Kazhdan-Lusztig polynomials

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 6 November 2012

Bar invariant bases

Let $(W,\le )$ be a poset such that for $u,v\in W$ the interval $[u,v]$ is finite. Let $M$ be the free $\mathbb{Z}[q,q^{-1}]$ modules with basis $\{T_w\mid w\in W\},$

$$M=\mathbb{Z}\left[q{q}^{-1}\right]\text{-span}\{T_w\mid w\in W\},$$

and let $‾:M\to M$ be a $\mathbb{Z}\text{-linear}$ involution such that

$$\bar{q}=q^{-1}\text{and}\overline{T_w}=T_w+\sum _{v<w}{a}_{wv}{T}_v,$$

where $a_{vw}\in \mathbb{Z}\left[q{q}^{-1}\right]\text{.}$ Then

1. There is a unique basis $\{C_w^{-}\mid w\in W\}$ such that $$\overline{C_w}=C_w\text{and}{C}_w^{-}=T_w+\sum _{v<w}{P}_{vw}^{-}{T}_v\text{with}{P}_{vw}^{-}\in q^{-1}\mathbb{Z}\left[q^{-1}\right]\text{.}$$
2. There is a unique basis $\{C_w^{-}\mid w\in W\}$ such that $$\overline{C_w}=C_w\text{and}{C}_w^{+}=T_w+\sum _{v<w}{P}_{vw}^{+}{T}_v\text{with}{P}_{vw}^{-}\in q\mathbb{Z}\left[q\right]\text{.}$$

		Proof.
	(a) The $p_{vw}^{-}$ are determined by induction: $$P_{ww}^{-}=1\text{and}{P}_{uw}^{-}=\sum _{k\in {\mathbb{Z}}_{<0}}{f}_k{q}^k$$ where $$f=\sum _{k\in \mathbb{Z}}{f}_k{q}^k=\sum _{u<z\le w}{a}_{uz}\overline{P_{zw}^{-}}(=P_{uw}^{-}-\overline{P_{uw}^{-}})\text{.}$$ (b) The $P_{vw}^{+}$ are determined by induction: $$P_{ww}^{+}=1\text{and}{P}_{uw}^{+}=\sum _{k\in {\mathbb{Z}}_{>0}}{f}_k{q}^k$$ where $$f=\sum _{k\in \mathbb{Z}}{f}_k{q}^k=\sum _{u<z\le w}{a}_{uz}\overline{P_{zw}^{+}}(=P_{uw}^{+}-\overline{P_{uw}^{+}})\text{.}$$ $\square$

The dual module

$$M^{*}={\text{Hom}}_{\mathbb{Z}[q,q^{-1}]}(M,\mathbb{Z}[q,q^{-1}])$$

is given a bar involution

$$‾:M^{*}\to M^{*}\text{defined by}⟨\bar{\phi },m⟩=\overline{⟨\phi ,\stackrel{‾}{m}⟩}$$

If $\{T^w\mid w\in W\}$ is the dual basis to $\{T_w\mid w\in W\}$ then

$$\overline{T^w}=\sum _v{b}_{vw}{T}^v\text{where}{b}_{vw}=⟨\overline{T^w},T_v⟩=\overline{⟨T^w,\stackrel{‾}{T_v}⟩}=\overline{⟨T^w,\sum _{z\le v}{a}_{zv}{T}_z⟩}$$

so that $B=\overline{A^t}\text{.}$ If $\{C^w\mid w\in W\}$ is the dual basis to $\{C_w\mid w\in W\}$ then

$$\overline{C^w}=C^w\text{since}⟨\overline{C^w},C_v⟩=\overline{⟨C^w,\stackrel{‾}{C_v}⟩}=\overline{⟨C^w,C_v⟩}={\delta }_{vw},$$

and

$$C^w=\sum P^{vw}{T}^v\text{where}{\delta }_{vw}⟨C^w,C_v⟩=⟨\sum _u{P}^{uw}{T}_u,\sum _z{P}_{zv}{T}_z⟩=\sum _u{P}^{uw}{P}_{uv}$$

so that

$$\left(P^{uw}\right)={\left({\left(P_{uv}\right)}^{-1}\right)}^t={\left(P^t\right)}^{-1}\text{.}$$

