,$

and that the process has not failed (to be defined subsequently). Then we construct ${R}_{s+1}$ as follows. Let ${z}_{j}\in {R}_{s}$ . If either:

1) there is some $1\le i\le n$ such that $f\left({z}_{j},{r}_{i}\right)\le -2$ , or

2) there is some $1\le i\le n$ such that ${z}_{j}\left(i\right)\ge 7$ ,

then the process is said to fail. Assume the process does not fail (so 1) and 2) do not occur for any ${z}_{j}\in {R}_{s}$ ). Let ${D}_{s}\subseteq {R}_{s}$ be the set of those roots ${z}_{j}$ with the property that there is some $1\le i\le n$ such that $f\left({z}_{j},{r}_{i}\right)=-1$ . If ${D}_{s}=\varphi$ , then ${R}_{s+1}:=\varphi$ and the process is said to be successful. If ${D}_{s}\ne \varphi$ , then

${R}_{s+1}:=\left\{{\sigma }_{i}\left({z}_{j}\right)|{z}_{j}\in {D}_{s}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}{r}_{i}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{such}\text{\hspace{0.17em}}\text{that}\text{\hspace{0.17em}}f\left({z}_{j},{r}_{i}\right)=-1\right\}.$

Remark: If ${R}_{s+1}\ne \varphi$ , then we apply the algorithm again to obtain ${R}_{s+2}$ , etc. The roots in ${R}_{s+1}$ (if ${R}_{s+1}\ne \varphi$ ) are all greater than the roots in ${R}_{s}$ . Thus condition 2) guarantees that this procedure is finite and if it is not successful then it will eventually fail.

Theorem 2.8 ([Blouin et al., 1995, Theorem 4])  . The unit form $f$ is weakly positive if and only if the above process is successful. If the process is successful with ${R}_{s+1}=\varphi$ , then ${R}_{1}\cup \cdots \cup {R}_{s}$ are all the positive roots of $f$ .

3. Main Results

Let $\mathfrak{g}$ be of type ${D}_{4}$ as follows.

Let $i=\left({i}_{1},\cdots ,{i}_{t}\right)\in {\Gamma }_{0}^{t}$ , $a=\left({a}_{1},\cdots ,{a}_{t}\right)\in {ℕ}^{t}$ . For convenience, we abbreviate a monomial ${E}_{{i}_{1}}^{\left({a}_{1}\right)}\cdots {E}_{{i}_{t}}^{\left({a}_{t}\right)}$ with any $\left({a}_{1},\cdots ,{a}_{t}\right)\in {ℕ}^{t}$ as a word ${i}_{1}\cdots {i}_{t}$ (1 as 0), an inequality ${a}_{{j}_{1}}+\cdots +{a}_{{j}_{p}}\le {a}_{{l}_{1}}+\cdots +{a}_{{l}_{q}}$ as ${j}_{1}\cdots {j}_{p}-{l}_{1}\cdots {l}_{q}$ . For example, a monomial ${E}_{1}^{\left({a}_{1}\right)}{E}_{2}^{\left({a}_{2}\right)}{E}_{3}^{\left({a}_{3}\right)}{E}_{4}^{\left({a}_{4}\right)}$ is abbreviated to a word 1234, and a monomial ${E}_{1}^{\left({a}_{1}\right)}{E}_{2}^{\left({a}_{2}\right)}{E}_{1}^{\left({a}_{3}\right)}\left({a}_{1}+{a}_{3}\le {a}_{2}\right)$ to a word 121 (13 − 2), etc.

If ${s}_{{i}_{1}}\cdots {s}_{{i}_{t}}$ is a reduced expression, we call ${i}_{1}\cdots {i}_{t}$ a reduced word. By Theorem 2.6 (b), we only consider those reduced tight words ${i}_{1}\cdots {i}_{t}$ with ${i}_{r}\ne {i}_{r+1},\forall 1\le r , in this case, ${i}_{1}\cdots {i}_{t}$ is called the word with t-value. If ${i}_{r}\cdot {i}_{r+1}=0$ for some $1\le r , we identify the word ${i}_{1}\cdots {i}_{r-1}{i}_{r}{i}_{r+1}{i}_{r+2}\cdots {i}_{t}$ with the word ${i}_{1}\cdots {i}_{r-1}{i}_{r+1}{i}_{r}{i}_{r+2}\cdots {i}_{t}$ . Denote the set of all words with t-value by ${M}_{t}$ . The non-trivial Dynkin diagram automorphism $\phi$ of $\mathfrak{g}$ is $\phi :1↦3,3↦4,4↦1,2↦2$ . Let us present the so called M − S word-procedure from t-value to $\left(t+1\right)$ -value.

Step 1. Take any ${i}_{1}\cdots {i}_{t}\in {M}_{t}$ , adding a number ${i}_{t+1}\in \left\{1,2,3,4\right\}$ different from ${i}_{1}$ (or ${i}_{t}$ ) in the front (or behind) of ${i}_{1}$ (or ${i}_{t}$ ), deleting the those words with t-value, getting all words with $\left(t+1\right)$ -value from ${i}_{1}\cdots {i}_{t}$ .

Step 2. Repeat step 1 until all words in ${M}_{t}$ are considered, deleting the non-tight words with $\left(t+1\right)$ -value, get ${M}_{t+1}$ .

Step 3. Use $\Phi$ and $\Psi$ , we have ${S}_{t+1}$ satisfying ${M}_{t+1}=\Phi \left({S}_{t+1}\right)\cup \Psi \Phi \left({S}_{t+1}\right)$ .

For example, applying the $M-S$ word-procedure to ${M}_{1}=\left\{1,2,3,4\right\}$ , get ${M}_{2}=\left\{12,13,14,21,23,24,32,34,42\right\}$ . Considering $\Phi ,\Psi$ , $24\stackrel{\phi }{←}23\stackrel{\phi }{←}21\stackrel{\Psi }{←}12\stackrel{\phi }{\to }32\stackrel{\phi }{\to }42$ and $13\stackrel{\phi }{\to }34\stackrel{\phi }{\to }14$ , so ${S}_{2}=\left\{12,13\right\}$ .

From now on, write ${M}_{t}=\Phi \left({S}_{t}\right)\cup \Psi \Phi \left({S}_{t}\right),t\ge 2$ .

Theorem 3.1. For the quantum group for type ${D}_{4}$ , we have the following results.

1) $t=0$ , ${M}_{0}=\left\{0\right\}$ , tight monomial has only one.

2) $t=1$ , ${M}_{1}=\Phi \left({S}_{1}\right)$ , tight monomials have 4 families, where ${S}_{1}=\left\{1,2\right\}$ .

3) $t=2$ , ${M}_{2}$ includes 9 families of tight monomials, where ${S}_{2}=\left\{12,13\right\}$ .

4) $t=3$ , ${M}_{3}$ includes 19 families of tight monomials, where ${S}_{3}={S}_{3}^{1}\cup {S}_{3}^{2}$ ,

${S}_{3}^{1}=\left\{123,132,134\right\}$ ; ${S}_{3}^{2}=\left\{121,212\left(13-2\right)\right\}$ .

5) $t=4$ , ${M}_{4}$ includes 35 families of tight monomials, where ${S}_{4}={S}_{4}^{1}\cup {S}_{4}^{2}\cup {S}_{4}^{3}$ ,

${S}_{4}^{1}=\left\{1234,1342\right\}$ ; ${S}_{4}^{2}=\left\{1213,1214,2123,2124\left(13-2\right)\right\}$ ; ${S}_{4}^{3}=\left\{2132\left(14-23\right)\right\}$ .

6) $t=5$ , M5 includes 58 families of tight monomials, where ${S}_{5}={S}_{5}^{1}\cup {S}_{5}^{2}\cup {S}_{5}^{3}$ ,

$\begin{array}{l}{S}_{5}^{1}=\left\{12134,24213\left(13-2\right);31214,12324\left(24-3\right);\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}12342\left(25-34\right);21342\left(15-234\right)\right\};\end{array}$

$\begin{array}{l}{S}_{5}^{2}=\left\{12312,12412\left(14-2,25-34\right);13213\left(14-3,25-3\right);\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}12321,12421\left(15-24-3\right)\right\};\end{array}$

${S}_{5}^{3}=\left\{21232\left(13-2,35-4\right)\right\}.$

7) $t=6$ , ${M}_{6}$ includes 93 families of tight monomials, where ${S}_{6}={S}_{6}^{1}\cup {S}_{6}^{2}\cup {S}_{6}^{3}\cup {S}_{6}^{4}$ ,

$\begin{array}{l}{S}_{6}^{1}=\left\{123124,124123\left(14-2,25-34\right);123214,124213\left(15-24-3\right);\\ \text{ }\text{ }132142,312342\left(14-3,36-45\right);132134\left(14-3,25-3\right);\\ \text{ }\text{ }123421\left(16-25-34\right);123142\left(14-2,26-345\right)\right\};\end{array}$

${S}_{6}^{2}=\left\{132132\left(14-3,25-3,36-45\right)\right\};$

${S}_{6}^{3}=\left\{212342\left(13-2,36-45\right);123242,124232\left(24-3,46-5\right)\right\};$

$\begin{array}{l}{S}_{6}^{4}=\left\{123121,124121\left(14-2,46-5,25-34\right);\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }123212,124212\left(46-5,15-24-3\right)\right\}.\end{array}$

8) $t=7$ , ${M}_{7}$ includes 133 families of tight monomials, where ${S}_{7}={S}_{7}^{1}\cup {S}_{7}^{2}\cup {S}_{7}^{3}\cup {S}_{7}^{4}$ ,

$\begin{array}{l}{S}_{7}^{1}=\left\{1234213,1243214\left(16-25-34,37-5\right);\\ \text{ }\text{ }1321342\left(14-3,25-3,37-456\right);1342134\left(15-4,26-4,37-4\right);\\ \text{ }\text{ }1234123\left(15-2,26-345,37-6\right);1324213\left(16-35-4,27-35\right);\\ \text{ }\text{ }1234234\left(36-5,47-5,25-34\right);1342132\left(15-4,26-4,47-56\right)\right\};\end{array}$

$\begin{array}{l}{S}_{7}^{2}=\left\{1231242,1241232\left(14-2,57-6,25-34\right);\\ \text{ }\text{ }1231241,1241231\left(25-34,14-2,47-5\right);\\ \text{ }\text{ }1324212,3124232\left(57-6,16-35-4\right);\\ \text{ }\text{ }1232142,1242132\left(15-24-3,47-56\right);\end{array}$

$\begin{array}{l}\text{ }\text{ }1232124,1242123\left(46-5,15-24-3\right);\\ \text{ }\text{ }1232421\left(17-246,24-3,46-5\right);1234212\left(57-6,16-25-34\right);\\ \text{ }\text{ }1231421\left(14-2,47-6,26-345\right);2132142\left(25-4,14-23,47-56\right)\right\};\end{array}$

${S}_{7}^{3}=\left\{2123242\left(13-2,35-4,57-6\right)\right\};$

${S}_{7}^{4}=\left\{2123212,2124212\left(13-2,26-35-4,57-6\right)\right\}.$

9) $t=8$ , ${M}_{8}$ includes 185 families of tight monomials, where ${S}_{8}={S}_{8}^{1}\cup {S}_{8}^{2}\cup {S}_{8}^{3}\cup {S}_{8}^{4}$ ,

$\begin{array}{l}{S}_{8}^{1}=\left\{12134234\left(13-2,26-345,47-6,58-6\right);\\ \text{ }\text{ }12342134\left(16-25-34,48-5,37-5\right);\\ \text{ }\text{ }13421342\left(15-4,48-567,26-4,37-4\right)\right\};\end{array}$

$\begin{array}{l}{S}_{8}^{2}=\left\{12132142,12142132\left(13-2,36-5,25-34,58-67\right);\\ \text{ }\text{ }12142321,12132421\left(13-2,38-57-6,25-34\right)\right\};\end{array}$

$\begin{array}{l}{S}_{8}^{3}=\left\{12132423,12142324\left(13-2,25-34,48-57-6\right);\\ \text{ }\text{ }12134213,12143214\left(13-2,37-6,26-345,48-6\right);\\ \text{ }\text{ }12321423,32123421\left(15-24-3,38-47-56\right);\\ \text{ }\text{ }12324213,12423214\left(17-246,24-3,38-46-5\right);\\ \text{ }\text{ }12342132,12432142\left(16-25-34,58-67,37-5\right);\end{array}$

$\begin{array}{l}\text{ }\text{ }13214232,31234212\left(14-3,27-36-45,68-7\right);\\ \text{ }\text{ }13214213,31234231\left(14-3,47-6,28-36-45\right);\\ \text{ }\text{ }13213242\left(14-3,36-45,68-7,25-3\right);\\ \text{ }\text{ }13242132\left(16-35-4,58-67,27-35\right);\\ \text{ }\text{ }21321342\left(25-4,14-23,36-4,48-567\right)\right\};\end{array}$

$\begin{array}{l}{S}_{8}^{4}=\left\{12321242,12421232\left(15-24-3,46-5,68-7\right);\\ \text{ }\text{ }12324212,12423212\left(17-246,24-3,46-5,68-7\right);\\ \text{ }\text{ }12324232,12423242\left(24-3,68-7,37-46-5\right);\\ \text{ }\text{ }21232142,21242132\left(13-2,58-67,26-35-4\right);\\ \text{ }\text{ }21234212\left(13-2,68-7,27-36-45\right)\right\}.\end{array}$

10) $t=9$ , ${M}_{9}$ includes 265 families of tight monomials, where ${S}_{9}={S}_{9}^{1}\cup {S}_{9}^{2}\cup {S}_{9}^{3}\cup {S}_{9}^{4}\cup {S}_{9}^{5}$ ,

$\begin{array}{l}{S}_{9}^{1}=\left\{121321423,121421324\left(13-2,36-5,25-34,49-58-67\right);\\ \text{ }\text{ }121342132,121432142\left(13-2,37-6,26-345,48-6,69-78\right);\\ \text{ }\text{ }121324213,121423214\left(13-2,38-57-6,49-57,25-34\right);\\ \text{ }\text{ }132142132,312342312\left(14-3,47-6,69-78,28-36-45\right);\end{array}$

$\begin{array}{l}\text{ }\text{ }124213214,123214213\left(15-24-3,39-47-56,58-7\right);\\ \text{ }\text{ }124213213,123214214\left(15-24-3,47-56,58-7,69-7\right);\\ \text{ }\text{ }132134213\left(14-3,25-3,48-7,59-7,37-456\right)\right\};\end{array}$

