22,32\rangle,\langle 32,22\rangle\Bigl{)}$ $\Bigl{(}\langle
31,33\rangle,\langle 33,31\rangle\Bigl{)}$ $\Bigl{(}\langle
32,33\rangle,\langle 33,32\rangle\Bigl{)}$

The ordered 1-state communicating classes follow:

$\Bigl{(}\langle 0,0\rangle\Bigl{)}$ $\Bigl{(}\langle 11,11\rangle\Bigl{)}$
$\Bigl{(}\langle 22,22\rangle\Bigl{)}$ $\Bigl{(}\langle 33,33\rangle\Bigl{)}$

###  7.3 Presenting $L_{Q}$

$L_{Q}$ is the direct sum of one $9\times 9$ block

$\mathcal{L}_{9}=$
$\left[\begin{array}[]{ccccccccc}*&D_{1}&D_{2}&D_{10}&D_{11}&D_{10}&D_{11}&D_{21}&D_{21}\\\
D_{3}&*&0&D_{12}&0&D_{13}&0&0&0\\\ D_{4}&0&*&0&D_{14}&0&D_{14}&0&0\\\
D_{5}&D_{6}&0&*&0&D_{15}&D_{16}&D_{22}&0\\\
D_{7}&0&D_{8}&0&*&D_{15}&D_{18}&0&D_{23}\\\
D_{5}&D_{6}&0&D_{15}&D_{16}&*&0&0&D_{22}\\\
D_{7}&0&D_{8}&D_{17}&D_{18}&0&*&D_{23}&0\\\ D_{9}&0&0&D_{19}&0&0&D_{20}&*&0\\\
D_{9}&0&0&0&D_{20}&D_{19}&0&0&*\\\ \end{array}\right]$

where

$D_{1}=\frac{q^{4n}+q^{4}-2q^{2n+2}}{(q^{12}+4q^{10}+6q^{8}+4q^{6}+q^{4})q^{2n}}\quad
D_{2}=\frac{q^{12}+q^{4n+8}-2q^{2n+10}}{(q^{8}+4q^{6}+6q^{4}+4q^{2}+1)q^{2n})}$

$D_{3}=\frac{q^{4n}-2q^{2n+2}+q^{4}}{q^{2n}}\quad
D_{4}=\frac{q^{4n}-2q^{2n+2}+q^{4}}{q^{2n+4}}$

$D_{5}=\frac{q^{4n}(2q^{2}-1)+2q^{2n}(q^{6}-q^{4}+q^{2})+2q^{6}-q^{8}}{(q^{4}+q^{2})q^{2n}}\quad
D_{6}=\frac{2q^{2n}-q^{2}+1}{(q^{4}+2q^{2}+1)q^{2n}}$

$D_{7}=\frac{q^{4n}(2q^{2}-1)+2q^{2n}(q^{6}-q^{4}+q^{2})+2q^{6}-q^{8}}{(q^{6}+q^{4})q^{2n}}\quad
D_{8}=\frac{q^{2n}(q^{4}-q^{2})+2q^{4}}{q^{4}+2q^{2}+1}$

$D_{9}=\frac{q^{4n}-2q^{2n+2}+q^{4}}{q^{2n+2}}\quad
D_{10}=\frac{2q^{2n}(q^{6}-q^{4}+q^{2})+q^{4n}(2q^{2}-1)+2q^{6}-q^{8}}{(q^{10}+3q^{8}+3q^{6}+q^{4})q^{2n}}$

$D_{11}=\frac{q^{4n}(2q^{4}-q^{2})+2q^{2n}(q^{8}-q^{6}+q^{4})+2q^{8}-q^{10}}{(q^{6}+3q^{4}+3q^{2}+1)q^{2n}}\quad
D_{12}=\frac{2q^{2n+2}-q^{2}-q^{4}}{q^{2n}}$

$D_{13}=\frac{2q^{2n+2}-q^{4}+q^{2}}{q^{2n}}\quad
D_{14}=\frac{q^{2n}(q^{2}-1)+2q^{2}}{q^{4}}$

$D_{15}=\frac{q^{4n}-2q^{2n+2}+q^{4}}{(q^{6}+2q^{4}+q^{2})q^{2n}}\quad
D_{16}=\frac{q^{4n+4}-2q^{2n+6}+q^{8}}{(q^{4}+2q^{2}+1)q^{2n}}$

$D_{17}=\frac{q^{4n}-2q^{2n+2}+q^{4}}{(q^{8}+2q^{6}+q^{4})q^{2n}}\quad
D_{18}=\frac{q^{4n+2}-2q^{2n+4}+q^{6}}{(q^{4}+2q^{2}+1)q^{2n}}$

$D_{19}=\frac{2q^{2n}+1-q^{2}}{q^{2n}}\quad
D_{20}=\frac{q^{2n}(q^{2}-1)+2q^{2}}{q^{2}}$

$D_{21}=\frac{q^{6}+q^{4n+2}-2q^{2n+4}}{(q^{8}+4q^{6}+6q^{4}+4q^{2}+1)q^{2n}}\quad
D_{22}=\frac{2q^{2n+6}+q^{6}-q^{8}}{(q^{4}+2q^{2}+1)q^{2n}}$

$D_{23}=\frac{(q^{2}-1)q^{2n}+2q^{2}}{q^{8}+2q^{6}+q^{4}}$

and $\mathcal{L}_{9}$ corresponds to the 9-state communicating class listed in
Section 7.2,

four $6\times 6$ blocks

$\mathcal{L}_{6}=$
$\left[\begin{array}[]{cccccc}*&D_{1}&D_{2}&D_{14}&D_{15}&D_{16}\\\
D_{3}&*&D_{4}&D_{17}&D_{18}&D_{19}\\\ D_{5}&D_{6}&*&0&D_{20}&0\\\
D_{7}&D_{8}&0&*&0&D_{21}\\\ D_{9}&D_{10}&D_{11}&0&*&0\\\
D_{12}&D_{13}&0&D_{22}&0&*\\\ \end{array}\right]$

where

$D_{1}=\frac{2q^{6}+(2q^{2}-1)q^{4n}+2q^{2n}(q^{6}-q^{4}+q^{2})-q^{8}}{(q^{4}+2q^{2}+1)q^{2n}}\quad
D_{2}=\frac{q^{4n}+2q^{2n+4}+q^{4}+q^{2}-q^{6}}{(q^{8}+3q^{6}+3q^{4}+q^{2})q^{2n}}$

$D_{3}=\frac{2q^{6}+(2q^{2}-1)q^{4n}+2q^{2n}(q^{6}-q^{4}+q^{2})-q^{8}}{(q^{8}+2q^{6}+q^{4})q^{2n}}\quad
D_{4}=\frac{q^{4}+q^{4n}-2q^{2n+2}}{(q^{10}+3q^{8}+3q^{6}+q^{4})q^{2n}}$

$D_{5}=\frac{q^{4n}+2q^{2n+4}+q^{4}+q^{2}-q^{6}}{(q^{2}+1)q^{2n}}\quad
D_{6}=\frac{q^{6}+q^{4n+2}-2q^{2n+4}}{(q^{2}+1)}$

$D_{7}=\frac{q^{4}+q^{4n}-2q^{2n+2}}{(q^{6}+q^{4})q^{2n}}\quad
D_{8}=\frac{q^{6}+(q^{4}+q^{2}-1)q^{4n}+2q^{2n+2}}{(q^{6}+q^{4})q^{2n}}$

$D_{9}=\frac{q^{4}+q^{4n}-2q^{2n+2}}{(q^{4}+q^{2})q^{2n}}\quad
D_{10}=\frac{q^{6}+(q^{4}+q^{2}-1)q^{4n}+2q^{2n+2}}{(q^{4}+q^{2})q^{2n}}$

$D_{11}=\frac{2q^{2n}-q^{2}+1}{(q^{2}+1)q^{2n}}\quad
D_{12}=\frac{q^{4n}+2q^{2n+4}}{(q^{4}+q^{2})q^{2n}}$

$D_{13}=\frac{q^{4n}-2q^{2n+2}+q^{4}}{(q^{2}+1)q^{2n}}\quad
D_{14}=\frac{q^{10}+q^{4n+6}-2q^{2n+8}}{(q^{6}+3q^{4}+3q^{2}+1)q^{2n}}$

$D_{15}=\frac{q^{4}+q^{4n}-2q^{2n+2}}{(q^{6}+3q^{4}+3q^{2}+1)q^{2n}}\quad
D_{16}=\frac{q^{8}+q^{6}+q^{4n+4}+2q^{2n+8}-q^{10}}{(q^{6}+3q^{4}+3q^{2}+1)q^{2n}}$

$D_{17}=\frac{q^{8}+(q^{6}+q^{4}-q^{2})q^{4n}+2q^{2n+4}}{(q^{6}+3q^{4}+3q^{2}+1)q^{2n}}\quad
D_{18}=\frac{q^{6}+(q^{4}+q^{2}-1)q^{4n}+2q^{2n+2}}{(q^{10}+3q^{8}+3q^{6}+q^{4})q^{2n})}$

$D_{19}=\frac{q^{6}+q^{4n+2}-2q^{2n+4}}{(q^{6}+3q^{4}+3q^{2}+1)q^{2n}}\quad
D_{20}=\frac{2q^{2n+4}+q^{4}-q^{6}}{(q^{2}+1)q^{2n}}$

$D_{21}=\frac{(q^{2}-1)q^{2n}+2q^{2}}{q^{6}+q^{4}}\quad
D_{22}=\frac{(q^{2}-1)q^{2n}+2q^{2}}{q^{2}+1}$

and $\mathcal{L}_{6}$ corresponds to the 6-state communicating classes listed
in Section 7.2,

four $4\times 4$ blocks

$\mathcal{L}_{4}=\frac{1}{(q^{4}+2q^{2}+1)q^{2n}}M$

where

$M=$ $\left[\begin{array}[]{cccc}*&D_{1}&D_{1}&D_{6}\\\ D_{2}&*&D_{3}&D_{7}\\\
D_{2}&D_{3}&*&D_{7}\\\ D_{4}&D_{5}&D_{5}&*\\\ \end{array}\right]$

where

$D_{1}=\frac{q^{6}+(q^{4}+q^{2}-1)q^{4n}+2q^{2n+2}}{q^{4}}\quad
D_{2}=q^{6}+(q^{4}+q^{2}-1)q^{4n}+2q^{2n+2}$

$D_{3}=q^{4}+q^{4n}-2q^{2n+2}\quad D_{4}=q^{8}+q^{4n+4}-2q^{2n+6}$

$D_{5}=q^{4n+2}+2q^{2n+6}+q^{6}+q^{4}-q^{8}\quad
D_{6}=\frac{q^{4}+q^{4n}-2q^{2n+2}}{q^{4}}$

$D_{7}=\frac{2q^{2n+4}+q^{4n}+q^{4}+q^{2}-q^{6}}{q^{2}}$

and $\mathcal{L}_{4}$ corresponds to the 4-state communicating classes listed
in Section 7.2,

four $3\times 3$ blocks

$\mathcal{L}_{3}=\
\begin{bmatrix}*&\frac{q^{2}+q^{4n}}{(q^{6}+2q^{4}+q^{2})q^{2n}}&\frac{q^{6}+q^{4n+4}}{(q^{4}+2q^{2}+1)q^{2n}}\\\
\frac{q^{2}+q^{4n}}{q^{2n}}&*&0\\\
\frac{q^{2}+q^{4n}}{q^{2n+2}}&0&*\end{bmatrix}$

corresponding to the 3-state communicating classes listed in Section 7.2,
eight $2\times 2$ blocks

$\mathcal{L}_{2}=\begin{bmatrix}*&\frac{q^{4}+q^{4n+2}}{(q^{2}+1)q^{2n}}\\\
\frac{q^{2}+q^{4n}}{(q^{4}+q^{2})q^{2n}}&*\end{bmatrix}$

corresponding to the 2-state communicating classes listed in Section 7.2,

and four $1\times 1$ blocks with entry 0 corresponding to the 1-state
communicating classes listed in Section 7.2. Here, the diagonal entries are
choosen so that the rows sum to 0. To summarize, the generator matrix is

$L^{D}_{Q}=\mathcal{L}_{9}\oplus\bigoplus_{i=1}^{4}\mathcal{L}_{6}\oplus\bigoplus_{i=1}^{4}\mathcal{L}_{4}\oplus\bigoplus_{i=1}^{4}\mathcal{L}_{3}\oplus\bigoplus_{i=1}^{8}\mathcal{L}_{2}\oplus\bigoplus_{i=1}^{4}[0]$

with respect to the ordered basis that is the concatenation of all of the
above communicating classes in the order they are presented.

###  7.4 Ordering of States and Permutations of $L_{m}$ and $L_{Q}$

####  7.4.1 $\Gamma$ State Ordering

The assumed ordering of states for four $\Gamma$ lattice sites is
$\Omega\otimes\Omega$. To be more concise we represent the state
$(x_{1},x_{2},x_{3},x_{4})$ as $x_{1}x_{2}x_{3}x_{4}$.

Assumed Ordering:

(3030, 3021, 3003, 3012, 3010, 3001, 3020, 3002, 3031, 3013, 3032, 3023, 3000,
3011, 3022, 3033, 2130, 2121, 2103, 2112, 2110, 2101, 2120, 2102, 2131, 2113,
2132, 2123, 2100, 2111, 2122, 2133, 0330, 0321, 0303, 0312, 0310, 0301, 0320,
0302, 0331, 0313, 0332, 0323, 0300, 0311, 0322, 0333, 1230, 1221, 1203, 1212,
1210, 1201, 1220, 1202, 1231, 1213, 1232, 1223, 1200, 1211, 1222, 1233, 1030,
1021, 1003, 1012, 1010, 1001, 1020, 1002, 1031, 1013, 1032, 1023, 1000, 1011,
1022, 1033, 0130, 0121, 0103, 0112, 0110, 0101, 0120, 0102, 0131, 0113, 0132,
0123, 0100, 0111, 0122, 0133, 2030, 2021, 2003, 2012, 2010, 2001, 2020, 2002,
2031, 2013, 2032, 2023, 2000, 2011, 2022, 2033, 0230, 0221, 0203, 0212, 0210,
0201, 0220, 0202, 0231, 0213, 0232, 0223, 0200, 0211, 0222, 0233, 3130, 3121,
3103, 3112, 3110, 3101, 3120, 3102, 3131, 3113, 3132, 3123, 3100, 3111, 3122,
3133, 1330, 1321, 1303, 1312, 1310, 1301, 1320, 1302, 1331, 1313, 1332, 1323,
1300, 1311, 1322, 1333, 3230, 3221, 3203, 3212, 3210, 3201, 3220, 3202, 3231,
3213, 3232, 3223, 3200, 3211, 3222, 3233, 2330, 2321, 2303, 2312, 2310, 2301,
2320, 2302, 2331, 2313, 2332, 2323, 2300, 2311, 2322, 2333, 0030, 0021, 0003,
0012, 0010, 0001, 0020, 0002, 0031, 0013, 0032, 0023, 0000, 0011, 0022, 0033,
1130, 1121, 1103, 1112, 1110, 1101, 1120, 1102, 1131, 1113, 1132, 1123, 1100,
1111, 1122, 1133, 2230, 2221, 2203, 2212, 2210, 2201, 2220, 2202, 2231, 2213,
2232, 2223, 2200, 2211, 2222, 2233, 3330, 3321, 3303, 3312, 3310, 3301, 3320,
3302, 3331, 3313, 3332, 3323, 3300, 3311, 3322, 3333 )

Diagonal Ordering:

(3300, 3210, 3030, 3120, 3100, 3010, 3200, 3020, 3310, 3130, 3320, 3230, 3000,
3110, 3220, 3330, 2301, 2211, 2031, 2121, 2101, 2011, 2201, 2021, 2311, 2131,
2321, 2231, 2001, 2111, 2221, 2331, 0303, 0213, 0033, 0123, 0103, 0013, 0203,
0023, 0313, 0133, 0323, 0233, 0003, 0113, 0223, 0333, 1302, 1212, 1032, 1122,
1102, 1012, 1202, 1022, 1312, 1132, 1322, 1232, 1002, 1112, 1222, 1332, 1300,
1210, 1030, 1120, 1100, 1010, 1200, 1020, 1310, 1130, 1320, 1230, 1000, 1110,
1220, 1330, 0301, 0211, 0031, 0121, 0101, 0011, 0201, 0021, 0311, 0131, 0321,
0231, 0001, 0111, 0221, 0331, 2300, 2210, 2030, 2120, 2100, 2010, 2200, 2020,
2310, 2130, 2320, 2230, 2000, 2110, 2220, 2330, 0302, 0212, 0032, 0122, 0102,
0012, 0202, 0022, 0312, 0132, 0322, 0232, 0002, 0112, 0222, 0332, 3301, 3211,
3031, 3121, 3101, 3011, 3201, 3021, 3311, 3131, 3321, 3231, 3001, 3111, 3221,
3331, 1303, 1213, 1033, 1123, 1103, 1013, 1203, 1023, 1313, 1133, 1323, 1233,
1003, 1113, 1223, 1333, 3302, 3212, 3032, 3122, 3102, 3012, 3202, 3022, 3312,
3132, 3322, 3232, 3002, 3112, 3222, 3332, 2303, 2213, 2033, 2123, 2103, 2013,
2203, 2023, 2313, 2133, 2323, 2233, 2003, 2113, 2223, 2333, 0300, 0210, 0030,
0120, 0100, 0010, 0200, 0020, 0310, 0130, 0320, 0230, 0000, 0110, 0220, 0330,
1301, 1211, 1031, 1121, 1101, 1011, 1201, 1021, 1311, 1131, 1321, 1231, 1001,
1111, 1221, 1331, 2302, 2212, 2032, 2122, 2102, 2012, 2202, 2022, 2312, 2132,
2322, 2232, 2002, 2112, 2222, 2332, 3303, 3213, 3033, 3123, 3103, 3013, 3203,
3023, 3313, 3133, 3323, 3233, 3003, 3113, 3223, 3333 )

####  7.4.2 $\gamma$ State Ordering

The assumed ordering of states for two $\gamma$ lattice sites is
$\chi\otimes\chi$. To be more concise we represent the state $(x_{1},x_{2})$
as $x_{1}\\_x_{2}$.

