The groups $G_{H,1,n}$

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 20 June 2010

The groups $G_{H,1,n}$

The semidirect product $G_{H,1,n}=H^n⋊S^n$ is the group of permutations with edges colored by elements of $h.$ The product is the usual product on permutations with the convention that elements of $H$ slide along the edges and mutliply when they collide. The group $G_{H,1,n}$ is generated by the elements

and the subgroup $H^n$ consists of the elements $$t_h=t_1\left(h_1\right)t_2\left(h_2\right)\dots t_n\left(h_n\right),\text{where }h=\left(h_1,\dots ,h_n\right),h_i\in H.$$ The operation in the semidirect product in the group $$G_{H,1,n}=H^n⋊S_n=\left\{t_{h}w|h\in H^n,w\in S_n\right\}$$ is determined by the product in $S_n,$ and $$t_h{t}_k=t_{hk},\text{ for }h,k\in H^n,$$ and $$w{t}_h=t_{wh}w,\text{where }w\left(h_1,\dots ,h_n\right)=\left(h_{w\left(1\right)},\dots ,h_{w\left(n\right)}\right),$$ for $w\in S_n$ and $h=\left(h_1,\dots ,h_n\right)\in H^n.$

Let $H^{*}$ be an index set for the conjugacy classes of $H$ and let $H_{\alpha },\alpha \in H^{*},$ be a set of conjugacy class representatives. Let $$G_{H,1,n}^{*}=\left\{\mu ={\left({\mu }^{\left(\alpha \right)}\right)}_{\alpha \in H^{*}}|\mu \text{ has }n\text{ boxes in total}\right\},$$ be the set of $H^{*}$-multipartitions, tuples of partitions with components indexed by the elements of $H^{*},$ such that the total number of boxes in the mutlipartition is $n.$ Then the elements $${\gamma }_{\mu }=\text{PICTURE},\text{for }\mu \in G_{H,1,n}^{*},$$ are a set of conjugacy class representatives for $G_{H,1,n}.$ The centraliser of ${\gamma }_{\mu }$ in $G_{H,1,n}$ is $$Z_G\left({\gamma }_{\mu }\right)=?$$ with $$\mathrm{Card}\left(Z_G\left(\mu \right)\right)=?.$$

Each element of $G\left(r,p,n\right)$ is conjugate by elements of $S_n$ to a disjoint product of cycles of the form $${\xi }_i^{{\lambda }_i}\dots {\xi }_k^{{\lambda }_k}\left(i,i+1,\dots ,k\right).$$ By conjugating this cycle by ${\xi }_i^{-c}{\xi }_{i+1}^{{\lambda }_i}{\xi }_{i+2}^{{\lambda }_i+{\lambda }_{i+1}}\dots {\xi }_k^{{\lambda }_i+\dots +{\lambda }_{k-1}}\in G\left(r,r,n\right),$ we have $${\xi }_i^{-c}{\xi }_k^{c+{\lambda }_i+\dots +{\lambda }_k}\left(i,\dots ,k\right),\text{where }c=\left(k-i\right){\lambda }_i+\left(k-i-1\right){\lambda }_{i+1}+\dots +{\lambda }_{k-1}.$$ If $i_1,i_2,\dots ,i_l$ denote the minimal indices of the cycles and $c_1,\dots ,c_l$ are the numbers $c$ for the various cycles, then after conjugating by ${\xi }_{i_1}^{c_1}\dots {\xi }_{i_{l-1}}^{c_{l-1}}{\xi }_{i_l}^{-\left(c_1+\dots +c_{l-1}\right)}\in G\left(r,r,n\right),$ each cycle becomes $${\xi }_k^{{\lambda }_i+\dots +{\lambda }_k}\left(i,\dots ,k\right)\text{except the last, which is }{\xi }_{i_l}^{-a}{\xi }_n^b\left(i_l,\dots ,n\right),$$ where $a=c_1+\dots +c_l$ and $b=a+{\lambda }_{i_l}+\dots +{\lambda }_n.$ If $k=n-i_l+1$ is the length of the last cycle, then conjugating the last cycle by ${\xi }_{i_l}^{k-1}{\xi }_{i_l+1}^{-1}\dots {\xi }_n^{-1}\in G\left(r,r,n\right)$ gives $${\xi }_{i_l}^{-a+k}{\xi }_n^{b-k}\left(i_l,\dots ,n\right).$$ If we conjugate the last cycle by ${\xi }_{i_l}^p\in G\left(r,p,n\right),$ we have $${\xi }_{i_l}^{-a+p}{\xi }_n^{b-p}\left(i_l,\dots ,n\right).$$ In summary, any element $g$ of $G\left(r,p,n\right)$ is conjugate to a product of disjoint cycles where each cycle is of the form $${\xi }_k^a\left(i,i+1,\dots ,k\right),0\le a\le r-1,$$ except possibly the last cycle, which is of the form $${\xi }_{i_l}^a{\xi }_n^b\left(i_l,i_l+1,\dots ,n\right)\text{with }0\le a\le \gcd \left(p,k\right)-1,$$ where $k=n-i_l+1$ is the length of the last cycle.

