A Ribbon Hopf Algebra Approach to the Irreducible Representations of Centralizer Algebras: The Brauer, Birman-Wenzl, and Type A Iwahori-Hecke Algebras

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

Last update: 13 March 2014

Appendix

Weight Polynomials

$$\begin{array}{c}{Q}_{\lambda }(z,q)=\prod _{(i,i)\in \lambda }\frac{[y+{\lambda }_i-{\lambda }_i^{\prime }]+\left[h_{\lambda }(i,i)\right]}{\left[h_{\lambda }(i,i)\right]}\prod _{\underset{i\ne j}{(i,j)\in \lambda }}\frac{[y+d_{\lambda }(i,j)]}{\left[h_{\lambda }(i,j)\right]}\\ P_{\lambda }\left(x\right)=\prod _{(i,j)\in \lambda }\frac{x-1+d_{\lambda }(i,j)}{h_{\lambda }(i,j)}\\ \begin{array}{ccc}\lambda & Q_{\lambda }(z,q)& P_{\lambda }\left(x\right)\\ \varnothing & 1& 1\\ \left(1\right)& \frac{[y+0]+\left[1\right]}{\left[1\right]}& x\\ \left(1^2\right)& \frac{[y-1]+\left[2\right]}{\left[2\right]}\frac{[y+0]}{\left[1\right]}& \frac{x(x-1)}{2}\\ \left(2\right)& \frac{[y+1]+\left[2\right]}{\left[2\right]}\frac{[y+0]}{\left[1\right]}& \frac{(x+2)(x-1)}{2}\\ \left(1^3\right)& \frac{[y-2]+\left[3\right]}{\left[3\right]}\frac{[y-1]}{\left[2\right]}\frac{[y+0]}{\left[1\right]}& \frac{x(x-2)(x-1)}{3!}\\ (2,1)& \frac{[y+0]+\left[3\right]}{\left[3\right]}\frac{[y+1]}{\left[2\right]}\frac{[y-1]}{\left[1\right]}& \frac{(x+2)x(x-2)}{3}\\ \left(3\right)& \frac{[y+2]+\left[3\right]}{\left[3\right]}\frac{[y+1]}{\left[2\right]}\frac{[y+0]}{\left[1\right]}& \frac{(x+4)x(x-1)}{3!}\\ \left(1^4\right)& \frac{[y-3]+\left[4\right]}{\left[4\right]}\frac{[y-2]}{\left[3\right]}\frac{[y-1]}{\left[2\right]}\frac{[y+0]}{\left[1\right]}& \frac{x(x-3)(x-2)(x-1)}{4!}\\ (2,1^2)& \frac{[y-1]+\left[4\right]}{\left[4\right]}\frac{[y+1]}{\left[1\right]}\frac{[y-2]}{\left[2\right]}\frac{[y+0]}{\left[0\right]}& \frac{(x+1)x(x-3)(x-1)}{4·2}\\ \left(2^2\right)& \frac{[y+0]+\left[3\right]}{\left[3\right]}\frac{[y+0]+\left[1\right]}{\left[1\right]}\frac{[y+0]}{\left[2\right]}\frac{[y+0]}{\left[2\right]}& \frac{x(x+2)(x+1)(x-3)}{3·2·2}\\ (3,1)& \frac{[y+1]+\left[4\right]}{\left[4\right]}\frac{[y+2]}{\left[2\right]}\frac{[y+0]}{\left[1\right]}\frac{[y-1]}{\left[1\right]}& \frac{(x+4)(x+1)(x-3)(x-2)}{4·2}\\ \left(4\right)& \frac{[y+3]+\left[4\right]}{\left[4\right]}\frac{[y+2]}{\left[3\right]}\frac{[y+1]}{\left[2\right]}\frac{[y+0]}{\left[1\right]}& \frac{(x+6)(x+1)x(x-1)}{4!}\end{array}\\ \begin{array}{cc}\begin{array}{ccc}\left[n\right]& =& \frac{q^n-q^{-n}}{q-q^{-1}}\\ [y+n]& =& \frac{z{q}^n-z^{-1}{q}^{-1}}{q-q^{-1}}\end{array}& \begin{array}{ccc}{h}_{\lambda }(i,j)& =& {\lambda }_i-i+{\lambda }_j^{\prime }-j+1\\ d_{\lambda }(i,j)& =& \{\begin{array}{cc}{\lambda }_i+{\lambda }_j-i-j+1& i\le j\\ -{\lambda }_i^{\prime }-{\lambda }_j^{\prime }+i+j-1& i>j\end{array}\end{array}\end{array}\\ \underset{q\to 1}{\text{lim}}{Q}_{\lambda }(q^{2r},q)=P_{\lambda }(2r+1)\end{array}$$

Representations of $\text{BW}(z,q)$

We give the representing matrices ${\pi }^{\lambda }\left(g_i\right)$ for the irreducible representations ${\pi }^{\lambda }$ of ${\text{BW}}_m(z,q),$ $m=2,3,4$ with respect to a given ordered basis $v_1,v_2,\dots ,v_{d_{\lambda }}$ of ${𝒵}^{\lambda }$ labeled by paths in the Bratelli diagram B. The representations ${\pi }^{\lambda }$ of ${\text{BW}}_m(z,q)$ such that $\left|\lambda \right|=m$ are the irreducible representations of the Iwahori-Hecke algebra $H_m\left(q^2\right)$ of type A.

