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

Centralizer Algebras of Tensor Powers of $V={\Lambda }_{{\omega }_{1}},$ Type ${B}_{r}$

The Bratteli diagram is given in Fig. 2. The shapes $\lambda \in {\stackrel{ˆ}{B}}_{m}$ of $B$ which are on level $m$ are the partitions of $m-2k,$ $0\le k\le ⌊m/2⌋\text{;}$ $Bˆm= { λ⊢m-2k,0≤k≤ ⌊m/2⌋ } .$ $∅ ∅ ∅ Figure 2$ A partition $\lambda \in {\stackrel{ˆ}{B}}_{m}$ is connected by an edge to a partition $\mu \in {\stackrel{ˆ}{B}}_{m+1}$ if $\mu$ can be obtained from $\lambda$ by adding a box to $\lambda$ or by removing a box from $\lambda \text{.}$ The diagram $B$ is a multiplicity free Bratteli diagram. The tableaux $T\in {𝒯}^{\lambda }$ in the Bratteli diagram $B$ are called up-down tableaux since they are sequences of partitions in which each partition differs from the previous one by either adding or removing a box.

The $r\text{-truncated}$ Bratteli diagram $B\left(r\right)$ is given by the sets $Bˆm(r)= { λ⊢m-2k,0≤k≤ ⌊m/2⌋ | l(λ) ≤r } .$ A partition $\lambda \in {\stackrel{ˆ}{B}}_{m}\left(r\right)$ is connected by an edge to a partition $\mu \in {\stackrel{ˆ}{B}}_{m+1}\left(r\right)$ if $\mu$ can be obtained by adding or removing a box from $\lambda \text{.}$ The Bratteli diagram $B\left(r\right)$ is a multiplicity free Bratteli diagram. It can be obtained from the Bratteli diagram $B$ by removing all the partitions with more than $r$ rows (and the edges connected to them). It is easy to see that tableaux in the $r\text{-truncated}$ Bratteli diagram $B\left(r\right)$ are up-down tableaux that never pass through a partition of length greater than $r\text{.}$ The Bratteli diagram $B$ can be viewed as the limit of the Bratteli diagrams $B\left(r\right),$ as $r$ goes to infinity.

For the remainder of this section let us fix $r,$ and, unless otherwise specified, all paths and tableaux shall be from the Bratteli diagram $B\left(r\right)\text{.}$

Fix $S=\left({\sigma }^{m-2},{\sigma }^{\left(m-1\right)},{\sigma }^{\left(m\right)}\right)\in {𝒯}_{m-2}^{m}$ and assume that ${\sigma }^{\left(m-2\right)}\ne {\sigma }^{\left(m\right)}$ as partitions. Then there is at most one $T\ne S$ such that $\left(S,T\right)\in {\Omega }_{m-2}^{m}\text{.}$

 Proof. Given a partition $\lambda$ let us write $\mu =\lambda +{\epsilon }_{k}$ (resp. $\mu =\lambda -{\epsilon }_{k}\text{)}$ to denote that $\mu$ is obtained by adding (resp. removing) a box to (resp. from) the $k\text{th}$ row of $\lambda \text{.}$ Fix $S=\left({\sigma }^{\left(m-2\right)},{\sigma }^{\left(m-1\right)},{\sigma }^{\left(m\right)}\right)\in {𝒯}_{m-2}^{m}$ and assume that ${\sigma }^{\left(m-2\right)}\ne {\sigma }^{\left(m\right)}$ as partitions. Suppose that ${\sigma }^{\left(m-1\right)}={\sigma }^{\left(m-2\right)}+{\delta }_{1}{\epsilon }_{k}$ and that ${\sigma }^{\left(m\right)}={\sigma }^{\left(m-1\right)}+{\delta }_{2}{\epsilon }_{l},$ where ${\delta }_{1}$ and ${\delta }_{2}$ are either $±1\text{.}$ If $T$ exists then $T=\left({\sigma }^{\left(m-2\right)},{\tau }^{\left(m-1\right)},{\sigma }^{\left(m\right)}\right)$ is given by ${\tau }^{\left(m-1\right)}={\sigma }^{\left(m-2\right)}+{\delta }_{2}{\epsilon }_{l}$ and ${\sigma }^{\left(m\right)}={\tau }^{\left(m-1\right)}+{\delta }_{1}{\epsilon }_{k}\text{.}$ The path $T$ exists when ${\tau }^{\left(m-1\right)}={\sigma }^{\left(m-2\right)}+{\delta }_{2}{\epsilon }_{l}$ is a partition and not equal to ${\sigma }^{\left(m-1\right)}\text{.}$ $\square$