The affine Hecke algebra

The affine Hecke algebra $\stackrel{\sim }{H}$ has $\mathbb{Z}[q,q^{-1}]$ basis $\{T_w\mid w\in \stackrel{\sim }{W}\},$

$$\stackrel{\sim }{H}=\mathbb{Z}[q,q^{-1}]\text{-span}\{T_w\mid w\in \stackrel{\sim }{W}\}$$

with relations

$$\begin{array}{cc}{T}_{w_1}{T}_{w_2}=T_{w_1{w}_2},& \text{if}\ell \left(w_1{w}_2\right)=\ell \left(w_1\right)+\ell \left(w_2\right),\\ T_{s_i}{T}_w=(q-q^{-1})T_w+T_{s_{i}w},& \text{if}\ell \left(s_{i}w\right)<\ell \left(w\right)(0\le i\le n)\text{.}\end{array}$$

The algebra $\stackrel{\sim }{H}$ also has bases

$$\{X^{\lambda }{T}_w\mid w\in W,\lambda \in P\}\text{and}\{T_v{X}^{\mu }\mid v\in W,\mu \in P\},$$

where

$$X^{\lambda }=T_{t_{\lambda }},\text{if}\lambda \in P^{+},\text{and}{X}^{\lambda }=X^{\mu }{\left(X^{\nu }\right)}^{-1},$$

if $\lambda =\mu -\nu$ with $\mu ,\nu \in P^{+}\text{.}$

The bar involution on $\stackrel{\sim }{H}$ is the $\mathbb{Z}\text{-linear}$ map $‾:\stackrel{\sim }{H}\to \stackrel{\sim }{H}$ given by

$$\bar{q}=q^{-1}\text{and}\overline{T_w}=T_{w^{-1}}^{-1}\text{for}w\in \stackrel{\sim }{W}\text{.}$$

Define elements $1_0,{\epsilon }_0\in H$ by

$$\begin{array}{ccc}{1}_0^2=1_0,& \text{and}& T_{s_i}{1}_0=q{1}_0,\text{for}1\le i\le n,\\ {\epsilon }_0^2={\epsilon }_0,& \text{and}& T_{s_i}{\epsilon }_0=q^{-1}{\epsilon }_0,\text{for}1\le i\le n,\end{array}$$

and let

$$A_{\mu }={\epsilon }_0{X}^{\mu }{1}_0,\text{for}\mu \in P\text{.}$$

1. $\overline{X^{\lambda }}=T_{w_0}{X}^{w_0\lambda }{T}_{w_0}^{-1},$ for $\lambda \in P,$
2. $\overline{1_0}=1_0$ and $\overline{{\epsilon }_0}={\epsilon }_0\text{.}$
3. If $z\in \mathbb{Z}{\left[P\right]}^W$ then $\bar{z}=z\text{.}$
4. $\overline{q^{-\ell \left(w_0\right)}{A}_{\lambda +\rho }}=q^{-\ell \left(w_0\right)}{A}_{\lambda +\rho }\text{.}$

The $\tau \text{-operators}$ are given by

$${\tau }_i=T_i-\frac{q-q^{-1}}{1-X^{-{\alpha }_i}}\text{.}$$

Then

1. $X^{\lambda }{\tau }_i={\tau }_i{X}^{s_i\lambda },$
2. ${\tau }_i^2=\frac{(q-q^{-1}{X}^{{\alpha }_i})(q-q^{-1}{X}^{-{\alpha }_i})}{(1-X^{{\alpha }_i})(1-X^{-{\alpha }_i})}$
3. $\underset{\underset{m_{ij}\text{factors}}{⏟}}{{\tau }_i{\tau }_j{\tau }_i\dots }=\underset{\underset{m_{ij}\text{factors}}{⏟}}{{\tau }_j{\tau }_i{\tau }_j\dots }$

The shift operator is

$$\Delta =\prod _{\alpha \in R^{+}}(q{X}^{\alpha /2}-q^{-1}{X}^{-\alpha /2})\text{.}$$

Then

$$\begin{array}{c}(T_i+q)\Delta =\left(s_i\Delta \right)(T_i-q)\text{and}\Delta ℂ{\left[P\right]}^W=\{h\in \stackrel{\sim }{H}\mid (t_i+q^{-1}h=0\text{for}1\le i\le n\}\\ {\epsilon }_0{E}_{\lambda }=\Delta 1_0{E}_{\lambda -\rho }\text{and}{⟨\Delta f,\Delta g⟩}_k=q^{Nk}{⟨f,g⟩}_{k+1}\text{.}\end{array}$$