${\text{BW}}_2(z,q)$

$$\begin{array}{ccc}{\pi }^{\varnothing }\left(g_1\right)=\left(\frac{z^{-1}}{[-y]}(1-\frac{Q_{\begin{array}{c}\end{array}}}{Q_{\varnothing }})\right)=\left(z^{-1}\right)& {\pi }^{\begin{array}{c}\end{array}}\left(g_1\right)=\left(\frac{q^{-1}}{[-1]}\right)& {\pi }^{\begin{array}{c}\end{array}}\left(g_1\right)=\left(\frac{q}{\left[1\right]}\right)\\ v_1=(\varnothing ,\begin{array}{c}\end{array},\varnothing )& v_1=(\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array})& v_1=(\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}$$

${\text{BW}}_3(z,q)$

$$\begin{array}{ccc}\begin{array}{c}{\pi }^{\begin{array}{c}\end{array}}\left(g_2\right)={\pi }^{\begin{array}{c}\end{array}}\left(g_1\right)=\left(\frac{q^{-1}}{[-1]}\right)\\ {\pi }^{\begin{array}{c}\end{array}}\left(g_2\right)={\pi }^{\begin{array}{c}\end{array}}\left(g_1\right)=\left(\frac{q}{\left[1\right]}\right)\end{array}& & \begin{array}{c}{v}_1=(\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_1=(\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\\ {\pi }^{\begin{array}{c}\end{array}}(g_1-)=\left(\begin{array}{cc}\frac{q^{-1}}{[-1]}& 0\\ 0& \frac{q}{\left[1\right]}\end{array}\right)& {\pi }^{\begin{array}{c}\end{array}}\left(g_2\right)=\left(\begin{array}{cc}\frac{q^2}{\left[2\right]}& \frac{\sqrt{\left[1\right]·\left[3\right]}}{\left[2\right]}\\ \frac{\sqrt{\left[1\right]·\left[3\right]}}{\left[2\right]}& \frac{q^{-2}}{[-2]}\end{array}\right)& \begin{array}{c}{v}_1=(\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_2=(\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\\ \multicolumn{2}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(g_1\right)=\left(\begin{array}{ccc}\frac{q^{-1}}{[-y]}(1-\frac{Q_{\begin{array}{c}\end{array}}}{Q_{\varnothing }})& 0& 0\\ 0& \frac{q^{-1}}{[-1]}& 0\\ 0& 0& \frac{q}{\left[1\right]}\end{array}\right)}\\ \multicolumn{2}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(g_2\right)=\left(\begin{array}{ccc}\frac{z}{\left[y\right]}(1-\frac{Q_{\varnothing }}{Q_{\begin{array}{c}\end{array}}})& -\frac{q}{\left[1\right]}\frac{\sqrt{Q_{\varnothing }{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& -\frac{q^{-1}}{[-1]}\frac{\sqrt{Q_{\varnothing }{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}\\ -\frac{q}{\left[1\right]}\frac{\sqrt{Q_{\varnothing }{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& \frac{z^{-1}{q}^2}{[-y+2]}(1-\frac{Q_{\begin{array}{c}\end{array}}}{Q_{\begin{array}{c}\end{array}}})& -\frac{z^{-1}}{[-y]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}\\ -\frac{q^{-1}}{[-1]}\frac{\sqrt{Q_{\varnothing }{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& -\frac{z^{-1}}{[-y]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& \frac{z^{-1}{q}^{-2}}{[-y-2]}(1-\frac{Q_{\begin{array}{c}\end{array}}}{Q_{\begin{array}{c}\end{array}}})\end{array}\right)}& \begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\varnothing ,\begin{array}{c}\end{array},)\\ v_2& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},)\\ v_3& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},)\end{array}\end{array}$$