The Centralizer Algebras ${𝒵}_{m}$

For the remainder of this section fix $𝔤$ to be a complex simple Lie algebra of type ${B}_{r},$ ${C}_{r}$ or ${D}_{r}$ and let $𝔘={𝔘}_{h}\left(𝔤\right)$ be the corresponding quantum group. We shall use the standard notations ([Bou1981], pp. 252-258) for the root systems of Types ${B}_{r},$ ${C}_{r},$ and ${D}_{r}$ so that ${\epsilon }_{1},\dots ,{\epsilon }_{r}$ are an orthonormal basis of ${𝔥}^{*}$ and the element $2\rho$ is given by $2ρ=∑k=1r2 ρkεj= ∑k=1r (y-2k+1) εk, (5.2)$ where $y= { 2r, in Type Br, 2r+1, in Type Cr, 2r-1, in Type Dr. (5.3)$ The finite dimensional irreducible representations of ${𝔘}_{h}\left(𝔤\right)$ which appear as irreducible summands in the tensor powers of $V={\Lambda }_{{\omega }_{1}}$ of ${𝔘}_{h}\left(𝔤\right)$ are indexed by the dominant integral weights in the set $𝔘ˆ= { λ=λ1ε1+⋯+ λrεr | λi∈ℤ,λ1≥⋯ ≥λr≥0 } .$ We shall identify each dominant integral weight $\lambda \in \stackrel{ˆ}{𝔘}$ with the partition $\lambda =\left({\lambda }_{1},\dots ,{\lambda }_{r}\right)\text{.}$ It will be helpful to note that, if $\lambda ={\lambda }_{1}{\epsilon }_{1}+\cdots +{\lambda }_{r}{\epsilon }_{r}\in \stackrel{ˆ}{𝔘}$ then $⟨λ,λ+2ρ⟩ =∑i=1rλi2 +∑i=1r2ρi λi. (5.4)$

Let $V={\Lambda }_{{\omega }_{1}}$ the irreducible $𝔘\text{-module}$ of highest weight ${\omega }_{1}\text{.}$ In type ${C}_{r},$ the decomposition rule for tensoring by $V$ is given by $Λλ⊗V≅ ⨁μ∈λ± Λμ, (5.5)$ where the sum is over all partitions $\mu \in \stackrel{ˆ}{𝔘}$ that are gotten by adding or removing a box from the partition $\lambda \text{.}$ It follows that in Type ${C}_{r}$ the Bratteli diagram for tensor powers of $V={\Lambda }_{{\omega }_{1}}$ is given by $B\left(r\right)\text{.}$ In Types ${B}_{r}$ and ${D}_{r}$ the tensor product rule given in (5.5) holds whenever $|\lambda | but must be modified slightly when $|\lambda |\ge r-1\text{.}$ In order to avoid this complication

(5.6) For the remainder of this section, we shall assume that in Types ${B}_{r}$ and ${D}_{r}$ we have that $r\gg 0\text{;}$ in particular, $m and $i whenever the constants $m$ and $i$ are used,