Assumed Ordering:

(0_0, 0_1, 0_2, 0_11, 0_3, 0_22, 0_31, 0_32, 0_33, 1_0, 1_1, 1_2, 1_11, 1_3,
1_22, 1_31, 1_32, 1_33, 2_0, 2_1, 2_2, 2_11, 2_3, 2_22, 2_31, 2_32, 2_33,
11_0, 11_1, 11_2, 11_11, 11_3, 11_22, 11_31, 11_32, 11_33, 3_0, 3_1, 3_2,
3_11, 3_3, 3_22, 3_31, 3_32, 3_33, 22_0, 22_1, 22_2, 22_11, 22_3, 22_22,
22_31, 22_32, 22_33, 31_0, 31_1, 31_2, 31_11, 31_3, 31_22, 31_31, 31_32,
31_33, 32_0, 32_1, 32_2, 32_11, 32_3, 32_22, 32_31, 32_32, 32_33, 33_0, 33_1,
33_2, 33_11, 33_3, 33_22, 33_31, 33_32, 33_33 )

Diagonal Ordering:

(3_3, 0_33, 33_0, 1_32, 32_1, 2_31, 31_2, 11_22, 22_11, 1_3, 3_1, 0_31, 31_0,
2_11, 11_2, 2_3, 3_2, 0_32, 32_0, 1_22, 22_1, 3_31, 31_3, 1_33, 33_1, 11_32,
32_11, 3_32, 32_3, 2_33, 33_2, 22_31, 31_22, 3_0, 1_2, 2_1, 0_3, 31_1, 11_3,
3_11, 1_31, 32_2, 22_3, 3_22, 2_32, 33_3, 31_32, 32_31, 3_33, 1_1, 0_11, 11_0,
2_2, 0_22, 22_0, 31_31, 11_33, 33_11, 32_32, 22_33, 33_22, 0_1, 1_0, 0_2, 2_0,
1_11, 11_1, 2_22, 22_2, 11_31, 31_11, 22_32, 32_22, 31_33, 33_31, 32_33,
33_32, 0_0, 11_11, 22_22, 33_33 )

###  7.5 Limits of Components of $L_{p}^{(2)}$

Recall that $L^{(2)}_{p}$ is given in section 2.2 “Probabilistic definitions”
of [RLY23] as the direct sum of three distinct block matrices $L_{1}$,
$L_{2}$, and four $1\times 1$ zero matrices.

We list the limits for $q^{-2n}L_{1}$ and $q^{-2n}L_{2}$ for $q>1$. The limits
follow similarly when multiplying $L_{1}$ and $L_{2}$ by $q^{2n}$ for $0<q<1$.

Take $q>1$.

$\lim_{n\to\infty}(q^{-2n}L_{1})=\begin{bmatrix}-\frac{2q^{2}-1}{q^{4}}&\frac{q^{2}-1}{q^{4}}&\frac{1}{q^{4}}&\frac{q^{2}-1}{q^{4}}\\\
\frac{q^{2}-1}{q^{2}}&-1&0&\frac{1}{q^{2}}\\\ 1&0&-1&0\\\
\frac{q^{2}-1}{q^{2}}&\frac{1}{q^{2}}&0&-1\\\ \end{bmatrix}$
$\lim_{n\to\infty}(q^{-2n}L_{2})=\begin{bmatrix}-\frac{1}{q^{2}}&\frac{1}{q^{2}}\\\
1&-1\\\ \end{bmatrix}$

###  7.6 Diagonalization for Communicating Classes

We show that for each communicating class $\mathcal{L}_{i}$ where
$i\in\\{3,4,6,9\\}$ the matrix of right eigenvectors is invertible for $q>0$
and $q\neq 1$ , and therefore that all the eigenvectors are linearly
independent which implies diagonalizability.

Consider $\mathcal{P}_{9}$,

$\mathcal{P}_{9}=$
$\begin{bmatrix}1&-\frac{q^{4}}{q^{8}+q^{6}+q^{2}+1}&-\frac{q^{4}}{q^{8}+q^{6}+q^{2}+1}&\frac{q^{2}}{q^{4}+2q^{2}+1}&\frac{q^{2}}{q^{4}+2q^{2}+1}&\frac{q^{2}}{q^{4}+2q^{2}+1}&\frac{q^{2}}{q^{4}+2q^{2}+1}&\frac{q^{2}}{q^{4}+2q^{2}+1}&\frac{-q^{4}+2q^{2}-1}{q^{4}+2q^{2}+1}\\\
1&\frac{q^{6}}{q^{6}+1}&\frac{q^{6}}{q^{6}+1}&\frac{q^{4}}{q^{4}-1}&\frac{q^{4}}{q^{4}-1}&q^{6}&-\frac{q^{6}}{q^{2}-1}&-\frac{q^{6}}{q^{2}-1}&-q^{4}\\\
1&\frac{1}{q^{6}+1}&\frac{1}{q^{6}+1}&-\frac{1}{q^{4}-1}&-\frac{1}{q^{4}-1}&\frac{1}{q^{6}}&\frac{1}{q^{6}-q^{4}}&\frac{1}{q^{6}-q^{4}}&-\frac{1}{q^{4}}\\\
1&\frac{q^{10}-q^{6}+q^{4}}{q^{10}-q^{6}+q^{4}-1}&-\frac{q^{6}}{q^{10}-q^{6}+q^{4}-1}&\frac{q^{6}+q^{4}-q^{2}}{q^{6}+q^{4}-q^{2}-1}&\frac{q^{4}}{q^{6}+q^{4}-q^{2}-1}&-\frac{q^{4}}{q^{2}+1}&\frac{-q^{8}+q^{6}+q^{4}}{q^{6}+q^{4}-q^{2}-1}&\frac{q^{4}}{q^{6}+q^{4}-q^{2}-1}&\frac{-q^{4}+q^{2}}{q^{2}+1}\\\
1&\frac{q^{4}}{q^{10}-q^{6}+q^{4}-1}&\frac{-q^{6}+q^{4}-1}{q^{10}-q^{6}+q^{4}-1}&-\frac{q^{2}}{q^{6}+q^{4}-q^{2}-1}&\frac{q^{4}-q^{2}-1}{q^{6}+q^{4}-q^{2}-1}&-\frac{1}{q^{4}+q^{2}}&-\frac{q^{2}}{q^{6}+q^{4}-q^{2}-1}&\frac{-q^{4}-q^{2}+1}{q^{8}+q^{6}-q^{4}-q^{2}}&\frac{q^{2}-1}{q^{4}+q^{2}}\\\
1&-\frac{q^{6}}{q^{10}-q^{6}+q^{4}-1}&\frac{q^{10}-q^{6}+q^{4}}{q^{10}-q^{6}+q^{4}-1}&\frac{q^{4}}{q^{6}+q^{4}-q^{2}-1}&\frac{q^{6}+q^{4}-q^{2}}{q^{6}+q^{4}-q^{2}-1}&-\frac{q^{4}}{q^{2}+1}&\frac{q^{4}}{q^{6}+q^{4}-q^{2}-1}&\frac{-q^{8}+q^{6}+q^{4}}{q^{6}+q^{4}-q^{2}-1}&\frac{-q^{4}+q^{2}}{q^{2}+1}\\\
1&\frac{-q^{6}+q^{4}-1}{q^{10}-q^{6}+q^{4}-1}&\frac{q^{4}}{q^{10}-q^{6}+q^{4}-1}&\frac{q^{4}-q^{2}-1}{q^{6}+q^{4}-q^{2}-1}&-\frac{q^{2}}{q^{6}+q^{4}-q^{2}-1}&-\frac{1}{q^{4}+q^{2}}&\frac{-q^{4}-q^{2}+1}{q^{8}+q^{6}-q^{4}-q^{2}}&-\frac{q^{2}}{q^{6}+q^{4}-q^{2}-1}&\frac{q^{2}-1}{q^{4}+q^{2}}\\\
1&1&0&1&0&1&1&0&1\\\ 1&0&1&0&1&1&0&1&1\end{bmatrix}$

It follows that

$\det(\mathcal{P}_{9})=\frac{\left(q^{4}+1\right)^{5}\left(q^{2}-q+1\right)^{5}\left(q^{2}+q+1\right)^{5}}{q^{6}\left(q-1\right)^{3}\left(q+1\right)^{3}\left(q^{2}+1\right)^{9}\left(q^{4}-q^{2}+1\right)}$

For $q>0$ and $q\neq 1$, the $\det(\mathcal{P}_{9})$ is not $0$ and is
defined. Therefore, $\mathcal{P}_{9}$ is invertible for $q>0$ and $q\neq 1$.

Consider $\mathcal{P}_{6}$,

$\mathcal{P}_{6}=\begin{bmatrix}1&-\frac{q^{4}}{q^{2}+1}&\frac{q^{4}}{q^{6}+q^{4}-q^{2}-1}&\frac{q^{6}+q^{4}-q^{2}}{q^{6}+q^{4}-q^{2}-1}&-\frac{q^{4}}{q^{2}+1}&\frac{-q^{4}+q^{2}}{q^{2}+1}\\\
1&-\frac{q^{4}}{q^{2}+1}&\frac{q^{4}-q^{2}-1}{q^{6}+q^{4}-q^{2}-1}&-\frac{q^{2}}{q^{6}+q^{4}-q^{2}-1}&\frac{1}{q^{2}+1}&\frac{q^{2}-1}{q^{4}+q^{2}}\\\
1&q^{6}&\frac{q^{4}}{q^{4}-1}&\frac{q^{4}}{q^{4}-1}&q^{6}&-q^{4}\\\
1&1&-\frac{1}{q^{4}-1}&-\frac{1}{q^{4}-1}&-\frac{1}{q^{4}}&-\frac{1}{q^{4}}\\\
1&q^{6}&1&0&-q^{2}&1\\\ 1&1&0&1&1&1\end{bmatrix}$

It follows that

$\det(\mathcal{P}_{6})=-\frac{(q^{4}+1)^{4}(q^{2}-q+1)^{2}(q^{2}+q+1)^{2}}{q^{4}(q-1)(q+1)(q^{2}+1)^{3}}$

For $q>0$ and $q\neq 1$, the $\det(\mathcal{P}_{6})$ is not $0$ and is
defined. Therefore, $\mathcal{P}_{6}$ is invertible for $q>0$ and $q\neq 1$.

Consider $\mathcal{P}_{4}$,

$\mathcal{P}_{4}=\begin{bmatrix}1&0&-\frac{1}{q^{4}}&\frac{1}{q^{8}}\\\
1&-1&\frac{q^{4}-1}{q^{4}}&-\frac{1}{q^{4}}\\\ 1&1&0&-\frac{1}{q^{4}}\\\
1&0&1&1\end{bmatrix}$

It follows that

$\det(\mathcal{P}_{4})=-\frac{q^{12}+3q^{8}+3q^{4}+1}{q^{12}}$

For $q>0$, the $\det(\mathcal{P}_{4})$ is not $0$. Therefore,
$\mathcal{P}_{4}$ is invertible for $q>0$.

Consider $\mathcal{P}_{3}$,

$\mathcal{P}_{3}=\begin{bmatrix}1&-\frac{q^{4}}{q^{2}+1}&\frac{q^{2}-q^{4}}{q^{2}+1}\\\
1&q^{6}&-q^{4}\\\ 1&1&1\end{bmatrix}$

It follows that

$\det(\mathcal{P}_{3})=\frac{q^{10}+q^{8}+2q^{6}+q^{4}+q^{2}}{q^{2}+1}$

For $q>0$, the $\det(\mathcal{P}_{3})$ is not $0$. Therefore,
$\mathcal{P}_{3}$ is invertible for $q>0$.

###  7.7 Ordering of Basis Vectors to Block $\pi_{W\otimes W}(C)$

Denote $v_{i}$ to be the $i^{th}$ basis vector of $W$. Then, the blocked
matrix $\pi_{W\otimes W}(C)$ is with respect to ordered basis:

$\displaystyle v_{1}\otimes v_{1},v_{1}\otimes v_{20},v_{2}\otimes
v_{19},v_{4}\otimes v_{17},v_{5}\otimes v_{16},v_{8}\otimes
v_{13},v_{9}\otimes v_{9},v_{3}\otimes v_{18},v_{6}\otimes v_{15},$
$\displaystyle v_{7}\otimes v_{14},v_{9}\otimes v_{11},v_{11}\otimes
v_{9},v_{11}\otimes v_{11},v_{13}\otimes v_{8},v_{10}\otimes
v_{12},v_{14}\otimes v_{7},v_{16}\otimes v_{5},v_{12}\otimes v_{10},$
$\displaystyle v_{15}\otimes v_{6},v_{17}\otimes v_{4},v_{18}\otimes
v_{3},v_{19}\otimes v_{2},v_{20}\otimes v_{1},v_{1}\otimes v_{13},v_{2}\otimes
v_{9},v_{2}\otimes v_{11},v_{4}\otimes v_{7},$ $\displaystyle v_{5}\otimes
v_{6},v_{8}\otimes v_{3},v_{9}\otimes v_{2},v_{3}\otimes v_{8},v_{6}\otimes
v_{5},v_{7}\otimes v_{4},v_{11}\otimes v_{2},v_{13}\otimes v_{1},v_{1}\otimes
v_{16},$ $\displaystyle v_{2}\otimes v_{14},v_{4}\otimes v_{9},v_{4}\otimes
v_{11},v_{5}\otimes v_{10},v_{8}\otimes v_{6},v_{9}\otimes v_{4},v_{6}\otimes
v_{8},v_{11}\otimes v_{4},v_{10}\otimes v_{5},$ $\displaystyle v_{14}\otimes
v_{2},v_{16}\otimes v_{1},v_{1}\otimes v_{17},v_{2}\otimes v_{15},v_{4}\otimes
v_{12},v_{5}\otimes v_{9},v_{5}\otimes v_{11},v_{8}\otimes v_{7},v_{9}\otimes
v_{5},$ $\displaystyle v_{7}\otimes v_{8},v_{11}\otimes v_{5},v_{12}\otimes
v_{4},v_{15}\otimes v_{2},v_{17}\otimes v_{1},v_{1}\otimes v_{19},v_{2}\otimes
v_{18},v_{4}\otimes v_{15},v_{5}\otimes v_{14},$ $\displaystyle v_{8}\otimes
v_{9},v_{8}\otimes v_{11},v_{9}\otimes v_{8},v_{11}\otimes v_{8},v_{14}\otimes
v_{5},v_{15}\otimes v_{4},v_{18}\otimes v_{2},v_{19}\otimes v_{1},v_{2}\otimes
v_{16},$ $\displaystyle v_{3}\otimes v_{14},v_{4}\otimes v_{13},v_{6}\otimes
v_{9},v_{6}\otimes v_{11},v_{7}\otimes v_{10},v_{9}\otimes v_{6},v_{11}\otimes
v_{6},v_{13}\otimes v_{4},v_{14}\otimes v_{3},$ $\displaystyle v_{16}\otimes
v_{2},v_{10}\otimes v_{7},v_{2}\otimes v_{17},v_{3}\otimes v_{15},v_{5}\otimes
v_{13},v_{6}\otimes v_{12},v_{7}\otimes v_{9},v_{7}\otimes v_{11},v_{9}\otimes
v_{7},$ $\displaystyle v_{11}\otimes v_{7},v_{13}\otimes v_{5},v_{12}\otimes
v_{6},v_{15}\otimes v_{3},v_{17}\otimes v_{2},v_{2}\otimes v_{20},v_{3}\otimes
v_{19},v_{6}\otimes v_{17},v_{7}\otimes v_{16},$ $\displaystyle v_{9}\otimes
v_{13},v_{11}\otimes v_{13},v_{13}\otimes v_{9},v_{13}\otimes
v_{11},v_{16}\otimes v_{7},v_{17}\otimes v_{6},v_{19}\otimes
v_{3},v_{20}\otimes v_{2},v_{4}\otimes v_{19},$ $\displaystyle v_{6}\otimes
v_{18},v_{8}\otimes v_{16},v_{9}\otimes v_{14},v_{10}\otimes
v_{15},v_{11}\otimes v_{14},v_{14}\otimes v_{9},v_{14}\otimes
v_{11},v_{16}\otimes v_{8},v_{18}\otimes v_{6},$ $\displaystyle v_{19}\otimes
v_{4},v_{15}\otimes v_{10},v_{4}\otimes v_{20},v_{6}\otimes
v_{19},v_{9}\otimes v_{16},v_{10}\otimes v_{17},v_{11}\otimes
v_{16},v_{14}\otimes v_{13},v_{16}\otimes v_{9},$ $\displaystyle v_{13}\otimes
v_{14},v_{16}\otimes v_{11},v_{17}\otimes v_{10},v_{19}\otimes
v_{6},v_{20}\otimes v_{4},v_{5}\otimes v_{19},v_{7}\otimes v_{18},v_{8}\otimes
v_{17},v_{9}\otimes v_{15},$ $\displaystyle v_{11}\otimes v_{15},v_{12}\otimes
v_{14},v_{15}\otimes v_{9},v_{15}\otimes v_{11},v_{17}\otimes
v_{8},v_{14}\otimes v_{12},v_{18}\otimes v_{7},v_{19}\otimes
v_{5},v_{5}\otimes v_{20},$ $\displaystyle v_{7}\otimes v_{19},v_{9}\otimes
v_{17},v_{11}\otimes v_{17},v_{12}\otimes v_{16},v_{15}\otimes
v_{13},v_{17}\otimes v_{9},v_{13}\otimes v_{15},v_{17}\otimes
v_{11},v_{16}\otimes v_{12},$ $\displaystyle v_{19}\otimes v_{7},v_{20}\otimes
v_{5},v_{8}\otimes v_{20},v_{9}\otimes v_{19},v_{11}\otimes
v_{19},v_{14}\otimes v_{17},v_{15}\otimes v_{16},v_{18}\otimes
v_{13},v_{19}\otimes v_{9},$ $\displaystyle v_{13}\otimes v_{18},v_{16}\otimes
v_{15},v_{17}\otimes v_{14},v_{19}\otimes v_{11},v_{20}\otimes
v_{8},v_{1}\otimes v_{9},v_{2}\otimes v_{8},v_{4}\otimes v_{5},v_{5}\otimes
v_{4},$ $\displaystyle v_{8}\otimes v_{2},v_{9}\otimes v_{1},v_{1}\otimes
v_{11},v_{11}\otimes v_{1},v_{2}\otimes v_{13},v_{3}\otimes v_{9},v_{3}\otimes
v_{11},v_{6}\otimes v_{7},v_{7}\otimes v_{6},$ $\displaystyle v_{9}\otimes
v_{3},v_{11}\otimes v_{3},v_{13}\otimes v_{2},v_{4}\otimes v_{16},v_{6}\otimes
v_{14},v_{9}\otimes v_{10},v_{10}\otimes v_{9},v_{10}\otimes
v_{11},v_{11}\otimes v_{10},$ $\displaystyle v_{14}\otimes v_{6},v_{16}\otimes
v_{4},v_{5}\otimes v_{17},v_{7}\otimes v_{15},v_{9}\otimes
v_{12},v_{11}\otimes v_{12},v_{12}\otimes v_{9},v_{12}\otimes
v_{11},v_{15}\otimes v_{7},$ $\displaystyle v_{17}\otimes v_{5},v_{8}\otimes
v_{19},v_{9}\otimes v_{18},v_{11}\otimes v_{18},v_{14}\otimes
v_{15},v_{15}\otimes v_{14},v_{18}\otimes v_{9},v_{18}\otimes
v_{11},v_{19}\otimes v_{8},$ $\displaystyle v_{9}\otimes v_{20},v_{13}\otimes
v_{19},v_{16}\otimes v_{17},v_{17}\otimes v_{16},v_{19}\otimes
v_{13},v_{20}\otimes v_{9},v_{11}\otimes v_{20},v_{20}\otimes
v_{11},v_{1}\otimes v_{6},$ $\displaystyle v_{2}\otimes v_{4},v_{4}\otimes
v_{2},v_{6}\otimes v_{1},v_{1}\otimes v_{7},v_{2}\otimes v_{5},v_{5}\otimes
v_{2},v_{7}\otimes v_{1},v_{1}\otimes v_{14},v_{4}\otimes v_{8},$
$\displaystyle v_{8}\otimes v_{4},v_{14}\otimes v_{1},v_{1}\otimes
v_{15},v_{5}\otimes v_{8},v_{8}\otimes v_{5},v_{15}\otimes v_{1},v_{2}\otimes
v_{6},v_{3}\otimes v_{4},v_{4}\otimes v_{3},$ $\displaystyle v_{6}\otimes
v_{2},v_{2}\otimes v_{7},v_{3}\otimes v_{5},v_{5}\otimes v_{3},v_{7}\otimes
v_{2},v_{2}\otimes v_{10},v_{4}\otimes v_{6},v_{6}\otimes v_{4},v_{10}\otimes
v_{2},$ $\displaystyle v_{2}\otimes v_{12},v_{5}\otimes v_{7},v_{7}\otimes
v_{5},v_{12}\otimes v_{2},v_{3}\otimes v_{16},v_{6}\otimes
v_{13},v_{13}\otimes v_{6},v_{16}\otimes v_{3},v_{3}\otimes v_{17},$
$\displaystyle v_{7}\otimes v_{13},v_{13}\otimes v_{7},v_{17}\otimes
v_{3},v_{4}\otimes v_{14},v_{8}\otimes v_{10},v_{10}\otimes
v_{8},v_{14}\otimes v_{4},v_{4}\otimes v_{18},v_{8}\otimes v_{14},$
$\displaystyle v_{14}\otimes v_{8},v_{18}\otimes v_{4},v_{5}\otimes
v_{15},v_{8}\otimes v_{12},v_{12}\otimes v_{8},v_{15}\otimes
v_{5},v_{5}\otimes v_{18},v_{8}\otimes v_{15},v_{15}\otimes v_{8},$
$\displaystyle v_{18}\otimes v_{5},v_{6}\otimes v_{16},v_{10}\otimes
v_{13},v_{13}\otimes v_{10},v_{16}\otimes v_{6},v_{6}\otimes
v_{20},v_{13}\otimes v_{16},v_{16}\otimes v_{13},v_{20}\otimes v_{6},$
$\displaystyle v_{7}\otimes v_{17},v_{12}\otimes v_{13},v_{13}\otimes
v_{12},v_{17}\otimes v_{7},v_{7}\otimes v_{20},v_{13}\otimes
v_{17},v_{17}\otimes v_{13},v_{20}\otimes v_{7},v_{10}\otimes v_{19},$
$\displaystyle v_{14}\otimes v_{16},v_{16}\otimes v_{14},v_{19}\otimes
v_{10},v_{12}\otimes v_{19},v_{15}\otimes v_{17},v_{17}\otimes
v_{15},v_{19}\otimes v_{12},v_{14}\otimes v_{19},v_{16}\otimes v_{18},$
$\displaystyle v_{18}\otimes v_{16},v_{19}\otimes v_{14},v_{14}\otimes
v_{20},v_{16}\otimes v_{19},v_{19}\otimes v_{16},v_{20}\otimes
v_{14},v_{15}\otimes v_{19},v_{17}\otimes v_{18},v_{18}\otimes v_{17},$
$\displaystyle v_{19}\otimes v_{15},v_{15}\otimes v_{20},v_{17}\otimes
v_{19},v_{19}\otimes v_{17},v_{20}\otimes v_{15},v_{1}\otimes
v_{3},v_{2}\otimes v_{2},v_{3}\otimes v_{1},v_{1}\otimes v_{10},$
$\displaystyle v_{4}\otimes v_{4},v_{10}\otimes v_{1},v_{1}\otimes
v_{12},v_{5}\otimes v_{5},v_{12}\otimes v_{1},v_{1}\otimes v_{18},v_{8}\otimes
v_{8},v_{18}\otimes v_{1},v_{3}\otimes v_{10},$ $\displaystyle v_{6}\otimes
v_{6},v_{10}\otimes v_{3},v_{3}\otimes v_{12},v_{7}\otimes v_{7},v_{12}\otimes
v_{3},v_{3}\otimes v_{20},v_{13}\otimes v_{13},v_{20}\otimes
v_{3},v_{10}\otimes v_{18},$ $\displaystyle v_{14}\otimes v_{14},v_{18}\otimes
v_{10},v_{10}\otimes v_{20},v_{16}\otimes v_{16},v_{20}\otimes
v_{10},v_{12}\otimes v_{18},v_{15}\otimes v_{15},v_{18}\otimes
v_{12},v_{12}\otimes v_{20},$ $\displaystyle v_{17}\otimes
v_{17},v_{20}\otimes v_{12},v_{18}\otimes v_{20},v_{19}\otimes
v_{19},v_{20}\otimes v_{18},v_{1}\otimes v_{2},v_{2}\otimes v_{1},v_{1}\otimes
v_{4},v_{4}\otimes v_{1},$ $\displaystyle v_{1}\otimes v_{5},v_{5}\otimes
v_{1},v_{1}\otimes v_{8},v_{8}\otimes v_{1},v_{2}\otimes v_{3},v_{3}\otimes
v_{2},v_{3}\otimes v_{6},v_{6}\otimes v_{3},v_{3}\otimes v_{7},$
$\displaystyle v_{7}\otimes v_{3},v_{3}\otimes v_{13},v_{13}\otimes
v_{3},v_{4}\otimes v_{10},v_{10}\otimes v_{4},v_{5}\otimes
v_{12},v_{12}\otimes v_{5},v_{6}\otimes v_{10},v_{10}\otimes v_{6},$
$\displaystyle v_{7}\otimes v_{12},v_{12}\otimes v_{7},v_{8}\otimes
v_{18},v_{18}\otimes v_{8},v_{10}\otimes v_{14},v_{14}\otimes
v_{10},v_{10}\otimes v_{16},v_{16}\otimes v_{10},v_{12}\otimes v_{15},$
$\displaystyle v_{15}\otimes v_{12},v_{12}\otimes v_{17},v_{17}\otimes
v_{12},v_{13}\otimes v_{20},v_{20}\otimes v_{13},v_{14}\otimes
v_{18},v_{18}\otimes v_{14},v_{15}\otimes v_{18},v_{18}\otimes v_{15},$
$\displaystyle v_{16}\otimes v_{20},v_{20}\otimes v_{16},v_{17}\otimes
v_{20},v_{20}\otimes v_{17},v_{18}\otimes v_{19},v_{19}\otimes
v_{18},v_{19}\otimes v_{20},v_{20}\otimes v_{19},v_{3}\otimes v_{3},$
$\displaystyle v_{10}\otimes v_{10},v_{12}\otimes v_{12},v_{18}\otimes
v_{18},v_{20}\otimes v_{20}$

###  7.8 Algebraically-produced Markov Generator

This generator is produced as described in Section 6.5.2, with the additional
step of deleting the $i$th row and column of the matrix
$a^{-1}G^{-1}\pi_{W\otimes W}(C)G-\textrm{Id}$ if $g_{i}=0$, or if row $i$ or
column $i$ of the matrix contains a negative off-diagonal entry, as those lack
a probabilistic interpretation.

The authors note that the polynomials below may be expressible as
$q$-hypergeometric series, but we do not investigate this possibility in this
paper. Recall that $a=q^{12}+q^{2}+q^{-2}+q^{-12}$, and let the matrix entry
$*$ denote, as in Section 7.3, $-1$ times the sum of the other elements in its
row. We may express the Markov generator produced algebraically as the direct
sum of one $9\times 9$ block:

$\displaystyle a^{-2}\begin{bmatrix}*&B_{1}&B_{1}&B_{2}&0&0&0&0&0\\\
B_{3}&*&B_{4}&B_{5}&B_{6}&B_{7}&0&0&0\\\
B_{3}&B_{4}&*&B_{5}&0&0&B_{6}&B_{7}&0\\\
B_{8}&B_{9}&B_{9}&*&B_{10}&B_{11}&B_{10}&B_{11}&B_{12}\\\
0&B_{13}&0&B_{14}&*&B_{15}&0&0&0\\\
0&B_{16}&0&B_{17}&B_{18}&*&0&B_{19}&B_{20}\\\
0&0&B_{13}&B_{14}&0&0&*&B_{15}&0\\\
0&0&B_{16}&B_{17}&0&B_{19}&B_{18}&*&B_{20}\\\
0&0&0&B_{21}&0&B_{22}&0&B_{22}&*\end{bmatrix}$

where

$\displaystyle B_{1}$
$\displaystyle=q^{22}-q^{20}-2q^{18}+2q^{16}+3q^{14}-q^{12}-4q^{10}+2q^{6}$
$\displaystyle B_{2}$
$\displaystyle=q^{20}-2q^{16}-2q^{14}+q^{12}+4q^{10}+q^{8}-2q^{6}-2q^{4}+1$
$\displaystyle B_{3}$
$\displaystyle=q^{24}-3q^{22}+3q^{20}-q^{18}+2q^{16}-4q^{14}+2q^{12}$
$\displaystyle B_{4}$
$\displaystyle=q^{22}-2q^{20}+q^{18}-2q^{16}+4q^{14}-2q^{12}+q^{10}-2q^{8}+q^{6}$
$\displaystyle B_{5}$
$\displaystyle=a^{2}\cdot\frac{2q^{16}-3q^{14}-q^{12}+2q^{10}+2q^{8}-q^{6}-3q^{4}+2q^{2}}{q^{20}-q^{16}+q^{12}+q^{10}+q^{8}-q^{4}+1}$
$\displaystyle B_{6}$
$\displaystyle=q^{18}-3q^{16}+3q^{14}-q^{12}+2q^{10}-4q^{8}+2q^{6}$
$\displaystyle B_{7}$
$\displaystyle=q^{16}-2q^{14}+q^{12}-2q^{10}+4q^{8}-2q^{6}+q^{4}-2q^{2}+1$
$\displaystyle B_{8}$ $\displaystyle=q^{24}-4q^{22}+6q^{20}-4q^{18}+q^{16}$
$\displaystyle B_{9}$
$\displaystyle=2q^{22}-5q^{20}+3q^{18}+q^{16}+q^{14}-4q^{12}+2q^{10}$
$\displaystyle B_{10}$
$\displaystyle=q^{18}-2q^{16}-q^{14}+4q^{12}-q^{10}-2q^{8}+q^{6}$
$\displaystyle B_{11}$
$\displaystyle=2q^{14}-4q^{12}+q^{10}+q^{8}+3q^{6}-5q^{4}+2q^{2}$
$\displaystyle B_{12}$ $\displaystyle=q^{8}-4q^{6}+6q^{4}-4q^{2}+1$
$\displaystyle B_{13}$
$\displaystyle=q^{24}-q^{22}-2q^{20}+2q^{18}+3q^{16}-q^{14}-4q^{12}+2q^{8}$
$\displaystyle B_{14}$
$\displaystyle=q^{22}-2q^{18}-2q^{16}+q^{14}+4q^{12}+q^{10}-2q^{8}-2q^{6}+q^{2}$
$\displaystyle B_{15}$
$\displaystyle=2q^{16}-4q^{12}-q^{10}+3q^{8}+2q^{6}-2q^{4}-q^{2}+1$
$\displaystyle B_{16}$
$\displaystyle=q^{24}-2q^{22}+q^{20}-2q^{18}+4q^{16}-2q^{14}+q^{12}-2q^{10}+q^{8}$
$\displaystyle B_{17}$
$\displaystyle=a^{2}\cdot\frac{2q^{18}-3q^{16}-q^{14}+2q^{12}+2q^{10}-q^{8}-3q^{6}+2q^{4}}{q^{20}-q^{16}+q^{12}+q^{10}+q^{8}-q^{4}+1}$
$\displaystyle B_{18}$
$\displaystyle=2q^{18}-4q^{16}+2q^{14}-q^{12}+3q^{10}-3q^{8}+q^{6}$
$\displaystyle B_{19}$
$\displaystyle=q^{18}-2q^{16}+q^{14}-2q^{12}+4q^{10}-2q^{8}+q^{6}-2q^{4}+q^{2}$
$\displaystyle B_{20}$
$\displaystyle=2q^{12}-4q^{10}+2q^{8}-q^{6}+3q^{4}-3q^{2}+1$ $\displaystyle
B_{21}$
$\displaystyle=q^{24}-2q^{20}-2q^{18}+q^{16}+4q^{14}+q^{12}-2q^{10}-2q^{8}+q^{4}$
$\displaystyle B_{22}$
$\displaystyle=2q^{18}-4q^{14}-q^{12}+3q^{10}+2q^{8}-2q^{6}-q^{4}+q^{2}$

a $6\times 6$ block:

$\displaystyle a^{-2}\begin{bmatrix}*&B_{1}&B_{2}&B_{3}&0&B_{4}\\\
B_{1}&*&B_{2}&0&B_{3}&B_{4}\\\ B_{5}&B_{5}&*&0&0&B_{6}\\\
B_{7}&0&0&*&B_{1}&B_{8}\\\ 0&B_{7}&0&B_{1}&*&B_{8}\\\
B_{9}&B_{9}&B_{10}&B_{11}&B_{11}&*\end{bmatrix}$

where

$\displaystyle B_{1}$
$\displaystyle=q^{20}-2q^{18}+q^{16}-2q^{14}+4q^{12}-2q^{10}+q^{8}-2q^{6}+q^{4}$
$\displaystyle B_{2}$
$\displaystyle=2q^{14}-4q^{12}+2q^{10}-q^{8}+3q^{6}-3q^{4}+q^{2}$
$\displaystyle B_{3}$ $\displaystyle=q^{20}-2q^{18}+q^{16}+q^{6}-2q^{4}+q^{2}$
$\displaystyle B_{4}$ $\displaystyle=q^{18}-2q^{16}+q^{14}+q^{4}-2q^{2}+1$
$\displaystyle B_{5}$
$\displaystyle=2q^{18}-2q^{16}-2q^{14}+q^{12}+2q^{10}-2q^{6}+q^{4}$
$\displaystyle B_{6}$
$\displaystyle=q^{20}-q^{18}-q^{16}+q^{14}+q^{6}-q^{4}-q^{2}+1$ $\displaystyle
B_{7}$ $\displaystyle=q^{22}-2q^{20}+q^{18}+q^{8}-2q^{6}+q^{4}$ $\displaystyle
B_{8}$
$\displaystyle=q^{18}-2q^{16}+3q^{14}-4q^{12}+2q^{10}-q^{8}+3q^{6}-2q^{4}-q^{2}+1$
$\displaystyle B_{9}$
$\displaystyle=q^{22}-2q^{20}+2q^{18}-3q^{16}+3q^{14}-q^{12}+q^{10}-2q^{8}+q^{6}$
$\displaystyle B_{10}$
$\displaystyle=-q^{18}+3q^{16}-3q^{14}+q^{12}-q^{10}+3q^{8}-3q^{6}+q^{4}$
$\displaystyle B_{11}$
$\displaystyle=q^{20}-2q^{18}+2q^{16}-3q^{14}+3q^{12}-q^{10}+q^{8}-2q^{6}+q^{4}$

two $6\times 6$ blocks:

$\displaystyle a^{-2}\begin{bmatrix}*&B_{1}&B_{2}&B_{3}&0&0\\\
B_{4}&*&B_{5}&B_{6}&0&0\\\ B_{7}&B_{8}&*&B_{9}&0&B_{3}\\\
B_{10}&B_{11}&B_{12}&*&B_{13}&B_{14}\\\ 0&0&0&B_{15}&*&B_{16}\\\
0&0&B_{17}&B_{18}&B_{19}&*\end{bmatrix}$

where

$\displaystyle B_{1}$
$\displaystyle=q^{20}-3q^{18}+3q^{16}-q^{14}+2q^{12}-4q^{10}+2q^{8}$
$\displaystyle B_{2}$
$\displaystyle=q^{18}-2q^{16}+q^{14}-2q^{12}+4q^{10}-2q^{8}+q^{6}-2q^{4}+q^{2}$
$\displaystyle B_{3}$ $\displaystyle=q^{18}-2q^{16}+q^{14}+q^{4}-2q^{2}+1$
$\displaystyle B_{4}$
$\displaystyle=q^{24}-2q^{22}+2q^{18}+q^{16}-2q^{14}-2q^{12}+2q^{10}$
$\displaystyle B_{5}$
$\displaystyle=2q^{16}-2q^{14}-2q^{12}+q^{10}+2q^{8}-2q^{4}+q^{2}$
$\displaystyle B_{6}$
$\displaystyle=q^{20}-q^{18}-q^{16}+q^{14}+q^{6}-q^{4}-q^{2}+1$ $\displaystyle
B_{7}$
$\displaystyle=q^{24}-2q^{22}+q^{20}-2q^{18}+4q^{16}-2q^{14}+q^{12}-2q^{10}+q^{8}$
$\displaystyle B_{8}$
$\displaystyle=2q^{18}-4q^{16}+2q^{14}-q^{12}+3q^{10}-3q^{8}+q^{6}$
$\displaystyle B_{9}$ $\displaystyle=q^{20}-2q^{18}+q^{16}+q^{6}-2q^{4}+q^{2}$
$\displaystyle B_{10}$
$\displaystyle=q^{24}-3q^{22}+3q^{20}-2q^{18}+3q^{16}-3q^{14}+q^{12}$
$\displaystyle B_{11}$
$\displaystyle=q^{22}-2q^{20}+2q^{18}-3q^{16}+3q^{14}-q^{12}+q^{10}-2q^{8}+q^{6}$
$\displaystyle B_{12}$
$\displaystyle=-q^{18}+3q^{16}-3q^{14}+q^{12}-q^{10}+3q^{8}-3q^{6}+q^{4}$
$\displaystyle B_{13}$
$\displaystyle=q^{18}-2q^{16}+q^{12}+q^{10}-2q^{6}+q^{4}$ $\displaystyle
B_{14}$ $\displaystyle=q^{14}-q^{12}-q^{10}+2q^{6}-2q^{2}+1$ $\displaystyle
B_{15}$
$\displaystyle=q^{20}-q^{18}-q^{16}-q^{14}+2q^{12}+2q^{10}-q^{8}-q^{6}-q^{4}+q^{2}$
$\displaystyle B_{16}$
$\displaystyle=2q^{14}-2q^{12}-2q^{10}+q^{8}+2q^{6}-2q^{2}+1$ $\displaystyle
B_{17}$ $\displaystyle=q^{22}-2q^{20}+q^{18}+q^{8}-2q^{6}+q^{4}$
$\displaystyle B_{18}$
$\displaystyle=q^{20}-2q^{18}+3q^{16}-4q^{14}+2q^{12}-q^{10}+3q^{8}-2q^{6}-q^{4}+q^{2}$
$\displaystyle B_{19}$
$\displaystyle=2q^{16}-4q^{14}+2q^{12}-q^{10}+3q^{8}-3q^{6}+q^{4}$

a $6\times 6$ block:

$\displaystyle a^{-2}\begin{bmatrix}*&B_{1}&B_{1}&0&0&0\\\
B_{2}&*&B_{3}&B_{4}&0&0\\\ B_{2}&B_{5}&*&0&B_{4}&0\\\
0&B_{6}&0&*&B_{7}&B_{8}\\\ 0&0&B_{6}&B_{7}&*&B_{8}\\\
0&0&0&B_{9}&B_{9}&*\end{bmatrix}$

where

$\displaystyle B_{1}$
$\displaystyle=q^{22}-2q^{20}+2q^{16}+q^{14}-2q^{12}-2q^{10}+2q^{8}$
$\displaystyle B_{2}$
$\displaystyle=q^{24}-3q^{22}+3q^{20}-q^{18}+2q^{16}-4q^{14}+2q^{12}$
$\displaystyle B_{3}$
$\displaystyle=q^{22}-2q^{20}+q^{18}-2q^{16}+4q^{14}-2q^{12}+q^{10}-2q^{8}+q^{6}$
$\displaystyle B_{4}$ $\displaystyle=q^{18}-2q^{16}+q^{14}+q^{4}-2q^{2}+1$
$\displaystyle B_{5}$
$\displaystyle=q^{22}-2q^{20}+q^{18}-2q^{16}+4q^{14}-2q^{12}+q^{10}-2q^{8}+q^{6}$
$\displaystyle B_{6}$
$\displaystyle=q^{24}-2q^{22}+q^{20}+q^{10}-2q^{8}+q^{6}$ $\displaystyle
B_{7}$
$\displaystyle=q^{18}-2q^{16}+q^{14}-2q^{12}+4q^{10}-2q^{8}+q^{6}-2q^{4}+q^{2}$
$\displaystyle B_{8}$
$\displaystyle=2q^{12}-4q^{10}+2q^{8}-q^{6}+3q^{4}-3q^{2}+1$ $\displaystyle
B_{9}$ $\displaystyle=2q^{16}-2q^{14}-2q^{12}+q^{10}+2q^{8}-2q^{4}+q^{2}$

four $6\times 6$ blocks:

$\displaystyle a^{-2}\begin{bmatrix}*&B_{1}&B_{2}&B_{3}&0&0\\\
B_{4}&*&B_{5}&B_{6}&0&0\\\ B_{7}&B_{8}&*&B_{9}&B_{10}&B_{11}\\\
B_{12}&B_{13}&B_{14}&*&B_{15}&B_{16}\\\ 0&0&B_{17}&B_{18}&*&B_{19}\\\
0&0&B_{20}&B_{21}&B_{22}&*\end{bmatrix}$

where

$\displaystyle B_{1}$
$\displaystyle=q^{20}-2q^{18}+2q^{14}+q^{12}-2q^{10}-2q^{8}+2q^{6}$
$\displaystyle B_{2}$
$\displaystyle=q^{22}-3q^{18}+q^{16}+2q^{14}+q^{12}-2q^{10}-q^{8}+q^{6}-q^{4}+q^{2}$
$\displaystyle B_{3}$
$\displaystyle=q^{18}-q^{16}-q^{14}-q^{12}+2q^{10}+2q^{8}-q^{6}-q^{4}-q^{2}+1$
$\displaystyle B_{4}$
$\displaystyle=q^{24}-2q^{22}+2q^{18}+q^{16}-2q^{14}-2q^{12}+2q^{10}$
$\displaystyle B_{5}$
$\displaystyle=q^{22}-q^{20}-q^{18}-q^{16}+2q^{14}+2q^{12}-q^{10}-q^{8}-q^{6}+q^{4}$
$\displaystyle B_{6}$
$\displaystyle=q^{20}-q^{18}+q^{16}-q^{14}-2q^{12}+q^{10}+2q^{8}+q^{6}-3q^{4}+1$
$\displaystyle B_{7}$
$\displaystyle=q^{24}-2q^{22}+2q^{18}-q^{14}-q^{12}+q^{10}$ $\displaystyle
B_{8}$ $\displaystyle=q^{20}-2q^{18}+q^{14}+q^{12}-2q^{8}+q^{6}$
$\displaystyle B_{9}$
$\displaystyle=q^{18}-q^{16}-2q^{12}+3q^{10}-q^{8}+2q^{6}-4q^{4}+2q^{2}$
$\displaystyle B_{10}$
$\displaystyle=q^{18}-2q^{16}+q^{14}-q^{12}+3q^{10}-3q^{8}+2q^{6}-2q^{4}+q^{2}$
$\displaystyle B_{11}$
$\displaystyle=q^{12}-3q^{10}+3q^{8}-2q^{6}+3q^{4}-3q^{2}+1$ $\displaystyle
B_{12}$ $\displaystyle=q^{24}-3q^{22}+3q^{20}-2q^{18}+3q^{16}-3q^{14}+q^{12}$
$\displaystyle B_{13}$
$\displaystyle=q^{22}-2q^{20}+2q^{18}-3q^{16}+3q^{14}-q^{12}+q^{10}-2q^{8}+q^{6}$
$\displaystyle B_{14}$
$\displaystyle=2q^{22}-4q^{20}+2q^{18}-q^{16}+3q^{14}-2q^{12}-q^{8}+q^{6}$
$\displaystyle B_{15}$
$\displaystyle=q^{18}-2q^{16}+q^{12}+q^{10}-2q^{6}+q^{4}$ $\displaystyle
B_{16}$ $\displaystyle=q^{14}-q^{12}-q^{10}+2q^{6}-2q^{2}+1$ $\displaystyle
B_{17}$
$\displaystyle=q^{24}-3q^{20}+q^{18}+2q^{16}+q^{14}-2q^{12}-q^{10}+q^{8}-q^{6}+q^{4}$
$\displaystyle B_{18}$
$\displaystyle=q^{20}-q^{18}-q^{16}-q^{14}+2q^{12}+2q^{10}-q^{8}-q^{6}-q^{4}+q^{2}$
$\displaystyle B_{19}$
$\displaystyle=2q^{14}-2q^{12}-2q^{10}+q^{8}+2q^{6}-2q^{2}+1$ $\displaystyle
B_{20}$
$\displaystyle=q^{24}-q^{22}-q^{20}-q^{18}+2q^{16}+2q^{14}-q^{12}-q^{10}-q^{8}+q^{6}$
$\displaystyle B_{21}$
$\displaystyle=q^{22}-q^{20}+q^{18}-q^{16}-2q^{14}+q^{12}+2q^{10}+q^{8}-3q^{6}+q^{2}$
$\displaystyle B_{22}$
$\displaystyle=2q^{18}-2q^{16}-2q^{14}+q^{12}+2q^{10}-2q^{6}+q^{4}$

a $4\times 4$ block:

$\displaystyle a^{-2}\begin{bmatrix}*&B_{1}&B_{2}&B_{3}\\\
B_{1}&*&B_{2}&B_{3}\\\ B_{4}&B_{4}&*&B_{3}\\\
B_{5}&B_{5}&B_{6}&*\end{bmatrix}$

where

$\displaystyle B_{1}$
$\displaystyle=q^{20}-2q^{18}+q^{16}-2q^{14}+4q^{12}-2q^{10}+q^{8}-2q^{6}+q^{4}$
$\displaystyle B_{2}$
$\displaystyle=2q^{14}-4q^{12}+2q^{10}-q^{8}+3q^{6}-3q^{4}+q^{2}$
$\displaystyle B_{3}$ $\displaystyle=q^{18}-2q^{16}+q^{14}+q^{4}-2q^{2}+1$
$\displaystyle B_{4}$
$\displaystyle=2q^{16}-4q^{14}+2q^{12}-q^{10}+3q^{8}-3q^{6}+q^{4}$
$\displaystyle B_{5}$
$\displaystyle=q^{22}-q^{20}-q^{16}+2q^{12}-q^{8}-q^{6}+q^{4}$ $\displaystyle
B_{6}$ $\displaystyle=-q^{18}+2q^{16}-2q^{12}+2q^{8}-2q^{4}+q^{2}$

two $4\times 4$ blocks:

$\displaystyle a^{-2}\begin{bmatrix}*&B_{1}&B_{1}&B_{2}\\\
B_{3}&*&B_{4}&B_{5}\\\ B_{3}&B_{4}&*&B_{5}\\\
B_{6}&B_{7}&B_{7}&*\end{bmatrix}$

where

$\displaystyle B_{1}$
$\displaystyle=q^{20}-q^{18}-q^{16}+2q^{12}-q^{8}-q^{4}+q^{2}$ $\displaystyle
B_{2}$ $\displaystyle=q^{14}-2q^{12}+q^{8}+q^{6}-2q^{2}+1$ $\displaystyle
B_{3}$
$\displaystyle=q^{24}-q^{22}-2q^{20}+3q^{18}-q^{16}+2q^{14}-4q^{12}+3q^{10}-2q^{8}+q^{6}$
$\displaystyle B_{4}$
$\displaystyle=q^{20}-2q^{18}+q^{16}-2q^{14}+4q^{12}-2q^{10}+q^{8}-2q^{6}+q^{4}$
$\displaystyle B_{5}$
$\displaystyle=q^{18}-2q^{16}+3q^{14}-4q^{12}+2q^{10}-q^{8}+3q^{6}-2q^{4}-q^{2}+1$
$\displaystyle B_{6}$
$\displaystyle=q^{24}-2q^{22}+q^{18}+q^{16}-2q^{12}+q^{10}$ $\displaystyle
B_{7}$ $\displaystyle=q^{22}-q^{20}-q^{16}+2q^{12}-q^{8}-q^{6}+q^{4}$

two $4\times 4$ blocks:

$\displaystyle a^{-2}\begin{bmatrix}*&B_{1}&B_{1}&B_{2}\\\
B_{3}&*&B_{4}&B_{5}\\\ B_{3}&B_{4}&*&B_{5}\\\
B_{6}&B_{7}&B_{7}&*\end{bmatrix}$

where

$\displaystyle B_{1}$
$\displaystyle=q^{20}-q^{18}-2q^{16}+3q^{14}-q^{12}+2q^{10}-4q^{8}+3q^{6}-2q^{4}+q^{2}$
$\displaystyle B_{2}$
$\displaystyle=q^{16}-2q^{14}+q^{12}-2q^{10}+4q^{8}-2q^{6}+q^{4}-2q^{2}+1$
$\displaystyle B_{3}$
$\displaystyle=q^{24}-q^{22}-2q^{20}+2q^{18}+2q^{16}-q^{14}-2q^{12}+q^{8}$
$\displaystyle B_{4}$
$\displaystyle=q^{20}-q^{18}-2q^{16}+q^{14}+2q^{12}+q^{10}-2q^{8}-q^{6}+q^{4}$
$\displaystyle B_{5}$
$\displaystyle=q^{16}-2q^{12}-q^{10}+2q^{8}+2q^{6}-2q^{4}-q^{2}+1$
$\displaystyle B_{6}$
$\displaystyle=q^{24}-2q^{22}+q^{20}-2q^{18}+4q^{16}-2q^{14}+q^{12}-2q^{10}+q^{8}$
$\displaystyle B_{7}$
$\displaystyle=q^{22}-2q^{20}+3q^{18}-4q^{16}+2q^{14}-q^{12}+3q^{10}-2q^{8}-q^{6}+q^{4}$

eight $4\times 4$ blocks:

$\displaystyle a^{-2}\begin{bmatrix}*&B_{1}&B_{2}&0\\\ B_{3}&*&B_{4}&B_{5}\\\
B_{3}&B_{6}&*&B_{5}\\\ 0&B_{7}&B_{1}&*\end{bmatrix}$

where

$\displaystyle B_{1}$
$\displaystyle=q^{22}-q^{20}-q^{18}+q^{16}+q^{8}-q^{6}-q^{4}+q^{2}$
$\displaystyle B_{2}$
$\displaystyle=q^{20}-q^{18}-q^{16}+q^{14}+q^{6}-q^{4}-q^{2}+1$ $\displaystyle
B_{3}$ $\displaystyle=q^{24}-2q^{22}+q^{20}+q^{10}-2q^{8}+q^{6}$
$\displaystyle B_{4}$ $\displaystyle=q^{20}-2q^{18}+q^{16}+q^{6}-2q^{4}+q^{2}$
$\displaystyle B_{5}$ $\displaystyle=q^{18}-2q^{16}+q^{14}+q^{4}-2q^{2}+1$
$\displaystyle B_{6}$ $\displaystyle=q^{22}-2q^{20}+q^{18}+q^{8}-2q^{6}+q^{4}$
$\displaystyle B_{7}$
$\displaystyle=q^{24}-q^{22}-q^{20}+q^{18}+q^{10}-q^{8}-q^{6}+q^{4}$

four $4\times 4$ blocks:

$\displaystyle a^{-2}\begin{bmatrix}*&B_{1}&B_{1}&B_{2}\\\ B_{3}&*&0&B_{4}\\\
B_{3}&0&*&B_{4}\\\ B_{5}&B_{6}&B_{6}&*\end{bmatrix}$

where

$\displaystyle B_{1}$ $\displaystyle=q^{20}-2q^{18}+q^{16}+q^{6}-2q^{4}+q^{2}$
$\displaystyle B_{2}$ $\displaystyle=q^{18}-2q^{16}+q^{14}+q^{4}-2q^{2}+1$
$\displaystyle B_{3}$
$\displaystyle=q^{24}-q^{22}-q^{20}+q^{18}+q^{10}-q^{8}-q^{6}+q^{4}$
$\displaystyle B_{4}$
$\displaystyle=q^{20}-q^{18}-q^{16}+q^{14}+q^{6}-q^{4}-q^{2}+1$ $\displaystyle
B_{5}$ $\displaystyle=q^{24}-2q^{22}+q^{20}+q^{10}-2q^{8}+q^{6}$
$\displaystyle B_{6}$ $\displaystyle=q^{22}-2q^{20}+q^{18}+q^{8}-2q^{6}+q^{4}$

eight $3\times 3$ blocks:

$\displaystyle a^{-2}\begin{bmatrix}B_{1}&-B_{1}&0\\\ B_{2}&B_{3}&B_{4}\\\
0&B_{5}&-B_{5}\end{bmatrix}$

where

$\displaystyle B_{1}$ $\displaystyle=-q^{22}+2q^{18}-q^{14}-q^{8}+2q^{4}-1$
$\displaystyle B_{2}$
$\displaystyle=q^{24}-2q^{22}+q^{20}+q^{10}-2q^{8}+q^{6}$ $\displaystyle
B_{3}$
$\displaystyle=-q^{24}+2q^{22}-q^{20}-q^{18}+2q^{16}-q^{14}-q^{10}+2q^{8}-q^{6}-q^{4}+2q^{2}-1$
$\displaystyle B_{4}$ $\displaystyle=q^{18}-2q^{16}+q^{14}+q^{4}-2q^{2}+1$
$\displaystyle B_{5}$
$\displaystyle=q^{24}-2q^{20}+q^{16}+q^{10}-2q^{6}+q^{2}$

sixteen $2\times 2$ blocks:

$\displaystyle a^{-2}\begin{bmatrix}B_{1}&-B_{1}\\\ B_{2}&-B_{2}\end{bmatrix}$

where

$\displaystyle B_{1}$
$\displaystyle=-q^{20}+q^{18}+q^{16}-q^{14}-q^{6}+q^{4}+q^{2}-1$
$\displaystyle B_{2}$
$\displaystyle=q^{24}-q^{22}-q^{20}+q^{18}+q^{10}-q^{8}-q^{6}+q^{4}$

and five $1\times 1$ blocks with entry $0$.

Note that multiplying the $3\times 3$ and $2\times 2$ blocks of the algebraic
generator by the scalar

$\displaystyle\frac{a^{2}}{q^{12}}\cdot\frac{q^{2}-1+q^{-2}}{(q-q^{-1})^{2}(q+q^{-1})(q^{7}+q^{-7})}$

yields the $3\times 3$ and $2\times 2$ blocks of $L_{Q}$ (as described in
Section 7.3) when $n=2$. Note additionally that no larger block of $L_{Q}$
(with $n=2$) is a constant multiple of a block of the algebraic generator.