The $??\text{-trace}$ on $\stackrel{\sim }{H}$ is the linear map $\tau :\stackrel{\sim }{H}\to ℂ$ given by

$$\text{tr}\left(h\right)=h{\mid }_1,\text{or, more precisely,}\text{tr}\left(T_w\right)={\delta }_{w1}\text{.}$$

Define an inner product $\stackrel{\sim }{H}$ by

$$⟨h_1,h_2⟩=\text{tr}\left(h_1{h}_2\right),$$

so that

$$⟨T_u,T_v⟩={\left[T_{u^{-1}}{T}_v\right]}_1\text{and}⟨T_u,T_v⟩=q^{\ell \left(u\right)??}{\delta }_{u{v}^{-1}}\text{.}$$

The generic degrees are $d_{\lambda }\left(q\right)$ given by

$$\text{tr}\sum _{\lambda \in \hat{H}}{d}_{\lambda }\left(q\right){\chi }_H^{\lambda }\text{.}$$

The Kazhdan-Lusztig basis is defined by

$$\{h\in H\mid ⟨h,h^{\#}⟩\in 1+q^{-1}\mathbb{Z}\left[q^{-1}\right],h=\bar{h}\}$$

or by the usual bar invariance and triangularity conditions.

Kazhdan-Lusztig polynomials

The Iwahori-Hecke algebra is the algebra over $\mathbb{Z}\left[q\right]$ given by generators $T_w,$ $w\in W$ and relations

$$\begin{array}{ccc}{T}_{s_i}{T}_w& =& \{\begin{array}{cc}{T}_{s_{i}w},& \text{if}{s}_{i}w>w,\\ q{T}_{s_{i}w}+(q-1)T_w,& \text{if}{s}_{i}w<w\text{.}\end{array}\end{array}$$

The bar involution on $H$ is the $\mathbb{Z}\text{-algebra}$ involution given by

$$\bar{q}=q^{-1}\text{and}\overline{T_w}=T_{w^{-1}}^{-1},$$

for $w\in W\text{.}$ The Kazhdan-Lusztig basis of $H$ is the basis $\{C_w\mid w\in W\}$ given by

1. $\overline{C_w}=C_w,$ and
2. $C_w=T_w+\sum _{v\le w}{p}_{vw}\left(q\right)T_v,\text{where}{p}_{vw}\left(q\right)\in q\mathbb{Z}\left[q\right]\text{.}$

Kazhdan-Lusztig polynomials

1. $P_{ww}\left(q\right)=1,$
2. $P_{xw}\left(q\right)=0,$ if $x\nless w,$
3. $\text{deg}\left(P_{xw}\left(q\right)\right)\le \frac{1}{2}(\ell \left(w\right)-\ell \left(x\right)-1),$ if $x\ne w\text{.}$

Define

$$\mu (x,w)=\text{coefficient of the highest degree term in}{P}_{xw}\left(q\right),$$

which is the term of degree $\frac{1}{2}(\ell \left(w\right)-\ell \left(x\right)-1)\text{.}$ Then, if $sw<w$

$$\begin{array}{ccc}{P}_{xw}\left(q\right)& =& \{\begin{array}{cc}{P}_{sx,w}\left(q\right),& \text{if}sx>x,\\ P_{sx,sw}\left(q\right)+q{P}_{x,sw}-\sum _{sz<z}{q}^{\frac{1}{2}(\ell \left(w\right)-\ell \left(z\right))}\mu (z,sw)P_{x,z},& \text{if}sx<x\text{.}\end{array}\end{array}$$

The $W\text{-graph}$ has

1. Vertices: $W$
2. Edges: $x\leftrightarrow y$ if $\mu [x,y]=\{\begin{array}{cc}\mu (x,y),& \text{if}x<y,\\ \mu (y,x),& \text{if}y<x,\end{array}$