${\text{BW}}_4(z,q)$

$$\begin{array}{ccc}\begin{array}{c}{\pi }^{\begin{array}{c}\end{array}}\left(g_3\right)={\pi }^{\begin{array}{c}\end{array}}\left(g_2\right)={\pi }^{\begin{array}{c}\end{array}}\left(g_1\right)=\left(\frac{q^{-1}}{[-1]}\right)\\ {\pi }^{\begin{array}{c}\end{array}}\left(g_3\right)={\pi }^{\begin{array}{c}\end{array}}\left(g_2\right)={\pi }^{\begin{array}{c}\end{array}}\left(g_1\right)=\left(\frac{q}{\left[1\right]}\right)\end{array}& & \begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_1& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\\ {\pi }^{\begin{array}{c}\end{array}}\left(g_1\right)=\left(\begin{array}{ccc}\frac{q^{-1}}{[-1]}& 0& 0\\ 0& \frac{q^{-1}}{[-1]}& 0\\ 0& 0& \frac{q}{\left[1\right]}\end{array}\right)& {\pi }^{\begin{array}{c}\end{array}}\left(g_2\right)=\left(\begin{array}{ccc}\frac{q^{-1}}{[-1]}& 0& 0\\ 0& \frac{q^2}{\left[2\right]}& \frac{\sqrt{\left[1\right]·\left[3\right]}}{\left[2\right]}\\ 0& \frac{\sqrt{\left[1\right]·\left[3\right]}}{\left[2\right]}& \frac{q^{-2}}{[-2]}\end{array}\right)& \begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_2& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_3& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\\ \multicolumn{2}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(g_3\right)=\left(\begin{array}{ccc}\frac{q^3}{\left[3\right]}& \frac{\sqrt{\left[2\right]·\left[4\right]}}{\left[3\right]}& 0\\ \frac{\sqrt{\left[2\right]·\left[4\right]}}{\left[3\right]}& \frac{q^{-3}}{[-3]}& 0\\ 0& 0& \frac{q^{-1}}{[-1]}\end{array}\right)}\\ {\pi }^{\begin{array}{c}\end{array}}\left(g_3\right)={\pi }^{\begin{array}{c}\end{array}}\left(g_1\right)=\left(\begin{array}{cc}\frac{q^{-1}}{[-1]}& 0\\ 0& \frac{q}{\left[1\right]}\end{array}\right)& {\pi }^{\begin{array}{c}\end{array}}\left(g_2\right)=\left(\begin{array}{cc}\frac{q^2}{\left[2\right]}& \frac{\sqrt{\left[1\right]·\left[3\right]}}{\left[2\right]}\\ \frac{\sqrt{\left[1\right]·\left[3\right]}}{\left[2\right]}& \frac{q^{-2}}{[-2]}\end{array}\right)& \begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_2& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\\ {\pi }^{\begin{array}{c}\end{array}}\left(g_1\right)=\left(\begin{array}{ccc}\frac{q^{-1}}{[-1]}& 0& 0\\ 0& \frac{q}{\left[1\right]}& 0\\ 0& 0& \frac{q}{\left[1\right]}\end{array}\right)& {\pi }^{\begin{array}{c}\end{array}}\left(g_2\right)=\left(\begin{array}{ccc}\frac{q^2}{\left[2\right]}& \frac{\sqrt{\left[1\right]·\left[3\right]}}{\left[2\right]}& 0\\ \frac{\sqrt{\left[1\right]·\left[3\right]}}{\left[2\right]}& \frac{q^{-2}}{[-2]}& 0\\ 0& 0& \frac{q}{\left[1\right]}\end{array}\right)& \begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_2& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_3& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\\ \multicolumn{2}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(g_3\right)=\left(\begin{array}{ccc}\frac{q}{\left[1\right]}& 0& 0\\ 0& \frac{q^3}{\left[3\right]}& \frac{\sqrt{\left[2\right]·\left[4\right]}}{\left[3\right]}\\ 0& \frac{\sqrt{\left[2\right]·\left[4\right]}}{\left[3\right]}& \frac{q^{-3}}{[-3]}\end{array}\right)}\\ \multicolumn{2}{c}{{\pi }^{\varnothing }\left(g_3\right)={\pi }^{\varnothing }\left(g_1\right)=\left(\begin{array}{ccc}\frac{z^{-1}}{[-y]}(1-\frac{Q_{\begin{array}{c}\end{array}}}{Q_{\varnothing }})& 0& 0\\ 0& \frac{q^{-1}}{[-1]}& 0\\ 0& 0& \frac{q}{\left[1\right]}\end{array}\right)}& \begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\varnothing ,\begin{array}{c}\end{array},\varnothing ,)\\ v_2& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\varnothing ,)\\ v_3& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\varnothing ,)\end{array}\\ \multicolumn{3}{c}{{\pi }^{\varnothing }\left(g_2\right)=\left(\begin{array}{ccc}\frac{z}{\left[y\right]}(1-\frac{Q_{\varnothing }}{Q_{\begin{array}{c}\end{array}}})& -\frac{q}{\left[1\right]}\frac{\sqrt{Q_{\varnothing }{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& -\frac{q^{-1}}{[-1]}\frac{\sqrt{Q_{\varnothing }{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}\\ -\frac{q}{\left[1\right]}\frac{\sqrt{Q_{\varnothing }{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& \frac{z^{-1}{q}^2}{[-y+2]}(1-\frac{Q_{\begin{array}{c}\end{array}}}{Q_{\begin{array}{c}\end{array}}})& -\frac{z^{-1}}{[-y]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}\\ -\frac{q^{-1}}{[-1]}\frac{\sqrt{Q_{\varnothing }{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& -\frac{z^{-1}}{[-y]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& \frac{z^{-1}{q}^{-2}}{[-y-2]}(1-\frac{Q_{\begin{array}{c}\end{array}}}{Q_{\begin{array}{c}\end{array}}})\end{array}\right)}\\ \multicolumn{3}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(g_1\right)=\left(\begin{array}{cccccc}\frac{z^{-1}}{[-y]}(1-\frac{Q_{\begin{array}{c}\end{array}}}{Q_{\varnothing }})& 0& 0& 0& 0& 0\\ 0& \frac{q^{-1}}{[-1]}& 0& 0& 0& 0\\ 0& 0& \frac{q}{\left[1\right]}& 0& 0& 0\\ 0& 0& 0& \frac{q^{-1}}{[-1]}& 0& 0\\ 0& 0& 0& 0& \frac{q^{-1}}{[-1]}& 0\\ 0& 0& 0& 0& 0& \frac{q}{\left[1\right]}\end{array}\right)}\\ \multicolumn{3}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(g_2\right)=\left(\begin{array}{cccccc}\frac{z}{\left[y\right]}(1-\frac{Q_{\varnothing }}{Q_{\begin{array}{c}\end{array}}})& -\frac{q}{\left[1\right]}\frac{\sqrt{Q_{\varnothing }{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& -\frac{q^{-1}}{[-1]}\frac{\sqrt{Q_{\varnothing }{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& 0& 0& 0\\ -\frac{q}{\left[1\right]}\frac{\sqrt{Q_{\varnothing }{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& \frac{z^{-1}{q}^2}{[-y+2]}(1-\frac{Q_{\begin{array}{c}\end{array}}}{Q_{\begin{array}{c}\end{array}}})& -\frac{z^{-1}}{[-y]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& 0& 0& 0\\ -\frac{q^{-1}}{[-1]}\frac{\sqrt{Q_{\varnothing }{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& -\frac{z^{-1}}{[-y]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& \frac{z^{-1}{q}^{-2}}{[-y-2]}(1-\frac{Q_{\begin{array}{c}\end{array}}}{Q_{\begin{array}{c}\end{array}}})& 0& 0& 0\\ 0& 0& 0& \frac{q^{-1}}{[-1]}& 0& 0\\ 0& 0& 0& 0& \frac{q^2}{\left[2\right]}& \frac{\sqrt{\left[1\right]·\left[3\right]}}{\left[2\right]}\\ 0& 0& 0& 0& \frac{\sqrt{\left[1\right]·\left[3\right]}}{\left[2\right]}& \frac{q^{-2}}{[-2]}\end{array}\right)}\\ \multicolumn{3}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(g_3\right)=\left(\begin{array}{cccccc}\frac{q^{-1}}{[-1]}& 0& 0& 0& 0& 0\\ 0& \frac{z{q}^{-2}}{[y-2]}(1-\frac{Q_{\begin{array}{c}\end{array}}}{Q_{\begin{array}{c}\end{array}}})& 0& -\frac{q}{\left[1\right]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& -\frac{q^{-2}}{[-2]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& 0\\ 0& 0& \frac{z}{\left[y\right]}& 0& 0& \frac{\sqrt{[y+1][y-1]}}{\left[y\right]}\\ 0& -\frac{q}{\left[1\right]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& 0& \frac{z^{-1}{q}^4}{[-y+4]}(1-\frac{Q_{\begin{array}{c}\end{array}}}{Q_{\begin{array}{c}\end{array}}})& -\frac{z^{-1}q}{[-y+1]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& 0\\ 0& -\frac{q^{-2}}{[-2]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& 0& -\frac{z^{-1}q}{[-y+1]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& \frac{z^{-1}{q}^{-2}}{[-y-2]}(1-\frac{Q_{\begin{array}{c}\end{array}}}{Q_{\begin{array}{c}\end{array}}})& 0\\ 0& 0& \frac{\sqrt{[y+1][y-1]}}{\left[y\right]}& 0& 0& \frac{z^{-1}}{[-y]}\end{array}\right)}\\ \multicolumn{3}{c}{\begin{array}{cc}\begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\varnothing ,\begin{array}{c}\end{array})\\ v_2& =& (\varnothing ,\begin{array}{c}\end{array},\varnothing ,\begin{array}{c}\end{array})\\ v_3& =& (\varnothing ,\begin{array}{c}\end{array},\varnothing ,\begin{array}{c}\end{array})\end{array}& \begin{array}{ccc}{v}_4& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_5& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_6& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\end{array}}\\ \multicolumn{3}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(g_1\right)=\left(\begin{array}{cccccc}\frac{z^{-1}}{[-y]}(1-Q_{\begin{array}{c}\end{array}}{Q}_{\varnothing })& 0& 0& 0& 0& 0\\ 0& \frac{q^{-1}}{[-1]}& 0& 0& 0& 0\\ 0& 0& \frac{q}{\left[1\right]}& 0& 0& 0\\ 0& 0& 0& \frac{q^{-1}}{[-1]}& 0& 0\\ 0& 0& 0& 0& \frac{q^{-1}}{\left[1\right]}& 0\\ 0& 0& 0& 0& 0& \frac{q}{\left[1\right]}\end{array}\right)}\\ \multicolumn{3}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(g_2\right)=\left(\begin{array}{cccccc}\frac{z}{\left[y\right]}(1-\frac{Q_{\varnothing }}{Q_{\begin{array}{c}\end{array}}})& -\frac{q}{\left[1\right]}\frac{\sqrt{Q_{\varnothing }{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& -\frac{q^{-1}}{[-1]}\frac{\sqrt{Q_{\varnothing }{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& 0& 0& 0\\ -\frac{q}{\left[1\right]}\frac{\sqrt{Q_{\varnothing }{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& \frac{z^{-1}{q}^2}{[-y+2]}(1-\frac{Q_{\begin{array}{c}\end{array}}}{Q_{\begin{array}{c}\end{array}}})& -\frac{z^{-1}}{[-y]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& 0& 0& 0\\ -\frac{q^{-1}}{[-1]}\frac{\sqrt{Q_{\varnothing }{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& -\frac{z^{-1}}{[-y]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& \frac{z^{-1}{q}^{-2}}{[-y-2]}(1-\frac{Q_{\begin{array}{c}\end{array}}}{Q_{\begin{array}{c}\end{array}}})& 0& 0& 0\\ 0& 0& 0& \frac{q^2}{\left[2\right]}& \frac{\sqrt{\left[1\right]·\left[3\right]}}{\left[2\right]}& 0\\ 0& 0& 0& \frac{\sqrt{\left[1\right]·\left[3\right]}}{\left[2\right]}& \frac{q^{-2}}{[-2]}& 0\\ 0& 0& 0& 0& 0& \frac{q}{\left[1\right]}\end{array}\right)}\\ \multicolumn{3}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(g_3\right)=\left(\begin{array}{cccccc}\frac{q}{\left[1\right]}& 0& 0& 0& 0& 0\\ 0& \frac{z}{\left[y\right]}& 0& \frac{\sqrt{[y+1][y-1]}}{\left[y\right]}& 0& 0\\ 0& 0& \frac{z{q}^2}{[y+2]}(1-\frac{Q_{\begin{array}{c}\end{array}}}{Q_{\begin{array}{c}\end{array}}})& 0& -\frac{q^2}{\left[2\right]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& -\frac{q^{-1}}{[-1]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}\\ 0& \frac{\sqrt{[y+1][y-1]}}{\left[y\right]}& 0& \frac{z^{-1}}{[-y]}& 0& 0\\ 0& 0& -\frac{q^2}{\left[2\right]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& 0& \frac{z^{-1}{q}^2}{[-y+2]}(1-\frac{Q_{\begin{array}{c}\end{array}}}{Q_{\begin{array}{c}\end{array}}})& -\frac{z^{-1}{q}^{-1}}{[-y-1]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}\\ 0& 0& -\frac{q^{-1}}{[-1]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& 0& -\frac{z^{-1}{q}^{-1}}{[-y-1]}\frac{\sqrt{Q_{\begin{array}{c}\end{array}}{Q}_{\begin{array}{c}\end{array}}}}{Q_{\begin{array}{c}\end{array}}}& \frac{z^{-1}{q}^{-4}}{[-y-4]}(1-\frac{Q_{\begin{array}{c}\end{array}}}{Q_{\begin{array}{c}\end{array}}})\end{array}\right)}\\ \multicolumn{3}{c}{\begin{array}{cc}\begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_2& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_3& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}& \begin{array}{ccc}{v}_4& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_5& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_6& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\end{array}}\end{array}$$