###  7.9 Type $A$ Ground State Transformation

The following notes were provided to us by Dr. Jeffrey Kuan to aid in our
calculation of the ground state transformation.

Suppose $C$ is a central element and $H=(\pi_{V}\otimes\pi_{V})(\Delta(C))$.
Let $v\in V$ be a highest-weight vector of a highest-weight representation
$V$. Then,

$H(v\otimes v)=av\otimes v$

for some constant $a$. By the definition of a central element,

$\Delta(F_{j}^{k})C=C\Delta(F_{j}^{k}).$

Given unit basis vectors $u_{i}\otimes u_{j}\in V\otimes V$, let
$(k_{1},k_{2},k_{3})\in\mathbb{Z}_{\geq 0}^{3}$ such that

$\langle u_{i}\otimes
u_{j}\mid\Delta(F_{3}^{k_{3}})\cdot\Delta(F_{2}^{k_{2}})\cdot\Delta(F_{1}^{k_{1}})\mid
v\otimes v\rangle\neq 0.$

$F_{3},F_{2},F_{1}$ may need to be in a different order, but the values of
k$=(k_{1},k_{2},k_{3})$ should be the same.

Set $G:V\otimes V\to V\otimes V$ by

$G(u_{i}\otimes u_{j})=\langle u_{i}\otimes
u_{j}\mid\Delta(F_{3}^{k_{3}})\cdot\Delta(F_{2}^{k_{2}})\cdot\Delta(F_{1}^{k_{1}})\mid
v\otimes v\rangle u_{i}\otimes u_{j}$

where k is the above term.

Now, set $S=G^{-1}HG$.

######  Theorem 5.

$a^{-1}S-\mathrm{Id}$ has rows that sum to $0$.

###### Proof.

For fixed $u_{i}\otimes u_{j}$, we have that the sum of the row indexed by
$u_{i}\otimes u_{j}$ equals

$\sum_{w_{1},w_{2}}\frac{\langle u_{i}\otimes u_{j}\mid H\mid w_{1}\otimes
w_{2}\rangle\langle w_{1}\otimes w_{2}\mid G\mid w_{1}\otimes
w_{2}\rangle}{\langle u_{i}\otimes u_{j}\mid G\mid u_{i}\otimes u_{j}\rangle}$

which then equals

$\sum_{w_{1},w_{2}}\left[\frac{\langle u_{i}\otimes u_{j}\mid H\mid
w_{1}\otimes w_{2}\rangle}{\langle u_{i}\otimes u_{j}\mid G\mid u_{i}\otimes
u_{j}\rangle}\sum_{\text{{k}}}\langle w_{1}\otimes
w_{2}\mid\Delta(F_{3}^{k_{3}})\cdot\Delta(F_{2}^{k_{2}})\cdot\Delta(F_{1}^{k_{1}})\mid
v\otimes v\rangle\right]$

By commutation between the coproduct and $H$, this must then equal

$\sum_{\text{{k}}}\frac{\langle u_{i}\otimes
u_{j}\mid\Delta(F_{3}^{k_{3}})\cdot\Delta(F_{2}^{k_{2}})\cdot\Delta(F_{1}^{k_{1}})H\mid
v\otimes v\rangle}{\langle u_{i}\otimes u_{j}\mid G\mid u_{i}\otimes
u_{j}\rangle}$

Finally, this equals

$\sum_{\text{{k}}}\frac{a\langle u_{i}\otimes
u_{j}\mid\Delta(F_{3}^{k_{3}})\cdot\Delta(F_{2}^{k_{2}})\cdot\Delta(F_{1}^{k_{1}})\mid
v\otimes v\rangle}{\langle u_{i}\otimes u_{j}\mid G\mid u_{i}\otimes
u_{j}\rangle}=a$

Multiplying by $a^{-1}$ and subtracting the identity, we obtain a matrix whose
rows sum to $0$. ∎

###  7.10 Explicit values of $E_{i},F_{i},q^{H_{i}}$ in $W\otimes W$

Now we give explicitly the values of $\pi_{W\otimes W}(E_{i}),\pi_{W\otimes
W}(F_{i}),\pi_{W\otimes W}(q^{H_{i}})$ for $1\leq i\leq 3$. When performing
the calculations, the decomposition of $W\otimes W$ into irreducible
representations and weight spaces significantly reduces the computational
effort, allowing all computations to be done by hand. The matrices below are
given for references only, and are strictly speaking not necessary for the
proof. In the expressions below, $E_{i,j}$ denotes the $400\times 400$ matrix
with all entries $0$ except for the $(i,j)$th entry, which is a $1$.

$\displaystyle\pi_{W\otimes W}(E_{1})=$
$\displaystyle(q^{3}+q)(E_{1,2}+E_{2,3}+E_{8,9}+E_{19,20}+E_{141,142}+E_{142,143}+E_{148,149}+E_{159,160}+E_{341,342}$
$\displaystyle+E_{342,343}+E_{348,349}+E_{359,360})$ $\displaystyle+$
$\displaystyle(q^{2}+1)(E_{61,62}+E_{62,63}+E_{68,69}+E_{79,80}+E_{81,82}+E_{82,83}+E_{88,89}+E_{99,100}+E_{261,262}$
$\displaystyle+E_{262,263}+E_{268,269}+E_{279,280}+E_{281,282}+E_{282,283}+E_{288,289}+E_{299,300})$
$\displaystyle+$
$\displaystyle(q^{2})(E_{4,6}+E_{5,7}+E_{8,11}+E_{9,13}+E_{14,16}+E_{15,17}+E_{18,19}+E_{144,146}+E_{145,147}+E_{148,151}$
$\displaystyle+E_{149,153}+E_{154,156}+E_{155,157}+E_{158,159}+E_{344,346}+E_{345,347}+E_{348,351}+E_{349,353}$
$\displaystyle+E_{354,356}+E_{355,357}+E_{358,359})$ $\displaystyle+$
$\displaystyle(q)(E_{64,66}+E_{65,67}+E_{68,71}+E_{69,73}+E_{74,76}+E_{75,77}+E_{78,79}+E_{84,86}+E_{85,87}+E_{88,91}$
$\displaystyle+E_{89,93}+E_{94,96}+E_{95,97}+E_{98,99}+E_{264,266}+E_{265,267}+E_{268,271}+E_{269,273}+E_{274,276}$
$\displaystyle+E_{275,277}+E_{278,279}+E_{284,286}+E_{285,287}+E_{288,291}+E_{289,293}+E_{294,296}+E_{295,297}+E_{298,299})$
$\displaystyle+$
$\displaystyle(q+q^{-1})\left(\sum_{i=1}^{40}E_{i,i+20}+\sum_{i=141}^{160}E_{i,i+20}+\sum_{361}^{400}E_{i,i+20}+E_{22,23}+E_{28,29}+E_{39,40}+E_{362,363}+E_{379,380}\right.$
$\displaystyle+E_{162,163}+E_{168,169}+E_{179,180}+E_{181,182}+E_{182,183}+E_{188,189}+E_{199,200}+E_{201,202}$
$\displaystyle\left.+E_{202,203}+E_{208,209}+E_{219,220}+E_{221,222}+E_{222,223}+E_{228,229}+E_{239,240}+E_{361,362}\right)$
$\displaystyle+$
$\displaystyle(1)\left(\sum_{i=61}^{100}E_{i,i+40}+\sum_{i=141}^{160}E_{i,i+60}+\sum_{i=161}^{180}E_{i,i+80}+\sum_{i=261}^{300}E_{i,i+40}+\sum_{i=341}^{360}E_{i,i+20}\right.$
$\displaystyle+E_{24,26}+E_{25,27}+E_{28,31}+E_{29,33}+E_{34,36}+E_{35,37}+E_{38,39}$
$\displaystyle+E_{164,166}+E_{165,167}+E_{169,173}+E_{174,176}+E_{175,177}+E_{178,179}$
$\displaystyle+E_{184,186}+E_{185,187}+E_{188,191}+E_{189,193}+E_{194,196}+E_{195,197}+E_{198,199}+E_{204,206}$
$\displaystyle+E_{205,207}+E_{208,211}+E_{209,213}+E_{214,216}+E_{215,217}+E_{218,219}+E_{224,226}+E_{225,227}$
$\displaystyle+E_{228,231}+E_{229,233}+E_{234,236}+E_{235,237}+E_{238,239}$
$\displaystyle\left.+E_{364,366}+E_{365,367}+E_{368,371}+E_{369,373}+E_{374,376}+E_{375,377}+E_{378,379})\right)$
$\displaystyle+$
$\displaystyle(q^{-1})(E_{104,106}+E_{105,107}+E_{108,111}+E_{109,113}+E_{114,116}+E_{115,117}+E_{118,119}+E_{124,126}$
$\displaystyle+E_{125,127}+E_{128,131}+E_{129,133}+E_{134,136}+E_{135,137}+E_{138,139}+E_{304,306}+E_{305,307}$
$\displaystyle+E_{308,311}+E_{309,313}+E_{314,316}+E_{315,317}+E_{318,319}+E_{324,326}+E_{325,327}+E_{328,331}$
$\displaystyle+E_{329,333}+E_{334,336}+E_{335,337}+E_{338,339})$
$\displaystyle+$
$\displaystyle(q^{-2})(E_{44,46}+E_{45,47}+E_{48,51}+E_{49,53}+E_{54,56}+E_{55,57}+E_{58,59}+E_{244,246}+E_{245,247}$
$\displaystyle+E_{248,251}+E_{249,253}+E_{254,256}+E_{255,257}+E_{258,259}+E_{384,386}+E_{385,387}+E_{388,391}$
$\displaystyle+E_{389,393}+E_{394,396}+E_{395,397}+E_{398,399})$
$\displaystyle+$
$\displaystyle(1+q^{-2})(E_{101,102}+E_{102,103}+E_{108,109}+E_{119,120}+E_{121,122}+E_{122,123}+E_{128,129}+E_{139,140}$
$\displaystyle+E_{301,302}+E_{302,303}+E_{308,309}+E_{319,320}+E_{321,322}+E_{322,323}+E_{328,329}+E_{339,340})$
$\displaystyle+$
$\displaystyle(q^{-1}+q^{-3})(E_{41,42}+E_{42,43}+E_{48,49}+E_{59,60}+E_{241,242}+E_{242,243}+E_{248,249}+E_{259,260}$
$\displaystyle+E_{381,382}+E_{382,383}+E_{388,389}+E_{399,400})$

$\displaystyle\pi_{W\otimes W}(E_{2})$
$\displaystyle=(q^{3}+q)(E_{46,50}+E_{47,51}+E_{51,54}+E_{52,55}+E_{53,56}+E_{55,58}+E_{126,130}+E_{127,131}+E_{131,134}$
$\displaystyle+E_{132,135}+E_{133,136}+E_{135,138}+E_{226,230}+E_{227,231}+E_{231,234}+E_{232,235}+E_{233,236}+E_{235,238})$
$\displaystyle+$
$\displaystyle(q^{2}+1)(E_{26,30}+E_{27,31}+E_{31,34}+E_{32,35}+E_{33,36}+E_{35,38}+E_{86,90}+E_{87,91}+E_{91,94}+E_{92,95}$
$\displaystyle+E_{93,96}+E_{95,98}+E_{246,250}+E_{247,251}+E_{251,254}+E_{252,255}+E_{253,256}+E_{255,258}+E_{326,330}$
$\displaystyle+E_{327,331}+E_{331,334}+E_{332,335}+E_{333,336}+E_{335,338})$
$\displaystyle+$
$\displaystyle(q^{2})(E_{42,44}+E_{43,46}+E_{45,48}+E_{47,49}+E_{57,59}+E_{122,124}+E_{123,126}+E_{125,128}+E_{127,129}$
$\displaystyle+E_{137,139}+E_{222,224}+E_{223,226}+E_{225,228}+E_{227,229}+E_{237,239})$
$\displaystyle+$
$\displaystyle(q)(E_{22,24}+E_{23,26}+E_{25,28}+E_{27,29}+E_{37,39}+E_{82,84}+E_{83,86}+E_{85,88}+E_{87,89}+E_{97,99}$
$\displaystyle+E_{242,244}+E_{243,246}+E_{245,248}+E_{247,249}+E_{257,259}+E_{322,324}+E_{323,326}+E_{325,328}$
$\displaystyle+E_{327,329}+E_{337,339})$ $\displaystyle+$
$\displaystyle(q+q^{-1})\left(\sum_{i=101}^{140}E_{i,i+80}+\sum_{i=201}^{260}E_{i,i+60}+\sum_{i=281}E_{i,i+60}+E_{6,10}+E_{7,11}+E_{11,14}+E_{12,15}+E_{13,16}\right.$
$\displaystyle+E_{15,18}+E_{106,110}+E_{107,111}+E_{111,114}+E_{112,115}+E_{113,116}+E_{115,118}+E_{166,170}+E_{167,171}$
$\displaystyle+E_{171,174}+E_{172,175}+E_{173,176}+E_{175,178}+E_{206,210}+E_{207,211}+E_{211,214}+E_{212,215}$
$\displaystyle+E_{213,216}+E_{215,218}+E_{286,290}+E_{287,291}+E_{291,294}+E_{292,295}+E_{293,296}+E_{295,298}$
$\displaystyle\left.+E_{386,390}+E_{387,391}+E_{391,394}+E_{392,395}+E_{393,396}+E_{395,398}\right)$
$\displaystyle+$
$\displaystyle(1)\left(\sum_{i=21}^{40}E_{i,i+40}+\sum_{i=41}^{60}E_{i,i+60}\sum_{i=81}^{100}E_{i,i+60}+\sum_{i=121}^{140}E_{i,i+40}+\sum_{i=321}^{340}E_{i,i+40}+E_{2,4}+E_{3,6}+E_{5,8}\right.$
$\displaystyle+E_{7,9}+E_{17,19}+E_{102,104}+E_{103,106}+E_{105,108}+E_{107,109}+E_{117,119}+E_{162,164}+E_{163,166}$
$\displaystyle+E_{165,168}+E_{167,169}+E_{177,179}+E_{202,204}+E_{203,206}+E_{205,208}+E_{207,209}+E_{217,219}+E_{282,284}$
$\displaystyle+E_{283,286}+E_{285,288}+E_{287,289}+E_{297,299}+E_{382,384}+E_{383,386}+E_{385,388}+E_{387,389}+E_{397,399})$
$\displaystyle+$
$\displaystyle(q^{-1})(E_{62,64}+E_{63,66}+E_{65,68}+E_{67,69}+E_{77,79}+E_{142,144}+E_{143,146}+E_{145,148}+E_{147,149}$
$\displaystyle+E_{157,159}+E_{302,304}+E_{303,306}+E_{305,308}+E_{307,309}+E_{317,319}+E_{362,364}+E_{363,366}$
$\displaystyle+E_{365,368}+E_{367,369}+E_{377,379})$ $\displaystyle+$
$\displaystyle(q^{-2})(E_{182,184}+E_{183,186}+E_{185,188}+E_{187,189}+E_{197,199}+E_{262,264}+E_{263,266}+E_{265,268}$
$\displaystyle+E_{267,269}+E_{277,279}+E_{342,344}+E_{343,346}+E_{345,348}+E_{347,349}+E_{357,359})$
$\displaystyle+$
$\displaystyle(1+q^{-2})(E_{66,70}+E_{67,71}+E_{71,74}+E_{72,75}+E_{73,76}+E_{75,78}+E_{146,150}+E_{147,151}+E_{151,154}$
$\displaystyle+E_{152,155}+E_{153,156}+E_{155,158}+E_{306,310}+E_{307,311}+E_{311,314}+E_{312,315}+E_{313,316}$
$\displaystyle+E_{315,318}+E_{366,370}+E_{367,371}+E_{371,374}+E_{372,375}+E_{373,376}+E_{375,378})$
$\displaystyle+$
$\displaystyle(q^{-1}+q^{-3})(E_{186,190}+E_{187,191}+E_{191,194}+E_{192,195}+E_{193,196}+E_{195,198}+E_{266,270}+E_{267,271}$
$\displaystyle+E_{271,274}+E_{272,275}+E_{273,276}+E_{275,278}+E_{346,350}+E_{347,351}+E_{351,354}+E_{352,355}$
$\displaystyle+E_{353,356}+E_{355,358})$