$\begin{array}{l}{S}_{9}^{2}=\left\{123214234,124213243\left(15-24-3,69-7,38-47-56\right);\\ \text{ }\text{ }121342134\left(13-2,37-6,48-6,59-6,26-345\right);\\ \text{ }\text{ }123421234\left(16-25-34,38-57-6,49-57\right);\\ \text{ }\text{ }123421324\left(16-25-34,37-5,49-58-67\right);\end{array}$

$\begin{array}{l}\text{ }\text{ }123421342\left(16-25-34,59-678,37-5,48-5\right);\\ \text{ }\text{ }132134214\left(14-3,48-7,37-456,25-3,69-7\right);\\ \text{ }\text{ }134213242\left(15-4,79-8,26-4,38-47-56\right);\\ \text{ }\text{ }213421342\left(26-5,15-234,59-678,37-5,48-5\right)\right\};\end{array}$

$\begin{array}{l}{S}_{9}^{3}=\left\{121324232,121423242\left(13-2,25-34,48-57-6,79-8\right);\\ \text{ }\text{ }123212423,124212324\left(15-24-3,46-5,68-7,39-468\right);\\ \text{ }\text{ }123214232,124213242\left(15-24-3,38-47-56,79-8\right);\\ \text{ }\text{ }123242132,124232142\left(17-246,24-3,69-78,38-46-5\right);\\ \text{ }\text{ }123242321,124232421\left(19-2468,24-3,68-7,37-46-5\right);\end{array}$

$\begin{array}{l}\text{ }\text{ }123421232,123421242\left(16-25-34,38-57-6,79-8\right);\\ \text{ }\text{ }132421232,312423212\left(16-35-4,57-6,79-8,28-357\right);\\ \text{ }\text{ }123242123\left(17-246,24-3,46-5,39-468,68-7\right);\\ \text{ }\text{ }212342132,212432142\left(13-2,27-36-45,48-6,69-78\right);\\ \text{ }\text{ }213242132\left(27-46-5,14-23,69-78,38-46\right)\right\};\end{array}$

$\begin{array}{l}{S}_{9}^{4}=\left\{121324212,121423212\left(13-2,25-34,38-57-6,79-8\right);\\ \text{ }\text{ }121321421\left(13-2,36-5,69-8,25-34,58-67\right);\\ \text{ }\text{ }123212421\left(15-24-3,59-68-7,46-5\right)\right\};\end{array}$

$\begin{array}{l}{S}_{9}^{5}=\left\{212321242,212421232\left(13-2,26-35-4,57-6,79-8\right);\\ \text{ }\text{ }212324212\left(13-2,28-357,35-4,57-6,79-8\right)\right\}.\end{array}$

4. Proof of Theorem 3.1

Consider the following quiver $Q=\left({Q}_{0},{Q}_{1}\right)$ of type ${D}_{4}$ where ${Q}_{1}=\left\{1\stackrel{{\rho }_{1}}{\to }2,3\stackrel{{\rho }_{2}}{\to }2,4\stackrel{{\rho }_{3}}{\to }2\right\}$ , ${Q}_{0}=\left\{1,2,3,4\right\}$ . Let $\sigma$ be the identity automorphism of Q such that $\sigma \left(i\right)=i,\sigma \left({\rho }_{i}\right)={\rho }_{i},i=1,2,3,4$ , then the valued quiver of $\left(Q,\sigma \right)$ is $\Gamma =\Gamma \left(Q,\sigma \right)=\left({\Gamma }_{0},{\Gamma }_{1}\right)=\left({Q}_{0},{Q}_{1}\right)$ , the valuation is given by ${d}_{i}=1,{m}_{{\rho }_{i}}=1,i=1,2,3,4$ . For $X={x}_{1}1+{x}_{2}2+{x}_{3}3+{x}_{4}4$ , $Y={y}_{1}1+{y}_{2}2+{y}_{3}3+{y}_{4}4\in ℤ\left[{\Gamma }_{0}\right]$ , Euler form $〈,〉$ on $\Gamma$ is

${〈X,Y〉}_{\Gamma }=\underset{i=1}{\overset{4}{\sum }}\text{ }\text{ }{d}_{i}{x}_{i}{y}_{i}-\underset{i=1}{\overset{3}{\sum }}\text{ }\text{ }{m}_{{\rho }_{i}}{x}_{{t}_{{\rho }_{i}}}{y}_{{h}_{{\rho }_{i}}}={x}_{1}{y}_{1}+{x}_{2}{y}_{2}+{x}_{3}{y}_{3}+{x}_{4}{y}_{4}-{x}_{1}{y}_{2}-{x}_{3}{y}_{2}-{x}_{4}{y}_{2}$ ,

symmetric Euler form $\cdot$ on $\Gamma$ is

$\begin{array}{l}X\cdot Y={〈X,Y〉}_{\Gamma }+{〈Y,X〉}_{\Gamma }=2{x}_{1}{y}_{1}+2{x}_{2}{y}_{2}+2{x}_{3}{y}_{3}\\ +2{x}_{4}{y}_{4}-{x}_{1}{y}_{2}-{x}_{2}{y}_{1}-{x}_{3}{y}_{2}-{x}_{2}{y}_{3}-{x}_{4}{y}_{2}-{x}_{2}{y}_{4}\end{array}$ .

By simple computation, we have $〈i,i〉=1,i=1,2,3,4$ ; $〈1,2〉=〈3,2〉=$ $〈4,2〉=-1$ , the other $〈i,j〉=0,i\ne j$ . So $i\cdot i=2,i=1,2,3,4$ ; $1\cdot 2=2\cdot 3=2\cdot 4=$ $-1$ , $1\cdot 3=1\cdot 4=3\cdot 4=0$ .

Let us prove Theorem 3.1. For any $t\in ℕ$ , let $i=\left({i}_{1},\cdots ,{i}_{t}\right)\in {\Gamma }_{0}^{t}$ , $a=\left({a}_{1},\cdots ,{a}_{t}\right)\in {ℕ}^{t}$ .

Case 1. $t\le 2$ . By Corollary 2.4, words with $t\le 2$ are all tight, so (1)~(3) hold.

Case 2. $t=3$ . Applying the $M-S$ word-procedure to ${M}_{2}$ , we get ${S}_{3}$ . Corollary 2.4 $⇒{S}_{3}^{1}$ . Consider ${S}_{3}^{2}$ , for $i\in \left\{\left(1,2,1\right),\left(2,1,2\right)\right\}$ , we have ${\mathcal{M}}_{i,a}=\left\{M={M}_{x}|x\in ℕ\right\}$ , where

$M=\left(\begin{array}{ccc}{a}_{1}-x& 0& x\\ 0& {a}_{2}& 0\\ x& 0& {a}_{3}-x\end{array}\right),$

and

$\begin{array}{c}q\left(M\right)=2〈{i}_{1},{i}_{1}〉\left({a}_{1}x-{x}^{2}\right)+2〈{i}_{3},{i}_{3}〉\left({a}_{3}x-{x}^{2}\right)+\left({i}_{1}\cdot {i}_{3}\right){x}^{2}+\left(\left({i}_{1}\cdot {i}_{2}\right)+\left({i}_{2}\cdot {i}_{3}\right)\right){a}_{2}x\\ =-2{x}^{2}+2\left({a}_{1}+{a}_{3}-{a}_{2}\right)x.\end{array}$

Obviously, $q\left(M\right)\le 0⇔{a}_{1}+{a}_{3}\le {a}_{2}$ . And $q\left(M\right)=0⇔{a}_{1}+{a}_{3}\le {a}_{2}$ , $x=0$ , so two words in ${S}_{3}^{2}$ are all tight by Theorem 2.3, (4) holds.

Case 3. $t=4$ . Applying the $M-S$ word-procedure to ${M}_{3}$ , we get ${S}_{4}\cup \left\{1212\right\}$ . Corollary 2.4 $⇒{S}_{4}^{1}$ . Corollary 2.5 and ${S}_{3}^{2}⇒{S}_{4}^{2}$ . Consider words 2132, 1212. For word 2132, we have ${\mathcal{M}}_{i,a}=\left\{M={M}_{x}|x\in ℕ\right\}$ , where

$M=\left(\begin{array}{cccc}{a}_{1}-x& 0& 0& x\\ 0& {a}_{2}& 0& 0\\ 0& 0& {a}_{3}& 0\\ x& 0& 0& {a}_{4}-x\end{array}\right),$

and $\begin{array}{c}q\left(M\right)=2〈{i}_{1},{i}_{1}〉\left({a}_{1}x+{a}_{4}x-2{x}^{2}\right)+\left({i}_{1}\cdot {i}_{1}\right){x}^{2}+2\left({i}_{1}\cdot {i}_{2}\right){a}_{2}x+2\left({i}_{1}\cdot {i}_{3}\right){a}_{3}x\\ =-2{x}^{2}+2\left({a}_{1}+{a}_{4}-{a}_{2}-{a}_{3}\right)x.\end{array}$

Obviously, $q\left(M\right)\le 0⇔{a}_{1}+{a}_{4}\le {a}_{2}+{a}_{3}$ . And $q\left(M\right)=0⇔{a}_{1}+{a}_{4}\le {a}_{2}+{a}_{3}$ , $x=0$ , so word 2132(14-23) is tight by Theorem 2.3.

For word 1212, we have ${\mathcal{M}}_{i,a}=\left\{M={M}_{x,y}|\left(x,y\right)\in {ℕ}^{2}\right\}$ , where

$M=\left(\begin{array}{cccc}{a}_{1}-x& 0& x& 0\\ 0& {a}_{2}-y& 0& y\\ x& 0& {a}_{3}-x& 0\\ 0& y& 0& {a}_{4}-y\end{array}\right),$

and $\begin{array}{c}q\left(M\right)=2〈{i}_{1},{i}_{1}〉\left({a}_{1}x+{a}_{3}x-2{x}^{2}\right)+2〈{i}_{2},{i}_{2}〉\left({a}_{2}y+{a}_{4}y-2{y}^{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({i}_{1}\cdot {i}_{1}\right){x}^{2}+\left({i}_{2}\cdot {i}_{2}\right){y}^{2}+2\left({i}_{1}\cdot {i}_{2}\right)\left({a}_{2}x+{a}_{3}y-xy\right)\\ =-{x}^{2}-{y}^{2}-{\left(x-y\right)}^{2}+2\left({a}_{1}+{a}_{3}-{a}_{2}\right)x+2\left({a}_{2}+{a}_{4}-{a}_{3}\right)y.\end{array}$

Obviously, $q\left(M\right)\le 0⇔{a}_{1}+{a}_{3}\le {a}_{2},{a}_{2}+{a}_{4}\le {a}_{3}$ . And $q\left(M\right)=0⇔{a}_{1}+{a}_{3}\le {a}_{2}$ , ${a}_{2}+{a}_{4}\le {a}_{3}$ , $x=y=0$ , but ${a}_{1}+{a}_{3}\le {a}_{2}$ , ${a}_{2}+{a}_{4}\le {a}_{3}⇒{a}_{1}+{a}_{4}\le 0$ , this is a contradiction, so word 1212 is not tight for any $a=\left({a}_{1},{a}_{2},{a}_{3},{a}_{4}\right)\in {ℕ}^{4}$ , (5) holds.

Case 4. $t=5$ . Applying the $M-S$ word-procedure to ${M}_{4}$ , deleting words including subwords in $\Phi \left(1212\right)\cup \Psi \Phi \left(1212\right)$ , we get ${S}_{5}$ . By Corollary 2.5 and ${S}_{3}^{2},{S}_{4}^{3}$ , words in ${S}_{5}^{1}$ besides 21342 are all tight. For word 21342, we have ${\mathcal{M}}_{i,a}=\left\{M={M}_{x}|x\in ℕ\right\}$ , where

$M=\left(\begin{array}{ccccc}{a}_{1}-x& 0& 0& 0& x\\ 0& {a}_{2}& 0& 0& 0\\ 0& 0& {a}_{3}& 0& 0\\ 0& 0& 0& {a}_{4}& 0\\ x& 0& 0& 0& {a}_{5}-x\end{array}\right),$

and $\begin{array}{c}q\left(M\right)=2〈{i}_{1},{i}_{1}〉\left({a}_{1}x+{a}_{5}x-2{x}^{2}\right)+\left({i}_{1}\cdot {i}_{1}\right){x}^{2}+2\left({i}_{1}\cdot {i}_{2}\right){a}_{2}x\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({i}_{1}\cdot {i}_{3}\right){a}_{3}x+2\left({i}_{1}\cdot {i}_{4}\right){a}_{4}x\\ =-2{x}^{2}+2\left({a}_{1}+{a}_{5}-{a}_{2}-{a}_{3}-{a}_{4}\right)x.\end{array}$

Obviously, $q\left(M\right)\le 0⇔{a}_{1}+{a}_{5}\le {a}_{2}+{a}_{3}+{a}_{4}$ . And $q\left(M\right)=0⇔{a}_{1}+{a}_{5}\le {a}_{2}+{a}_{3}+{a}_{4}$ , $x=0$ , so word 21342(15-234) is tight by Theorem 2.3.

