Actions of Tantalizers

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

Last updates: 15 October 2012

Actions of tantalizers

Let $𝔤$ be a complex semisimple Lie algebra and let be the universal enveloping algebra. Let $C$ be the center of and let $\kappa$ be the Casimir in $C$. Let $M$ and $V$ be $𝔤$-modules.

1. Let ${𝔹}_k$ be the degenerate affine braid group. Then $M\otimes V^{\otimes k}$ is a ${𝔹}_k$-module with action given by $$\begin{array}{rcrcl}\Phi & :& {𝔹}_k& \to & {End}_U\left(M\otimes V^{\otimes k}\right),\\ & & t_{s_i}& \mapsto & {id}_M\otimes {id}_V^{\otimes i-1}\otimes s_1\otimes {id}_V^{\otimes k-i-1},\\ & & c_i& \mapsto & {\kappa }_{M\otimes V^{\otimes i}}\otimes {id}_V^{\otimes k-i}\\ & & z& \mapsto & z\otimes {id}_V^{\otimes k},\end{array}$$ for $z\in C$ and where $s_1\cdot \left(u\otimes v\right)=v\otimes u,$ for $u,v\in V$ and ${\kappa }_{M\otimes V^{\otimes i}}$ is $\kappa$ acting on $M$ and the first $i$ factors of $V$. This action commutes with the $𝔤$-action on $M\otimes V^{\otimes k}$.
2. Let $𝔤$ be ${𝔰𝔬}_{2r+1}$, ${𝔰𝔬}_{2r}$ or ${𝔰𝔭}_{2r}$. Using notations for irreducible representations as in (5.27), let $$\epsilon =\left\{\begin{array}{ll}1,& \text{if}𝔤\text{is}{𝔰𝔬}_{2r+1}\text{or}{𝔰𝔬}_{2r},\\ -1,& \text{if}𝔤\text{is}{𝔰𝔭}_{2r},\end{array}\right.V=L\left({ϵ}_1\right),y=⟨{ϵ}_1,{ϵ}_1+2\rho ⟩,$$ and $$z_V^{\left(𝓁\right)}=\epsilon \left(id\otimes {tr}_V\right)\left({\left(y+t\right)}^{𝓁}\right),\text{for}𝓁\in {\mathbb{Z}}_{\ge 0}.$$ Then the algebra homomorphism $\Phi :{𝔹}_k\to {End}_U\left(M\otimes V^{\otimes k}\right)$ is a representation of the degenerate affine BMW algebra ${𝒲}_k$.
3. Let $𝔤=𝔰{𝔩}_{r+1}$ and $V=L\left({ϵ}_1\right)$. Then $\Phi :{𝔹}_k\to {End}_U\left(M\otimes V^{\otimes k}\right)$ is a representation of the graded Hecke algebra.