The Elements ${Ě}_{i}$

The weights of the Markov traces on ${𝒵}_{m}={\text{End}}_{𝔘}\left({V}^{\otimes m}\right)$ are given by $wtm(λ)= dimq(Λλ) (dimq(V))m ,λ∈𝔘ˆ (5.7)$ where the quantum dimension of $V={\Lambda }_{{\omega }_{1}}$ is given by $dimq(V)= { [2r]+1, in Type Br, [2r+1]-1, in Type Cr, [2r-1]+1, in Type Dr.$ Since all automorphisms of the Dynkin diagram corresponding to $𝔤$ fix the node corresponding to the fundamental weight ${\omega }_{1}$ it follows that $V={\Lambda }_{{\omega }_{1}}\cong {V}^{*}={\left({\Lambda }_{{\omega }_{1}}\right)}^{*}\text{.}$ As in (3.2), define $ě\in {\text{End}}_{𝔘}\left(V\otimes V\right)$ to be the $𝔘\text{-invariant}$ projection onto the invariants, ${\Lambda }_{\left(0\right)}\subseteq V\otimes V\text{.}$ Define $Ěi=δdimq(V) ( id⊗⋯⊗id⊗ě⊗id⊗ ⋯⊗id ) ∈End𝔘(V⊗m),$ where the factor $ě$ appears as a transformation on the $i\text{th}$ and the $\left(i+1\right)\text{st}$ tensor factors and $δ= { 1, in Types Br and Dr, -1, in Type Cr. (5.8)$ By Theorem (3.12), there is a natural identification of the centralizer algebras ${𝒵}_{m}$ with the path algebras corresponding to the Bratteli diagram $B\left(r\right)$ so that $Ěm-1= ∑(S,T)∈Ωm-1m (Ěm-1)ST EST$ where, if $S=\left({\sigma }^{\left(m-2\right)},{\sigma }^{\left(m-1\right)},{\sigma }^{\left(m\right)}\right)$ and $T=\left({\sigma }^{\left(m-2\right)},{\tau }^{\left(m-1\right)},{\sigma }^{\left(m\right)}\right),$ then $(Ěm-1)ST= { δdimq(Λσ(m-1))·dimq(Λτ(m-1)) dimq(Λσ(m-2)) if σ(m-2)= σ(m), 0 otherwise (5.9)$ where we have replaced the weights of the Markov trace by $q\text{-dimensions.}$

Let $V={\Lambda }_{{\omega }_{1}}$ be the irreducible $𝔘={𝔘}_{h}\left(𝔤\right)\text{-module}$ indexed by the fundamental weight ${\omega }_{1}\text{.}$ The matrices ${Ř}_{i}$ and ${Ě}_{i}$ in ${\text{End}}_{𝔘}\left({V}^{\otimes m}\right)$ satisfy the relations

 (a) ${Ř}_{i}{Ř}_{j}={Ř}_{j}{Ř}_{i},\phantom{\rule{2em}{0ex}}|i-j|>1,$ (b) ${Ř}_{i}{Ř}_{i+1}{Ř}_{i}={Ř}_{i+1}{Ř}_{i}{Ř}_{i+1},\phantom{\rule{2em}{0ex}}1\le i\le m-2,$ (c) $\left({Ř}_{i}-{z}^{-1}\right)\left({Ř}_{i}-q\right)\left({Ř}_{i}+{q}^{-1}\right)=0,\phantom{\rule{2em}{0ex}}1\le i\le m-1\text{.}$ (d) ${Ě}_{i}{Ř}_{i-1}^{±1}{Ě}_{i}={z}^{±1}{Ě}_{i}$ and ${Ě}_{i}{Ř}_{i+1}^{±1}{Ě}_{i}={z}^{±1}{Ě}_{i},$ (e) ${Ř}_{i}-{Ř}_{i}^{-1}=\left(q-{q}^{-1}\right)\left(1-{Ě}_{i}\right),$ (f) ${Ě}_{i}{Ř}_{i}^{±1}={Ř}_{i}^{±1}{Ě}_{i}={z}^{\mp 1}{Ě}_{i},$
where $z=δqy= { q2r, in Type Br, -q2r+1, in Type Cr, q2r-1, in Type Dr.$