Consider ${S}_{5}^{2}$ . For words 12312, 12412, 13213, we have ${\mathcal{M}}_{i,a}=\left\{M={M}_{x,y}|\left(x,y\right)\in {ℕ}^{2}\right\}$ , where

$M=\left(\begin{array}{ccccc}{a}_{1}-x& 0& 0& x& 0\\ 0& {a}_{2}-y& 0& 0& y\\ 0& 0& {a}_{3}& 0& 0\\ x& 0& 0& {a}_{4}-x& 0\\ 0& y& 0& 0& {a}_{5}-y\end{array}\right),$

and

$\begin{array}{l}q\left(M\right)=2〈{i}_{1},{i}_{1}〉\left({a}_{1}x+{a}_{4}x-2{x}^{2}\right)+2〈{i}_{2},{i}_{2}〉\left({a}_{2}y+{a}_{5}y-2{y}^{2}\right)+\left({i}_{1}\cdot {i}_{1}\right){x}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({i}_{2}\cdot {i}_{2}\right){y}^{2}+2\left({i}_{1}\cdot {i}_{3}\right){a}_{3}x+2\left({i}_{2}\cdot {i}_{3}\right){a}_{3}y+2\left({i}_{1}\cdot {i}_{2}\right)\left({a}_{2}x+{a}_{4}y-xy\right)\\ =\left\{\begin{array}{l}-{\left(x-y\right)}^{2}-{x}^{2}-{y}^{2}+2\left({a}_{1}+{a}_{4}-{a}_{2}\right)x+2\left({a}_{2}+{a}_{5}-{a}_{3}-{a}_{4}\right)y,\text{\hspace{0.17em}}12312,12412,\\ -2{x}^{2}-2{y}^{2}+2\left({a}_{1}+{a}_{4}-{a}_{3}\right)x+2\left({a}_{2}+{a}_{5}-{a}_{3}\right)y,\text{\hspace{0.17em}}13213.\end{array}\end{array}$

Obviously, $q\left(M\right)\le 0⇔\left\{\begin{array}{l}{a}_{1}+{a}_{4}\le {a}_{2},{a}_{2}+{a}_{5}\le {a}_{3}+{a}_{4},\text{\hspace{0.17em}}12312,12412,\\ {a}_{1}+{a}_{4}\le {a}_{3},{a}_{2}+{a}_{5}\le {a}_{3},\text{\hspace{0.17em}}13213\end{array}$ . And $q\left(M\right)=0⇔\left\{\begin{array}{l}{a}_{1}+{a}_{4}\le {a}_{2},{a}_{2}+{a}_{5}\le {a}_{3}+{a}_{4},\text{\hspace{0.17em}}12312,12412,\\ {a}_{1}+{a}_{4}\le {a}_{3},{a}_{2}+{a}_{5}\le {a}_{3},\text{\hspace{0.17em}}13213\end{array}$ and $x=y=0$ , so

words 12312, 12412(14-2, 25-34), 13213(14-3, 25-3) are tight by Theorem 2.3.

For words 12321, 12421, we have ${\mathcal{M}}_{i,a}=\left\{M={M}_{x,y}|\left(x,y\right)\in {ℕ}^{2}\right\}$ , where

$M=\left(\begin{array}{ccccc}{a}_{1}-x& 0& 0& 0& x\\ 0& {a}_{2}-y& 0& y& 0\\ 0& 0& {a}_{3}& 0& 0\\ 0& y& 0& {a}_{4}-y& 0\\ x& 0& 0& 0& {a}_{5}-x\end{array}\right),$

and $\begin{array}{c}q\left(M\right)=2〈{i}_{1},{i}_{1}〉\left({a}_{1}x+{a}_{5}x-2{x}^{2}\right)+2〈{i}_{2},{i}_{2}〉\left({a}_{2}y+{a}_{4}y-2{y}^{2}\right)+\left({i}_{1}\cdot {i}_{1}\right){x}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({i}_{2}\cdot {i}_{2}\right){y}^{2}+2\left({i}_{1}\cdot {i}_{2}\right)\left({a}_{2}x+{a}_{4}x\right)+2\left({i}_{1}\cdot {i}_{3}\right){a}_{3}x+2\left({i}_{2}\cdot {i}_{3}\right){a}_{3}y\\ =-2{x}^{2}-2{y}^{2}+2\left({a}_{1}+{a}_{5}-{a}_{2}-{a}_{4}\right)x+2\left({a}_{2}+{a}_{4}-{a}_{3}\right)y\end{array}$

Obviously, $q\left(M\right)\le 0⇔{a}_{1}+{a}_{5}\le {a}_{2}+{a}_{4}\le {a}_{3}$ . And $q\left(M\right)=0⇔{a}_{1}+{a}_{5}\le {a}_{2}+{a}_{4}\le {a}_{3}$ and $x=y=0$ , so words 12321, 12421(14-25-3) are tight by Theorem 2.3.

Now let us consider ${S}_{5}^{3}$ , we have ${\mathcal{M}}_{i,a}=\left\{M={M}_{x,{x}_{1},{x}_{2},{x}_{3}}|\left(x,{x}_{1},{x}_{2},{x}_{3}\right)\in {ℕ}^{4}\right\}$ , where

$M=\left(\begin{array}{ccccc}{a}_{1}-x-{x}_{1}& 0& x& 0& {x}_{1}\\ 0& {a}_{2}& 0& 0& 0\\ {x}_{2}& 0& {a}_{3}-{x}_{2}-{x}_{3}& 0& {x}_{3}\\ 0& 0& 0& {a}_{4}& 0\\ x+{x}_{1}-{x}_{2}& 0& {x}_{2}+{x}_{3}-x& 0& {a}_{5}-{x}_{1}-{x}_{3}\end{array}\right),$

and

$\begin{array}{c}q\left(M\right)=2〈{i}_{1},{i}_{1}〉\left({a}_{1}x+{a}_{1}{x}_{1}+{a}_{3}{x}_{2}+{a}_{3}{x}_{3}+{a}_{5}{x}_{1}+{a}_{5}{x}_{3}-2{x}^{2}-2{x}_{1}^{2}-2{x}_{2}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{x}_{3}^{2}-2{x}_{1}x+2{x}_{2}x+{x}_{3}x+{x}_{1}{x}_{2}-{x}_{1}{x}_{3}-2{x}_{2}{x}_{3}\right)+\left({i}_{1}\cdot {i}_{1}\right)\left({a}_{3}x\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2{a}_{3}{x}_{1}-{a}_{3}{x}_{2}+{x}^{2}+{x}_{1}^{2}+{x}_{2}^{2}+{x}_{3}^{2}+{x}_{1}x-{x}_{2}x-{x}_{3}x-{x}_{1}{x}_{2}+{x}_{2}{x}_{3}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({i}_{1}\cdot {i}_{2}\right)\left({a}_{2}x+{a}_{2}{x}_{1}\right)+2\left({i}_{1}\cdot {i}_{4}\right)\left({a}_{4}{x}_{1}+{a}_{4}{x}_{3}\right)\end{array}$

$\begin{array}{c}=-{\left(x-{x}_{2}\right)}^{2}-{x}^{2}-2{x}_{1}^{2}-{x}_{2}^{2}-2{x}_{3}^{2}-2{x}_{1}x-2{x}_{1}{x}_{3}-2{x}_{2}{x}_{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({a}_{1}+{a}_{3}-{a}_{2}\right)x+2\left({a}_{1}+2{a}_{3}+{a}_{5}-{a}_{2}-{a}_{4}\right){x}_{1}+2\left({a}_{3}+{a}_{5}-{a}_{4}\right){x}_{3}.\end{array}$

Obviously, $q\left(M\right)\le 0⇔{a}_{1}+{a}_{3}\le {a}_{2}$ , ${a}_{3}+{a}_{5}\le {a}_{4}$ . And $q\left(M\right)=0⇔{a}_{1}+{a}_{3}\le {a}_{2}$ , ${a}_{3}+{a}_{5}\le {a}_{4}$ , $x={x}_{1}={x}_{2}={x}_{3}=0$ , so word 21232(13-2, 35-4) is tight. So (6) holds.

Case 5. $t=6$ . Applying the $M-S$ word-procedure to ${M}_{5}$ , deleting words including subwords in $\Phi \left(1212\right)\cup \Psi \Phi \left(1212\right)$ , we get ${S}_{6}\cup \left\{123123\right\}$ .

Firstly, as ${S}_{5}^{2},{S}_{5}^{3}$ , we can prove that words in ${S}_{6}^{1},{S}_{6}^{3}$ are all tight.

Secondly, consider words 123123, 132132, we have ${\mathcal{M}}_{i,a}=\left\{M={M}_{x,y,z}|\left(x,y,z\right)\in {ℕ}^{3}\right\}$ , where

$M=\left(\begin{array}{cccccc}{a}_{1}-x& 0& 0& x& 0& 0\\ 0& {a}_{2}-y& 0& 0& y& 0\\ 0& 0& {a}_{3}-z& 0& 0& z\\ x& 0& 0& {a}_{4}-x& 0& 0\\ 0& y& 0& 0& {a}_{5}-y& 0\\ 0& 0& z& 0& 0& {a}_{6}-z\end{array}\right)$

and

$\begin{array}{c}q\left(M\right)=2〈{i}_{1},{i}_{1}〉\left({a}_{1}x+{a}_{4}x-2{x}^{2}\right)+2〈{i}_{2},{i}_{2}〉\left({a}_{2}y+{a}_{5}y-2{y}^{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2〈{i}_{3},{i}_{3}〉\left({a}_{3}z+{a}_{6}z-2{z}^{2}\right)+\left({i}_{1}\cdot {i}_{1}\right){x}^{2}+\left({i}_{2}\cdot {i}_{2}\right){y}^{2}+\left({i}_{3}\cdot {i}_{3}\right){z}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({i}_{1}\cdot {i}_{2}\right)\left({a}_{2}x+{a}_{4}y-xy\right)+2\left({i}_{1}\cdot {i}_{3}\right)\left({a}_{3}x+{a}_{4}z-xz\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({i}_{2}\cdot {i}_{3}\right)\left({a}_{3}y+{a}_{5}z-yz\right)\end{array}$

$=\left\{\begin{array}{l}-{\left(x-y\right)}^{2}-{\left(z-y\right)}^{2}-{x}^{2}-{z}^{2}+2\left({a}_{1}+{a}_{4}-{a}_{2}\right)x\\ +\text{\hspace{0.17em}}2\left({a}_{2}+{a}_{5}-{a}_{3}-{a}_{4}\right)y+2\left({a}_{3}+{a}_{6}-{a}_{5}\right)z,\text{\hspace{0.17em}}\text{\hspace{0.17em}}123123,\\ -{\left(x-z\right)}^{2}-{\left(z-y\right)}^{2}-{x}^{2}-{y}^{2}+2\left({a}_{1}+{a}_{4}-{a}_{3}\right)x\\ +\text{\hspace{0.17em}}2\left({a}_{2}+{a}_{5}-{a}_{3}\right)y+2\left({a}_{3}+{a}_{6}-{a}_{4}-{a}_{5}\right)z,\text{\hspace{0.17em}}\text{\hspace{0.17em}}132132.\end{array}$

Obviously, $q\left(M\right)\le 0⇔\left\{\begin{array}{l}{a}_{1}+{a}_{4}\le {a}_{2},{a}_{2}+{a}_{5}\le {a}_{3}+{a}_{4},{a}_{3}+{a}_{6}\le {a}_{5},123123,\\ {a}_{1}+{a}_{4}\le {a}_{3},{a}_{2}+{a}_{5}\le {a}_{3},{a}_{3}+{a}_{6}\le {a}_{4}+{a}_{5},132132.\end{array}$ And $q\left(M\right)=0⇔\left\{\begin{array}{l}{a}_{1}+{a}_{4}\le {a}_{2},{a}_{2}+{a}_{5}\le {a}_{3}+{a}_{4},{a}_{3}+{a}_{6}\le {a}_{5},\text{\hspace{0.17em}}123123,\\ {a}_{1}+{a}_{4}\le {a}_{3},{a}_{2}+{a}_{5}\le {a}_{3},{a}_{3}+{a}_{6}\le {a}_{4}+{a}_{5},\text{\hspace{0.17em}}132132,\end{array}$ and

$x=y=z=0$ , but ${a}_{1}+{a}_{4}\le {a}_{2}$ , ${a}_{2}+{a}_{5}\le {a}_{3}+{a}_{4}$ , ${a}_{3}+{a}_{6}\le {a}_{5}⇒{a}_{1}+{a}_{6}\le 0$ , this is a contradiction, so word 123123 is not tight for any $\left({a}_{1},\cdots ,{a}_{6}\right)\in {ℕ}^{6}$ , and word 132132(14-3, 25-3, 36-45) is tight by Theorem 2.3.

Lastly, let us consider ${S}_{6}^{4}$ . Take word 123121 as an example, we have ${\mathcal{M}}_{i,a}=\left\{M={M}_{x,{x}_{1},{x}_{2},{x}_{3},y}|\left(x,{x}_{1},{x}_{2},{x}_{3},y\right)\in {ℕ}^{5}\right\}$ , where

$M=\left(\begin{array}{cccccc}{a}_{1}-x-{x}_{1}& 0& 0& x& 0& {x}_{1}\\ 0& {a}_{2}-y& 0& 0& y& 0\\ 0& 0& {a}_{3}& 0& 0& 0\\ {x}_{2}& 0& 0& {a}_{4}-{x}_{2}-{x}_{3}& 0& {x}_{3}\\ 0& y& 0& 0& {a}_{5}-y& 0\\ x+{x}_{1}-{x}_{2}& 0& 0& {x}_{2}+{x}_{3}-x& 0& {a}_{6}-{x}_{1}-{x}_{3}\end{array}\right)$

and

$\begin{array}{l}q\left(M\right)=2〈{i}_{1},{i}_{1}〉\left({a}_{1}x+{a}_{1}{x}_{1}+{a}_{4}{x}_{2}+{a}_{4}{x}_{3}+{a}_{6}{x}_{1}+{a}_{6}{x}_{3}-2{x}^{2}-2{x}_{1}^{2}-2{x}_{2}^{2}-2{x}_{3}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{x}_{1}x+2{x}_{2}x+{x}_{3}x+{x}_{1}{x}_{2}-{x}_{1}{x}_{3}-2{x}_{2}{x}_{3}\right)+2〈{i}_{2},{i}_{2}〉\left({a}_{2}y+{a}_{5}y-2{y}^{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({i}_{1}\cdot {i}_{1}\right)\left({a}_{4}x+2{a}_{4}{x}_{1}-{a}_{4}{x}_{2}+{x}^{2}+{x}_{1}^{2}+{x}_{2}^{2}+{x}_{3}^{2}+{x}_{1}x-{x}_{2}x-{x}_{3}x-{x}_{1}{x}_{2}+{x}_{2}{x}_{3}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({i}_{2}\cdot {i}_{2}\right){y}^{2}+2\left({i}_{1}\cdot {i}_{2}\right)\left({a}_{2}x+{a}_{2}{x}_{1}+{a}_{5}{x}_{1}+{a}_{5}{x}_{3}+{a}_{4}y-{x}_{2}y-{x}_{3}y\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({i}_{1}\cdot {i}_{3}\right)\left({a}_{3}x+{a}_{3}{x}_{1}\right)+2\left({i}_{2}\cdot {i}_{3}\right){a}_{3}y\end{array}$

$\begin{array}{l}=-{\left(x-{x}_{2}\right)}^{2}-\left({x}_{2}-y\right){\right)}^{2}-{\left({x}_{3}-y\right)}^{2}-{x}^{2}-2{x}_{1}^{2}-{x}_{3}^{2}-2{x}_{1}x-2{x}_{1}{x}_{3}-2{x}_{2}{x}_{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({a}_{1}+{a}_{4}-{a}_{2}\right)x+2\left({a}_{1}+2{a}_{4}+{a}_{6}-{a}_{2}-{a}_{5}\right){x}_{1}+2\left({a}_{4}+{a}_{6}-{a}_{5}\right){x}_{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({a}_{2}+{a}_{5}-{a}_{3}-{a}_{4}\right)y.\end{array}$

Obviously, $q\left(M\right)\le 0⇔{a}_{1}+{a}_{4}\le {a}_{2}$ , ${a}_{4}+{a}_{6}\le {a}_{5}$ , ${a}_{2}+{a}_{5}\le {a}_{3}+{a}_{4}$ . And $q\left(M\right)=0⇔{a}_{1}+{a}_{4}\le {a}_{2}$ , ${a}_{4}+{a}_{6}\le {a}_{5}$ , ${a}_{2}+{a}_{5}\le {a}_{3}+{a}_{4}$ , $x={x}_{1}={x}_{2}={x}_{3}=y=0$ , so word 123121(14-2, 46-5, 25-34) is tight by Theorem 2.3. So (7) holds.