		Proof.
	Since the $t_{s_i}$ act as simple transpositions, they generate an action of $S_k$ on $M\otimes V^{\otimes k}$. This action commutes with the $𝔤$-action. Since $\kappa \in C$, $c_i\in {End}_{𝔤}\left(M\otimes V^{\otimes i}\right)\subseteq {End}_{𝔤}\left(M\otimes V^{\otimes j}\right)$ for $i\le j$. Then, since $\kappa \in U𝔤$, $c_i$ commutes with $c_j$ for $i\le j$. So $c_0,\dots ,c_k$ all commute with each other. By (ctoy), $y_1,\dots ,y_k$ commute with each other. Since $c_1,\dots ,c_{i-1}$ act as the identity on the $i$th and $\left(i+1\right)$st components of $M\otimes V^{\otimes \left(i+1\right)}$, $t_{s_i}{c}_j=c_j{t}_{s_i}$ for $j<i$. Since $t_{s_i}$ commutes with the action of $U𝔤$ on $M\otimes V^{\otimes j}$, it follows that $t_{s_i}{c}_j=c_j{t}_{s_i}$ for $j>i$. Hence the relations in (gbp3), and thus in (gbp1), are satisfied. Let $t=\sum b\otimes b^{*}$ as in (tdefn). If $$\begin{array}{l}{\kappa }_i\text{ is the operator }\kappa \text{ acting in the }i\text{th factor of }V\text{ in }M\otimes V^{\otimes k},\text{ and}\\ t_{l,m}\text{ is the operator }t\text{ acting in the }l\text{th and }m\text{th factor of }V\text{ in }M\otimes V^{\otimes k},\end{array}$$ (ktop) (so that ${\kappa }_0$ is $\kappa$ acting on $M$) then the relations in (gbp4) hold. Applying the coproduct to compute the action of $\kappa$ on $m\otimes v_1\otimes \cdots \otimes v_j$ gives the first identity in (cytokt) and so the relations in (gbp2) hold. By (5.20), the computations in (5.30) determine the action of $t$ on the components of $V\otimes V$. The decompositions in (5.31) and (5.32) determine the action of $t_{s_1}$ on $V\otimes V$. The operator $\Phi \left(e_1\right)$ is determined from $\Phi \left(t_{s_1}\right)$ and $\Phi \left(t\right)$ via (gbp6), $$\Phi \left(t\right)\Phi \left(t_{s_1}\right)=1-\Phi \left(e_1\right).$$ Then $\Phi \left(t_{s_1}\right)$, $\Phi \left(e_1\right)$, and $\Phi \left(t\right)$ act on the components of $V\otimes V$ by $$\begin{array}{lccc}& L\left(0\right)& L\left(2{ϵ}_1\right)& L({ϵ}_1+{ϵ}_2)\\ \Phi \left(t\right)& -y& 1& -1\\ \Phi \left(t_{s_1}\right)& \epsilon & 1& -1\\ \Phi \left(e_1\right)& 1+\epsilon y& 0& 0\end{array}$$ where $y$ and $t$ are as in (5.24) and (5.17) respectively. The first relation in (dbw1) follows. Since ${dim}_q\left(V\right)=\epsilon +y$, the first identity in (5.15) gives that $$\Phi \left(e_1\right)=\epsilon E_V.$$ By (5.12), the second identity in (5.15), and (5.4), $$\begin{array}{rcl}\Phi \left(e_i{t}_{s_{i-1}}{e}_i\right)& =& \epsilon (1\otimes E_V)(R_{VV}\otimes 1)(1\otimes E_V)\epsilon \\ & =& (\mathrm{id}\otimes {\mathrm{tr}}_V)\left(R_{VV}\right)\otimes E_V\\ & =& C_V^{-1}\otimes E_V=id\otimes E_V=\epsilon \Phi \left(e_i\right),\end{array}$$ which establishes the second relation in (dbw1). By (cytokt), $\Phi \left(y_1\right)=\frac{1}{2}{\kappa }_1+t_{0,1}=\frac{1}{2}y+t_{0,1},$ and by (5.12), $$\begin{array}{rcl}\Phi \left(e_1{y}_1^{𝓁}{e}_1\right)& =& \epsilon \left(id\otimes E_V\right){\left(\frac{1}{2}y+t\right)}^{𝓁}\epsilon \left(id\otimes E_V\right)\\ & =& \left(id\otimes {tr}_V\right)\left({\left(\frac{1}{2}y+t\right)}^{𝓁}\right)\otimes E_V\\ & =& \epsilon \left(id\otimes {tr}_V\right)\left({\left(\frac{1}{2}y+t\right)}^{𝓁}\right)\Phi \left(e_1\right)\\ & =& z_V^{\left(𝓁\right)}\Phi \left(e_1\right),\end{array}$$ which gives the first relation in (dbw2). Since the $y_i$ commute and $t_{s_i}(y_i+y_{i+1})=(y_i+y_{i+1})t_{s_i},$ $$e_i(y_i+y_{i+1})=(t_{s_i}{y}_i-y_{i+1}{t}_{s_i}+1)(y_i+y_{i+1})=(y_i+y_{i+1})(t_{s_i}{y}_i-y_{i+1}{t}_{s_i}+1)=(y_i+y_{i+1})e_i.$$ Use notations similar to that in (ktop), so that, for an element $b\in U𝔤$ or $U𝔤\otimes U𝔤$, $b_i$ and $b_{i,i+1}$ denote the action of an element $b$ on the $i$th, respectively $i$th and $(i+1)$st, factors of $V$ in $M\otimes V^{\otimes (i+1)}$. Then $$\begin{array}{rcl}(y_i+y_{i+1})e_i& =& \left(\frac{1}{2}{\kappa }_i+\underset{r=0}{\sum ^{i-1}}{t}_{r,i}+\frac{1}{2}{\kappa }_{i+1}+\underset{r=0}{\sum ^i}{t}_{r,i+1}\right)e_i\\ & =& \left(\frac{1}{2}{\kappa }_{i,i+1}+\sum _{r=0}^{i-1}(t_{r,i}+t_{r,i+1})\right)e_i\\ & =& \left(\frac{1}{2}{\kappa }_{i,i+1}+\sum _{r=0}^{i-1}\sum _{b}b\otimes \Delta {\left(b^{*}\right)}_{i,i+1}\right)e_i\\ & =& 0,\end{array}$$ because the action of $b^{*}$ and $\kappa$ on $L\left(0\right)$ is $0$. In this case, $$V\otimes V=L\left(2{\epsilon }_1\right)\oplus L({\epsilon }_1+{\epsilon }_2)\text{with}{\Lambda }^2\left(V\right)=L({\epsilon }_1+{\epsilon }_2)$$ so that $$\begin{array}{lcc}& L\left(2{\epsilon }_1\right)& L({\epsilon }_1+{\epsilon }_2)\\ \Phi \left(t\right)& 1& -1\\ \Phi \left(t_{s_1}\right)& 1& -1\end{array}\text{and}\Phi \left(e_1\right)=\Phi \left(t\right)-\Phi \left(t_{s_1}\right)=0.$$ $\square$