Representations of $B\left(x\right)$

We give the representing matrices ${\pi }^{\lambda }\left(s_i\right)$ and ${\pi }^{\lambda }\left(e_i\right)$ for the irreducible representations of ${\pi }^{\lambda }$ of $B_m\left(x\right),$ $m=2,3,4$ with respect to a given ordered basis $v_1,v_2,\dots ,v_{d_{\lambda }}$ of $Z^{\lambda }$ labeled by paths in the Bratelli diagram B. The representations ${\pi }^{\lambda }$ of $B_m\left(x\right)$ such that $\left|\lambda \right|=m$ are the irreducible representations of the group algebra of the symmetric group $S_m\text{.}$ In these cases ${\pi }^{\lambda }\left(e_i\right)=0$ for all $1\le i\le m-1\text{.}$

$B_2\left(x\right)$

$$\begin{array}{ccc}{\pi }^{\varnothing }\left(s_1\right)=\left(\frac{1}{1-x}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\varnothing }})\right)=\left(1\right)& {\pi }^{\begin{array}{c}\end{array}}\left(s_1\right)=(-1)& {\pi }^{\begin{array}{c}\end{array}}\left(s_1\right)=\left(1\right)\\ {\pi }^{\varnothing }\left(e_1\right)=\left(\frac{P_{\begin{array}{c}\end{array}}}{P_{\varnothing }}\right)=\left(x\right)& v_1=(\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array})& v_1=(\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_1=(\varnothing ,\begin{array}{c}\end{array},\varnothing )\end{array}$$