$\displaystyle\pi_{W\otimes W}(E_{3})$
$\displaystyle=(-q^{3}-q)(E_{46,51}+E_{47,52}+E_{50,54}+E_{51,55}+E_{53,57}+E_{54,58}+E_{106,111}+E_{107,112}+E_{110,114}$
$\displaystyle+E_{111,115}+E_{113,117}+E_{114,118}+E_{186,191}+E_{187,192}+E_{190,194}+E_{191,195}+E_{193,197}+E_{194,198})$
$\displaystyle+$
$\displaystyle(-q^{2}-1)(E_{26,31}+E_{27,32}+E_{30,34}+E_{31,35}+E_{33,37}+E_{34,38}+E_{66,71}+E_{67,72}+E_{70,74}+E_{71,75}$
$\displaystyle+E_{73,77}+E_{74,78}+E_{246,251}+E_{247,252}+E_{250,254}+E_{251,255}+E_{253,257}+E_{254,258}+E_{306,311}$
$\displaystyle+E_{307,312}+E_{310,314}+E_{311,315}+E_{313,317}+E_{314,318})$
$\displaystyle+$
$\displaystyle(-q^{2})(E_{42,45}+E_{43,47}+E_{44,48}+E_{46,49}+E_{56,59}+E_{102,105}+E_{103,107}+E_{104,108}+E_{106,109}$
$\displaystyle+E_{116,119}+E_{182,185}+E_{183,187}+E_{184,188}+E_{186,189}+E_{196,199})$
$\displaystyle+$
$\displaystyle(-q)(E_{22,25}+E_{23,27}+E_{24,28}+E_{26,29}+E_{36,39}+E_{62,65}+E_{63,67}+E_{64,68}+E_{66,69}+E_{76,79}$
$\displaystyle+E_{242,245}+E_{243,247}+E_{244,248}+E_{246,249}+E_{256,259}+E_{302,305}+E_{303,307}+E_{304,308}$
$\displaystyle+E_{306,309}+E_{316,319})$ $\displaystyle+$
$\displaystyle(-q-q^{-1})\left(\sum_{i=101}^{140}E_{i,i+100}+\sum_{i=181}^{220}E_{i,i+80}+\sum_{i=241}^{280}E_{i,i+80}+E_{6,11}+E_{7,12}+E_{10,14}+E_{11,15}\right.$
$\displaystyle+E_{13,17}+E_{14,18}+E_{126,131}+E_{127,132}+E_{130,134}+E_{131,135}+E_{133,137}+E_{134,138}+E_{166,171}$
$\displaystyle+E_{167,172}+E_{170,174}+E_{171,175}+E_{173,177}+E_{174,178}+E_{206,211}+E_{207,212}+E_{210,214}+E_{211,215}$
$\displaystyle+E_{213,217}+E_{214,218}+E_{267,272}+E_{270,274}+E_{271,275}+E_{273,277}+E_{274,278}+E_{386,391}+E_{387,392}$
$\displaystyle+E_{390,394}+E_{391,395}+E_{393,397}+E_{394,398})$
$\displaystyle+$
$\displaystyle(-1)\left(\sum_{i=21}^{40}+\sum_{i=41}E_{i,i+80}+E_{i,i+60}^{80}+\sum_{i=101}^{120}E_{i,i+60}+\sum_{i=301}^{320}E_{i,i+60}+E_{2,5}+E_{3,7}+E_{4,8}+E_{6,9}\right.$
$\displaystyle++E_{16,19}+E_{122,125}+E_{123,127}+E_{124,128}+E_{126,129}+E_{136,139}+E_{162,165}+E_{163,167}$
$\displaystyle+E_{164,168}+E_{166,169}+E_{176,179}+E_{202,205}+E_{203,207}+E_{204,208}+E_{206,209}+E_{216,219}+E_{262,265}$
$\displaystyle+E_{263,267}+E_{264,268}+E_{266,269}+E_{276,279}+E_{382,385}+E_{383,387}+E_{384,388}+E_{386,389}+E_{396,399})$
$\displaystyle+$
$\displaystyle(-q^{-1})(E_{82,85}+E_{83,87}+E_{84,88}+E_{86,89}+E_{96,99}+E_{142,145}+E_{143,147}+E_{144,148}+E_{146,149}$
$\displaystyle+E_{156,159}+E_{322,325}+E_{323,327}+E_{324,328}+E_{326,329}+E_{336,339}+E_{362,365}+E_{363,367}$
$\displaystyle+E_{364,368}+E_{366,369}+E_{376,379})$ $\displaystyle+$
$\displaystyle(-q^{-2})(E_{222,225}+E_{223,227}+E_{224,228}+E_{226,229}+E_{236,239}+E_{282,285}+E_{283,287}+E_{284,288}$
$\displaystyle+E_{286,289}+E_{296,299}+E_{342,345}+E_{343,347}+E_{344,348}+E_{346,349}+E_{356,359})$
$\displaystyle+$
$\displaystyle(-1-q^{-2})(E_{86,91}+E_{87,92}+E_{90,94}+E_{91,95}+E_{93,97}+E_{94,98}+E_{146,151}+E_{147,152}+E_{150,154}$
$\displaystyle+E_{151,155}+E_{153,157}+E_{154,158}+E_{326,331}+E_{327,332}+E_{330,334}+E_{331,335}+E_{333,337}$
$\displaystyle+E_{334,338}+E_{366,371}+E_{367,372}+E_{370,374}+E_{371,375}+E_{373,377}+E_{374,378})$
$\displaystyle+$
$\displaystyle(-q^{-1}-q^{-3})(E_{226,231}+E_{227,232}+E_{230,234}+E_{231,235}+E_{233,237}+E_{234,238}+E_{286,291}+E_{287,292}$
$\displaystyle+E_{290,294}+E_{291,295}+E_{293,297}+E_{294,298}+E_{346,351}+E_{347,352}+E_{350,354}+E_{351,355}$
$\displaystyle+E_{353,357}+E_{354,358})$

$\displaystyle\pi_{W\otimes W}(F_{1})=$
$\displaystyle(q^{3}+q)(E_{243,163}+E_{253,173}+E_{260,180}+E_{363,343}+E_{373,353}+E_{380,360})$
$\displaystyle+$
$\displaystyle(q^{2}+1)(E_{246,166}+E_{247,167}+E_{256,176}+E_{257,177}+E_{366,346}+E_{367,347}+E_{376,356}+E_{377,357})$
$\displaystyle+$
$\displaystyle(q^{2})(E_{23,3}+E_{33,13}+E_{40,20}+E_{43,23}+E_{53,33}+E_{60,40}+E_{103,63}+E_{113,73}+E_{120,80}+E_{123,83}$
$\displaystyle+E_{133,93}+E_{140,100}+E_{163,143}+E_{173,153}+E_{180,160}+E_{243,203}+E_{253,213}+E_{260,220}+E_{303,263}$
$\displaystyle+E_{313,273}+E_{320,280}+E_{323,283}+E_{333,293}+E_{340,300}+E_{383,363}+E_{393,373}+E_{400,380})$
$\displaystyle+$
$\displaystyle(q)\left(\sum_{i=1}^{4}E_{10i+16,10i-4}+E_{10i+17,10i-3}\right.$
$\displaystyle\left.+E_{106,66}+E_{107,67}+E_{116,76}\vphantom{\sum_{i=1}^{4}}\right)$
$\displaystyle+$
$\displaystyle(q+q^{-1})\left(\sum_{i=1}^{20}E_{20i-7,20i-11}+E_{20i-1,20i-2}\right.$
$\displaystyle+\sum_{i=1}^{2}E_{120i+122,180i-18}+E_{120i+129,180i-11}+E_{120i+130,180i-10}+E_{120i+131,180i-9}$
$\displaystyle\hskip
50.00008pt\left.+E_{120i+132,180i-8}+E_{120i+139,180i-1}\vphantom{\sum_{i=1}^{20}}\right)$
$\displaystyle+$
$\displaystyle(1)\left(\sum_{i=1}^{20}E_{20i-18,20i-19}+E_{20i-17,20i-18}+E_{20i-11,20i-12}+E_{20i,20i-1}+E_{20i-14,20i-16}\right.$
$\displaystyle\hskip
50.00008pt+E_{20i-13,20i-15}+E_{20i-7,20i-9}+E_{20i-4,20i-6}+E_{20i-3,20i-5}$
$\displaystyle+\sum_{i\in\\{1,2,8,19\\}}E_{20i+2,20i-18}+E_{20i+9,20i-11}+E_{20i+10,20i-10}+E_{20i+11,20i-9}+E_{20i+12,20i-8}$
$\displaystyle\hskip 50.00008pt+E_{20i+19,20i-1}$
$\displaystyle+\sum_{i\in\\{1,2,8,11,12\\}}E_{20i+82,20i+42}+E_{20i+89,20i+49}+E_{20i+90,20i+50}+E_{20i+91,20i+51}+E_{20i+92,20i+52}$
$\displaystyle\hskip
50.00008pt\left.+E_{20i+99,20i+59}\vphantom{\sum_{i=1}^{20}}\right)$
$\displaystyle+$
$\displaystyle(q^{-1})\left(\sum_{i\in\\{1,2,3,4,15,16,37,38\\}}E_{10i+14,10i-6}+E_{10i+15,10i-5}\right.$
$\displaystyle\left.+\sum_{i\in\\{1,2,3,4,15,16,21,22,23,24\\}}E_{10i+94,10i+54}+E_{10i+95,10i+55}\right)$
$\displaystyle+$
$\displaystyle(q^{-2})(E_{21,1}+E_{28,8}+E_{38,18}+E_{41,21}+E_{48,28}+E_{58,38}+E_{101,61}+E_{108,68}+E_{118,78}+E_{121,81}$
$\displaystyle+E_{128,88}+E_{138,98}+E_{161,141}+E_{168,148}+E_{178,158}+E_{241,201}+E_{248,208}+E_{258,218}+E_{301,261}$
$\displaystyle+E_{308,268}+E_{318,278}+E_{321,281}+E_{328,288}+E_{338,298}+E_{381,361}+E_{388,368}+E_{398,378})$
$\displaystyle+$
$\displaystyle(1+q^{-2})(E_{244,164}+E_{245,165}+E_{254,174}+E_{255,175}+E_{364,344}+E_{365,345}+E_{374,354}+E_{375,355})$
$\displaystyle+$
$\displaystyle(q^{-1}+q^{-3})(E_{241,161}+E_{248,168}+E_{258,178}+E_{361,341}+E_{368,348}+E_{378,358})$

$\displaystyle\pi_{W\otimes W}(F_{2})=$
$\displaystyle(q^{3}+q)(E_{110,50}+E_{114,54}+E_{118,58})$ $\displaystyle+$
$\displaystyle(q^{2}+1)(E_{104,44}+E_{108,48}+E_{116,56}+E_{119,59})$
$\displaystyle+$
$\displaystyle(q^{2})\left(\sum_{i\in\\{1,16\\}}E_{20i+50,20i+10}+E_{20i+54,20i+14}+E_{20i+58,20i+18}\right.$
$\displaystyle+\sum_{i\in\\{4,10,11,14\\}}E_{20i+70,20i+10}+E_{20i+74,20i+14}+E_{20i+78,20i+18}$
$\displaystyle\left.+\sum_{i\in\\{5,6\\}}E_{20i+90,20i+10}+E_{20i+94,20i+14}+E_{20i+98,20i+18}\right)$
$\displaystyle+$
$\displaystyle(q)\left(\sum_{i\in\\{1,16\\}}E_{20i+44,20i+4}+E_{20i+48,20i+8}+E_{20i+56,20i+16}+E_{E_{20i+59,20i+19}}\right.$
$\displaystyle+\sum_{i\in\\{4,10,11,14\\}}E_{20i+64,20i+4}+E_{20i+68,20i+8}+E_{20i+76,20i+16}+E_{E_{20i+79,20i+19}}$
$\displaystyle\left.+\sum_{i\in\\{5,6\\}}E_{20i+84,20i+4}+E_{20i+88,20i+8}+E_{20i+96,20i+16}+E_{E_{20i+99,20i+19}}\right)$
$\displaystyle+$
$\displaystyle(q+q^{-1})\left(\sum_{i=1}^{20}E_{20i-14,20i-17}\vphantom{\sum_{j\in\\{1,6,9,11,15,20\\}}}\right.$
$\displaystyle\left.+\sum_{j\in\\{1,6,9,11,15,20\\}}E_{j+100,j+40}\right)$
$\displaystyle+$
$\displaystyle(1)\left(\sum_{i=1}^{20}E_{20i-16,20i-18}+E_{20i-12,20i-15}+E_{20i-10,20i-14}+E_{20i-9,20i-13}+E_{20i-6,20i-9}\right.$
$\displaystyle\hskip
50.00008pt+E_{20i-5,20i-8}+E_{20i-2,20i-5}+E_{20i-1,20i-3}$
$\displaystyle+\sum_{i\in\\{1,16\\}}E_{20i+41,20i+1}+E_{20i+46,20i+6}+E_{20i+49,20i+9}+E_{20i+51,20i+11}+E_{20i+55,20i+15}$
$\displaystyle\hskip 50.00008pt+E_{20i+60,20i+20}$
$\displaystyle+\sum_{i\in\\{4,10,11,14\\}}E_{20i+61,20i+1}+E_{20i+66,20i+6}+E_{20i+69,20i+9}+E_{20i+71,20i+11}+E_{20i+75,20i+15}$
$\displaystyle\hskip 50.00008pt+E_{20i+80,20i+20}$
$\displaystyle+\sum_{i\in\\{5,6\\}}E_{20i+81,20i+1}+E_{20i+86,20i+6}+E_{20i+89,20i+9}+E_{20i+91,20i+11}+E_{20i+95,20i+15}$
$\displaystyle\hskip
50.00008pt\left.+E_{20i+100,20i+20}\vphantom{\sum_{i=1}^{20}}\right)$
$\displaystyle+$
$\displaystyle(q^{-1})\left(\sum_{i\in\\{1,16\\}}E_{20i+42,20i+2}+E_{20i+45,20i+5}+E_{20i+53,20i+13}+E_{20i+57,20i+17}\right.$
$\displaystyle+\sum_{i\in\\{4,10,11,14\\}}E_{20i+62,20i+2}+E_{20i+65,20i+5}+E_{20i+73,20i+13}+E_{20i+77,20i+17}$
$\displaystyle\left.+\sum_{i\in\\{5,6\\}}E_{20i+82,20i+2}+E_{20i+85,20i+5}+E_{20i+93,20i+13}+E_{20i+97,20i+17}\right)$
$\displaystyle+$
$\displaystyle(q^{-2})\left(\sum_{i\in\\{1,16\\}}E_{20i+43,20i+3}+E_{20i+47,20i+7}+E_{20i+52,20i+12}\right.$
$\displaystyle+\sum_{i\in\\{4,10,11,14\\}}E_{20i+63,20i+3}+E_{20i+67,20i+7}+E_{20i+72,20i+12}$
$\displaystyle\left.+\sum_{i\in\\{5,6\\}}E_{20i+83,20i+3}+E_{20i+87,20i+7}+E_{20i+92,20i+12}\right)$
$\displaystyle+$
$\displaystyle(1+q^{-2})(E_{102,42}+E_{105,45}+E_{113,53}+E_{117,57})$
$\displaystyle+$
$\displaystyle(q^{-1}+q^{-3})(E_{103,43}+E_{107,47}+E_{112,52})$
$\displaystyle+$
$\displaystyle\left(\frac{q^{2}}{q+q^{-1}}\right)(E_{270,170}+E_{274,174}+E_{278,178}+E_{310,250}+E_{314,254}+E_{318,258})$
$\displaystyle+$
$\displaystyle\left(\frac{q}{q+q^{-1}}\right)(E_{264,164}+E_{268,168}+E_{276,176}+E_{279,179}+E_{304,244}+E_{308,248}+E_{316,256}+E_{319,259})$
$\displaystyle+$
$\displaystyle\left(\frac{1}{q+q^{-1}}\right)\left(\sum_{i=1}^{20}E_{20i-6,20i-11}+E_{20i-4,20i-7}\vphantom{\sum_{j\in\\{1,6,9,11,15,20\\}}}\right.$
$\displaystyle\left.+\sum_{j\in\\{1,6,9,11,15,20\\}}E_{j+300,j+240}+E_{j+260,j+160}\right)$
$\displaystyle+$
$\displaystyle\left(\frac{q^{-1}}{q+q^{-1}}\right)(E_{262,162}+E_{265,165}+E_{273,173}+E_{277,177}+E_{302,242}+E_{305,245}+E_{313,253}+E_{317,257})$
$\displaystyle+$
$\displaystyle\left(\frac{q^{-2}}{q+q^{-1}}\right)(E_{263,163}+E_{267,167}+E_{272,172}+E_{303,243}+E_{307,247}+E_{312,252})$

$\displaystyle\pi_{W\otimes W}(F_{3})=-$
$\displaystyle(q^{3}+q)(E_{132,52}+E_{135,55}+E_{138,58})$ $\displaystyle-$
$\displaystyle(q^{2}+1)(E_{125,45}+E_{128,48}+E_{137,57}+E_{139,59})$
$\displaystyle-$
$\displaystyle(q^{2})\left(\sum_{i\in\\{1,15\\}}E_{20i+72,20i+12}+E_{20i+75,20i+15}+E_{20i+78,20i+18}\right.$
$\displaystyle+\sum_{i\in\\{3,9,10,13\\}}E_{20i+92,20i+12}+E_{20i+95,20i+15}+E_{20i+98,20i+18}$
$\displaystyle\left.+\sum_{i\in\\{5,6\\}}E_{20i+112,20i+12}+E_{20i+115,20i+15}+E_{20i+118,20i+18}\right)$
$\displaystyle-$
$\displaystyle(q)\left(\sum_{i\in\\{1,15\\}}E_{20i+65,20i+5}+E_{20i+68,20i+8}+E_{20i+77,20i+17}+E_{20i+79,20i+19}\right.$
$\displaystyle+\sum_{i\in\\{3,9,10,13\\}}E_{20i+85,20i+5}+E_{20i+88,20i+8}+E_{20i+97,20i+17}+E_{20i+99,20i+19}$
$\displaystyle\left.+\sum_{i\in\\{5,6\\}}E_{20i+105,20i+5}+E_{20i+108,20i+8}+E_{20i+117,20i+17}+E_{20i+119,20i+19}\right)$
$\displaystyle-$
$\displaystyle(q+q^{-1})\left(\sum_{i=1}^{20}E_{20i-13,20i-17}\vphantom{\sum_{j\in\\{1,7,9,11,14,20\\}}}\right.$
$\displaystyle\left.+\sum_{j\in\\{1,7,9,11,14,20\\}}E_{j+120,j+40}\right)$
$\displaystyle-(1)\left(\sum_{i=1}^{20}E_{20i-15,20i-18}+E_{20i-12,20i-16}+E_{20i-9,20i-14}+E_{20i-8,20i-13}+E_{20i-6,20i-10}\right.$
$\displaystyle\hskip
50.00008pt+E_{20i-5,20i-9}+E_{20i-2,20i-6}+E_{20i-1,20i-4}$
$\displaystyle+\sum_{i\in\\{1,15\\}}E_{20i+61,20i+1}+E_{20i+67,20i+7}+E_{20i+69,20i+9}+E_{20i+71,20i+11}+E_{20i+74,20i+14}$
$\displaystyle\hskip 50.00008pt+E_{20i+80,20i+20}$
$\displaystyle+\sum_{i\in\\{3,9,10,13\\}}E_{20i+81,20i+1}+E_{20i+87,20i+7}+E_{20i+89,20i+9}+E_{20i+91,20i+11}+E_{20i+94,20i+14}$
$\displaystyle\hskip 50.00008pt+E_{20i+80,20i+20}$
$\displaystyle+\sum_{i\in\\{5,6\\}}E_{20i+101,20i+1}+E_{20i+107,20i+7}+E_{20i+109,20i+9}+E_{20i+111,20i+11}+E_{20i+114,20i+14}$
$\displaystyle\left.\hskip
50.00008pt+E_{20i+120,20i+20}\vphantom{\sum_{i\in\\{5,6\\}}}\right)$
$\displaystyle-$
$\displaystyle(q^{-1})\left(\sum_{i\in\\{1,15\\}}E_{20i+62,20i+2}+E_{20i+64,20i+4}+E_{20i+73,20i+13}+E_{20i+76,20i+16}\right.$
$\displaystyle+\sum_{i\in\\{3,9,10,13\\}}E_{20i+82,20i+2}+E_{20i+84,20i+4}+E_{20i+93,20i+13}+E_{20i+96,20i+16}$
$\displaystyle\left.+\sum_{i\in\\{5,6\\}}E_{20i+102,20i+2}+E_{20i+104,20i+4}+E_{20i+113,20i+13}+E_{20i+116,20i+16}\right)$
$\displaystyle-$
$\displaystyle(q^{-2})\left(\sum_{i\in\\{1,15\\}}E_{20i+63,20i+3}+E_{20i+66,20i+6}+E_{20i+70,20i+10}\right.$
$\displaystyle+\sum_{i\in\\{3,9,10,13\\}}E_{20i+83,20i+3}+E_{20i+86,20i+6}+E_{20i+90,20i+10}$
$\displaystyle\left.+\sum_{i\in\\{1,15\\}}E_{20i+103,20i+3}+E_{20i+106,20i+6}+E_{20i+110,20i+10}\right)$
$\displaystyle-$
$\displaystyle(1+q^{-2})(E_{122,42}+E_{124,44}+E_{133,53}+E_{136,56})$
$\displaystyle-$
$\displaystyle(q^{-1}+q^{-3})(E_{123,43}+E_{126,46}+E_{130,50})$
$\displaystyle-$
$\displaystyle\left(\frac{q^{2}}{q+q^{-1}}\right)(E_{292,172}+E_{295,175}+E_{298,178}+E_{332,252}+E_{335,255}+E_{338,258})$
$\displaystyle-$
$\displaystyle\left(\frac{q}{q+q^{-1}}\right)(E_{285,165}+E_{288,168}+E_{297,177}+E_{299,179}+E_{325,245}+E_{328,248}+E_{337,257}+E_{339,259})$
$\displaystyle-$
$\displaystyle\left(\frac{1}{q+q^{-1}}\right)\left(\sum_{i=1}^{20}E_{20i-5,20i-11}+E_{20i-3,20i-7}\vphantom{\sum_{j\in\\{1,7,9,11,14,20\\}}}\right.$
$\displaystyle\left.+\sum_{j\in\\{1,7,9,11,14,20\\}}E_{j+280,j+160}+E_{j+320,j+240}\right)$
$\displaystyle-$
$\displaystyle\left(\frac{q^{-1}}{q+q^{-1}}\right)(E_{282,162}+E_{284,164}+E_{293,173}+E_{296,176}+E_{322,242}+E_{324,244}+E_{333,253}+E_{336,256})$
$\displaystyle-$
$\displaystyle\left(\frac{q^{-2}}{q+q^{-1}}\right)(E_{283,163}+E_{286,166}+E_{290,170}+E_{323,243}+E_{326,246}+E_{330,250})$