Say I wish to discuss irreducible representations of SU(3). I think it should be perfectly acceptable to point out that: $\otimes Fundamental\; Representation=Adjoint\; Representation\oplus Rank-3 Symmetric Tensor Representation$.

Remark. In Theorem 1.1, if $\Phi \left(y_1\right)$ has eigenvalues $u_1,\dots ,u_r$, then $\Phi$ is a representation of the degenerate cyclotomic BMW algebra ${𝒲}_{r,k}(u_1,\dots ,u_r)$.

Pictorially,

$Adjoint Representation$

STUFF   STUFF

$$\text{PICTURE GOES HERE.}$$ By (5.20) and (2.3), the eigenvalues of $y_j$ are related to the eigenvalues of the Casimir.

The Schur functors are the functors $$\begin{array}{ccccc}{F}_V^{\lambda }& :& \left\{U𝔤\text{-modules}\right\}& \to & \left\{{𝔹}_k\text{-modules}\right\}\\ & & M& \mapsto & {Hom}_{U𝔤}\left(M(\lambda ),M\otimes V^{\otimes k}\right)\end{array}$$ where ${Hom}_{U𝔤}\left(M(\lambda ),M\otimes V^{\otimes k}\right)$ is the vector space of highest weight $\lambda$ in $M\otimes V^{\otimes k}$.

Let $𝔤$ be a complex semisimple Lie algebra and let $U=U_h𝔤$ be the corresponding Drinfel'd-Jimbo quantum group. Let $M$ and $V$ be $U$-modules.

1. Then $M\otimes V^{\otimes k}$ is a $C{B}_k$-module with action given by

   $$\begin{array}{rcrcl}\Phi & :& C{B}_k& \to & {End}_U\left(M\otimes V^{\otimes k}\right),\\ & & T_i& \mapsto & {\check{R}}_i,& 1\le k\le k-1,\\ & & X^{{\epsilon }_1}& \mapsto & {\check{R}}_0^2,\\ & & z& \mapsto & z_M,\end{array}$$

   phidefn

   where $z_M=z\otimes {id}_V^{\otimes k},$ $${\check{R}}_i={id}_M\otimes {id}_V^{\otimes (i-1)}{\check{R}}_{VV}\otimes {id}_V^{\otimes (k-i-1)}\text{and}{\check{R}}_0^2=\left({\check{R}}_{MV}{\check{R}}_{VM}\right)\otimes {id}_V^{\otimes (k-1)},$$ with ${\check{R}}_{MV}$ as in INSERTREF. The $C{B}_k$-action commutes with the $U$-action on $M\otimes V^{\otimes k}$.
2. Let $𝔤$ be ${𝔰𝔬}_{2r+1}$, ${𝔰𝔬}_{2r}$ or ${𝔰𝔭}_{2r}$. Let $$\epsilon =\left\{\begin{array}{ll}1,& \text{if}𝔤\text{is}{𝔰𝔬}_{2r+1}\text{or}{𝔰𝔬}_{2r},\\ -1,& \text{if}𝔤\text{is}{𝔰𝔬}_{2r},\end{array}\right.V=L\left({ϵ}_1\right),z=\epsilon q^{⟨ϵ,ϵ+2\rho ⟩},$$ and $$Z_V^{\left(l\right)}=\epsilon \left(id\otimes {tr}_V\right)\left({\left(z{ℛ}_{21}ℛ\right)}^l\right),\text{for}l\in {\mathbb{Z}}_{\ge 0}.$$ Then the algebra homomorphism $\Phi :C{B}_k\to {End}_U\left(M\otimes V^{\otimes k}\right)$ is a representation of the affine BMW algebra ${𝒲}_k$.
3. Let $𝔤=𝔰{𝔩}_{r+1}$ and $V=L\left({\epsilon }_1\right)$. Then $\Phi :C{B}_k\to {End}_U\left(M\otimes V^{\otimes k}\right)$ is a representation of the affine Hecke algebra.