Case 6. $t=7$ . Applying the $M-S$ word-procedure to ${M}_{6}$ , deleting words including subwords in $\Phi \left(\left\{1212,123123\right\}\right)\cup \Psi \Phi \left(\left\{1212,123123\right\}\right)$ , we get ${S}_{7}\cup \left\{2132132,1232123\right\}$ .

As ${S}_{6}^{2},{S}_{6}^{4}$ , we can prove ${S}_{7}^{1},{S}_{7}^{2}$ . Consider word 2123242 in ${S}_{7}^{3}$ , we have ${\mathcal{M}}_{i,a}=\left\{M={M}_{x,{x}_{1},\cdots ,{x}_{8}}|\left(x,{x}_{1},\cdots ,{x}_{8}\right)\in {ℕ}^{9}\right\}$ , where

$M=\left(\begin{array}{ccccccc}{b}_{11}& 0& x& 0& {x}_{1}& 0& {x}_{2}\\ 0& {a}_{2}& 0& 0& 0& 0& 0\\ {x}_{3}& 0& {b}_{33}& 0& {x}_{4}& 0& {x}_{5}\\ 0& 0& 0& {a}_{4}& 0& 0& 0\\ {x}_{6}& 0& {x}_{7}& 0& {b}_{55}& 0& {x}_{8}\\ 0& 0& 0& 0& 0& {a}_{6}& 0\\ {b}_{71}& 0& {b}_{73}& 0& {b}_{75}& 0& {b}_{77}\end{array}\right),$

$\begin{array}{l}{b}_{11}={a}_{1}-x-{x}_{1}-{x}_{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{33}={a}_{3}-{x}_{3}-{x}_{4}-{x}_{5},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{55}={a}_{5}-{x}_{6}-{x}_{7}-{x}_{8},\\ {b}_{77}={a}_{7}-{x}_{2}-{x}_{5}-{x}_{8},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{71}=x+{x}_{1}+{x}_{2}-{x}_{3}-{x}_{6},\\ {b}_{73}={x}_{3}+{x}_{4}+{x}_{5}-x-{x}_{7},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{75}={x}_{6}+{x}_{7}+{x}_{8}-{x}_{1}-{x}_{4},\end{array}$

$\begin{array}{l}q\left(M\right)=2〈{i}_{1},{i}_{1}〉\left({a}_{1}x+{a}_{1}{x}_{1}+{a}_{1}{x}_{2}+{a}_{3}{x}_{3}+{a}_{3}{x}_{4}+{a}_{3}{x}_{5}+{a}_{5}{x}_{6}+{a}_{5}{x}_{7}+{a}_{5}{x}_{8}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{a}_{7}{x}_{2}+{a}_{7}{x}_{5}+{a}_{7}{x}_{8}-2{x}^{2}-2{x}_{1}^{2}-2{x}_{2}^{2}-2{x}_{3}^{2}-2{x}_{4}^{2}-2{x}_{5}^{2}-2{x}_{6}^{2}-2{x}_{7}^{2}-2{x}_{8}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{x}_{1}x-2{x}_{2}x+2{x}_{3}x+{x}_{4}x+{x}_{5}x+{x}_{6}x-{x}_{7}x-2{x}_{1}{x}_{2}+{x}_{1}{x}_{3}-{x}_{1}{x}_{4}+2{x}_{1}{x}_{6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}_{1}{x}_{7}+{x}_{1}{x}_{8}+{x}_{2}{x}_{3}-{x}_{2}{x}_{5}+{x}_{2}{x}_{6}-{x}_{2}{x}_{8}-2{x}_{3}{x}_{4}-2{x}_{3}{x}_{5}-{x}_{3}{x}_{6}+{x}_{3}{x}_{7}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{x}_{4}{x}_{5}+{x}_{4}{x}_{6}+2{x}_{4}{x}_{7}+{x}_{4}{x}_{8}+{x}_{5}{x}_{7}-{x}_{5}{x}_{8}-2{x}_{6}{x}_{7}-2{x}_{6}{x}_{8}-2{x}_{7}{x}_{8}\right)\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({i}_{1}\cdot {i}_{1}\right)\left({a}_{3}x+2{a}_{3}{x}_{1}+2{a}_{3}{x}_{2}-{a}_{3}{x}_{3}+{a}_{5}{x}_{1}+2{a}_{5}{x}_{2}-{a}_{5}{x}_{6}+{a}_{5}{x}_{4}+2{a}_{5}{x}_{5}-{a}_{5}{x}_{7}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{2}+{x}_{1}^{2}+{x}_{2}^{2}+{x}_{3}^{2}+{x}_{4}^{2}+{x}_{5}^{2}+{x}_{6}^{2}+{x}_{7}^{2}+{x}_{8}^{2}+{x}_{1}x+{x}_{2}x-{x}_{3}x-{x}_{4}x-{x}_{5}x+{x}_{7}x\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}_{1}{x}_{2}-{x}_{1}{x}_{3}-{x}_{1}{x}_{5}-{x}_{1}{x}_{6}-{x}_{1}{x}_{8}-{x}_{2}{x}_{3}-{x}_{2}{x}_{6}+{x}_{3}{x}_{4}+{x}_{3}{x}_{5}-{x}_{3}{x}_{7}+{x}_{4}{x}_{5}-{x}_{4}{x}_{6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{x}_{4}{x}_{7}-{x}_{4}{x}_{8}-{x}_{5}{x}_{6}-{x}_{5}{x}_{7}+{x}_{6}{x}_{7}+{x}_{6}{x}_{8}+{x}_{7}{x}_{8}\right)+2\left({i}_{1}\cdot {i}_{2}\right)\left({a}_{2}x+{a}_{2}{x}_{1}+{a}_{2}{x}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({i}_{1}\cdot {i}_{4}\right)\left({a}_{4}{x}_{1}+{a}_{4}{x}_{2}+{a}_{4}{x}_{4}+{a}_{4}{x}_{5}\right)+2\left({i}_{1}\cdot {i}_{6}\right)\left({a}_{6}{x}_{2}+{a}_{6}{x}_{5}+{a}_{6}{x}_{8}\right)\end{array}$

$\begin{array}{l}=-2f\left(x,{x}_{1},\cdots ,{x}_{8}\right)+2\left({a}_{1}+{a}_{3}-{a}_{2}\right)x+2\left({a}_{1}+2{a}_{3}+{a}_{5}-{a}_{2}-{a}_{4}\right){x}_{1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({a}_{1}+2{a}_{3}+2{a}_{5}+{a}_{7}-{a}_{2}-{a}_{4}-{a}_{6}\right){x}_{2}+2\left({a}_{3}+{a}_{5}-{a}_{4}\right){x}_{4}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({a}_{3}+2{a}_{5}+{a}_{7}-{a}_{4}-{a}_{6}\right){x}_{5}+2\left({a}_{5}+{a}_{7}-{a}_{6}\right){x}_{8},\end{array}$

where $f\left(x,{x}_{1},\cdots ,{x}_{8}\right)=\frac{1}{2}X{A}_{f}{X}^{\text{T}}$ , $X=\left(x,{x}_{1},\cdots ,{x}_{8}\right)$ symmetric matrix ${A}_{f}$ is

as follows

${A}_{f}=\left(\begin{array}{ccccccccc}2& 1& 1& -1& 0& 0& -1& 0& 0\\ 1& 2& 1& 0& 1& 1& -1& -1& 0\\ 1& 1& 2& 0& 0& 1& 0& 0& 1\\ -1& 0& 0& 2& 1& 1& 1& 0& 0\\ 0& 1& 0& 1& 2& 1& 0& -1& 0\\ 0& 1& 1& 1& 1& 2& 1& 0& 1\\ -1& -1& 0& 1& 0& 1& 2& 1& 1\\ 0& -1& 0& 0& -1& 0& 1& 2& 1\\ 0& 0& 1& 0& 0& 1& 1& 1& 2\end{array}\right)$

Using the above algorithm in §2, we have

$\begin{array}{l}{R}_{1}=\left\{{r}_{1},\cdots ,{r}_{9}\right\};\\ {R}_{2}=\left\{{r}_{1}+{r}_{4},{r}_{1}+{r}_{7},{r}_{2}+{r}_{7},{r}_{2}+{r}_{8},{r}_{5}+{r}_{8}\right\};\\ {R}_{3}=\varphi .\end{array}$

By Theorem 2.8, the unit form $f\left(x,{x}_{1},\cdots ,{x}_{8}\right)$ is weakly positive, i.e., $f\left(x,{x}_{1},\cdots ,{x}_{8}\right)\ge 0$ for any $\left(x,{x}_{1},\cdots ,{x}_{8}\right)\in {ℕ}^{9}$ , and $f\left(x,{x}_{1},\cdots ,{x}_{8}\right)=0⇔$ $x={x}_{1}=\cdots ={x}_{8}=0$ . So, $q\left(M\right)\le 0⇔{a}_{1}+{a}_{3}\le {a}_{2}$ , ${a}_{5}+{a}_{7}\le {a}_{6}$ , ${a}_{3}+{a}_{5}\le {a}_{4}$ . And $q\left(M\right)=0⇔{a}_{1}+{a}_{3}\le {a}_{2}$ , ${a}_{5}+{a}_{7}\le {a}_{6}$ , ${a}_{3}+{a}_{5}\le {a}_{4}$ , $x={x}_{1}=\cdots ={x}_{8}=0$ , so word 2123242(13-2, 57-6, 35-4) is tight by Theorem 2.3.

At last, let us see ${S}_{7}^{4}$ , for words 2123212, 2124212 in ${S}_{7}^{4}$ , we have ${\mathcal{M}}_{i,a}=\left\{M={M}_{x,{x}_{1},\cdots ,{x}_{8},y}|\left(x,{x}_{1},\cdots ,{x}_{8},y\right)\in {ℕ}^{10}\right\}$ , where

$M=\left(\begin{array}{ccccccc}{b}_{11}& 0& x& 0& {x}_{1}& 0& {x}_{2}\\ 0& {a}_{2}-y& 0& 0& 0& y& 0\\ {x}_{3}& 0& {b}_{33}& 0& {x}_{4}& 0& {x}_{5}\\ 0& 0& 0& {a}_{4}& 0& 0& 0\\ {x}_{6}& 0& {x}_{7}& 0& {b}_{55}& 0& {x}_{8}\\ 0& y& 0& 0& 0& {a}_{6}-y& 0\\ {b}_{71}& 0& {b}_{73}& 0& {b}_{75}& 0& {b}_{77}\end{array}\right)$

$\begin{array}{l}{b}_{11}={a}_{1}-x-{x}_{1}-{x}_{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{33}={a}_{3}-{x}_{3}-{x}_{4}-{x}_{5},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{55}={a}_{5}-{x}_{6}-{x}_{7}-{x}_{8},\\ {b}_{77}={a}_{7}-{x}_{2}-{x}_{5}-{x}_{8},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{71}=x+{x}_{1}+{x}_{2}-{x}_{3}-{x}_{6},\\ {b}_{73}={x}_{3}+{x}_{4}+{x}_{5}-x-{x}_{7},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{75}={x}_{6}+{x}_{7}+{x}_{8}-{x}_{1}-{x}_{4},\end{array}$

