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

Last update: 13 March 2014

## Appendix

### Weight Polynomials

$Qλ(z,q)= ∏(i,i)∈λ [y+λi-λi′] +[hλ(i,i)] [hλ(i,i)] ∏(i,j)∈λi≠j [y+dλ(i,j)] [hλ(i,j)] Pλ(x)= ∏(i,j)∈λ x-1+dλ(i,j) hλ(i,j) λ Qλ(z,q) Pλ(x) ∅ 1 1 (1) [y+0]+[1] [1] x (12) [y-1]+[2][2] [y+0][1] x(x-1)2 (2) [y+1]+[2][2] [y+0][1] (x+2)(x-1)2 (13) [y-2]+[3][3] [y-1][2] [y+0][1] x(x-2)(x-1)3! (2,1) [y+0]+[3][3] [y+1][2] [y-1][1] (x+2)x(x-2)3 (3) [y+2]+[3][3] [y+1][2] [y+0][1] (x+4)x(x-1)3! (14) [y-3]+[4][4] [y-2][3] [y-1][2] [y+0][1] x(x-3)(x-2)(x-1)4! (2,12) [y-1]+[4][4] [y+1][1] [y-2][2] [y+0][0] (x+1)x(x-3)(x-1)4·2 (22) [y+0]+[3][3] [y+0]+[1][1] [y+0][2] [y+0][2] x(x+2)(x+1)(x-3)3·2·2 (3,1) [y+1]+[4][4] [y+2][2] [y+0][1] [y-1][1] (x+4)(x+1)(x-3)(x-2)4·2 (4) [y+3]+[4][4] [y+2][3] [y+1][2] [y+0][1] (x+6)(x+1)x(x-1)4! [n] = qn-q-n q-q-1 [y+n] = zqn-z-1q-1 q-q-1 hλ(i,j) = λi-i+λj′-j+1 dλ(i,j) = { λi+λj-i-j+1 i≤j -λi′- λj′+i+j-1 i>j limq→1Qλ (q2r,q)= Pλ(2r+1)$

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

We give the representing matrices ${\pi }^{\lambda }\left({g}_{i}\right)$ for the irreducible representations ${\pi }^{\lambda }$ of ${\text{BW}}_{m}\left(z,q\right),$ $m=2,3,4$ 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 ${\text{BW}}_{m}\left(z,q\right)$ such that $|\lambda |=m$ are the irreducible representations of the Iwahori-Hecke algebra ${H}_{m}\left({q}^{2}\right)$ of type A.

${\text{BW}}_{2}\left(z,q\right)$

$π∅(g1)= ( z-1[-y] (1-QQ∅) ) =(z-1) π (g1)=(q-1[-1]) π (g1)=(q[1]) v1= ( ∅, ,∅ ) v1= ( ∅, , ) v1= ( ∅, , )$

${\text{BW}}_{3}\left(z,q\right)$

$π(g2)= π(g1)= (q-1[-1]) π(g2)= π(g1)= (q[1]) v1= ( ∅, , , ) v1= ( ∅, , , ) π(g1-)= ( q-1[-1] 0 0 q[1] ) π(g2)= ( q2[2] [1]·[3][2] [1]·[3][2] q-2[-2] ) v1= ( ∅, , , ) v2= ( ∅, , , ) π(g1)= ( q-1[-y](1-QQ∅) 0 0 0 q-1[-1] 0 0 0 q[1] ) π(g2)= ( z[y] (1-Q∅Q) -q[1] Q∅QQ -q-1[-1] Q∅QQ -q[1] Q∅QQ z-1q2[-y+2] (1-QQ) -z-1[-y] QQQ -q-1[-1] Q∅QQ -z-1[-y] QQQ z-1q-2[-y-2] (1-QQ) ) v1 = ( ∅, , ∅, , ) v2 = ( ∅, , , , ) v3 = ( ∅, , , , )$