 Proof. (a) and (b) follow from Proposition (2.18). From (5.5), we have $V⊗V≅Λ∅ ⊕Λ(12) ⊕Λ(2)= Λ0⊕ Λ2ω1⊕ Λω1.$ Use (5.4) to show that $⟨0,0+2ρ⟩ =0, ⟨ε1+ε2,ε1+ε2+2ρ⟩ =2y-2, ⟨2ε1,2ε1+2ρ⟩ =2y+2,and ⟨ε1,ε1+2ρ⟩ =y.$ It follows that $q(1/2)⟨0,0+2ρ⟩-⟨ε1,ε1+2ρ⟩ = q-y, q(1/2)⟨ε1+ε2,ε1+ε2+2ρ⟩-⟨ε1,ε1+2ρ⟩ = q-1, and q(1/2)⟨2ε1,2ε1+2ρ⟩-⟨ε1,ε1+2ρ⟩ = q.$ Relation (c) now follows from Corollary (2.22); the signs of the eigenvalues of ${Ř}_{i}$ are determined by which summands are in ${\bigwedge }^{2}\left(V\right),$ $⋀2(V)= { Λ(12), in Types Br and Dr, Λ(12) ⊕Λ0, in Type Cr.$ (d) follows from Proposition (3.11) part (2) and the fact that ${q}^{⟨{\epsilon }_{1},{\epsilon }_{1}+2\rho ⟩}={q}^{y}\text{.}$ Let us prove (e). By Corollary (2.22), ${Ř}_{1}$ acts by the eigenvalue ${z}^{-1}$ on the irreducible summand ${\Lambda }_{0}$ in $V\otimes V\text{.}$ Thus, it follows from relation (c) that $Ěi=δdimq(V) (Ři-q) (Ři+q-1) (z-1-q) (z-1+q-1) .$ Using this formula and the relation $δdimq(V)= z-z-1 q-q-1 +1,$ it can be easily checked that relation (c) is equivalent to relation (e). The relation ${Ř}_{1}{Ě}_{1}=z{Ě}_{1},$ follows by noting that, except for the constant $\delta {\text{dim}}_{q}\left(V\right),$ ${Ě}_{i}$ is the projection onto the invariants ${\Lambda }_{0}\subseteq V\otimes V$ and that ${Ř}_{i}$ acts by constant ${z}^{-1}$ on ${\Lambda }_{0}\text{.}$ All of the relations in (f) follow silimlarly. $\square$

A Path Algebra Formula for ${Ř}_{i}$

Let $B\left(r\right)$ be the Bratteli diagram for tensor powers of $V={\Lambda }_{{\omega }_{1}}$ (with the assumptions in (5.6)). Identify the centralizer algebras ${𝒵}_{m}={\text{End}}_{𝔘}\left({V}^{\otimes m}\right)$ with the path algebras corresponding to the Bratteli diagram $B\left(r\right)\text{.}$ Recall that the path algebras have a natural basis ${E}_{ST},$ $\left(S,T\right)\in {\Omega }^{m}$ of matrix units.

For each tableau $S=\left({\sigma }^{\left(0\right)},\dots ,{\sigma }^{\left(m\right)}\right)\in {𝒯}^{m}$ define $∇i(S)= ⟨σ(i),σ(i)+2ρ⟩ -⟨σ(i-1),σ(i-1)+2ρ⟩ -⟨ω1,ω1+2ρ⟩. (5.11)$ Let $\left(S,T\right)$ be a pair of tableaux $S=(σ(0),…,σ(i-1),σ(i),σ(i+1),…,σ(m)),$ and $T=(σ(0),…,σ(i-1),τ(i),σ(i+1),…,σ(m)),$ in ${𝒯}^{m}$ such that $S$ and $T$ are the same except possibly at the shape at level $i\text{.}$ In other words the pair $\left(S,T\right)\in {\Omega }_{i-1}^{i+1}\text{.}$ Define $⋄i(S,T)=12 (∇i+1(S)-∇i(T)) . (5.12)$ These constants are defined so that, if ${D}_{m}={Ř}_{m-1}{Ř}_{m-2}\cdots {Ř}_{2}{Ř}_{1}{Ř}_{1}{Ř}_{2}\cdots {Ř}_{m-2}{Ř}_{m-1},$ then $Dm=∑S∈𝒯m (Dm)SS ESS,where (Dm)SS= q∇m(S),$ and $q-2⋄m-1(S,T) =(Dm-1)SS (Dm-1)TT.$ The first of these formulas is a consequence of Corollary (2.25).

Let $y$ be as given in (5.3).