$\displaystyle\pi_{W\otimes W}(q^{H_{1}})$
$\displaystyle=(q^{4})(E_{1,1}+E_{8,8}+E_{18,18}+E_{141,141}+E_{148,148}+E_{158,158}+E_{341,341}+E_{348,348}+E_{358,358})$
$\displaystyle+$
$\displaystyle(q^{3})(E_{4,4}+E_{5,5}+E_{14,14}+E_{15,15}+E_{61,61}+E_{68,68}+E_{78,78}+E_{81,81}+E_{88,88}+E_{98,98}+E_{144,144}$
$\displaystyle+E_{145,145}+E_{154,154}+E_{155,155}+E_{261,261}+E_{268,268}+E_{278,278}+E_{281,281}+E_{288,288}+E_{298,298}$
$\displaystyle+E_{344,344}+E_{345,345}+E_{354,354}+E_{355,355})$
$\displaystyle+$
$\displaystyle(q^{2})(E_{2,2}+E_{9,9}+E_{10,10}+E_{11,11}+E_{12,12}+E_{19,19}+E_{21,21}+E_{28,28}+E_{38,38}+E_{64,64}+E_{65,65}$
$\displaystyle+E_{74,74}+E_{75,75}+E_{84,84}+E_{85,85}+E_{94,94}+E_{95,95}+E_{142,142}+E_{149,149}+E_{150,150}+E_{151,151}$
$\displaystyle+E_{152,152}+E_{159,159}+E_{161,161}+E_{168,168}+E_{178,178}+E_{181,181}+E_{188,188}+E_{198,198}+E_{201,201}$
$\displaystyle+E_{208,208}+E_{218,218}+E_{221,221}+E_{228,228}+E_{238,238}+E_{264,264}+E_{265,265}+E_{274,274}+E_{275,275}$
$\displaystyle+E_{284,284}+E_{285,285}+E_{294,294}+E_{295,295}+E_{342,342}+E_{349,349}+E_{350,350}+E_{351,351}+E_{352,352}$
$\displaystyle+E_{359,359}+E_{361,361}+E_{368,368}+E_{378,378})$
$\displaystyle+$
$\displaystyle(q)(E_{6,6}+E_{7,7}+E_{16,16}+E_{17,17}+E_{24,24}+E_{25,25}+E_{34,34}+E_{35,35}+E_{62,62}+E_{69,69}+E_{70,70}$
$\displaystyle+E_{71,71}+E_{72,72}+E_{79,79}+E_{82,82}+E_{89,89}+E_{90,90}+E_{91,91}+E_{92,92}+E_{99,99}+E_{101,101}$
$\displaystyle+E_{108,108}+E_{118,118}+E_{121,121}+E_{128,128}+E_{138,138}+E_{146,146}+E_{147,147}+E_{156,156}+E_{157,157}$
$\displaystyle+E_{164,164}+E_{165,165}+E_{174,174}+E_{175,175}+E_{184,184}+E_{185,185}+E_{194,194}+E_{195,195}+E_{204,204}$
$\displaystyle+E_{205,205}+E_{214,214}+E_{215,215}+E_{224,224}+E_{225,225}+E_{234,234}+E_{235,235}+E_{262,262}+E_{269,269}$
$\displaystyle+E_{270,270}+E_{271,271}+E_{272,272}+E_{279,279}+E_{282,282}+E_{289,289}+E_{290,290}+E_{291,291}+E_{292,292}$
$\displaystyle+E_{299,299}+E_{301,301}+E_{308,308}+E_{318,318}+E_{321,321}+E_{328,328}+E_{338,338}+E_{346,346}+E_{347,347}$
$\displaystyle+E_{356,356}+E_{357,357}+E_{364,364}+E_{365,365}+E_{374,374}+E_{375,375})$
$\displaystyle+$
$\displaystyle(1)(E_{3,3}+E_{13,13}+E_{20,20}+E_{22,22}+E_{29,29}+E_{30,30}+E_{31,31}+E_{32,32}+E_{39,39}+E_{41,41}+E_{48,48}$
$\displaystyle+E_{58,58}+E_{66,66}+E_{67,67}+E_{76,76}+E_{77,77}+E_{86,86}+E_{87,87}+E_{96,96}+E_{97,97}+E_{104,104}$
$\displaystyle+E_{105,105}+E_{114,114}+E_{115,115}+E_{124,124}+E_{125,125}+E_{134,134}+E_{135,135}+E_{143,143}+E_{153,153}$
$\displaystyle+E_{160,160}+E_{162,162}+E_{169,169}+E_{170,170}+E_{171,171}+E_{172,172}+E_{179,179}+E_{182,182}+E_{189,189}$
$\displaystyle+E_{190,190}+E_{191,191}+E_{192,192}+E_{199,199}+E_{202,202}+E_{209,209}+E_{210,210}+E_{211,211}+E_{212,212}$
$\displaystyle+E_{219,219}+E_{222,222}+E_{229,229}+E_{230,230}+E_{231,231}+E_{232,232}+E_{239,239}+E_{241,241}+E_{248,248}$
$\displaystyle+E_{258,258}+E_{266,266}+E_{267,267}+E_{276,276}+E_{277,277}+E_{286,286}+E_{287,287}+E_{296,296}+E_{297,297}$
$\displaystyle+E_{304,304}+E_{305,305}+E_{314,314}+E_{315,315}+E_{324,324}+E_{325,325}+E_{334,334}+E_{335,335}+E_{343,343}$
$\displaystyle+E_{353,353}+E_{360,360}+E_{362,362}+E_{369,369}+E_{370,370}+E_{371,371}+E_{372,372}+E_{379,379}+E_{381,381}$
$\displaystyle+E_{388,388}+E_{398,398})$ $\displaystyle+$
$\displaystyle(q^{-1})(E_{26,26}+E_{27,27}+E_{36,36}+E_{37,37}+E_{44,44}+E_{45,45}+E_{54,54}+E_{55,55}+E_{63,63}+E_{73,73}$
$\displaystyle+E_{80,80}+E_{83,83}+E_{93,93}+E_{100,100}+E_{102,102}+E_{109,109}+E_{110,110}+E_{111,111}+E_{112,112}$
$\displaystyle+E_{119,119}+E_{122,122}+E_{129,129}+E_{130,130}+E_{131,131}+E_{132,132}+E_{139,139}+E_{166,166}+E_{167,167}$
$\displaystyle+E_{176,176}+E_{177,177}+E_{186,186}+E_{187,187}+E_{196,196}+E_{197,197}+E_{206,206}+E_{207,207}+E_{216,216}$
$\displaystyle+E_{217,217}+E_{226,226}+E_{227,227}+E_{236,236}+E_{237,237}+E_{244,244}+E_{245,245}+E_{254,254}+E_{255,255}$
$\displaystyle+E_{263,263}+E_{273,273}+E_{280,280}+E_{283,283}+E_{293,293}+E_{300,300}+E_{302,302}+E_{309,309}+E_{310,310}$
$\displaystyle+E_{311,311}+E_{312,312}+E_{319,319}+E_{322,322}+E_{329,329}+E_{330,330}+E_{331,331}+E_{332,332}+E_{339,339}$
$\displaystyle+E_{366,366}+E_{367,367}+E_{376,376}+E_{377,377}+E_{384,384}+E_{385,385}+E_{394,394}+E_{395,395})$
$\displaystyle+$
$\displaystyle(q^{-2})(E_{23,23}+E_{33,33}+E_{40,40}+E_{42,42}+E_{49,49}+E_{50,50}+E_{51,51}+E_{52,52}+E_{59,59}+E_{106,106}$
$\displaystyle+E_{107,107}+E_{116,116}+E_{117,117}+E_{126,126}+E_{127,127}+E_{136,136}+E_{137,137}+E_{163,163}+E_{173,173}$
$\displaystyle+E_{180,180}+E_{183,183}+E_{193,193}+E_{200,200}+E_{203,203}+E_{213,213}+E_{220,220}+E_{223,223}+E_{233,233}$
$\displaystyle+E_{240,240}+E_{242,242}+E_{249,249}+E_{250,250}+E_{251,251}+E_{252,252}+E_{259,259}+E_{306,306}+E_{307,307}$
$\displaystyle+E_{316,316}+E_{317,317}+E_{326,326}+E_{327,327}+E_{336,336}+E_{337,337}+E_{363,363}+E_{373,373}+E_{380,380}$
$\displaystyle+E_{382,382}+E_{389,389}+E_{390,390}+E_{391,391}+E_{392,392}+E_{399,399})$
$\displaystyle+$
$\displaystyle(q^{-3})(E_{46,46}+E_{47,47}+E_{56,56}+E_{57,57}+E_{103,103}+E_{113,113}+E_{120,120}+E_{123,123}+E_{133,133}$
$\displaystyle+E_{140,140}+E_{246,246}+E_{247,247}+E_{256,256}+E_{257,257}+E_{303,303}+E_{313,313}+E_{320,320}$
$\displaystyle+E_{323,323}+E_{333,333}+E_{340,340}+E_{386,386}+E_{387,387}+E_{396,396}+E_{397,397})$
$\displaystyle+$
$\displaystyle(q^{-4})(E_{43,43}+E_{53,53}+E_{60,60}+E_{243,243}+E_{253,253}+E_{260,260}+E_{383,383}+E_{393,393}+E_{400,400})$

$\displaystyle\pi_{W\otimes W}(q^{H_{2}})$
$\displaystyle=(q^{4})(E_{43,43}+E_{47,47}+E_{52,52}+E_{123,123}+E_{127,127}+E_{132,132}+E_{223,223}+E_{227,227}+E_{232,232})$
$\displaystyle+$
$\displaystyle(q^{3})(E_{23,23}+E_{27,27}+E_{32,32}+E_{42,42}+E_{45,45}+E_{53,53}+E_{57,57}+E_{83,83}+E_{87,87}+E_{92,92}+E_{122,122}$
$\displaystyle+E_{125,125}+E_{133,133}+E_{137,137}+E_{222,222}+E_{225,225}+E_{233,233}+E_{237,237}+E_{243,243}+E_{247,247}$
$\displaystyle+E_{252,252}+E_{323,323}+E_{327,327}+E_{332,332})$
$\displaystyle+$
$\displaystyle(q^{2})(E_{3,3}+E_{7,7}+E_{12,12}+E_{22,22}+E_{25,25}+E_{33,33}+E_{37,37}+E_{41,41}+E_{46,46}+E_{49,49}+E_{51,51}$
$\displaystyle+E_{55,55}+E_{60,60}+E_{82,82}+E_{85,85}+E_{93,93}+E_{97,97}+E_{103,103}+E_{107,107}+E_{112,112}+E_{121,121}$
$\displaystyle+E_{126,126}+E_{129,129}+E_{131,131}+E_{135,135}+E_{140,140}+E_{163,163}+E_{167,167}+E_{172,172}+E_{203,203}$
$\displaystyle+E_{207,207}+E_{212,212}+E_{221,221}+E_{226,226}+E_{229,229}+E_{231,231}+E_{235,235}+E_{240,240}+E_{242,242}$
$\displaystyle+E_{245,245}+E_{253,253}+E_{257,257}+E_{283,283}+E_{287,287}+E_{292,292}+E_{322,322}+E_{325,325}+E_{333,333}$
$\displaystyle+E_{337,337}+E_{383,383}+E_{387,387}+E_{392,392})$
$\displaystyle+$
$\displaystyle(q)(E_{2,2}+E_{5,5}+E_{13,13}+E_{17,17}+E_{21,21}+E_{26,26}+E_{29,29}+E_{31,31}+E_{35,35}+E_{40,40}+E_{44,44}$
$\displaystyle+E_{48,48}+E_{56,56}+E_{59,59}+E_{63,63}+E_{67,67}+E_{72,72}+E_{81,81}+E_{86,86}+E_{89,89}+E_{91,91}$
$\displaystyle+E_{95,95}+E_{100,100}+E_{102,102}+E_{105,105}+E_{113,113}+E_{117,117}+E_{124,124}+E_{128,128}+E_{136,136}$
$\displaystyle+E_{139,139}+E_{143,143}+E_{147,147}+E_{152,152}+E_{162,162}+E_{165,165}+E_{173,173}+E_{177,177}+E_{202,202}$
$\displaystyle+E_{205,205}+E_{213,213}+E_{217,217}+E_{224,224}+E_{228,228}+E_{236,236}+E_{239,239}+E_{241,241}+E_{246,246}$
$\displaystyle+E_{249,249}+E_{251,251}+E_{255,255}+E_{260,260}+E_{282,282}+E_{285,285}+E_{293,293}+E_{297,297}+E_{303,303}$
$\displaystyle+E_{307,307}+E_{312,312}+E_{321,321}+E_{326,326}+E_{329,329}+E_{331,331}+E_{335,335}+E_{340,340}+E_{363,363}$
$\displaystyle+E_{367,367}+E_{372,372}+E_{382,382}+E_{385,385}+E_{393,393}+E_{397,397})$
$\displaystyle+$
$\displaystyle(1)(E_{1,1}+E_{6,6}+E_{9,9}+E_{11,11}+E_{15,15}+E_{20,20}+E_{24,24}+E_{28,28}+E_{36,36}+E_{39,39}+E_{50,50}$
$\displaystyle+E_{54,54}+E_{58,58}+E_{62,62}+E_{65,65}+E_{73,73}+E_{77,77}+E_{84,84}+E_{88,88}+E_{96,96}+E_{99,99}$
$\displaystyle+E_{101,101}+E_{106,106}+E_{109,109}+E_{111,111}+E_{115,115}+E_{120,120}+E_{130,130}+E_{134,134}+E_{138,138}$
$\displaystyle+E_{142,142}+E_{145,145}+E_{153,153}+E_{157,157}+E_{161,161}+E_{166,166}+E_{169,169}+E_{171,171}+E_{175,175}$
$\displaystyle+E_{180,180}+E_{183,183}+E_{187,187}+E_{192,192}+E_{201,201}+E_{206,206}+E_{209,209}+E_{211,211}+E_{215,215}$
$\displaystyle+E_{220,220}+E_{230,230}+E_{234,234}+E_{238,238}+E_{244,244}+E_{248,248}+E_{256,256}+E_{259,259}+E_{263,263}$
$\displaystyle+E_{267,267}+E_{272,272}+E_{281,281}+E_{286,286}+E_{289,289}+E_{291,291}+E_{295,295}+E_{300,300}+E_{302,302}$
$\displaystyle+E_{305,305}+E_{313,313}+E_{317,317}+E_{324,324}+E_{328,328}+E_{336,336}+E_{339,339}+E_{343,343}+E_{347,347}$
$\displaystyle+E_{352,352}+E_{362,362}+E_{365,365}+E_{373,373}+E_{377,377}+E_{381,381}+E_{386,386}+E_{389,389}+E_{391,391}$
$\displaystyle+E_{395,395}+E_{400,400})$ $\displaystyle+$
$\displaystyle(q^{-1})(E_{4,4}+E_{8,8}+E_{16,16}+E_{19,19}+E_{30,30}+E_{34,34}+E_{38,38}+E_{61,61}+E_{66,66}+E_{69,69}+E_{71,71}$
$\displaystyle+E_{75,75}+E_{80,80}+E_{90,90}+E_{94,94}+E_{98,98}+E_{104,104}+E_{108,108}+E_{116,116}+E_{119,119}+E_{141,141}$
$\displaystyle+E_{146,146}+E_{149,149}+E_{151,151}+E_{155,155}+E_{160,160}+E_{164,164}+E_{168,168}+E_{176,176}+E_{179,179}$
$\displaystyle+E_{182,182}+E_{185,185}+E_{193,193}+E_{197,197}+E_{204,204}+E_{208,208}+E_{216,216}+E_{219,219}+E_{250,250}$
$\displaystyle+E_{254,254}+E_{258,258}+E_{262,262}+E_{265,265}+E_{273,273}+E_{277,277}+E_{284,284}+E_{288,288}+E_{296,296}$
$\displaystyle+E_{299,299}+E_{301,301}+E_{306,306}+E_{309,309}+E_{311,311}+E_{315,315}+E_{320,320}+E_{330,330}+E_{334,334}$
$\displaystyle+E_{338,338}+E_{342,342}+E_{345,345}+E_{353,353}+E_{357,357}+E_{361,361}+E_{366,366}+E_{369,369}+E_{371,371}$
$\displaystyle+E_{375,375}+E_{380,380}+E_{384,384}+E_{388,388}+E_{396,396}+E_{399,399})$
$\displaystyle+$
$\displaystyle(q^{-2})(E_{10,10}+E_{14,14}+E_{18,18}+E_{64,64}+E_{68,68}+E_{76,76}+E_{79,79}+E_{110,110}+E_{114,114}+E_{118,118}$
$\displaystyle+E_{144,144}+E_{148,148}+E_{156,156}+E_{159,159}+E_{170,170}+E_{174,174}+E_{178,178}+E_{181,181}+E_{186,186}$
$\displaystyle+E_{189,189}+E_{191,191}+E_{195,195}+E_{200,200}+E_{210,210}+E_{214,214}+E_{218,218}+E_{261,261}+E_{266,266}$
$\displaystyle+E_{269,269}+E_{271,271}+E_{275,275}+E_{280,280}+E_{290,290}+E_{294,294}+E_{298,298}+E_{304,304}+E_{308,308}$
$\displaystyle+E_{316,316}+E_{319,319}+E_{341,341}+E_{346,346}+E_{349,349}+E_{351,351}+E_{355,355}+E_{360,360}+E_{364,364}$
$\displaystyle+E_{368,368}+E_{376,376}+E_{379,379}+E_{390,390}+E_{394,394}+E_{398,398})$
$\displaystyle+$
$\displaystyle(q^{-3})(E_{70,70}+E_{74,74}+E_{78,78}+E_{150,150}+E_{154,154}+E_{158,158}+E_{184,184}+E_{188,188}+E_{196,196}$
$\displaystyle+E_{199,199}+E_{264,264}+E_{268,268}+E_{276,276}+E_{279,279}+E_{310,310}+E_{314,314}+E_{318,318}+E_{344,344}$
$\displaystyle+E_{348,348}+E_{356,356}+E_{359,359}+E_{370,370}+E_{374,374}+E_{378,378})$
$\displaystyle+$
$\displaystyle(q^{-4})(E_{190,190}+E_{194,194}+E_{198,198}+E_{270,270}+E_{274,274}+E_{278,278}+E_{350,350}+E_{354,354}+E_{358,358})$