$B_3\left(x\right)$

$$\begin{array}{cc}\begin{array}{c}{\pi }^{\begin{array}{c}\end{array}}\left(s_2\right)={\pi }^{\begin{array}{c}\end{array}}\left(s_1\right)=(-1)\\ {\pi }^{\begin{array}{c}\end{array}}\left(s_2\right)={\pi }^{\begin{array}{c}\end{array}}\left(s_1\right)=\left(1\right)\end{array}& \begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_1& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\\ {\pi }^{\begin{array}{c}\end{array}}\left(s_1\right)=\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right){\pi }^{\begin{array}{c}\end{array}}\left(s_2\right)=\left(\begin{array}{cc}\frac{1}{2}& \frac{\sqrt{1·3}}{2}\\ \frac{\sqrt{1·3}}{2}& \frac{1}{-2}\end{array}\right)& \begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_2& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\\ {\pi }^{\begin{array}{c}\end{array}}\left(s_1\right)=\left(\begin{array}{ccc}\frac{1}{1-x}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\varnothing }})& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right){\pi }^{\begin{array}{c}\end{array}}\left(e_1\right)=\left(\begin{array}{ccc}\frac{P_{\begin{array}{c}\end{array}}}{P_{\varnothing }}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\\ {\pi }^{\begin{array}{c}\end{array}}\left(s_2\right)=\left(\begin{array}{ccc}\frac{1}{x-1}(1-\frac{P_{\varnothing }}{P_{\begin{array}{c}\end{array}}})& -\frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}\\ -\frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{1}{3-x}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}})& -\frac{1}{1-x}\frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}\\ \frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& -\frac{1}{1-x}\frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{1}{-(x+1)}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}})\end{array}\right)& \begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\varnothing ,\begin{array}{c}\end{array})\\ v_2& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_3& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\\ {\pi }^{\begin{array}{c}\end{array}}\left(e_2\right)=\left(\begin{array}{ccc}\frac{P_{\varnothing }}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}\\ \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\varnothing }}}{P_{\begin{array}{c}\end{array}}}& \frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}\\ \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\varnothing }}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}}\end{array}\right)\\ \begin{array}{c}{\pi }^{\begin{array}{c}\end{array}}\left(s_3\right)={\pi }^{\begin{array}{c}\end{array}}\left(s_2\right)={\pi }^{\begin{array}{c}\end{array}}\left(s_1\right)=(-1)\\ {\pi }^{\begin{array}{c}\end{array}}\left(s_3\right)={\pi }^{\begin{array}{c}\end{array}}\left(s_2\right)={\pi }^{\begin{array}{c}\end{array}}\left(s_1\right)=\left(1\right)\end{array}& \begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_1& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\\ {\pi }^{\begin{array}{c}\end{array}}\left(s_1\right)=\left(\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& 0\end{array}\right){\pi }^{\begin{array}{c}\end{array}}\left(s_2\right)=\left(\begin{array}{ccc}-1& 0& 0\\ 0& \frac{1}{2}& \frac{\sqrt{1·3}}{2}\\ 0& \frac{\sqrt{1·3}}{2}& \frac{1}{-2}\end{array}\right)& \begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_1& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_1& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\\ {\pi }^{\begin{array}{c}\end{array}}\left(s_3\right)=\left(\begin{array}{ccc}1/3& \frac{\sqrt{2·4}}{3}& 0\\ \frac{\sqrt{2·4}}{3}& \frac{1}{-3}& 0\\ 0& 0& -1\end{array}\right)\\ {\pi }^{\begin{array}{c}\end{array}}\left(s_3\right)={\pi }^{\begin{array}{c}\end{array}}\left(s_1\right)=\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right){\pi }^{\begin{array}{c}\end{array}}\left(s_2\right)=\left(\begin{array}{cc}\frac{1}{2}& \frac{\sqrt{1·3}}{2}\\ \frac{\sqrt{1·3}}{2}& \frac{1}{-2}\end{array}\right)& \begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_2& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\\ {\pi }^{\begin{array}{c}\end{array}}\left(s_1\right)=\left(\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right){\pi }^{\begin{array}{c}\end{array}}\left(s_2\right)=\left(\begin{array}{ccc}\frac{1}{2}& \frac{\sqrt{1·3}}{2}& 0\\ \frac{\sqrt{1·3}}{2}& \frac{1}{-2}& 0\\ 0& 0& 1\end{array}\right)& \begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_2& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array}\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_3& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array}\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\\ {\pi }^{\begin{array}{c}\end{array}}\left(s_3\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1/3& \frac{\sqrt{2·4}}{3}\\ 0& \frac{\sqrt{2·4}}{3}& \frac{1}{-3}\end{array}\right)\\ {\pi }^{\varnothing }\left(s_3\right)={\pi }^{\varnothing }\left(s_1\right)=\left(\begin{array}{ccc}\frac{1}{1-x}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\varnothing }})& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right){\pi }^{\varnothing }\left(e_3\right)={\pi }^{\varnothing }\left(e_1\right)=\left(\begin{array}{ccc}\frac{P_{\begin{array}{c}\end{array}}}{P_{\varnothing }}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\\ {\pi }^{\varnothing }\left(s_2\right)=\left(\begin{array}{ccc}\frac{1}{x-1}(1-\frac{P_{\varnothing }}{P_{\begin{array}{c}\end{array}}})& -\frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}\\ -\frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{1}{3-x}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}})& -\frac{1}{1-x}\frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}\\ \frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& -\frac{1}{1-x}\frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{1}{-(x+1)}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}})\end{array}\right)& \begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\varnothing ,\begin{array}{c}\end{array},\varnothing )\\ v_2& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\varnothing )\\ v_3& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\varnothing )\end{array}\\ {\pi }^{\varnothing }\left(e_2\right)=\left(\begin{array}{ccc}\frac{P_{\varnothing }}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}\\ \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\varnothing }}}{P_{\begin{array}{c}\end{array}}}& \frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}\\ \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\varnothing }}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}}\end{array}\right)\\ \multicolumn{2}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(s_1\right)=\left(\begin{array}{cccccc}\frac{1}{1-x}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\varnothing }})& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right)}\\ \multicolumn{2}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(s_2\right)=\left(\begin{array}{cccccc}\frac{1}{x-1}(1-\frac{P_{\varnothing }}{P_{\begin{array}{c}\end{array}}})& -\frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0& 0& 0\\ -\frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{1}{3-x}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}})& -\frac{1}{1-x}\frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0& 0& 0\\ \frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& -\frac{1}{1-x}\frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{1}{-(x+1)}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}})& 0& 0& 0\\ 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& \frac{1}{2}& \frac{\sqrt{1·3}}{2}\\ 0& 0& 0& 0& \frac{\sqrt{1·3}}{2}& \frac{1}{-2}\end{array}\right)}\\ \multicolumn{2}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(s_3\right)=\left(\begin{array}{cccccc}-1& 0& 0& 0& 0& 0\\ 