$\begin{array}{l}q\left(M\right)=2〈{i}_{1},{i}_{1}〉\left({a}_{1}x+{a}_{1}{x}_{1}+{a}_{1}{x}_{2}+{a}_{3}{x}_{3}+{a}_{3}{x}_{4}+{a}_{3}{x}_{5}+{a}_{5}{x}_{6}+{a}_{5}{x}_{7}+{a}_{5}{x}_{8}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{a}_{7}{x}_{2}+{a}_{7}{x}_{5}+{a}_{7}{x}_{8}-2{x}^{2}-2{x}_{1}^{2}-2{x}_{2}^{2}-2{x}_{3}^{2}-2{x}_{4}^{2}-2{x}_{5}^{2}-2{x}_{6}^{2}-2{x}_{7}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{x}_{8}^{2}-2{x}_{2}x+2{x}_{3}x+{x}_{4}x+{x}_{5}x+{x}_{6}x-{x}_{7}x-2{x}_{1}{x}_{2}+{x}_{1}{x}_{3}-{x}_{1}{x}_{4}+2{x}_{1}{x}_{6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}_{1}{x}_{7}-2{x}_{1}x+{x}_{1}{x}_{8}+{x}_{2}{x}_{3}-{x}_{2}{x}_{5}+{x}_{2}{x}_{6}-{x}_{2}{x}_{8}-2{x}_{3}{x}_{4}-2{x}_{3}{x}_{5}-{x}_{3}{x}_{6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}_{3}{x}_{7}-2{x}_{4}{x}_{5}+{x}_{4}{x}_{6}+2{x}_{4}{x}_{7}+{x}_{4}{x}_{8}+{x}_{5}{x}_{7}-{x}_{5}{x}_{8}-2{x}_{6}{x}_{7}-2{x}_{6}{x}_{8}-2{x}_{7}{x}_{8}\right)\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+2〈{i}_{2},{i}_{2}〉\left({a}_{2}y+{a}_{6}y-2{y}^{2}\right)+\left({i}_{1}\cdot {i}_{1}\right)\left({a}_{3}x+2{a}_{3}{x}_{1}+2{a}_{3}{x}_{2}-{a}_{3}{x}_{3}+{a}_{5}{x}_{1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2{a}_{5}{x}_{2}-{a}_{5}{x}_{6}+{a}_{5}{x}_{4}+2{a}_{5}{x}_{5}-{a}_{5}{x}_{7}+{x}^{2}+{x}_{1}^{2}+{x}_{2}^{2}+{x}_{3}^{2}+{x}_{4}^{2}+{x}_{5}^{2}+{x}_{6}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}_{7}^{2}+{x}_{8}^{2}+{x}_{1}x+{x}_{2}x-{x}_{3}x-{x}_{4}x-{x}_{5}x+{x}_{7}x+{x}_{1}{x}_{2}-{x}_{1}{x}_{3}-{x}_{1}{x}_{5}-{x}_{1}{x}_{6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{x}_{1}{x}_{8}-{x}_{2}{x}_{3}-{x}_{2}{x}_{6}+{x}_{3}{x}_{4}+{x}_{3}{x}_{5}-{x}_{3}{x}_{7}+{x}_{4}{x}_{5}-{x}_{4}{x}_{6}-{x}_{4}{x}_{7}-{x}_{4}{x}_{8}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{x}_{5}{x}_{6}-{x}_{5}{x}_{7}+{x}_{6}{x}_{7}+{x}_{6}{x}_{8}+{x}_{7}{x}_{8}\right)+\left({i}_{2}\cdot {i}_{2}\right){y}^{2}+2\left({i}_{1}\cdot {i}_{2}\right)\left({a}_{2}x+{a}_{2}{x}_{1}\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{a}_{2}{x}_{2}+{a}_{6}{x}_{2}+{a}_{3}y+{a}_{5}y+{a}_{6}{x}_{5}+{a}_{6}{x}_{8}-{x}_{3}y-{x}_{5}y-{x}_{6}y-{x}_{8}y\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({i}_{1}\cdot {i}_{4}\right)\left({a}_{4}{x}_{1}+{a}_{4}{x}_{2}+{a}_{4}{x}_{4}+{a}_{4}{x}_{5}\right)\\ =-2f\left(x,{x}_{1},\cdots ,{x}_{8},y\right)+2\left({a}_{1}+{a}_{3}-{a}_{2}\right)x+2\left({a}_{1}+2{a}_{3}+{a}_{5}-{a}_{2}-{a}_{4}\right){x}_{1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({a}_{1}+2{a}_{3}+2{a}_{5}+{a}_{7}-{a}_{2}-{a}_{4}-{a}_{6}\right){x}_{2}+2\left({a}_{3}+{a}_{5}-{a}_{4}\right){x}_{4}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({a}_{3}+2{a}_{5}+{a}_{7}-{a}_{4}-{a}_{6}\right){x}_{5}+2\left({a}_{5}+{a}_{7}-{a}_{6}\right){x}_{8}+2\left({a}_{2}+{a}_{6}-{a}_{3}-{a}_{5}\right)y,\end{array}$

where $f\left(x,{x}_{1},\cdots ,{x}_{8},y\right)=\frac{1}{2}X{A}_{f}{X}^{\text{T}}$ , $X=\left(x,{x}_{1},\cdots ,{x}_{8},y\right)$ , symmetric matrix

${A}_{f}$ is as follows

${A}_{f}=\left(\begin{array}{cccccccccc}2& 1& 1& -1& 0& 0& -1& 0& 0& 0\\ 1& 2& 1& 0& 1& 1& -1& -1& 0& 0\\ 1& 1& 2& 0& 0& 1& 0& 0& 1& 0\\ -1& 0& 0& 2& 1& 1& 1& 0& 0& -1\\ 0& 1& 0& 1& 2& 1& 0& -1& 0& 0\\ 0& 1& 1& 1& 1& 2& 1& 0& 1& -1\\ -1& -1& 0& 1& 0& 1& 2& 1& 1& -1\\ 0& -1& 0& 0& -1& 0& 1& 2& 1& 0\\ 0& 0& 1& 0& 0& 1& 1& 1& 2& -1\\ 0& 0& 0& -1& 0& -1& -1& 0& -1& 2\end{array}\right)$

Using the above algorithm in §2, we have

$\begin{array}{l}{R}_{1}=\left\{{r}_{1},\cdots ,{r}_{10}\right\};\\ {R}_{2}=\left\{{r}_{1}+{r}_{4},{r}_{1}+{r}_{7},{r}_{2}+{r}_{7},{r}_{2}+{r}_{8},{r}_{4}+{r}_{10},{r}_{5}+{r}_{8},{r}_{6}+{r}_{10},{r}_{7}+{r}_{10},{r}_{9}+{r}_{10}\right\};\end{array}$

$\begin{array}{l}{R}_{3}=\left\{{r}_{1}+{r}_{4}+{r}_{10},{r}_{1}+{r}_{7}+{r}_{10},{r}_{2}+{r}_{7}+{r}_{10},{r}_{4}+{r}_{9}+{r}_{10}\right\};\\ {R}_{4}=\left\{{r}_{1}+{r}_{4}+{r}_{7}+{r}_{10},{r}_{1}+{r}_{4}+{r}_{9}+{r}_{10}\right\};\\ {R}_{5}=\varphi .\end{array}$

By Theorem 2.8, the unit form $f\left(x,{x}_{1},\cdots ,{x}_{9},y\right)$ is weakly positive, i.e., $f\left(x,{x}_{1},\cdots ,{x}_{9},y\right)\ge 0$ for any $\left(x,{x}_{1},\cdots ,{x}_{9},y\right)\in {ℕ}^{10}$ , and $f\left(x,{x}_{1},\cdots ,{x}_{9},y\right)=$ $0⇔x={x}_{1}=\cdots ={x}_{8}=y=0$ . So, $q\left(M\right)\le 0⇔{a}_{1}+{a}_{3}\le {a}_{2}$ , ${a}_{5}+{a}_{7}\le {a}_{6}$ , ${a}_{2}+{a}_{6}\le {a}_{3}+{a}_{5}\le {a}_{4}$ . And $q\left(M\right)=0⇔{a}_{1}+{a}_{3}\le {a}_{2}$ , ${a}_{5}+{a}_{7}\le {a}_{6}$ , ${a}_{2}+{a}_{6}\le {a}_{3}+{a}_{5}\le {a}_{4}$ and $x={x}_{1}=\cdots ={x}_{8}=y=0$ , so words 2123212, 2124212(13-2, 57-6, 26-35-4) are all tight by Theorem 2.3. So (8) holds.

Case 7. $t=8$ . Applying the $M-S$ word-procedure to ${M}_{7}$ , deleting words including subwords in $\Phi \left(\left\{1212,123123,1232123,2132132\right\}\right)\cup \Psi \Phi \left(\left\{1212,123123,1232123,2132132\right\}\right)$ , we get ${S}_{8}$ .

As ${S}_{7}^{4}$ , we can prove that words in ${S}_{8}^{4}$ are all tight.

For ${S}_{8}^{1}$ , only consider word 12134234, we have ${\mathcal{M}}_{i,a}=\left\{M={M}_{x,y,z,u}|\left(x,y,z,u\right)\in {ℕ}^{4}\right\}$ , where

$M=\left(\begin{array}{cccccccc}{a}_{1}-x& 0& x& 0& 0& 0& 0& 0\\ 0& {a}_{2}-y& 0& 0& 0& y& 0& 0\\ x& 0& {a}_{3}-x& 0& 0& 0& 0& 0\\ 0& 0& 0& {a}_{4}-z& 0& 0& z& 0\\ 0& 0& 0& 0& {a}_{5}-u& 0& 0& u\\ 0& y& 0& 0& 0& {a}_{6}-y& 0& 0\\ 0& 0& 0& z& 0& 0& {a}_{7}-z& 0\\ 0& 0& 0& 0& u& 0& 0& {a}_{8}-u\end{array}\right),$

and $\begin{array}{l}q\left(M\right)=2〈{i}_{1},{i}_{1}〉\left({a}_{1}x+{a}_{3}x-2{x}^{2}\right)+2〈{i}_{2},{i}_{2}〉\left({a}_{2}y+{a}_{6}y-2{y}^{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2〈{i}_{4},{i}_{4}〉\left({a}_{4}z+{a}_{7}z-2{z}^{2}\right)+2〈{i}_{5},{i}_{5}〉\left({a}_{5}u+{a}_{8}u-2{u}^{2}\right)+\left({i}_{1}\cdot {i}_{1}\right){x}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({i}_{2}\cdot {i}_{2}\right){y}^{2}+\left({i}_{4}\cdot {i}_{4}\right){z}^{2}+\left({i}_{5}\cdot {i}_{5}\right){u}^{2}+2\left({i}_{1}\cdot {i}_{2}\right)\left({a}_{2}x+{a}_{3}y-xy\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({i}_{2}\cdot {i}_{4}\right)\left({a}_{4}y+{a}_{6}z-yz\right)+2\left({i}_{2}\cdot {i}_{5}\right)\left({a}_{5}y+{a}_{6}u-yu\right)\\ =-2f\left(x,y,z,u\right)+2\left({a}_{1}+{a}_{3}-{a}_{2}\right)x+2\left({a}_{2}+{a}_{6}-{a}_{3}-{a}_{4}-{a}_{5}\right)y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({a}_{4}+{a}_{7}-{a}_{6}\right)z+2\left({a}_{5}+{a}_{8}-{a}_{6}\right)u,\end{array}$

where $f\left(x,y,z,u\right)=\frac{1}{2}X{A}_{f}{X}^{\text{T}},X=\left(x,y,z,u\right)$ , symmetric matrix ${A}_{f}$ is as

follows

${A}_{f}=\left(\begin{array}{cccc}2& -1& 0& 0\\ -1& 2& -1& -1\\ 0& -1& 2& 0\\ 0& -1& 0& 2\end{array}\right)$

Using the above algorithm in §2, we have

$\begin{array}{l}{R}_{1}=\left\{{r}_{1},{r}_{2},{r}_{3},{r}_{4}\right\};\\ {R}_{2}=\left\{{r}_{1}+{r}_{2},{r}_{2}+{r}_{3},{r}_{2}+{r}_{4}\right\};\end{array}$

$\begin{array}{l}{R}_{3}=\left\{{r}_{1}+{r}_{2}+{r}_{3},{r}_{1}+{r}_{2}+{r}_{4},{r}_{2}+{r}_{3}+{r}_{4}\right\};\\ {R}_{4}=\left\{{r}_{1}+{r}_{2}+{r}_{3}+{r}_{4}\right\};\\ {R}_{5}=\left\{{r}_{1}+2{r}_{2}+{r}_{3}+{r}_{4}\right\};\\ {R}_{6}=\varphi .\end{array}$

By Theorem 2.8, the unit form $f\left(x,y,z,u\right)$ is weakly positive, i.e., $f\left(x,y,z,u\right)\ge 0$ for any $\left(x,y,z,u\right)\in {ℕ}^{4}$ , and $f\left(x,y,z,u\right)=0⇔x=y=z=$ $u=0$ . So, $q\left(M\right)\le 0⇔{a}_{1}+{a}_{3}\le {a}_{2}$ , ${a}_{4}+{a}_{7}\le {a}_{6}$ , ${a}_{5}+{a}_{8}\le {a}_{6}$ , ${a}_{2}+{a}_{6}\le {a}_{3}+{a}_{4}+{a}_{5}$ . And $q\left(M\right)=0⇔{a}_{1}+{a}_{3}\le {a}_{2}$ , ${a}_{4}+{a}_{7}\le {a}_{6}$ , ${a}_{5}+{a}_{8}\le {a}_{6}$ , ${a}_{2}+{a}_{6}\le {a}_{3}+{a}_{4}+{a}_{5}$ and $x=y=z=u=0$ , word 12134234(13-2, 47-6, 58-6, 26-345) is tight by Theorem 2.3.

Consider words $12132142,12142132$ in ${S}_{8}^{2}$ , we have

${\mathcal{M}}_{i,a}=\left\{M={M}_{x,{x}_{1},{x}_{2},{x}_{3},y,{y}_{1},{y}_{2},{y}_{3}}|\left(x,{x}_{1},{x}_{2},{x}_{3},y,{y}_{1},{y}_{2},{y}_{3}\right)\in {ℕ}^{8}\right\}$ , where

$M=\left(\begin{array}{cccccccc}{b}_{11}& 0& x& 0& 0& {x}_{1}& 0& 0\\ 0& {b}_{22}& 0& 0& y& 0& 0& {y}_{1}\\ {x}_{2}& 0& {b}_{33}& 0& 0& {x}_{3}& 0& 0\\ 0& 0& 0& {a}_{4}& 0& 0& 0& 0\\ 0& {y}_{2}& 0& 0& {b}_{55}& 0& 0& {y}_{3}\\ {b}_{61}& 0& {b}_{63}& 0& 0& {b}_{66}& 0& 0\\ 0& 0& 0& 0& 0& 0& {a}_{7}& 0\\ 0& {b}_{82}& 0& 0& {b}_{85}& 0& 0& {b}_{88}\end{array}\right)$

$\begin{array}{l}{b}_{11}={a}_{1}-x-{x}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{22}={a}_{2}-y-{y}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{33}={a}_{3}-{x}_{2}-{x}_{3},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{55}={a}_{5}-{y}_{2}-{y}_{3},\\ {b}_{61}=x+{x}_{1}-{x}_{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{63}={x}_{2}+{x}_{3}-x,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{66}={a}_{6}-{x}_{1}-{x}_{3},\\ {b}_{82}=y+{y}_{1}-{y}_{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{85}={y}_{2}+{y}_{3}-y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{88}={a}_{8}-{y}_{1}-{y}_{3}.\end{array}$