 (a) Let $S=\left({\sigma }^{\left(0\right)},\dots ,{\sigma }^{\left(m\right)}\right)$ be a tableau in the Bratteli diagram $B\left(r\right)\text{.}$ Then $∇m(S)= { 2(σk(m-1)-k+1), whenσ(m)= σ(m-1)+εk, 2(-σk(m-1)-y+k), whenσ(m)= σ(m-1)-εk.$ (b) Let $S=\left({\sigma }^{\left(m-2\right)},{\sigma }^{\left(m-1\right)},{\sigma }^{\left(m\right)}\right)$ and $T=\left({\sigma }^{\left(m-2\right)},{\tau }^{\left(m-1\right)},{\sigma }^{\left(m\right)}\right)$ be such that $\left(S,T\right)\in {\Omega }_{m-2}^{m}\text{.}$ Then $⋄m-1(S,T)= { ±(σk(m)-k-τl(m-1)+l), ifτ(m-1) =τ(m-2)±εl and σ(m) =σ(m-1)±εk, ±(τl(m-1)-l+σk(m)-k+y+1), ifσ(m)= σ(m-1)±εk and τ(m-1) =τ(m-2)∓εl.$

 Proof. Let $S=\left({\sigma }^{\left(0\right)},\dots ,{\sigma }^{\left(m\right)}\right)$ be a tableau in the Bratteli diagram $B\left(r\right)\text{.}$ Then, since ${\sigma }^{\left(m\right)}$ differs from ${\sigma }^{\left(m-1\right)}$ by either adding or removing a box in the $k\text{th}$ row, $σ(m-1)= ∑j=1r σj(m-1) εjand σ(m)= (σk(m-1)±1) εk+∑1≤j≤rj≠k σj(m-1)εj.$ Using (5.4) to compute ${\nabla }_{m}\left(S\right)$ we get $∇m(S) = ⟨σ(m),σ(m)+2ρ⟩- ⟨σ(m-1),σ(m-1)+2ρ⟩ -⟨ε1,ε1+2ρ⟩ = (∑j≠k(σj(m-1))2) +(σk(m-1)±1)2 +(∑j≠k2σj(m-1)ρj) +2(σk(m-1)±1) ρk- (∑j≠k(σj(m-1))2) -(σk(m-1))2 -(∑j≠k2σj(m-1)ρj) -2σk(m-1)ρk -1-2ρ1 = ±2σk(m-1)+ 2(±ρk-ρ1).$ The formula for ${\nabla }_{m}\left(S\right)$ follows since ${\rho }_{k}-{\rho }_{1}=\left(y-2k+1\right)-\left(y-1\right)=2\left(-k+1\right)$ and $-{\rho }_{k}-{\rho }_{1}=-\left(y-2k+1\right)-\left(y-1\right)=2\left(-y+k\right)\text{.}$ (b) The formulas for ${\diamond }_{m-1}\left(S,T\right)$ now follow from the definition of ${\diamond }_{m-1}$ and the formula for ${\nabla }_{m}$ in (a). $\square$

One can choose the identification (Section 1) of the centralizer algebras ${𝒵}_{m}$ with the path algebras corresponding to the Bratteli diagram $B\left(r\right)$ (with the assumption in (5.6)) so that the matrices ${Ř}_{i}$ are given by the formula $Ři=∑(S,T)∈Ωi-1i+1 (Ři)ST EST,$ where, for each $S=\left({\sigma }^{\left(i-1\right)},{\sigma }^{\left(i\right)},{\sigma }^{\left(i+1\right)}\right),$ $(Ři)SS= { q⋄i(S,S) [⋄i(S,S)] , ifσ(i-1) ≠σ(i+1), q⋄i(S,S) [⋄i(S,S)] ( 1- δdimq(Λσ(i)) dimq(Λσ(i-1)) ) , ifσ(i-1) =σ(i+1),$ and for each pair $\left(S,T\right)=\left(\left({\sigma }^{\left(i-1\right)},{\sigma }^{\left(i\right)},{\sigma }^{\left(i+1\right)}\right),\left({\sigma }^{\left(i-1\right)},{\tau }^{\left(i\right)},{\sigma }^{\left(i+1\right)}\right)\right)\in {\Omega }_{i-1}^{i+1}$ such that $S\ne T,$ $(Ři)ST= { [⋄i(S,S)-1] [⋄i(S,S)+1] [|⋄i(S,S)|] , if σ(i-1)≠ σ(i+1), - q⋄i(S,T) [⋄i(S,T)] δdimq(Λσ(i))·dimq(Λτ(i)) dimq(Λσ(i-1)) , ifσ(i-1) =σ(i+1),$ where ${\diamond }_{i}\left(S,T\right)$ and $\delta$ are given by (5.11) and (5.12) respectively.