${\text{BW}}_{4}\left(z,q\right)$

$π(g3)= π(g2)= π(g1)= (q-1[-1]) π(g3)= π(g2)= π(g1)= (q[1]) v1 = ( ∅, , , , ) v1 = ( ∅, , , , ) π(g1)= ( q-1[-1] 0 0 0 q-1[-1] 0 0 0 q[1] ) π(g2)= ( q-1[-1] 0 0 0 q2[2] [1]·[3][2] 0 [1]·[3][2] q-2[-2] ) v1 = ( ∅, , , , ) v2 = ( ∅, , , , ) v3 = ( ∅, , , , ) π(g3)= ( q3[3] [2]·[4][3] 0 [2]·[4][3] q-3[-3] 0 0 0 q-1[-1] ) π(g3)= π(g1)= ( q-1[-1] 0 0 q[1] ) π(g2)= ( q2[2] [1]·[3][2] [1]·[3][2] q-2[-2] ) v1 = ( ∅, , , , ) v2 = ( ∅, , , , ) π(g1)= ( q-1[-1] 0 0 0 q[1] 0 0 0 q[1] ) π(g2)= ( q2[2] [1]·[3][2] 0 [1]·[3] [2] q-2[-2] 0 0 0 q[1] ) v1 = ( ∅, , , , ) v2 = ( ∅, , , , ) v3 = ( ∅, , , , ) π(g3)= ( q[1] 0 0 0 q3[3] [2]·[4][3] 0 [2]·[4][3] q-3[-3] ) π∅(g3)= π∅(g1)= ( z-1[-y] (1-QQ∅) 0 0 0 q-1[-1] 0 0 0 q[1] ) v1 = ( ∅, , ∅, , ∅, ) v2 = ( ∅, , , , ∅, ) v3 = ( ∅, , , , ∅, ) π∅(g2)= ( z[y] (1-Q∅Q) -q[1] Q∅QQ -q-1[-1] Q∅QQ -q[1] Q∅QQ z-1q2[-y+2] (1-QQ) -z-1[-y] QQQ -q-1[-1] Q∅QQ -z-1[-y] QQQ z-1q-2[-y-2] (1-QQ) ) π(g1)= ( z-1[-y] (1-QQ∅) 0 0 0 0 0 0 q-1[-1] 0 0 0 0 0 0 q[1] 0 0 0 0 0 0 q-1[-1] 0 0 0 0 0 0 q-1[-1] 0 0 0 0 0 0 q[1] ) π(g2)= ( z[y] (1-Q∅Q) -q[1] Q∅QQ -q-1[-1] Q∅QQ 0 0 0 -q[1] Q∅QQ z-1q2[-y+2] (1-QQ) -z-1[-y] QQQ 0 0 0 -q-1[-1] Q∅QQ -z-1[-y] QQQ z-1q-2[-y-2] (1-QQ) 0 0 0 0 0 0 q-1[-1] 0 0 0 0 0 0 q2[2] [1]·[3][2] 0 0 0 0 [1]·[3][2] q-2[-2] ) π(g3)= ( q-1[-1] 0 0 0 0 0 0 zq-2[y-2] (1-QQ) 0 -q[1] QQQ -q-2[-2] QQQ 0 0 0 z[y] 0 0 [y+1][y-1][y] 0 -q[1] QQQ 0 z-1q4[-y+4] (1-QQ) -z-1q[-y+1] QQQ 0 0 -q-2[-2] QQQ 0 -z-1q[-y+1] QQQ z-1q-2[-y-2] (1-QQ) 0 0 0 [y+1][y-1][y] 0 0 z-1[-y] ) v1 = ( ∅, , ∅, ) v2 = ( ∅, , ∅, ) v3 = ( ∅, , ∅, ) v4 = (∅, , , , ) v5 = (∅, , , , ) v6 = (∅, , , , ) π(g1)= ( z-1[-y] (1-QQ∅) 0 0 0 0 0 0 q-1[-1] 0 0 0 0 0 0 q[1] 0 0 0 0 0 0 q-1[-1] 0 0 0 0 0 0 q-1[1] 0 0 0 0 0 0 q[1] ) π(g2)= ( z[y] (1-Q∅Q) -q[1] Q∅QQ -q-1[-1] Q∅QQ 0 0 0 -q[1] Q∅QQ z-1q2[-y+2] (1-QQ) -z-1[-y] QQQ 0 0 0 -q-1[-1] Q∅QQ -z-1[-y] QQQ z-1q-2[-y-2] (1-QQ) 0 0 0 0 0 0 q2[2] [1]·[3][2] 0 0 0 0 [1]·[3][2] q-2[-2] 0 0 0 0 0 0 q[1] ) π(g3)= ( q[1] 0 0 0 0 0 0 z[y] 0 [y+1][y-1][y] 0 0 0 0 zq2[y+2] (1-QQ) 0 -q2[2] QQQ -q-1[-1] QQQ 0 [y+1][y-1][y] 0 z-1[-y] 0 0 0 0 -q2[2] QQQ 0 z-1q2[-y+2] (1-QQ) -z-1q-1[-y-1] QQQ 0 0 -q-1[-1] QQQ 0 -z-1q-1[-y-1] QQQ z-1q-4[-y-4] (1-QQ) ) v1 = ( ∅, , ∅, , ) v2 = ( ∅, , , , ) v3 = ( ∅, , , , ) v4 = ( ∅, , , , ) v5 = ( ∅, , , , ) v6 = ( ∅, , , , )$