So, we have

$\begin{array}{l}q\left(M\right)=2〈{i}_{1},{i}_{1}〉\left({a}_{1}x+{a}_{1}{x}_{1}+{a}_{3}{x}_{2}+{a}_{3}{x}_{3}+{a}_{6}{x}_{1}+{a}_{6}{x}_{3}-2{x}^{2}-2{x}_{1}^{2}-2{x}_{2}^{2}-2{x}_{3}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{x}_{1}x+2{x}_{2}x+{x}_{3}x+{x}_{1}{x}_{2}-{x}_{1}{x}_{3}-2{x}_{2}{x}_{3}\right)+2〈{i}_{2},{i}_{2}〉\left({a}_{2}y+{a}_{2}{y}_{1}+{a}_{5}{y}_{2}+{a}_{5}{y}_{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{a}_{8}{y}_{1}+{a}_{8}{y}_{3}-2{y}^{2}-2{y}_{1}^{2}-2{y}_{2}^{2}-2{y}_{3}^{2}-2{y}_{1}y+2{y}_{2}y+{y}_{3}y+{y}_{1}{y}_{2}-{y}_{1}{y}_{3}-2{y}_{2}{y}_{3}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({i}_{1}\cdot {i}_{1}\right)\left({a}_{3}x+2{a}_{3}{x}_{1}-{a}_{3}{x}_{2}+{x}^{2}+{x}_{1}^{2}+{x}_{2}^{2}+{x}_{3}^{2}+{x}_{1}x-{x}_{2}x-{x}_{3}x-{x}_{1}{x}_{2}+{x}_{2}{x}_{3}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({i}_{2}\cdot {i}_{2}\right)\left({a}_{5}y+2{a}_{5}{y}_{1}-{a}_{5}{y}_{2}+{y}^{2}+{y}_{1}^{2}+{y}_{2}^{2}+{y}_{3}^{2}+{y}_{1}y-{y}_{2}y-{y}_{3}y-{y}_{1}{y}_{2}+{y}_{2}{y}_{3}\right)\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({i}_{1}\cdot {i}_{2}\right)\left({a}_{2}x+{a}_{2}{x}_{1}+{a}_{5}{x}_{1}+{a}_{5}{x}_{3}+{a}_{3}y+{a}_{3}{y}_{1}+{a}_{6}{y}_{1}+{a}_{6}{y}_{3}-xy-x{y}_{1}-{x}_{1}{y}_{1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{x}_{2}{y}_{2}+x{y}_{2}-{x}_{3}{y}_{2}-{x}_{1}{y}_{3}-{x}_{3}{y}_{3}\right)+2\left({i}_{2}\cdot {i}_{4}\right)\left({a}_{4}y+{a}_{4}{y}_{1}\right)+2\left({i}_{2}\cdot {i}_{7}\right)\left({a}_{7}{y}_{1}+{a}_{7}{y}_{3}\right)\\ =-2f\left(x,{x}_{1},{x}_{2},{x}_{3},y,{y}_{1},{y}_{2},{y}_{3}\right)+2\left({a}_{1}+{a}_{3}-{a}_{2}\right)x+2\left({a}_{1}+2{a}_{3}+{a}_{6}-{a}_{2}-{a}_{5}\right){x}_{1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({a}_{3}+{a}_{6}-{a}_{5}\right){x}_{3}+2\left({a}_{2}+{a}_{5}-{a}_{3}-{a}_{4}\right)y+2\left({a}_{2}+2{a}_{5}+{a}_{8}-{a}_{3}-{a}_{4}-{a}_{6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{a}_{7}\right){y}_{1}+2\left({a}_{5}+{a}_{8}-{a}_{6}-{a}_{7}\right){y}_{3},\end{array}$

where $f\left(x,{x}_{1},{x}_{2},{x}_{3},y,{y}_{1},{y}_{2},{y}_{3}\right)=\frac{1}{2}X{A}_{f}{X}^{\text{T}}$ , $X=\left(x,{x}_{1},{x}_{2},{x}_{3},y,{y}_{1},{y}_{2},{y}_{3}\right)$ ,

symmetric matrix ${A}_{f}$ is as follows

${A}_{f}=\left(\begin{array}{cccccccc}2& 1& -1& 0& -1& -1& 1& 0\\ 1& 2& 0& 1& 0& -1& 0& -1\\ -1& 0& 2& 1& 0& 0& -1& 0\\ 0& 1& 1& 2& 0& 0& -1& -1\\ -1& 0& 0& 0& 2& 1& -1& 0\\ -1& -1& 0& 0& 1& 2& 0& 1\\ 1& 0& -1& -1& -1& 0& 2& 1\\ 0& -1& 0& -1& 0& 1& 1& 2\end{array}\right)$

Using the above algorithm in §2, we have

$\begin{array}{l}{R}_{1}=\left\{{r}_{1},\cdots ,{r}_{8}\right\};\\ {R}_{2}=\left\{{r}_{1}+{r}_{3},{r}_{1}+{r}_{5},{r}_{1}+{r}_{6},{r}_{2}+{r}_{6},{r}_{2}+{r}_{8},{r}_{3}+{r}_{7},{r}_{4}+{r}_{7},{r}_{4}+{r}_{8},{r}_{5}+{r}_{7}\right\};\\ {R}_{3}=\left\{{r}_{1}+{r}_{3}+{r}_{5},{r}_{1}+{r}_{3}+{r}_{6},{r}_{3}+{r}_{5}+{r}_{7},{r}_{4}+{r}_{5}+{r}_{7}\right\};\\ {R}_{4}=\left\{{r}_{1}+{r}_{3}+{r}_{5}+{r}_{7}\right\};\\ {R}_{5}=\varphi .\end{array}$

By Theorem 2.8, the unit form $f\left(x,{x}_{1},{x}_{2},{x}_{3},y,{y}_{1},{y}_{2},{y}_{3}\right)$ is weakly positive, i.e., $f\left(x,{x}_{1},{x}_{2},{x}_{3},y,{y}_{1},{y}_{2},{y}_{3}\right)\ge 0$ for any $\left(x,{x}_{1},{x}_{2},{x}_{3},y,{y}_{1},{y}_{2},{y}_{3}\right)\in {ℕ}^{8}$ , and $f\left(x,{x}_{1},{x}_{2},{x}_{3},y,{y}_{1},{y}_{2},{y}_{3}\right)=0⇔x={x}_{1}={x}_{2}={x}_{3}=y={y}_{1}={y}_{2}={y}_{3}=0$ . So, $q\left(M\right)\le 0⇔{a}_{1}+{a}_{3}\le {a}_{2}$ , ${a}_{3}+{a}_{6}\le {a}_{5}$ , ${a}_{5}+{a}_{8}\le {a}_{6}+{a}_{7}$ , ${a}_{2}+{a}_{5}\le {a}_{3}+{a}_{4}$ . And $q\left(M\right)=0⇔{a}_{1}+{a}_{3}\le {a}_{2}$ , ${a}_{3}+{a}_{6}\le {a}_{5}$ , ${a}_{5}+{a}_{8}\le {a}_{6}+{a}_{7}$ , ${a}_{2}+{a}_{5}\le {a}_{3}+{a}_{4}$ and $x={x}_{1}={x}_{2}={x}_{3}=y={y}_{1}={y}_{2}={y}_{3}=0$ , words 12132142, 12142132(13-2, 36-5, 58-67, 25-34) are all tight by Theorem 2.3.

For ${S}_{8}^{3}$ , only consider word 12132423, we have ${\mathcal{M}}_{i,a}=\left\{M={M}_{x,y,{y}_{1},{y}_{2},{y}_{3},z}|\left(x,y,{y}_{1},{y}_{2},{y}_{3},z\right)\in {ℕ}^{6}\right\}$ , where

$M=\left(\begin{array}{cccccccc}{a}_{1}-x& 0& x& 0& 0& 0& 0& 0\\ 0& {a}_{2}-y-{y}_{1}& 0& 0& y& 0& {y}_{1}& 0\\ x& 0& {a}_{3}-x& 0& 0& 0& 0& 0\\ 0& 0& 0& {a}_{4}-z& 0& 0& 0& z\\ 0& {y}_{2}& 0& 0& {a}_{5}-{y}_{2}-{y}_{3}& 0& {y}_{3}& 0\\ 0& 0& 0& 0& 0& {a}_{6}& 0& 0\\ 0& y+{y}_{1}-{y}_{2}& 0& 0& {y}_{2}+{y}_{3}-y& 0& {a}_{7}-{y}_{1}-{y}_{3}& 0\\ 0& 0& 0& z& 0& 0& 0& {a}_{8}-z\end{array}\right),$

and

$\begin{array}{l}q\left(M\right)=2〈{i}_{1},{i}_{1}〉\left({a}_{1}x+{a}_{3}x-2{x}^{2}\right)+2〈{i}_{2},{i}_{2}〉\left({a}_{2}y+{a}_{2}{y}_{1}+{a}_{5}{y}_{2}+{a}_{5}{y}_{3}+{a}_{7}{y}_{1}+{a}_{7}{y}_{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{y}^{2}-2{y}_{1}^{2}-2{y}_{2}^{2}-2{y}_{3}^{2}-2{y}_{1}y+2{y}_{2}y+{y}_{3}y+{y}_{1}{y}_{2}-{y}_{1}{y}_{3}-2{y}_{2}{y}_{3}\right)+2〈{i}_{4},{i}_{4}〉\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left({a}_{4}z+{a}_{8}z-2{z}^{2}\right)+\left({i}_{1}\cdot {i}_{1}\right){x}^{2}+\left({i}_{2}\cdot {i}_{2}\right)\left({a}_{5}y+2{a}_{5}{y}_{1}-{a}_{5}{y}_{2}+{y}^{2}+{y}_{1}^{2}+{y}_{2}^{2}+{y}_{3}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{y}_{1}y-{y}_{2}y-{y}_{3}y-{y}_{1}{y}_{2}+{y}_{2}{y}_{3}\right)+\left({i}_{4}\cdot {i}_{4}\right){z}^{2}+2\left({i}_{1}\cdot {i}_{2}\right)\left({a}_{2}x+{a}_{3}y+{a}_{3}{y}_{1}-xy\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-x{y}_{1}\right)+2\left({i}_{2}\cdot {i}_{4}\right)\left({a}_{4}y+{a}_{4}{y}_{1}+{a}_{5}z+{a}_{7}z-yz-{y}_{1}z\right)+2\left({i}_{2}\cdot {i}_{6}\right)\left({a}_{6}{y}_{1}+{a}_{6}{y}_{3}\right)\end{array}$

$\begin{array}{l}=-2f\left(x,y,{y}_{1},{y}_{2},{y}_{3},z\right)+2\left({a}_{1}+{a}_{3}-{a}_{2}\right)x+2\left({a}_{2}+{a}_{5}-{a}_{3}-{a}_{4}\right)y+2\left({a}_{2}+2{a}_{5}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{a}_{7}-{a}_{3}-{a}_{4}-{a}_{6}\right){y}_{1}+2\left({a}_{5}+{a}_{7}-{a}_{6}\right){y}_{3}+2\left({a}_{4}+{a}_{8}-{a}_{5}-{a}_{7}\right)z,\end{array}$

where $f\left(x,y,{y}_{1},{y}_{2},{y}_{3},z\right)=\frac{1}{2}X{A}_{f}{X}^{\text{T}},X=\left(x,y,{y}_{1},{y}_{2},{y}_{3},z\right)$ , symmetric matrix

${A}_{f}$ is as follows

${A}_{f}=\left(\begin{array}{cccccc}2& -1& -1& 0& 0& 0\\ -1& 2& 1& -1& 0& -1\\ -1& 1& 2& 0& 1& -1\\ 0& -1& 0& 2& 1& 0\\ 0& 0& 1& 1& 2& 0\\ 0& -1& -1& 0& 0& 2\end{array}\right)$

Using the above algorithm in §2, we have

$\begin{array}{l}{R}_{1}=\left\{{r}_{1},\cdots ,{r}_{6}\right\};\\ {R}_{2}=\left\{{r}_{1}+{r}_{2},{r}_{1}+{r}_{3},{r}_{2}+{r}_{4},{r}_{2}+{r}_{6},{r}_{3}+{r}_{6}\right\};\\ {R}_{3}=\left\{{r}_{1}+{r}_{2}+{r}_{4},{r}_{1}+{r}_{2}+{r}_{6},{r}_{1}+{r}_{3}+{r}_{6},{r}_{2}+{r}_{4}+{r}_{6}\right\};\\ {R}_{4}=\left\{{r}_{1}+{r}_{2}+{r}_{4}+{r}_{6},{r}_{1}+{r}_{2}+{r}_{3}+{r}_{6}\right\};\\ {R}_{5}=\left\{{r}_{1}+2{r}_{2}+{r}_{4}+{r}_{6},{r}_{1}+{r}_{2}+2{r}_{3}+{r}_{6},{r}_{1}+{r}_{2}+{r}_{3}+{r}_{4}+{r}_{6}\right\};\end{array}$

$\begin{array}{l}{R}_{6}=\left\{2{r}_{1}+{r}_{2}+2{r}_{3}+{r}_{6},{r}_{1}+{r}_{2}+2{r}_{3}+{r}_{4}+{r}_{6},{r}_{1}+{r}_{2}+2{r}_{3}+2{r}_{6}\right\};\\ {R}_{7}=\left\{2{r}_{1}+{r}_{2}+2{r}_{3}+{r}_{4}+{r}_{6},2{r}_{1}+{r}_{2}+2{r}_{3}+2{r}_{6},{r}_{1}+{r}_{2}+2{r}_{3}+{r}_{4}+2{r}_{6}\right\};\\ {R}_{8}=\left\{2{r}_{1}+{r}_{2}+2{r}_{3}+{r}_{4}+2{r}_{6}\right\};\\ {R}_{9}=\left\{2{r}_{1}+2{r}_{2}+2{r}_{3}+{r}_{4}+2{r}_{6}\right\};\\ {R}_{10}=\varphi .\end{array}$

By Theorem 2.8, the unit form $f\left(x,y,{y}_{1},{y}_{2},{y}_{3},z\right)$ is weakly positive, i.e., $f\left(x,y,{y}_{1},{y}_{2},{y}_{3},z\right)\ge 0$ for any $\left(x,y,{y}_{1},{y}_{2},{y}_{3},z\right)\in {ℕ}^{6}$ , and $f\left(x,y,{y}_{1},{y}_{2},{y}_{3},z\right)=0⇔x=y={y}_{1}={y}_{2}={y}_{3}=z=0$ . So, $q\left(M\right)\le 0⇔{a}_{1}+{a}_{3}\le {a}_{2}$ , ${a}_{2}+{a}_{5}\le {a}_{3}+{a}_{4}$ , ${a}_{4}+{a}_{8}\le {a}_{5}+{a}_{7}\le {a}_{6}$ . And $q\left(M\right)=0⇔{a}_{1}+{a}_{3}\le {a}_{2}$ , ${a}_{2}+{a}_{5}\le {a}_{3}+{a}_{4}$ , ${a}_{4}+{a}_{8}\le {a}_{5}+{a}_{7}\le {a}_{6}$ and $x=y={y}_{1}={y}_{2}={y}_{3}=z=0$ , word 12132423(13-2, 25-34, 48-57-6) is tight by Theorem 2.3. So (9) holds.