Then

$$\begin{array}{ccc}KL{\left(s\right)}_{xx}& =& \{\begin{array}{cc}-1,& \text{if}sx<x,\\ 1,& \text{if}sx>x,\end{array}\\ KL{\left(s\right)}_{xy}& =& \{\begin{array}{cc}\mu [x,y],& \text{if}sx<x,sy>y\text{and}x\leftrightarrow y,\\ 0,& \text{otherwise,}\end{array}\end{array}$$

Define a relation ${\le }_L$ by taking the closure of the relation

$$x{\le }_{L}y\text{if}{D}_{\ell }\left(x\right)⊈D_{\ell }\left(y\right)\text{and}x\leftrightarrow y\text{is an edge.}$$

and define

$$x{=}_{L}y\text{if}x{\le }_{L}y\text{and}y{\le }_{L}x\text{.}$$

The case of dihedral groups

In type $A_1,$

$$H=\text{span}\{1,T_1\}\text{with}{T}_1^2=(q-1)T_1+q\text{.}$$

So

$$\overline{T_1}=T_1^{-1}=(q^{-1}-1)+q^{-1}{T}_1,\text{and}{C}_1=q^{-\frac{1}{2}}(1+T_1),$$

since

$$\overline{q^{-\frac{1}{2}}(1+T_1)}=q^{\frac{1}{2}}(1+T_1^{-1})=q^{\frac{1}{2}}(1+q^{-1}{T}_1+(q^{-1}-1))=q^{\frac{1}{2}}{q}^{-1}(1+T_1)=q^{-\frac{1}{2}}(1+T_1)\text{.}$$

In type $A_2,$ $H=\text{span}\{1,T_1,T_2,T_1{T}_2,T_2{T}_1,T_1{T}_2{T}_2\}$ and

$$\begin{array}{ccc}{C}_1& =& q^{-\frac{1}{2}}(1+T_1),\\ C_2& =& q^{-\frac{1}{2}}(1+T_2),\\ C_1{C}_2& =& q^{-1}(1+T_1+T_2+T_1{T}_2=C_{12},)\\ C_2{C}_1& =& q^{-1}(1+T_1+T_2+T_2{T}_1=C_{21},)\\ C_1{C}_{21}& =& q^{-\frac{3}{2}}(T_1{T}_2{T}_1+T_1{T}_2+(q-1)T_1+q+T_1+T_2{T}_1+T_1+T_2+1),\\ & =& q^{-\frac{3}{2}}(T_1{T}_2{T}_1+T_1{T}_2+T_2{T}_1+T_1+T_2+1)+C_1,\end{array}$$

so that

$$C_{121}=C_1{C}_{12}-C_1=q^{-\frac{3}{2}}(T_1{T}_2{T}_1+T_1{T}_2+T_2{T}_1+T_1+T_2+1)\text{.}$$

Note that $C_1^2=(q^{\frac{1}{2}}+q^{-\frac{1}{2}})C_1\text{.}$ Then, using that $T_i=q^{\frac{1}{2}}{C}_i-1,$ to produce the matrices for the regular representation in the KL-basis,

$$\rho \left(T_1\right)=\left(\begin{array}{cccccc}-1& 0& 0& 0& 0& 0\\ q^{\frac{1}{2}}& q& q^{\frac{1}{2}}& 0& 0& 0\\ 0& 0& -1& 0& 0& 0\\ 0& 0& 0& -1& 0& 0\\ 0& 0& 0& q^{\frac{1}{2}}& q& 0\\ 0& 0& q^{\frac{1}{2}}& 0& 0& q\end{array}\right)\text{and}\rho \left(T_2\right)=\left(\begin{array}{cccccc}-1& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0\\ 0& q^{\frac{1}{2}}& q& 0& 0& 0\\ q^{\frac{1}{2}}& 0& 0& q& q^{\frac{1}{2}}& 0\\ 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 0& q^{\frac{1}{2}}& q\end{array}\right)$$

with rows and columns indexed by $1,C_1,C_{21},C_2,C_{12},C_{121}\text{.}$

In type $B_2,$ $H=\text{span}\{1,T_1,T_2,T_1{T}_2,T_2{T}_1,T_1{T}_2{T}_1,T_2{T}_1{T}_2,T_1{T}_2{T}_1{T}_2\},$ and