### 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 $|\lambda |=m$ 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 $1\le i\le m-1\text{.}$

${B}_{2}\left(x\right)$

$π∅(s1)= ( 11-x (1-PP∅) ) =(1) π (s1)=(-1) π (s1)=(1) π∅(e1)= (PP∅) =(x) v1= ( ∅, , ) v1= ( ∅, , ) v1= ( ∅, , ∅ )$

${B}_{3}\left(x\right)$

$π(s2)= π(s1)= (-1) π(s2)= π(s1)= (1) v1 = ( ∅, , , ) v1 = ( ∅, , , ) π(s1)= (-1001) π(s2)= ( 12 1·32 1·32 1-2 ) v1 = ( ∅, , , ) v2 = ( ∅, , , ) π(s1)= ( 11-x(1-PP∅) 0 0 0 -1 0 0 0 1 ) π(e1)= ( PP∅ 0 0 0 0 0 0 0 0 ) π(s2)= ( 1x-1(1-P∅P) -P∅PP P∅PP -P∅PP 13-x(1-PP) -11-xPPP P∅PP -11-xPPP 1-(x+1)(1-PP) ) v1 = ( ∅, , ∅, ) v2 = ( ∅, , , ) v3 = ( ∅, , , ) π(e2)= ( P∅P P∅PP P∅PP PP∅P PP PPP PP∅P PPP PP ) π(s3)= π(s2)= π(s1)= (-1) π(s3)= π(s2)= π(s1)= (1) v1 = ( ∅, , , , ) v1 = ( ∅, , , , ) π(s1)= ( -100 0-10 000 ) π(s2)= ( -1 0 0 0 12 1·32 0 1·32 1-2 ) v1 = ( ∅, , , , ) v1 = ( ∅, , , , ) v1 = ( ∅, , , , ) π(s3)= ( 1/3 2·43 0 2·43 1-3 0 0 0 -1 ) π(s3)= π(s1)= (-1001) π(s2)= ( 12 1·32 1·32 1-2 ) v1 = ( ∅, , , , ) v2 = ( ∅, , , , ) π(s1)= ( -1 0 0 0 1 0 0 0 1 ) π(s2)= ( 12 1·32 0 1·32 1-2 0 0 0 1 ) v1 = ( ∅, , , , ) v2 = ( ∅, , , ) v3 = ( ∅, , , ) π(s3)= ( 1 0 0 0 1/3 2·43 0 2·43 1-3 ) π∅(s3)= π∅(s1)= ( 11-x(1-PP∅) 0 0 0 -1 0 0 0 1 ) π∅(e3)= π∅(e1)= ( PP∅00 000 000 ) π∅(s2)= ( 1x-1(1-P∅P) -P∅PP P∅PP -P∅PP 13-x(1-PP) -11-xPPP P∅PP -11-xPPP 1-(x+1)(1-PP) ) v1 = ( ∅, , ∅, , ∅ ) v2 = ( ∅, , , , ∅ ) v3 = ( ∅, , , , ∅ ) π∅(e2)= ( P∅P P∅PP P∅PP PP∅P PP PPP PP∅P PPP PP ) π(s1)= ( 11-x(1-PP∅) 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 ) π(s2)= ( 1x-1(1-P∅P) -P∅PP P∅PP 0 0 0 -P∅PP 13-x(1-PP) -11-xPPP 0 0 0 P∅PP -11-xPPP 