Case 8. $t=9$ . Applying the $M-S$ word-procedure to ${M}_{8}$ , deleting words including subwords in $\Phi \left(\left\{1212,123123,1232123,2132132\right\}\right)\cup \Psi \Phi \left(\left\{1212,123123,1232123,2132132\right\}\right)$ , we get ${S}_{9}\cup \Phi \left(\left\{121342342\right\}\right)\cup \Psi \Phi \left(\left\{121342342\right\}\right)$ .

As ${S}_{8}^{2},{S}_{8}^{3},{S}_{8}^{4}$ , words in ${S}_{9}^{1},{S}_{9}^{2},{S}_{9}^{3}$ are all tight.

Consider ${S}_{9}^{4}$ , it is found that there is four 2, three 1 (or 2), one 3, and one 4 in every word, only consider word 121324212, we have ${\mathcal{M}}_{i,a}=\left\{M={M}_{{x}_{1},\cdots {x}_{4},{y}_{1},\cdots ,{y}_{9}}|\left({x}_{1},\cdots {x}_{4},{y}_{1},\cdots ,{y}_{9}\right)\in {ℕ}^{13}\right\}$ , where

$M=\left(\begin{array}{ccccccccc}{b}_{11}& 0& {x}_{1}& 0& 0& 0& 0& {x}_{2}& 0\\ 0& {b}_{22}& 0& 0& {y}_{1}& 0& {y}_{2}& 0& {y}_{3}\\ {x}_{3}& 0& {b}_{33}& 0& 0& 0& 0& {x}_{4}& 0\\ 0& 0& 0& {a}_{4}& 0& 0& 0& 0& 0\\ 0& {y}_{4}& 0& 0& {b}_{55}& 0& {y}_{5}& 0& {y}_{6}\\ 0& 0& 0& 0& 0& {a}_{6}& 0& 0& 0\\ 0& {y}_{7}& 0& 0& {y}_{8}& 0& {b}_{77}& 0& {y}_{9}\\ {b}_{81}& 0& {b}_{83}& 0& 0& 0& 0& {b}_{88}& 0\\ 0& {b}_{92}& 0& 0& {b}_{95}& 0& {b}_{97}& 0& {b}_{99}\end{array}\right)$

$\begin{array}{l}{b}_{11}={a}_{1}-{x}_{1}-{x}_{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{22}={a}_{2}-{y}_{1}-{y}_{2}-{y}_{3},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{33}={a}_{3}-{x}_{3}-{x}_{4},\\ {b}_{55}={a}_{5}-{y}_{4}-{y}_{5}-{y}_{6},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{77}={a}_{7}-{y}_{7}-{y}_{8}-{y}_{9},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{81}={x}_{1}+{x}_{2}-{x}_{3},\\ {b}_{83}={x}_{3}+{x}_{4}-{x}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{88}={a}_{8}-{x}_{2}-{x}_{4},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{92}={y}_{1}+{y}_{2}+{y}_{3}-{y}_{4}-{y}_{7},\\ {b}_{95}={y}_{4}+{y}_{5}+{y}_{6}-{y}_{1}-{y}_{8},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{97}={y}_{7}+{y}_{8}+{y}_{9}-{y}_{2}-{y}_{5},\\ {b}_{99}={a}_{9}-{y}_{3}-{y}_{6}-{y}_{9},\end{array}$

and

$\begin{array}{l}q\left(M\right)=2〈{i}_{1},{i}_{1}〉\left({a}_{1}{x}_{1}+{a}_{1}{x}_{2}+{a}_{3}{x}_{3}+{a}_{3}{x}_{4}+{a}_{8}{x}_{2}+{a}_{8}{x}_{4}-2{x}_{1}^{2}-2{x}_{2}^{2}-2{x}_{3}^{2}-2{x}_{4}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{x}_{1}{x}_{2}+2{x}_{1}{x}_{3}+{x}_{1}{x}_{4}+{x}_{2}{x}_{3}-{x}_{2}{x}_{4}-2{x}_{3}{x}_{4}\right)+2〈{i}_{2},{i}_{2}〉\left({a}_{2}{y}_{1}+{a}_{2}{y}_{2}+{a}_{2}{y}_{3}+{a}_{5}{y}_{4}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{a}_{5}{y}_{5}+{a}_{5}{y}_{6}+{a}_{7}{y}_{7}+{a}_{7}{y}_{8}+{a}_{7}{y}_{9}+{a}_{9}{y}_{3}+{a}_{9}{y}_{6}+{a}_{9}{y}_{9}-2{y}_{1}^{2}-2{y}_{2}^{2}-2{y}_{3}^{2}-2{y}_{4}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{y}_{5}^{2}-2{y}_{6}^{2}-2{y}_{7}^{2}-2{y}_{8}^{2}-2{y}_{9}^{2}-2{y}_{1}{y}_{2}-2{y}_{1}{y}_{3}+2{y}_{1}{y}_{4}+{y}_{1}{y}_{5}+{y}_{1}{y}_{6}+{y}_{1}{y}_{7}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{y}_{1}{y}_{8}-2{y}_{2}{y}_{3}+{y}_{2}{y}_{4}-{y}_{2}{y}_{5}+2{y}_{2}{y}_{7}+{y}_{2}{y}_{8}+{y}_{2}{y}_{9}+{y}_{3}{y}_{4}-{y}_{3}{y}_{6}+{y}_{3}{y}_{7}-{y}_{3}{y}_{9}\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{y}_{4}{y}_{5}-2{y}_{4}{y}_{6}-{y}_{4}{y}_{7}+{y}_{4}{y}_{8}-2{y}_{5}{y}_{6}+{y}_{5}{y}_{7}+2{y}_{5}{y}_{8}+{y}_{5}{y}_{9}+{y}_{6}{y}_{8}-{y}_{6}{y}_{9}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{y}_{7}{y}_{8}-2{y}_{7}{y}_{9}-2{y}_{8}{y}_{9}\right)+\left({i}_{1}\cdot {i}_{1}\right)\left({a}_{3}{x}_{1}+2{a}_{3}{x}_{2}-{a}_{3}{x}_{3}+{x}_{1}^{2}+{x}_{2}^{2}+{x}_{3}^{2}+{x}_{4}^{2}+{x}_{1}{x}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}_{3}{x}_{4}-{x}_{1}{x}_{3}-{x}_{1}{x}_{4}-{x}_{2}{x}_{3}\right)+\left({i}_{2}\cdot {i}_{2}\right)\left({a}_{5}{y}_{1}+2{a}_{5}{y}_{2}+2{a}_{5}{y}_{3}-{a}_{5}{y}_{4}+{a}_{7}{y}_{2}+2{a}_{7}{y}_{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{a}_{7}{y}_{7}+{a}_{7}{y}_{5}+2{a}_{7}{y}_{6}-{a}_{7}{y}_{8}+{y}_{1}^{2}+{y}_{2}^{2}+{y}_{3}^{2}+{y}_{4}^{2}+{y}_{5}^{2}+{y}_{6}^{2}+{y}_{7}^{2}+{y}_{8}^{2}+{y}_{9}^{2}+{y}_{1}{y}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{y}_{1}{y}_{3}-{y}_{1}{y}_{4}-{y}_{1}{y}_{5}-{y}_{1}{y}_{6}+{y}_{1}{y}_{8}+{y}_{2}{y}_{3}-{y}_{2}{y}_{4}-{y}_{2}{y}_{6}-{y}_{2}{y}_{7}-{y}_{2}{y}_{9}-{y}_{3}{y}_{4}\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-{y}_{3}{y}_{7}+{y}_{4}{y}_{5}+{y}_{4}{y}_{6}-{y}_{4}{y}_{8}+{y}_{5}{y}_{6}-{y}_{5}{y}_{7}-{y}_{5}{y}_{8}-{y}_{5}{y}_{9}-{y}_{6}{y}_{7}-{y}_{6}{y}_{8}+{y}_{7}{y}_{8}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{y}_{7}{y}_{9}+{y}_{8}{y}_{9}\right)+2\left({i}_{1}\cdot {i}_{2}\right)\left({a}_{2}{x}_{1}+{a}_{2}{x}_{2}+{a}_{5}{x}_{2}+{a}_{5}{x}_{4}+{a}_{7}{x}_{2}+{a}_{7}{x}_{4}+{a}_{3}{y}_{1}+{a}_{3}{y}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{a}_{3}{y}_{3}+{a}_{8}{y}_{3}+{a}_{8}{y}_{6}+{a}_{8}{y}_{9}-{x}_{1}{y}_{1}-{x}_{1}{y}_{2}-{x}_{1}{y}_{3}+{x}_{1}{y}_{4}+{x}_{1}{y}_{7}-{x}_{2}{y}_{3}-{x}_{2}{y}_{6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{x}_{2}{y}_{9}-{x}_{3}{y}_{4}-{x}_{3}{y}_{7}-{x}_{4}{y}_{4}-{x}_{4}{y}_{6}-{x}_{4}{y}_{7}-{x}_{4}{y}_{9}\right)+2\left({i}_{2}\cdot {i}_{4}\right)\left({a}_{4}{y}_{1}+{a}_{4}{y}_{2}+{a}_{4}{y}_{3}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({i}_{2}\cdot {i}_{6}\right)\left({a}_{6}{y}_{2}+{a}_{6}{y}_{3}+{a}_{6}{y}_{5}+{a}_{6}{y}_{6}\right)\end{array}$

$\begin{array}{l}=-2f\left({x}_{1},\cdots ,{x}_{4},{y}_{1},\cdots ,{y}_{9}\right)+2\left({a}_{1}+{a}_{3}-{a}_{2}\right){x}_{1}+2\left({a}_{1}+2{a}_{3}+{a}_{8}-{a}_{2}-{a}_{5}-{a}_{7}\right){x}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({a}_{3}+{a}_{8}-{a}_{5}-{a}_{7}\right){x}_{4}+2\left({a}_{2}+{a}_{5}-{a}_{3}-{a}_{4}\right){y}_{1}+2\left({a}_{2}+2{a}_{5}+{a}_{7}-{a}_{3}-{a}_{4}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{a}_{6}\right){y}_{2}+2\left({a}_{2}+2{a}_{5}+2{a}_{7}+{a}_{9}-{a}_{3}-{a}_{4}-{a}_{6}-{a}_{8}\right){y}_{3}+2\left({a}_{5}+{a}_{7}-{a}_{6}\right){y}_{5}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({a}_{5}+2{a}_{7}+{a}_{9}-{a}_{6}-{a}_{8}\right){y}_{6}+2\left({a}_{7}+{a}_{9}-{a}_{8}\right){y}_{9},\end{array}$

where $f\left({x}_{1},\cdots ,{x}_{4},{y}_{1},\cdots ,{y}_{9}\right)=\frac{1}{2}X{A}_{f}{X}^{\text{T}}$ , $X=\left({x}_{1},\cdots ,{x}_{4},{y}_{1},\cdots ,{y}_{9}\right)$ symmetric

matrix ${A}_{f}$ is as follows

${A}_{f}=\left(\begin{array}{ccccccccccccc}2& 1& -1& 0& -1& -1& -1& 1& 0& 0& 1& 0& 0\\ 1& 2& 0& 1& 0& 0& -1& 0& 0& -1& 0& 0& -1\\ -1& 0& 2& 1& 0& 0& 0& -1& 0& 0& -1& 0& 0\\ 0& 1& 1& 2& 0& 0& 0& -1& 0& -1& -1& 0& -1\\ -1& 0& 0& 0& 2& 1& 1& -1& 0& 0& -1& 0& 0\\ -1& 0& 0& 0& 1& 2& 1& 0& 1& 1& -1& -1& 0\\ -1& -1& 0& 0& 1& 1& 2& 0& 0& 1& 0& 0& 1\\ 1& 0& -1& -1& -1& 0& 0& 2& 1& 1& 1& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 1& 2& 1& 0& -1& 0\\ 0& -1& 0& -1& 0& 1& 1& 1& 1& 2& 1& 0& 1\\ 1& 0& -1& -1& -1& -1& 0& 1& 0& 1& 2& 1& 1\\ 0& 0& 0& 0& 0& -1& 0& 0& -1& 0& 1& 2& 1\\ 0& -1& 0& -1& 0& 0& 1& 0& 0& 1& 1& 1& 2\end{array}\right)$

Using the above algorithm in §2, we have

$\begin{array}{l}{R}_{1}=\left\{{r}_{1},\cdots ,{r}_{13}\right\};\\ {R}_{2}=\left\{{r}_{1}+{r}_{3},{r}_{1}+{r}_{5},{r}_{1}+{r}_{6},{r}_{1}+{r}_{7},{r}_{2}+{r}_{7},{r}_{2}+{r}_{10},{r}_{2}+{r}_{13},{r}_{3}+{r}_{8},{r}_{3}+{r}_{11},\\ \text{ }\text{ }{r}_{4}+{r}_{8},{r}_{4}+{r}_{10},{r}_{4}+{r}_{11},{r}_{4}+{r}_{13},{r}_{5}+{r}_{8},{r}_{5}+{r}_{11},{r}_{6}+{r}_{11},{r}_{6}+{r}_{12},{r}_{9}+{r}_{12}\right\};\\ {R}_{3}=\left\{{r}_{1}+{r}_{3}+{r}_{5},{r}_{1}+{r}_{3}+{r}_{6},{r}_{1}+{r}_{3}+{r}_{7},{r}_{1}+{r}_{6}+{r}_{12},{r}_{3}+{r}_{5}+{r}_{8},{r}_{3}+{r}_{5}+{r}_{11},\\ \text{ }\text{ }{r}_{3}+{r}_{6}+{r}_{11},{r}_{4}+{r}_{5}+{r}_{8},{r}_{4}+{r}_{8}+{r}_{13},{r}_{4}+{r}_{5}+{r}_{11},{r}_{4}+{r}_{6}+{r}_{11}\right\};\end{array}$