$\displaystyle\pi_{W\otimes W}(q^{H_{3}})$
$\displaystyle=(q^{4})(E_{43,43}+E_{46,46}+E_{50,50}+E_{103,103}+E_{106,106}+E_{110,110}+E_{183,183}+E_{186,186}+E_{190,190})$
$\displaystyle+$
$\displaystyle(q^{3})(E_{23,23}+E_{26,26}+E_{30,30}+E_{42,42}+E_{44,44}+E_{53,53}+E_{56,56}+E_{63,63}+E_{66,66}+E_{70,70}+E_{102,102}$
$\displaystyle+E_{104,104}+E_{113,113}+E_{116,116}+E_{182,182}+E_{184,184}+E_{193,193}+E_{196,196}+E_{243,243}+E_{246,246}$
$\displaystyle+E_{250,250}+E_{303,303}+E_{306,306}+E_{310,310})$
$\displaystyle+$
$\displaystyle(q^{2})(E_{3,3}+E_{6,6}+E_{10,10}+E_{22,22}+E_{24,24}+E_{33,33}+E_{36,36}+E_{41,41}+E_{47,47}+E_{49,49}+E_{51,51}$
$\displaystyle+E_{54,54}+E_{60,60}+E_{62,62}+E_{64,64}+E_{73,73}+E_{76,76}+E_{101,101}+E_{107,107}+E_{109,109}+E_{111,111}$
$\displaystyle+E_{114,114}+E_{120,120}+E_{123,123}+E_{126,126}+E_{130,130}+E_{163,163}+E_{166,166}+E_{170,170}+E_{181,181}$
$\displaystyle+E_{187,187}+E_{189,189}+E_{191,191}+E_{194,194}+E_{200,200}+E_{203,203}+E_{206,206}+E_{210,210}+E_{242,242}$
$\displaystyle+E_{244,244}+E_{253,253}+E_{256,256}+E_{263,263}+E_{266,266}+E_{270,270}+E_{302,302}+E_{304,304}+E_{313,313}$
$\displaystyle+E_{316,316}+E_{383,383}+E_{386,386}+E_{390,390})$
$\displaystyle+$
$\displaystyle(q)(E_{2,2}+E_{4,4}+E_{13,13}+E_{16,16}+E_{21,21}+E_{27,27}+E_{29,29}+E_{31,31}+E_{34,34}+E_{40,40}+E_{45,45}$
$\displaystyle+E_{48,48}+E_{57,57}+E_{59,59}+E_{61,61}+E_{67,67}+E_{69,69}+E_{71,71}+E_{74,74}+E_{80,80}+E_{83,83}$
$\displaystyle+E_{86,86}+E_{90,90}+E_{105,105}+E_{108,108}+E_{117,117}+E_{119,119}+E_{122,122}+E_{124,124}+E_{133,133}$
$\displaystyle+E_{136,136}+E_{143,143}+E_{146,146}+E_{150,150}+E_{162,162}+E_{164,164}+E_{173,173}+E_{176,176}+E_{185,185}$
$\displaystyle+E_{188,188}+E_{197,197}+E_{199,199}+E_{202,202}+E_{204,204}+E_{213,213}+E_{216,216}+E_{241,241}+E_{247,247}$
$\displaystyle+E_{249,249}+E_{251,251}+E_{254,254}+E_{260,260}+E_{262,262}+E_{264,264}+E_{273,273}+E_{276,276}+E_{301,301}$
$\displaystyle+E_{307,307}+E_{309,309}+E_{311,311}+E_{314,314}+E_{320,320}+E_{323,323}+E_{326,326}+E_{330,330}+E_{363,363}$
$\displaystyle+E_{366,366}+E_{370,370}+E_{382,382}+E_{384,384}+E_{393,393}+E_{396,396})$
$\displaystyle+$
$\displaystyle(1)(E_{1,1}+E_{7,7}+E_{9,9}+E_{11,11}+E_{14,14}+E_{20,20}+E_{25,25}+E_{28,28}+E_{37,37}+E_{39,39}+E_{52,52}$
$\displaystyle+E_{55,55}+E_{58,58}+E_{65,65}+E_{68,68}+E_{77,77}+E_{79,79}+E_{82,82}+E_{84,84}+E_{93,93}+E_{96,96}$
$\displaystyle+E_{112,112}+E_{115,115}+E_{118,118}+E_{121,121}+E_{127,127}+E_{129,129}+E_{131,131}+E_{134,134}+E_{140,140}$
$\displaystyle+E_{142,142}+E_{144,144}+E_{153,153}+E_{156,156}+E_{161,161}+E_{167,167}+E_{169,169}+E_{171,171}+E_{174,174}$
$\displaystyle+E_{180,180}+E_{192,192}+E_{195,195}+E_{198,198}+E_{201,201}+E_{207,207}+E_{209,209}+E_{211,211}+E_{214,214}$
$\displaystyle+E_{220,220}+E_{223,223}+E_{226,226}+E_{230,230}+E_{245,245}+E_{248,248}+E_{257,257}+E_{259,259}+E_{261,261}$
$\displaystyle+E_{267,267}+E_{269,269}+E_{271,271}+E_{274,274}+E_{280,280}+E_{283,283}+E_{286,286}+E_{290,290}+E_{305,305}$
$\displaystyle+E_{308,308}+E_{317,317}+E_{319,319}+E_{322,322}+E_{324,324}+E_{333,333}+E_{336,336}+E_{343,343}+E_{346,346}$
$\displaystyle+E_{350,350}+E_{362,362}+E_{364,364}+E_{373,373}+E_{376,376}+E_{381,381}+E_{387,387}+E_{389,389}+E_{391,391}$
$\displaystyle+E_{394,394}+E_{400,400})$ $\displaystyle+$
$\displaystyle(q^{-1})(E_{5,5}+E_{8,8}+E_{17,17}+E_{19,19}+E_{32,32}+E_{35,35}+E_{38,38}+E_{72,72}+E_{75,75}+E_{78,78}+E_{81,81}$
$\displaystyle+E_{87,87}+E_{89,89}+E_{91,91}+E_{94,94}+E_{100,100}+E_{125,125}+E_{128,128}+E_{137,137}+E_{139,139}+E_{141,141}$
$\displaystyle+E_{147,147}+E_{149,149}+E_{151,151}+E_{154,154}+E_{160,160}+E_{165,165}+E_{168,168}+E_{177,177}+E_{179,179}$
$\displaystyle+E_{205,205}+E_{208,208}+E_{217,217}+E_{219,219}+E_{222,222}+E_{224,224}+E_{233,233}+E_{236,236}+E_{252,252}$
$\displaystyle+E_{255,255}+E_{258,258}+E_{265,265}+E_{268,268}+E_{277,277}+E_{279,279}+E_{282,282}+E_{284,284}+E_{293,293}$
$\displaystyle+E_{296,296}+E_{312,312}+E_{315,315}+E_{318,318}+E_{321,321}+E_{327,327}+E_{329,329}+E_{331,331}+E_{334,334}$
$\displaystyle+E_{340,340}+E_{342,342}+E_{344,344}+E_{353,353}+E_{356,356}+E_{361,361}+E_{367,367}+E_{369,369}+E_{371,371}$
$\displaystyle+E_{374,374}+E_{380,380}+E_{385,385}+E_{388,388}+E_{397,397}+E_{399,399})$
$\displaystyle+$
$\displaystyle(q^{-2})(E_{12,12}+E_{15,15}+E_{18,18}+E_{85,85}+E_{88,88}+E_{97,97}+E_{99,99}+E_{132,132}+E_{135,135}+E_{138,138}$
$\displaystyle+E_{145,145}+E_{148,148}+E_{157,157}+E_{159,159}+E_{172,172}+E_{175,175}+E_{178,178}+E_{212,212}+E_{215,215}$
$\displaystyle+E_{218,218}+E_{221,221}+E_{227,227}+E_{229,229}+E_{231,231}+E_{234,234}+E_{240,240}+E_{272,272}+E_{275,275}$
$\displaystyle+E_{278,278}+E_{281,281}+E_{287,287}+E_{289,289}+E_{291,291}+E_{294,294}+E_{300,300}+E_{325,325}+E_{328,328}$
$\displaystyle+E_{337,337}+E_{339,339}+E_{341,341}+E_{347,347}+E_{349,349}+E_{351,351}+E_{354,354}+E_{360,360}+E_{365,365}$
$\displaystyle+E_{368,368}+E_{377,377}+E_{379,379}+E_{392,392}+E_{395,395}+E_{398,398})$
$\displaystyle+$
$\displaystyle(q^{-3})(E_{92,92}+E_{95,95}+E_{98,98}+E_{152,152}+E_{155,155}+E_{158,158}+E_{225,225}+E_{228,228}+E_{237,237}$
$\displaystyle+E_{239,239}+E_{285,285}+E_{288,288}+E_{297,297}+E_{299,299}+E_{332,332}+E_{335,335}+E_{338,338}+E_{345,345}$
$\displaystyle+E_{348,348}+E_{357,357}+E_{359,359}+E_{372,372}+E_{375,375}+E_{378,378})$
$\displaystyle+$
$\displaystyle(q^{-4})(E_{232,232}+E_{235,235}+E_{238,238}+E_{292,292}+E_{295,295}+E_{298,298}+E_{352,352}+E_{355,355}+E_{358,358})$

###  7.11 Listing $4\times 4$ Blocks of $\pi_{W\otimes W}(C)$

The twenty-four $4\times 4$ blocks of $\pi_{W\otimes W}(C)$ take the following
forms:

Ten blocks take the form

$\displaystyle\begin{bmatrix}B_{1}&B_{2}&B_{3}&0\\\ B_{4}&B_{5}&B_{3}&B_{6}\\\
B_{7}&B_{3}&B_{8}&B_{9}\\\ 0&B_{10}&B_{11}&B_{12}\end{bmatrix}$

Four blocks take the form

$\displaystyle\begin{bmatrix}B_{1}&B_{7}&B_{6}&0\\\
B_{13}&B_{5}&B_{3}&B_{3}\\\ B_{2}&B_{3}&B_{8}&B_{10}\\\
0&B_{9}&B_{14}&B_{12}\end{bmatrix}$

Four blocks take the form

$\displaystyle\begin{bmatrix}B_{15}&B_{7}&B_{7}&B_{3}\\\
B_{3}&B_{16}&0&B_{10}\\\ B_{3}&0&B_{16}&B_{10}\\\
B_{3}&B_{6}&B_{6}&B_{17}\end{bmatrix}$

Four blocks take the form

$\displaystyle\begin{bmatrix}B_{15}&B_{2}&B_{2}&B_{3}\\\
B_{6}&B_{16}&0&B_{9}\\\ B_{6}&0&B_{16}&B_{9}\\\
B_{3}&B_{3}&B_{3}&B_{17}\end{bmatrix}$

And two blocks take the form

$\displaystyle\begin{bmatrix}B_{1}&B_{18}&B_{19}&0\\\
B_{20}&B_{5}&B_{3}&B_{21}\\\ B_{22}&B_{3}&B_{8}&B_{23}\\\
0&B_{24}&B_{25}&B_{12}\end{bmatrix}$

Where

$\displaystyle B_{1}$
$\displaystyle=q^{12}-q^{10}+2q^{6}+2+q^{-2}-q^{-4}+2q^{-8}$ $\displaystyle
B_{2}$ $\displaystyle=q^{10}-2q^{8}+q^{6}+q^{-4}-2q^{-6}+q^{-8}$
$\displaystyle B_{3}$ $\displaystyle=q^{9}-2q^{7}+q^{5}+q^{-5}-2q^{-7}+q^{-9}$
$\displaystyle B_{4}$
$\displaystyle=q^{12}-q^{10}-q^{8}+q^{6}+q^{-2}-q^{-4}-q^{-6}+q^{-8}$
$\displaystyle B_{5}$
$\displaystyle=2q^{10}-2q^{8}+q^{6}+q^{4}+2+2q^{-4}-2q^{-6}+q^{-8}+q^{-10}$
$\displaystyle B_{6}$
$\displaystyle=q^{10}-q^{8}-q^{6}+q^{4}+q^{-4}-q^{-6}-q^{-8}+q^{-10}$
$\displaystyle B_{7}$
$\displaystyle=q^{11}-q^{9}-q^{7}+q^{5}+q^{-3}-q^{-5}-q^{-7}+q^{-9}$
$\displaystyle B_{8}$
$\displaystyle=q^{10}+q^{8}-2q^{6}+2q^{4}+2+q^{-4}+q^{-6}-2q^{-8}+2q^{-10}$
$\displaystyle B_{9}$
$\displaystyle=q^{9}-q^{7}-q^{5}+q^{3}+q^{-5}-q^{-7}-q^{-9}+q^{-11}$
$\displaystyle B_{10}$
$\displaystyle=q^{8}-2q^{6}+q^{4}+q^{-6}-2q^{-8}+q^{-10}$ $\displaystyle
B_{11}$ $\displaystyle=q^{7}-2q^{5}+q^{3}+q^{-7}-2q^{-9}+q^{-11}$
$\displaystyle B_{12}$
$\displaystyle=2q^{8}-q^{4}+q^{2}+2+2q^{-6}-q^{-10}+q^{-12}$ $\displaystyle
B_{13}$ $\displaystyle=q^{11}-2q^{9}+q^{7}+q^{-3}-2q^{-5}+q^{-7}$
$\displaystyle B_{14}$
$\displaystyle=q^{8}-q^{6}-q^{4}+q^{2}+q^{-6}-q^{-8}-q^{-10}+q^{-12}$
$\displaystyle B_{15}$
$\displaystyle=q^{12}-2q^{8}+3q^{6}+2+q^{-2}-2q^{-6}+3q^{-8}$ $\displaystyle
B_{16}$ $\displaystyle=q^{10}+q^{4}+2+q^{-4}+q^{-10}$ $\displaystyle B_{17}$
$\displaystyle=3q^{8}-2q^{6}+q^{2}+2+3q^{-6}-2q^{-8}+q^{-12}$ $\displaystyle
B_{18}$
$\displaystyle=q^{9}-3q^{7}+4q^{5}-4q^{3}+4q-4q^{-4/q^{-}}-3q^{-5}+q^{-7}$
$\displaystyle B_{19}$
$\displaystyle=q^{8}-3q^{6}+4q^{4}-4q^{2}+4-4q^{-2}+4q^{-4}-3q^{-6}+q^{-8}$
$\displaystyle B_{20}$ $\displaystyle=q^{13}-2q^{9}+q^{5}+q^{-2/q^{-}}+q^{-9}$
$\displaystyle B_{21}$
$\displaystyle=q^{11}-2q^{7}+q^{3}+q^{-3}-2q^{-7}+q^{-11}$ $\displaystyle
B_{22}$ $\displaystyle=q^{12}-2q^{8}+q^{4}+q^{-2}-2q^{-6}+q^{-10}$
$\displaystyle B_{23}$
$\displaystyle=q^{10}-2q^{6}+q^{2}+q^{-4}-2q^{-8}+q^{-12}$ $\displaystyle
B_{24}$
$\displaystyle=q^{7}-3q^{5}+4q^{3}-4q+4q^{-4/q^{-}}+4q^{-5}-3q^{-7}+q^{-9}$
$\displaystyle B_{25}$
$\displaystyle=q^{6}-3q^{4}+4q^{2}-4+4q^{-2}-4q^{-4}+4q^{-6}-3q^{-8}+q^{-10}$

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