$$C_1{C}_2=C_{12},C_2{C}_1=C_{21},C_1{C}_{21}=C_{121}+C_1,C_2{C}_{12}=C_{212}+C_2,C_2{C}_{121}=C_{2121}+C_{21}\text{.}$$

where

$$\begin{array}{ccc}{C}_1& =& q^{-\frac{1}{2}}(1+T_1),\\ C_2& =& q^{-\frac{1}{2}}(1+T_2),\\ C_{12}& =& q^{-1}(1+T_1+T_2+T_1{T}_2,)\\ C_{21}& =& q^{-1}(1+T_1+T_2+T_2{T}_1,)\\ C_{121}& =& q^{-\frac{3}{2}}(1+T_1+T_2+T_1{T}_2+T_2{T}_1+T_1{T}_2{T}_1),\\ C_{212}& =& q^{-\frac{3}{2}}(1+T_1+T_2+T_1{T}_2+T_2{T}_1+T_2{T}_1{T}_2),\\ C_{1212}& =& q^{-\frac{3}{2}}(1+T_1+T_2+T_1{T}_2+T_2{T}_1+T_1{T}_2{T}_1+T_2{T}_1{T}_2+T_2{T}_1{T}_2{T}_1)\text{.}\end{array}$$

and the matrices of the regular representation in the KL-basis are

$$\begin{array}{ccc}\rho \left(T_1\right)& =& \left(\begin{array}{cccccccc}-1& 0& 0& 0& 0& 0& 0& 0\\ q^{\frac{1}{2}}& q& q^{\frac{1}{2}}& 0& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0& 0& 0\\ 0& 0& q^{\frac{1}{2}}& q& 0& 0& 0& 0\\ 0& 0& 0& 0& -1& 0& 0& 0\\ 0& 0& 0& 0& q^{\frac{1}{2}}& q& q^{\frac{1}{2}}& 0\\ 0& 0& 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 0& 0& q^{\frac{1}{2}}& q\end{array}\right)\\ \rho \left(T_2\right)& =& \left(\begin{array}{cccccccc}-1& 0& 0& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0& 0& 0\\ 0& q^{\frac{1}{2}}& q& q^{\frac{1}{2}}& 0& 0& 0& 0\\ 0& 0& 0& -1& 0& 0& 0& 0\\ q^{\frac{1}{2}}& 0& 0& 0& q& q^{\frac{1}{2}}& 0& 0\\ 0& 0& 0& 0& 0& -1& 0& 0\\ 0& 0& 0& q& 0& q^{\frac{1}{2}}& q& 0\\ 0& 0& 0& q^{\frac{1}{2}}& 0& 0& 0& q\end{array}\right)\end{array}$$

with rows and columns indexed by $1,C_1,C_{21},C_{121},C_2,C_{12},C_{212},C_{1212}\text{.}$

Let $W$ be the dihedral group of order $2m\text{.}$ Then

$$C_w=q^{-\frac{1}{2}\ell \left(w\right)}\left(\sum _{v\le w}{T}_v\right),\text{so that}{p}_{vw}\left(q\right)=1,\text{for all}v\le w\text{.}$$