 Proof. Since ${Ř}_{i}\in {𝒵}_{i}={\text{End}}_{𝔘}\left({V}^{\otimes \left(i+1\right)}\right)$ commutes with all elements of ${𝒵}_{i-1}={\text{End}}_{𝔘}\left({V}^{\otimes \left(i-1\right)}\right)$ it follows from Proposition Corollary (1.5) that $Ři=∑(S,T)∈Ωi-1i+1 (Ři)STEST,$ for some constants ${\left({Ř}_{i}\right)}_{ST}\text{.}$ In view of the imbeddings ${𝒵}_{0}\subseteq {𝒵}_{1}\subseteq \cdots \subseteq {𝒵}_{m},$ it is sufficient to show that the formulas for ${Ř}_{i}$ hold for $i=m-1\text{.}$ By definition ${D}_{m}={Ř}_{m-1}{Ř}_{m-2}\cdots {Ř}_{2}{Ř}_{1}{Ř}_{1}{Ř}_{2}\cdots {Ř}_{m-2}{Ř}_{m-1}$ and it follows that ${Ř}_{m-1}^{-1}={D}_{m}^{-1}{Ř}_{m-1}{D}_{m-1}\text{.}$ Thus, we may rewrite the relation (5.10e) in the form $Řm-1-Dm-1 Řm-1Dm-1= (q-q-1) (1-Ěm-1). (5.15)$ Let $\left(S,T\right)\in {\Omega }^{m}$ and view (5.15) as an equation in the path algebra. Since the matrices ${D}_{m}$ and ${D}_{m-1}$ are diagonal, taking the ${E}_{ST}\text{-entry}$ of this equation yields $(Řm-1)ST- (Dm-1)SS (Řm-1)ST (Dm-1)TT= (q-q-1) (δST-(Ěm-1)ST),$ or, equivalently, $(1-(Dm-1)SS(Dm-1)TT) (Řm-1)ST= (q-q-1) (δST-(Ěm-1)ST), (5.16)$ Hence, $(Řm-1)ST= (q-q-1) (δST-(Ěm-1)ST) 1-(Dm-1)SS(Dm-1)TT ,if1- (Dm-1)SS (Dm-1)TT ≠0.$ Plugging in the following $q-q-1 1-(Dm-1)SS(Dm-1)TT = q-q-1 1-q-2⋄m-1(S,T) = q⋄m-1(S,T) (q-q-1) q⋄m-1(S,T) -q-⋄m-1(S,T) = q⋄m-1(S,T) [⋄m-1(S,T)]$ we get $(Řm-1)ST= q⋄m-1(S,T) [⋄m-1(S,T)] (δST-(Ěm-1)ST), if1- (Dm-1)SS (Dm-1)TT ≠0.$ All except the last of the formulas in Theorem (5.14) now follow immediately from (5.9) and the following lemma. $\square$

Let $\left(S,T\right)\in {\Omega }_{m-2}^{m}m\text{.}$ If $S=T$ or if ${\sigma }^{\left(m-2\right)}\ne {\sigma }^{\left(m\right)}$ then $1-{\left({D}_{m}^{-1}\right)}_{SS}{\left({D}_{m-1}\right)}_{TT}\ne 0\text{.}$