0& \frac{1}{x-3}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}})& 0& -\frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& -\frac{1}{-2}\frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0\\ 0& 0& \frac{1}{x-1}& 0& 0& \frac{\sqrt{x(x-2)}}{x-1}\\ 0& -\frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0& \frac{1}{5-x}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}})& -\frac{1}{2-x}\frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0\\ 0& -\frac{1}{-2}\frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0& -\frac{1}{2-x}\frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{1}{-(x+1)}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}})& 0\\ 0& 0& \frac{\sqrt{x(x-2)}}{x-1}& 0& 0& \frac{1}{1-x}\end{array}\right)}\\ \multicolumn{2}{c}{\begin{array}{cc}\begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_2& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_2& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}& \begin{array}{ccc}{v}_4& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_5& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_6& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\end{array}}\\ \multicolumn{2}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(e_1\right)=\left(\begin{array}{ccccc}\frac{P_{\begin{array}{c}\end{array}}}{P_{\varnothing }}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right)}\\ \multicolumn{2}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(e_2\right)=\left(\begin{array}{cccccc}\frac{P_{\varnothing }}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0& 0& 0\\ \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\varnothing }}}{P_{\begin{array}{c}\end{array}}}& \frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0& 0& 0\\ \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\varnothing }}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right)}\\ \multicolumn{2}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(e_3\right)=\left(\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& \frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}}& 0& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0\\ 0& 0& 0& 0& 0& 0\\ 0& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0& \frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0\\ 0& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}}& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right)}\\ \multicolumn{2}{c}{\begin{array}{cc}\begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\varnothing ,\begin{array}{c}\end{array})\\ v_2& =& (\varnothing ,\begin{array}{c}\end{array},\varnothing ,\begin{array}{c}\end{array})\\ v_3& =& (\varnothing ,\begin{array}{c}\end{array},\varnothing ,\begin{array}{c}\end{array})\end{array}& \begin{array}{ccc}{v}_4& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_5& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_6& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\end{array}}\\ \multicolumn{2}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(s_1\right)=\left(\begin{array}{cccccc}\frac{1}{1-x}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\varnothing }})& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right)}\\ \multicolumn{2}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(s_2\right)=\left(\begin{array}{cccccc}\frac{1}{x-1}(1-\frac{P_{\varnothing }}{P_{\begin{array}{c}\end{array}}})& -\frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0& 0& 0\\ -\frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{1}{3-x}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}})& -\frac{1}{1-x}\frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0& 0& 0\\ \frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& -\frac{1}{1-x}\frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{1}{-(x+1)}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}})& 0& 0& 0\\ 0& 0& 0& \frac{\sqrt{1·3}}{2}& \frac{1}{-2}& 0\\ 0& 0& 0& \frac{1}{2}& \frac{\sqrt{1·3}}{2}& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right)}\\ \multicolumn{2}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(s_3\right)=\left(\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& \frac{1}{x-1}& 0& \frac{\sqrt{x(x-2)}}{x-1}& 0& 0\\ 0& 0& \frac{1}{x+1}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}})& 0& -\frac{1}{2}\frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}\\ 0& \frac{\sqrt{x(x-2)}}{x-1}& 0& \frac{1}{1-x}& 0& 0\\ 0& 0& -\frac{1}{2}\frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0& \frac{1}{3-x}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}})& -\frac{1}{-x}\frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}\\ 0& 0& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0& -\frac{1}{-x}\frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{1}{-(x+3)}(1-\frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}})\end{array}\right)}\\ \multicolumn{2}{c}{\begin{array}{cc}\begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_2& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_3& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}& \begin{array}{ccc}{v}_4& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_5& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_6& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\end{array}}\\ \multicolumn{2}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(e_1\right)=\left(\begin{array}{cccccc}\frac{P_{\begin{array}{c}\end{array}}}{P_{\varnothing }}& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right)}\\ \multicolumn{2}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(e_2\right)=\left(\begin{array}{cccccc}\frac{P_{\varnothing }}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\varnothing }{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0& 0& 0\\ \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\varnothing }}}{P_{\begin{array}{c}\end{array}}}& \frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0& 0& 0\\ \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\varnothing }}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right)}\\ \multicolumn{2}{c}{{\pi }^{\begin{array}{c}\end{array}}\left(e_3\right)=\left(\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& \frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}}& 0& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}\\ 0& 0& 0& 0& 0& 0\\ 0& 0& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0& \frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}}& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}\\ 0& 0& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& 0& \frac{\sqrt{P_{\begin{array}{c}\end{array}}{P}_{\begin{array}{c}\end{array}}}}{P_{\begin{array}{c}\end{array}}}& \frac{P_{\begin{array}{c}\end{array}}}{P_{\begin{array}{c}\end{array}}}\end{array}\right)}\\ \multicolumn{2}{c}{\begin{array}{cc}\begin{array}{ccc}{v}_1& =& (\varnothing ,\begin{array}{c}\end{array},\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_2& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_3& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}& \begin{array}{ccc}{v}_4& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_5& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\\ v_6& =& (\varnothing ,\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array})\end{array}\end{array}}\end{array}$$

Notes and References

This is a typed version of the paper A Ribbon Hopf Algebra Approach to the Irreducible Representations of Centralizer Algebras: The Brauer, Birman-Wenzl, and Type A Iwahori-Hecke Algebras by Robert ${\text{Leduc}}^{*}$ and Arun ${\text{Ram}}^{†}\text{.}$

The paper was received June 24, 1994; accepted September 12, 1994.

${}^{*}$Supported in part by National Science Foundation Grant DMS-9300523 to the University of Wisconsin.
${}^{†}$Supported in part by National Science Foundation Postdoctoral Fellowship DMS-9107863.

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