		Proof.
	Let $C_w$ be defined by the formula in the statement of the Theorem. If $s_{1}w>w$ so that $w=s_2{s}_1{s}_2{s}_1\dots$ then $$\begin{array}{ccc}{C}_{s_1}{C}_w& =& q^{-\ell \left(w\right)/2}{q}^{-\frac{1}{2}}(\sum _{v\le w}{T}_v+\sum _{\underset{s_{1}v<v}{v\le s_{1}w}}{T}_v+(q-1)\sum _{\underset{s_{1}v<v}{v<w}}{T}_v+q\sum _{\underset{s_{2}v<v}{v<w}}{T}_v)\\ & =& q^{-\ell \left(w\right)/2}{q}^{-\frac{1}{2}}(\sum _{\underset{s_{2}v<v}{v\le s_{1}w}}{T}_v+\sum _{\underset{s_{1}v<v}{v<w}}{T}_v+\sum _{\underset{s_{1}v<v}{v\le s_{1}w}}{T}_v-\sum _{\underset{s_{1}v<v}{v<w}}{T}_v+q\sum _{v\le s_{2}w}{T}_v)\\ & =& C_{s_{1}v}+q^{-\ell \left(w\right)/2}{q}^{1/2}\left(\sum _{v\le s_{2}w}{T}_v\right)\\ & =& C_{s_{1}w}+C_{s_{2}w},\end{array}$$ and, if $s_{1}w<w$ so that $w=s_1{s}_2{s}_1{s}_2\dots$ then let $w^{\prime }=s_{1}w$ and $w^{\prime \prime }=s_2{s}_{1}w$ so that $$\begin{array}{ccc}{C}_{s_1}{C}_w& =& C_{s_1}{C}_{s_1{w}^{\prime }}=C_{s_1}(C_{s_1}{C}_{w^{\prime }}-C_{s_2{w}^{\prime }})\\ & =& C_{s_1}(C_{s_1}{C}_{w^{\prime }}-C_{w^{\prime \prime }})=(q^{1/2}+q^{-1/2})C_{s_1}{C}_{w^{\prime }}-C_{s_1}{C}_{w^{\prime \prime }}\\ & =& (q^{1/2}+q^{-1/2})C_{s_1}{C}_{w^{\prime }}-(q^{1/2}+q^{-1/2})C_{w^{\prime \prime }},\text{by induction,}\\ & =& (q^{1/2}+q^{-1/2})(C_{s_1}{C}_{w^{\prime }}-C_{w^{\prime \prime }})=(q^{1/2}+q^{-1/2})C_w\text{.}\end{array}$$ So, $$\begin{array}{cc}{C}_{s_1}{C}_w=\{\begin{array}{cc}{C}_{s_{1}w}+C_{s_{2}w},& \text{if}{s}_{1}w<w,\text{i.e.}w=s_2{s}_1{s}_2{s}_1\dots ,\\ (q^{\frac{1}{2}}+q^{-\frac{1}{2}})C_w,& \text{if}{s}_{1}w<w,\text{i.e.}w=s_1{s}_2{s}_1{s}_2\dots \text{.}\end{array}& \text{(3.1)}\end{array}$$ In the first case, $\ell \left(s_{2}w\right)<\ell \left(w\right)$ and so, by induction, $C_{s_{1}w}=C_{s_1}{C}_w-C_{s_{2}w}$ is bar invariant. $\square$

From equation (???)

$$T_1{C}_w=\{\begin{array}{cc}q{C}_w,& \text{if}{s}_{1}w<w,\text{i.e.}w=s_1{s}_2{s}_1{s}_2\dots ,\\ q^{\frac{1}{2}}{C}_{s_{1}w}-C_w+q^{\frac{1}{2}}{C}_{s_{2}w},& \text{if}{s}_{1}w>w,\text{i.e.}w=s_2{s}_1{s}_2{s}_1\dots \text{.}\end{array}$$

For example, in the case $I_2\left(5\right),$

$$\begin{array}{c}{C}_2{C}_1=C_{21},C_1{C}_{21}=C_{121}+C_1,C_2{C}_{121}=C_{2121}+C_{21},C_1{C}_{2121}=C_{12121}+C_{121},\\ C_1{C}_2=C_{12},C_2{C}_{12}=C_{212}+C_2,C_1{C}_{212}=C_{1212}+C_{12},C_2{C}_{1212}=C_{21212}+C_{212},\end{array}$$

and the matrices of the regular representation in the KL-basis are

$$\begin{array}{ccc}\rho \left(T_1\right)& =& \left(\begin{array}{cccccccccc}-1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ q^{\frac{1}{2}}& q& q^{\frac{1}{2}}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& q^{\frac{1}{2}}& q& q^{\frac{1}{2}}& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& q^{\frac{1}{2}}& q& q^{\frac{1}{2}}& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& q^{\frac{1}{2}}& q& 0\\ 0& 0& 0& 0& q^{\frac{1}{2}}& 0& 0& 0& 0& q\end{array}\right)\\ \rho \left(T_2\right)& =& \left(\begin{array}{cccccccccc}-1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& q^{\frac{1}{2}}& q& q^{\frac{1}{2}}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& q^{\frac{1}{2}}& q& 0& 0& 0& 0& 0\\ q^{\frac{1}{2}}& 0& 0& 0& 0& q& q^{\frac{1}{2}}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& -1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& q^{\frac{1}{2}}& q& q^{\frac{1}{2}}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& q^{\frac{1}{2}}& q\end{array}\right)\end{array}$$

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