 Proof. Consider the equation (5.16). Case 1. If $S\ne T$ and ${\sigma }^{\left(m-2\right)}={\sigma }^{\left(m\right)}$ then ${\delta }_{ST}=0$ and ${\left({Ě}_{i}\right)}_{ST}\ne 0$ since the weights ${\text{wt}}_{k}\left(\mu \right)$ are all nonzero. Thus the right hand side of (5.16) is nonzero. This implies that $1-{\left({D}_{m}^{-1}\right)}_{SS}{\left({D}_{m-1}\right)}_{TT}$ is nonzero. Case 2. If $S=T$ and ${\sigma }^{\left(m-2\right)}\ne {\sigma }^{\left(m\right)}$ then ${\left({Ě}_{i}\right)}_{ST}=0$ and ${\delta }_{ST}\ne 0\text{.}$ Thus the right hand side of (5.16) is nonzero. This implies that $1-{\left({D}_{m}^{-1}\right)}_{SS}{\left({D}_{m-1}\right)}_{TT}$ is nonzero. Case 3. Suppose $S=T$ and ${\sigma }^{\left(m-2\right)}={\sigma }^{\left(m\right)}\text{.}$ Clearly $1-{\left({D}_{m}^{-1}\right)}_{SS}{\left({D}_{m-1}\right)}_{SS}$ is nonzero if and only if ${\diamond }_{m}\left(S,S\right)\ne 0\text{.}$ Then, by Proposition (5.13), there is some $k$ such that ${\diamond }_{m}\left(S,S\right)=±\left(2{\sigma }_{k}^{\left(m-1\right)}-2k+y\right)\text{.}$ This value is nonzero in Types ${C}_{r}$ and ${D}_{r}$ since $y$ is odd, and is nonzero in type ${B}_{r}$ since, by the assumption in (5.6), $2k and ${\sigma }_{k}^{\left(m-1\right)}\ge 0\text{.}$ $\square$

Now let us prove the last formula in Theorem (5.14). Let $S\in {𝒯}_{m-2}^{m}$ and suppose that $T\in {𝒯}_{m-2}^{m}$ is such that $\left(S,T\right)\in {\Omega }_{m-2}^{m}$ and $T\ne S\text{.}$ By Lemma (5.1), $T$ is unique. It follows that $(Řm-12)SS = ∑L∈𝒯m (Řm-1)SL (Řm-1)LS = ((Řm-1)SS)2 +(Řm-1)ST (Řm-1)TS (5.18)$ On the other hand $Řm-12 = Řm-1 (Řm-1-Řm-1-1) = Řm-1 (q-q-1) (1-Ěm-1) = (q-q-1) ( Řm-1- z-1Ěm-1 ) ,$ since ${Ě}_{m-1}{Ř}_{m-1}={z}^{-1}{Ě}_{m-1}\text{.}$ Since ${\sigma }^{\left(m-2\right)}\ne {\sigma }^{\left(m\right)},$ it follows that ${\left({Ě}_{m-1}\right)}_{SS}=0$ and thus that $(Řm-12)SS =(q-q-1) (Řm-1)SS+1. (5.19)$ Equating (5.18) and (5.19) and using the formula for ${\left({Ř}_{m-1}\right)}_{SS}$ gives $(Řm-1)ST (Řm-1)TS = (q-q-1) (Řm-1)SS +1-(Řm-12)SS = [⋄m-1(S,S)-1] [⋄m-1(S,S)+1] [⋄m-1(S,S)]2$ exactly as in the proof of Theorem (4.8). It follows from the remarks at the end of Section 1 that we can choose the normalization of the elements ${E}_{ST}$ so that ${\left({Ř}_{m-1}\right)}_{ST}$ and ${\left({Ř}_{m-1}\right)}_{TS}$ are as given in the theorem.

Matrix Units

Given a tableau $T=\left({\tau }^{\left(0\right)},\dots ,{\tau }^{\left(m\right)}\right)\in {𝒯}^{m}$ let $T\prime$ denote the tableau $T\prime =\left({\tau }^{\left(0\right)},\dots ,{\tau }^{\left(m-1\right)}\right)\in {𝒯}^{m-1}\text{.}$ Let ${\left(T\prime \right)}^{+}$ denote the set of all extensions of $T\prime \text{;}$ $(T′)+= {S∈𝒯m | S′=T′}.$ Given tableaux $S=\left({\sigma }^{\left(0\right)},\dots ,{\sigma }^{\left(m\right)}\right)$ and $T=\left({\tau }^{\left(0\right)},\dots ,{\tau }^{\left(m\right)}\right)$ in ${𝒯}^{m}$ let ${\left({Ř}_{m-1}\right)}_{ST}$ be the constant given by Theorem (5.14) in the case that $\left(\left({\sigma }^{\left(m-2\right)},{\sigma }^{\left(m-1\right)},{\sigma }^{\left(m\right)}\right),\left({\tau }^{\left(m-2\right)},{\tau }^{\left(m-1\right)},{\tau }^{\left(m\right)}\right)\right)\in {\Omega }_{m-2}^{m}$ and let ${\left({Ř}_{m-1}\right)}_{ST}=0$ otherwise.