$\begin{array}{l}{R}_{4}=\left\{{r}_{1}+{r}_{3}+{r}_{5}+{r}_{8},{r}_{1}+{r}_{3}+{r}_{5}+{r}_{11},{r}_{1}+{r}_{3}+{r}_{6}+{r}_{11},{r}_{1}+{r}_{3}+{r}_{6}+{r}_{12},\\ \text{ }\text{ }{r}_{3}+{r}_{5}+{r}_{8}+{r}_{11},{r}_{4}+{r}_{5}+{r}_{8}+{r}_{11},{r}_{4}+{r}_{5}+{r}_{8}+{r}_{13}\right\};\\ {R}_{5}=\left\{{r}_{1}+{r}_{3}+{r}_{5}+{r}_{6}+{r}_{11},{r}_{3}+{r}_{4}+{r}_{5}+{r}_{8}+{r}_{11}\right\};\\ {R}_{6}=\varphi .\end{array}$

By Theorem 2.8, the unit form $f\left({x}_{1},\cdots ,{x}_{4},{y}_{1},\cdots ,{y}_{9}\right)$ is weakly positive, i.e., for any $\left({x}_{1},\cdots ,{x}_{4},{y}_{1},\cdots ,{y}_{9}\right)\in {ℕ}^{13}$ , $f\left({x}_{1},\cdots ,{x}_{4},{y}_{1},\cdots ,{y}_{9}\right)\ge 0$ , and $f\left({x}_{1},\cdots ,{x}_{4},{y}_{1},\cdots ,{y}_{9}\right)=0⇔{x}_{1}=\cdots ={x}_{4}={y}_{1}=\cdots ={y}_{9}=0$ . So, we have q(M) ≤ 0 if and only if

${a}_{1}+{a}_{3}\le {a}_{2},{a}_{3}+{a}_{8}\le {a}_{5}+{a}_{7}\le {a}_{6},{a}_{7}+{a}_{9}\le {a}_{8},{a}_{2}+{a}_{5}\le {a}_{3}+{a}_{4}.$ (2)

$q\left(M\right)=0$ if and only if (2) hold and ${x}_{1}=\cdots ={x}_{4}={y}_{1}=\cdots ={y}_{9}=0$ , so word 121324212(13-2, 38-57-6, 79-8, 25-34) is tight by Theorem 2.3.

At last, let us see ${S}_{9}^{5}$ , we find that there is five 2, two 1, one 2, and one 4 in every word, so it suffices to consider word 212321242, we have ${\mathcal{M}}_{i,a}=\left\{M={M}_{{x}_{1},\cdots {x}_{16},y}|\left({x}_{1},\cdots ,{x}_{16},y\right)\in {ℕ}^{17}\right\}$ , where

$M=\left(\begin{array}{ccccccccc}{b}_{11}& 0& {x}_{1}& 0& {x}_{2}& 0& {x}_{3}& 0& {x}_{4}\\ 0& {a}_{2}-y& 0& 0& 0& y& 0& 0& 0\\ {x}_{5}& 0& {b}_{33}& 0& {x}_{6}& 0& {x}_{7}& 0& {x}_{8}\\ 0& 0& 0& {a}_{4}& 0& 0& 0& 0& 0\\ {x}_{9}& 0& {x}_{10}& 0& {b}_{55}& 0& {x}_{11}& 0& {x}_{12}\\ 0& y& 0& 0& 0& {a}_{6}-y& 0& 0& 0\\ {x}_{13}& 0& {x}_{14}& 0& {x}_{15}& 0& {b}_{77}& 0& {x}_{16}\\ 0& 0& 0& 0& 0& 0& 0& {a}_{8}& 0\\ {b}_{91}& 0& {b}_{93}& 0& {b}_{95}& 0& {b}_{97}& 0& {b}_{99}\end{array}\right)$

$\begin{array}{l}{b}_{11}={a}_{1}-{x}_{1}-{x}_{2}-{x}_{3}-{x}_{4},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{33}={a}_{3}-{x}_{5}-{x}_{6}-{x}_{7}-{x}_{8},\\ {b}_{55}={a}_{5}-{x}_{9}-{x}_{10}-{x}_{11}-{x}_{12},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{77}={a}_{7}-{x}_{13}-{x}_{14}-{x}_{15}-{x}_{16},\\ {b}_{91}={x}_{1}+{x}_{2}+{x}_{3}+{x}_{4}-{x}_{5}-{x}_{9}-{x}_{13},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{93}={x}_{5}+{x}_{6}+{x}_{7}+{x}_{8}-{x}_{1}-{x}_{10}-{x}_{14},\\ {b}_{95}={x}_{9}+{x}_{10}+{x}_{11}+{x}_{12}-{x}_{2}-{x}_{6}-{x}_{15},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{97}={x}_{13}+{x}_{14}+{x}_{15}+{x}_{16}-{x}_{3}-{x}_{7}-{x}_{11},\\ {b}_{99}={a}_{9}-{x}_{4}-{x}_{8}-{x}_{12}-{x}_{16}.\end{array}$

So, we have

$\begin{array}{l}q\left(M\right)=2〈{i}_{1},{i}_{1}〉\left({a}_{1}{x}_{1}+{a}_{1}{x}_{2}+{a}_{1}{x}_{3}+{a}_{1}{x}_{4}+{a}_{3}{x}_{5}+{a}_{3}{x}_{6}+{a}_{3}{x}_{7}+{a}_{3}{x}_{8}+{a}_{5}{x}_{9}+{a}_{5}{x}_{10}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{a}_{5}{x}_{11}+{a}_{5}{x}_{12}+{a}_{7}{x}_{13}+{a}_{7}{x}_{14}+{a}_{7}{x}_{15}+{a}_{7}{x}_{16}+{a}_{9}{x}_{4}+{a}_{9}{x}_{8}+{a}_{9}{x}_{12}+{a}_{9}{x}_{16}-2{x}_{1}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{x}_{2}^{2}-2{x}_{3}^{2}-2{x}_{4}^{2}-2{x}_{5}^{2}-2{x}_{6}^{2}-2{x}_{7}^{2}-2{x}_{8}^{2}-2{x}_{9}^{2}-2{x}_{10}^{2}-2{x}_{11}^{2}-2{x}_{12}^{2}-2{x}_{13}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{x}_{14}^{2}-2{x}_{15}^{2}-2{x}_{16}^{2}-2{x}_{1}{x}_{2}-2{x}_{1}{x}_{3}-2{x}_{1}{x}_{4}+2{x}_{1}{x}_{5}+{x}_{1}{x}_{6}+{x}_{1}{x}_{7}+{x}_{1}{x}_{8}+{x}_{1}{x}_{9}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{x}_{1}{x}_{10}+{x}_{1}{x}_{13}-{x}_{1}{x}_{14}-2{x}_{2}{x}_{3}-2{x}_{2}{x}_{4}+{x}_{2}{x}_{5}-{x}_{2}{x}_{6}+2{x}_{2}{x}_{9}+{x}_{2}{x}_{10}+{x}_{2}{x}_{11}\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}_{2}{x}_{12}+{x}_{2}{x}_{13}-{x}_{2}{x}_{15}-2{x}_{3}{x}_{4}+{x}_{3}{x}_{5}-{x}_{3}{x}_{7}+{x}_{3}{x}_{9}-{x}_{3}{x}_{11}+2{x}_{3}{x}_{13}+{x}_{3}{x}_{14}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}_{3}{x}_{15}+{x}_{3}{x}_{16}+{x}_{4}{x}_{5}-{x}_{4}{x}_{8}+{x}_{4}{x}_{9}-{x}_{4}{x}_{12}+{x}_{4}{x}_{13}-{x}_{4}{x}_{16}-2{x}_{5}{x}_{6}-2{x}_{5}{x}_{7}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{x}_{5}{x}_{8}-{x}_{5}{x}_{9}+{x}_{5}{x}_{10}-{x}_{5}{x}_{13}+{x}_{5}{x}_{14}-2{x}_{6}{x}_{7}-2{x}_{6}{x}_{8}+{x}_{6}{x}_{9}+2{x}_{6}{x}_{10}+{x}_{6}{x}_{11}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}_{6}{x}_{12}+{x}_{6}{x}_{14}-{x}_{6}{x}_{15}-2{x}_{7}{x}_{8}+{x}_{7}{x}_{10}-{x}_{7}{x}_{11}+{x}_{7}{x}_{13}+2{x}_{7}{x}_{14}+{x}_{7}{x}_{15}+{x}_{7}{x}_{16}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}_{8}{x}_{10}-{x}_{8}{x}_{12}+{x}_{8}{x}_{14}-{x}_{8}{x}_{16}-2{x}_{9}{x}_{10}-2{x}_{9}{x}_{11}-2{x}_{9}{x}_{12}-{x}_{9}{x}_{13}+{x}_{9}{x}_{15}-2{x}_{10}{x}_{11}\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-2{x}_{10}{x}_{12}-{x}_{10}{x}_{14}+{x}_{10}{x}_{15}-2{x}_{11}{x}_{12}+{x}_{11}{x}_{13}+{x}_{11}{x}_{14}+2{x}_{11}{x}_{15}+{x}_{11}{x}_{16}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}_{12}{x}_{15}-{x}_{12}{x}_{16}-2{x}_{13}{x}_{14}-2{x}_{13}{x}_{15}-2{x}_{13}{x}_{16}-2{x}_{14}{x}_{15}-2{x}_{14}{x}_{16}-2{x}_{15}{x}_{16}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2〈{i}_{2},{i}_{2}〉\left({a}_{2}y+{a}_{6}y-2{y}^{2}\right)+2\left({i}_{1}\cdot {i}_{2}\right)\left({a}_{2}{x}_{1}+{a}_{2}{x}_{2}+{a}_{2}{x}_{3}+{a}_{2}{x}_{4}+{a}_{6}{x}_{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{a}_{6}{x}_{4}+{a}_{6}{x}_{7}+{a}_{6}{x}_{8}+{a}_{6}{x}_{11}+{a}_{6}{x}_{12}+{a}_{3}y+{a}_{5}y-{x}_{5}y-{x}_{7}y-{x}_{8}y-{x}_{9}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{x}_{11}y-{x}_{12}y\right)+2\left({i}_{1}\cdot {i}_{4}\right)\left({a}_{4}{x}_{2}+{a}_{4}{x}_{3}+{a}_{4}{x}_{4}+{a}_{4}{x}_{6}+{a}_{4}{x}_{7}+{a}_{4}{x}_{8}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({i}_{1}\cdot {i}_{8}\right)\left({a}_{8}{x}_{4}+{a}_{8}{x}_{8}+{a}_{8}{x}_{12}+{a}_{8}{x}_{16}\right)\end{array}$

$\begin{array}{l}=-2f\left({x}_{1},\cdots ,{x}_{16},y\right)+2\left({a}_{1}+{a}_{3}-{a}_{2}\right){x}_{1}+2\left({a}_{1}+2{a}_{3}+{a}_{5}-{a}_{2}-{a}_{4}\right){x}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({a}_{1}+2{a}_{3}+2{a}_{5}+{a}_{7}-{a}_{2}-{a}_{4}-{a}_{6}\right){x}_{3}+2\left({a}_{1}+2{a}_{3}+2{a}_{5}+2{a}_{7}+{a}_{9}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{a}_{2}-{a}_{4}-{a}_{6}-{a}_{8}\right)\right){x}_{4}+2\left({a}_{3}+{a}_{5}-{a}_{4}\right){x}_{6}+2\left({a}_{3}+2{a}_{5}+{a}_{7}-{a}_{4}-{a}_{6}\right){x}_{7}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left({a}_{3}+2{a}_{5}+2{a}_{7}+{a}_{9}-{a}_{4}-{a}_{6}-{a}_{8}\right){x}_{8}+2\left({a}_{5}+{a}_{7}-{a}_{6}\right){x}_{11}+2\left({a}_{5}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2{a}_{7}+{a}_{9}-{a}_{6}-{a}_{8}\right){x}_{12}+2\left({a}_{7}+{a}_{9}-{a}_{9}\right){x}_{16}+2\left({a}_{2}+{a}_{6}-{a}_{3}-{a}_{5}\right)y,\end{array}$

where $f\left({x}_{1},\cdots ,{x}_{16},y\right)=\frac{1}{2}X{A}_{f}{X}^{\text{T}},X=\left({x}_{1},\cdots ,{x}_{16},y\right)$ , symmetric matrix ${A}_{f}$ is

as follows

${A}_{f}=\left(\begin{array}{ccccccccccccccccc}2& 1& 1& 1& -1& 0& 0& 0& -1& 0& 0& 0& -1& 0& 0& 0& 0\\ 1& 2& 1& 1& 0& 1& 1& 1& -1& -1& 0& 0& -1& -1& 0& 0& 0\\ 1& 1& 2& 1& 0& 0& 1& 1& 0& 0& 1& 1& -1& -1& -1& 0& 0\\ 1& 1& 1& 2& 0& 0& 0& 1& 0& 0& 0& 1& 0& 0& 0& 1& 0\\ -1& 0& 0& 0& 2& 1& 1& 1& 1& 0& 0& 0& 1& 0& 0& 0& -1\\ 0& 1& 0& 0& 1& 2& 1& 1& 0& -1& 0& 0& 0& -1& 0& 0& 0\\ 0& 1& 1& 0& 1& 1& 2& 1& 1& 0& 1& 1& 0& -1& -1& 0& -1\\ 0& 1& 1& 1& 1& 1& 1& 2& 1& 0& 0& 1& 1& 0& 0& 1& -1\\ -1& -1& 0& 0& 1& 0& 1& 1& 2& 1& 1& 1& 1& 1& 0& 0& -1\\ 0& -1& 0& 0& 0& -1& 0& 0& 1& 2& 1& 1& 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 1& 0& 1& 1& 2& 1& 0& 0& -1& 0& -1\\ 0& 0& 1& 1& 0& 0& 1& 1& 1& 1& 1& 2& 1& 1& 0& 1& -1\\ -1& -1& -1& 0& 1& 0& 0& 1& 1& 0& 0& 1& 2& 1& 1& 1& 0\\ 0& -1& -1& 0& 0& -1& -1& 0& 1& 1& 0& 1& 1& 2& 1& 1& 0\\ 0& 0& -1& 0& 0& 0& -1& 0& 0& 0& -1& 0& 1& 1& 2& 1& 0\\ 0& 0& 0& 1& 0& 0& 0& \end{array}$