1-(x+1)(1-PP) 0 0 0 0 0 0 -1 0 0 0 0 0 0 12 1·32 0 0 0 0 1·32 1-2 ) π(s3)= ( -1 0 0 0 0 0 0 1x-3(1-PP) 0 -PPP -1-2PPP 0 0 0 1x-1 0 0 x(x-2)x-1 0 -PPP 0 15-x(1-PP) -12-xPPP 0 0 -1-2PPP 0 -12-xPPP 1-(x+1)(1-PP) 0 0 0 x(x-2)x-1 0 0 11-x ) v1 = ( ∅, , ∅, , ) v2 = ( ∅, , , , ) v2 = ( ∅, , , , ) v4 = ( ∅, , , , ) v5 = ( ∅, , , , ) v6 = ( ∅, , , , ) π(e1)= ( PP∅ 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 ) π(e2)= ( P∅P P∅PP P∅PP 0 0 0 PP∅P PP PPP 0 0 0 PP∅P PPP PP 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) π(e3)= ( 0 0 0 0 0 0 0 PP 0 PPP PPP 0 0 0 0 0 0 0 0 PPP 0 PP PPP 0 0 PPP 0 PPP PP 0 0 0 0 0 0 0 ) v1 = ( ∅, , ∅, ) v2 = ( ∅, , ∅, ) v3 = ( ∅, , ∅, ) v4 = ( ∅, , , , ) v5 = ( ∅, , , , ) v6 = ( ∅, , , , ) π(s1)= ( 11-x(1-PP∅) 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 ) π(s2)= ( 1x-1(1-P∅P) -P∅PP P∅PP 0 0 0 -P∅PP 13-x(1-PP) -11-xPPP 0 0 0 P∅PP -11-xPPP 1-(x+1)(1-PP) 0 0 0 0 0 0 1·32 1-2 0 0 0 0 12 1·32 0 0 0 0 0 0 1 ) π(s3)= ( 1 0 0 0 0 0 0 1x-1 0 x(x-2)x-1 0 0 0 0 1x+1(1-PP) 0 -12PPP PPP 0 x(x-2)x-1 0 11-x 0 0 0 0 -12PPP 0 13-x(1-PP) -1-xPPP 0 0 PPP 0 -1-xPPP 1-(x+3)(1-PP) ) v1 = ( ∅, , ∅, , ) v2 = ( ∅, , , , ) v3 = ( ∅, , , , ) v4 = ( ∅, , , , ) v5 = ( ∅, , , , ) v6 = ( ∅, , , , ) π(e1)= ( PP∅ 0 0 0 0 0 000000 000000 000000 000000 000000 ) π(e2)= ( P∅P P∅PP P∅PP 0 0 0 PP∅P PP PPP 0 0 0 PP∅P PPP PP 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) π(e3)= ( 0 0 0 0 0 0 0 0 0 0 0 0 0 0 PP 0 PPP PPP 0 0 0 0 0 0 0 0 PPP 0 PP PPP 0 0 PPP 0 PPP PP ) v1 = ( ∅, , ∅, , ) v2 = ( ∅, , , , ) v3 = ( ∅, , , , ) v4 = ( ∅, , , , ) v5 = ( ∅, , , ) v6 = ( ∅, , , )$

## 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.