Let $T\prime =\left({\tau }^{\left(0\right)},\dots ,{\tau }^{\left(m-1\right)}\right)\in {𝒯}^{m-1}$ and let ${\left(T\prime \right)}^{+}$ be the set of extensions of $T\prime \text{.}$ Then the values ${\nabla }_{m}\left(S\right)$ are different as $S$ ranges over all elements of ${\left(T\prime \right)}^{+}\text{.}$

 Proof. Let $S=\left({\tau }^{\left(0\right)},\dots ,{\tau }^{\left(m-1\right)},{\sigma }^{\left(m\right)}\right)\in {\left(T\prime \right)}^{+}\text{.}$ By Proposition (5.13), $∇m(S)=2 (τk(m-1)-k+1) or ∇m(S)=2 (-τk(m-1)-y+k),$ for some positive integer $k\text{.}$ Let $l$ be the largest value of $k$ such that $S\in {\left(T\prime \right)}^{+}\text{.}$ By the assumption in (5.6), $2l-1 Since $τ1(m-1)≥⋯≥ τk(m-1)≥⋯≥ τl(m-1),and -τl(m-1)≥⋯≥ -τk(m-1)≥⋯≥ -τ1(m-1),$ it follows that $τ1(m-1)>⋯> τk(m-1)-k+1 >⋯>τl(m-1) -l+1,and -τl(m-1)-y+l >⋯>-τk(m-1) -y+k>⋯>- τ1(m-1)-y+1.$ Since ${\tau }_{l}^{\left(m-1\right)}\ge -{\tau }_{l}^{\left(m-1\right)}$ and $-l+1>-y+l,$ it follows that $τl(m-1)-l+1> -τl(m-1)-y+l.$ The result follows. $\square$

The proofs of the following results are essentially the same as the proofs of Theorem (4.14) and Corollary (4.15).

The matrix units ${E}_{ST}\in {𝒵}_{m},$ $\left(S,T\right)\in {\Omega }^{m}$ are given in terms of the ${Ř}_{i},$ $1\le i\le m-1,$ inductively, by the following formulas.

 (1) Let $T\in {𝒯}^{m}\text{.}$ Then ${E}_{TT}=\prod _{S\in {𝒯}_{m},S\ne T,S\prime =T\prime }\left({E}_{T\prime T\prime }{Ř}_{m-1}{E}_{T\prime T\prime }-{\left({Ř}_{m-1}\right)}_{SS}{E}_{T\prime T\prime }\right)/\left({\left({Ř}_{m-1}\right)}_{TT}-{\left({Ř}_{m-1}\right)}_{SS}\right)$ (2) Let $\left(S,T\right)\in {\Omega }^{m}\text{.}$ If $\text{shp}\left(S\prime \right)=\text{shp}\left(T\prime \right)$ then ${E}_{ST}={E}_{S\prime T\prime }{E}_{TT}$ where ${E}_{TT}$ is given by (1). (3) Let $\left(S,T\right)\in {\Omega }^{m}\text{.}$ If $\text{shp}\left(S\prime \right)\ne \text{shp}\left(T\prime \right)$ then $EST=1(Řm-1)MN ES′M′Řm-1 EN′T′ETT,$ where $M$ and $N$ are tableaux in ${𝒯}^{m}$ of the form $M=\left({\mu }^{\left(0\right)},\dots ,{\mu }^{\left(m-2\right)},\text{shp}\left(S\prime \right),\text{shp}\left(S\right)\right)$ and $N=\left({\mu }^{\left(0\right)},\dots ,{\mu }^{\left(m-2\right)},\text{shp}\left(T\prime \right),\text{shp}\left(S\right)\right)\text{.}$

The centralizer ${𝒵}_{m}={\text{End}}_{𝔘}\left({V}^{\otimes m}\right)$ is generated by the matrices ${Ř}_{i},$ $1\le i\le m-1\text{.}$

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.