		Proof.
	The relations (2.26) and (2.29) are consequences of the definition of the action of $T_i$ and $X^{{\epsilon }_1}$. The relations (2.27) and (2.28) follow from the following computations: $${\check{R}}_i{\check{R}}_{i+1}{\check{R}}_i=\text{PICTURE}=\text{PICTURE}={\check{R}}_{i+1}{\check{R}}_i{\check{R}}_{i+1}$$ and $${\check{R}}_0^2{\check{R}}_1{\check{R}}_0^2{\check{R}}_1=\text{PICTURE}=\text{PICTURE}={\check{R}}_1{\check{R}}_0^2{\check{R}}_1{\check{R}}_0^2.$$ By (5.23), the computations in (5.30) determine the action of ${\check{R}}_{VV}^2$ on the components of $V\otimes V$. The operator $\Phi \left(T_1\right)={\check{R}}_{VV}$ is the square root of ${\check{R}}_{VV}^2$ and, at $q=1$, specializes to $t_{s_1}$, the operator that switches the factors in $V\otimes V$. Thus equations (5.31) and (5.32) determine the sign of $\Phi \left(T_1\right)$ on each component. The operator $\Phi \left(E_1\right)$ is determined from $\Phi \left(T_1\right)$ via the first identity in (2.39), $$\Phi \left(E_1\right)=1-{\displaystyle \frac{\Phi \left(T_1\right)-\Phi \left(T_1^{-1}\right)}{q-q^{-1}}.}$$ Then ${\check{R}}_{VV}^2,$ $\Phi \left(T_1\right)$ and $\Phi \left(E_1\right)$ act on the components of $V\otimes V$ by $$\begin{array}{lccc}& L\left(0\right)& L\left(2{\epsilon }_1\right)& L\left({\epsilon }_1+{\epsilon }_2\right)\\ {\check{R}}_{VV}^2& q^{-2\epsilon }& q^2& q^{-2}\\ \Phi \left(T_1\right)& \epsilon q^{-y}& q& -q^{-1}\\ \Phi \left(E_1\right)& 1+\epsilon \left[y\right]& 0& 0\end{array}\text{where}\left[y\right]={\displaystyle \frac{q^y-q^{-y}}{q-q^{-1}}.}$$ The first relation in (phidefn) follows from $$\Phi \left(E_1{T}_1\right)=\epsilon y^{\mathrm{-}y}\Phi \left(E_1\right)=z^{-1}\Phi \left(E_1\right).$$ Since ${dim}_q\left(V\right)=\epsilon +\left[y\right]$, the first identity in (5.15) gives that $$\Phi \left(E_1\right)=\epsilon E_V.$$ LABEL By (5.12), the second identity in (5.15), and (5.22), $$\begin{array}{rcl}\Phi \left(E_i{T}_{i-1}{E}_i\right)& =& \epsilon \left(1\otimes E_V\right)\left({\check{R}}_{VV}\otimes 1\right)\left(1\otimes E_V\right)\epsilon =\left(id\otimes {qtr}_V\right)\left({\check{R}}_{VV}\right)\otimes E_V\\ & =& C_V^{-1}\otimes E_V=q^{⟨{\epsilon }_1,{\epsilon }_1+2\rho ⟩}id\otimes E_V=q^y\Phi \left(E_i\right),\end{array}$$ LABEL which establishes the second relation in (2.36). By (5.12), $$\begin{array}{rcl}\Phi \left(E_1{Y}_1^l{E}_1\right)& =& \epsilon \left(id\otimes E_V\right){\left(z{ℛ}_{21}ℛ\right)}^l\epsilon \left(id\otimes E_V\right)=\left(id\otimes {qtr}_V\right)\left({\left(z{ℛ}_{21}ℛ\right)}^l\right)\otimes E_V\\ & =& \epsilon \left(id\otimes {qtr}_V\right)\left({\left(z{ℛ}_{21}ℛ\right)}^l\right)\Phi \left(E_1\right)=Z_V^{\left(l\right)}\Phi \left(E_1\right),\end{array}$$ LABEL which gives the first relation in (2.37). Since the $Y_i$ commute, and $T_i{Y}_i{Y}_{i+1}=Y_i{Y}_{i+1}{T}_i,$ $$E_i{Y}_i{Y}_{i+1}=\left(1-\frac{T_i-T_i^{-1}}{q-q^{-1}}\right)Y_i{Y}_{i+1}=Y_i{Y}_{i+1}\left(1-\frac{T_i-T_i^{-1}}{q-q^{-1}}\right)=Y_i{Y}_{i+1}{E}_i.$$ The proof that $E_i{Y}_i{Y}_{i+1}=E_i$ is exactly as in the proof of [OR, Thm. 6.1(c)]: Since $\Phi \left(E_1\right)=\epsilon E_V,$ using $E_1{T}_1=z^{-1}{E}_1,$ the pictorial equalites $$\begin{array}{cccccc}\epsilon z^2·& & =\epsilon z^2·& & =\epsilon z^2{z}^{-1}·& \end{array}$$ it follows that $\Phi \left(E_1{Y}_1{Y}_2{T}_1^{-1}\right)=\epsilon \left(1\otimes E_V\right)\Phi \left(z{X}^{{\epsilon }_1}\right)\Phi \left(z{T}_1{X}^{{\epsilon }_1}\right)$ acts as $\epsilon z^2{z}^{-1}\cdot {\check{R}}_{L\left(0\right),M}{\check{R}}_{M,L\left(0\right)}\left({id}_M\otimes E_V\right).$ By (5.10), this is equal to $$\epsilon z\left(C_M\otimes C_{L\left(0\right)}\right)C_{M\otimes L\left(0\right)}^{-1}\left({id}_M\otimes E_V\right)=\epsilon z\cdot C_M{C}_M^{-1}\left({id}_M\otimes E_V\right)=z\cdot \Phi \left(D_1\right)=\Phi \left(E_1{T}_1^{-1}\right),$$ so that $\left(E_1{Y}_1{T}_1^{-1}\right)=\Phi \left(E_1{T}_1^{-1}\right).$ This establishes the second relation in (2.37). When $𝔤=𝔰{𝔩}_r$, $V\otimes V=L\left(2{\epsilon }_1\right)\oplus L\left({\epsilon }_1+{\epsilon }_2\right)$ with $S^2\left(V\right)=L\left(2{\epsilon }_1\right)$ and ${\wedge }^2\left(V\right)=L\left({\epsilon }_1+{\epsilon }_2\right)$, and $$\begin{array}{ccc}& L\left(2{\epsilon }_1\right)& L\left({\epsilon }_1+{\epsilon }_2\right)\\ \Phi \left({\check{R}}_{VV}^2\right)& q^2& q^{-2}\\ \Phi \left(T_1\right)& q& -q^{-1}\end{array}\text{so that}\Phi \left(E_1\right)={\displaystyle }\frac{\Phi \left(T_1\right)-\Phi \left(T_1^{-1}\right)}{q-q^{-1}}=0.$$ Thus the relations (2.44), (2.36) and (2.37) are satisfied. $\square$

Remark. In Theorem 1.2, if $\Phi \left(Y_1\right)$ has eigenvalues $u_1,\dots ,u_r$, then $\Phi$ is a representation of the cyclotomic BMW algebra ${𝒲}_{r,k}\left(u_1,\dots ,u_r\right)$.

By the definition of $Y_i$ in (2.33), $$\Phi \left(Y_i\right)=z{\check{R}}_{M\otimes V^{\otimes (i-1)},V}{\check{R}}_{V,M\otimes V^{\otimes (i-1)}}=z\text{PICTURE.}$$

The Schur functors are the functors $$\begin{array}{ccccc}{F}_V^{\lambda }& :& \left\{U𝔤\text{-modules}\right\}& \to & \left\{{\widetilde{ℬ}}_k\text{-modules}\right\}\\ & & M& \mapsto & {Hom}_{U𝔤}\left(M(\lambda ),M\otimes V^{\otimes k}\right)\end{array}$$ where ${Hom}_{U𝔤}\left(M(\lambda ),M\otimes V^{\otimes k}\right)$ is the vector space of highest weight $\lambda$ in $M\otimes V^{\otimes k}$.

References

[AMR] S. Ariki, A. Mathas and H. Rui Cyclotomic Nazarov Wenzl algebras, Nagoya Math. J. 182, (2006), 47-134. MR2235339 (2007d:20005)

[DRV] Z. Daugherty, A. Ram, and R. Virk, Affine and graded BMW algebras, in preparation.

[OR] R. Orellana and A. Ram, Affine braids, Markov traces and the category $𝒪$, Algebraic groups and homogeneous spaces, 423-473, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai, 2007. MR2348913 (2008m:17034)

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