Let $Z_{G\left(r,p,n\right)}\left(g\right)=\left\{h\in G\left(r,p,n\right)|hg=gh\right\}$ denote the centraliser of $g\in G\left(r,p,n\right).$ Since $G\left(r,p,n\right)$ is a subgroup of $G\left(r,1,n\right),$ $$Z_{G\left(r,p,n\right)}\left(g\right)=Z_{G\left(r,1,n\right)}\left(g\right)\cap G\left(r,p,n\right),$$ for any element $g\in G\left(r,p,n\right).$ Suppose that $g$ is an element of $G\left(r,1,n\right)$ which is a product of disjoint cycles of the form ${\xi }_k^a\left(i,\dots ,k\right)$ and that $h\in G\left(r,1,n\right)$ commutes with $g.$ Conjugating $g$ by $h$ effects some combination of the following operations on the cycles of $g:$

1. permuting cycles of the same type, ${\xi }_k^a\left(i,\dots ,k\right)$ and ${\xi }_m^b\left(j,\dots ,m\right)$ with $b=a$ and $k-i=m-j,$
2. conjugating a single cycle ${\xi }_k^a\left(i,\dots ,k\right)$ by powers of itself, and
3. conjugating a single cycle ${\xi }_k^a\left(i,\dots ,k\right)$ by ${\xi }_i^b\dots {\xi }_k^b,$ for any $0\le b\le r-1.$

Furthermore, the elements of $G\left(r,1,n\right)$ which commute with $g$ are determined by how they "rearrange" the cycles of $g$ and a count (see [Mac, p170]) of the number of such operations shows that if $g\in G\left(r,1,n\right)$ and $m_{a,k}$ is the number of cycles of type ${\xi }_{i+k}^a\left(i,i+1,\dots ,i+k\right)$ for $g,$ then $$\mathrm{Card}\left(Z_{G\left(r,,,1,,,n\right)}\left(g\right)\right)=\prod _{a,k}\left(m_{a,k}!.k^{m_{a,k}}.r\right).$$

Let $\hat{H}$ be an index set for the irreducible $H$ modules. If $\gamma \in \hat{H}$ then let ${\hat{H}}^{\gamma }$ be an index set for a basis of $H^{\gamma }$ so that $$H^{\gamma }\text{ has basis }\left\{m_P|P\in {\hat{H}}^{\gamma }\right\},\text{with }H\text{-action }h{m}_P=\sum _{Q\in {\hat{H}}^{\gamma }}{h}_{QP}{m}_Q,$$ for appropriater constants $h_{QP}\in ℂ.$

Let $\lambda ={\left({\lambda }^{\left(\alpha \right)}\right)}_{\alpha \in \hat{H}}$ be a $\hat{H}$-tuple of partitions with $n$ boxes in total. A standard tableau of shape $\lambda$ is a filling of the boxes of $\lambda$ with $1,2,\dots ,n$ such that, in each partition ${\lambda }^{\left(\alpha \right)},$

1. the rows increase from left to right,
2. the columns increase from top to bottom.

The rows and columns of each partition ${\lambda }^{\left(\alpha \right)}$ are numbered as for matrices and

1. $T\left(i\right)$ is the box containing $i$ in $T,$
2. $c\left(b\right)=j-i,$ if $b$ is in position $\left(i,j\right),$ and
3. $s\left(b\right)=\alpha ,$ if $b$ is in ${\lambda }^{\left(\alpha \right)}.$

The numbers $c\left(b\right)$ and $s\left(b\right)$ are the content and the $\hat{H}$-type of the box $b,$ respectively.

PICTURE

Use notations as in (???) and (???).

1. The irreducible representations $G_n^{\lambda }$ of the group $G_{H,1,n}=H^n⋉S_n$ are indexed by the set of $\hat{H}$ multipartitions with $n$ boxes in total.
2. $\dim G_n^{\lambda }=\sum _{\alpha \in \hat{H}}\dim \left(H^{\alpha }\right)\dim \left(S_n^{{\lambda }^{\left(\alpha \right)}}\right).$
3. The irreducible $G_{H,1,n}$ module $$G_{r,1,n}^{\lambda }\text{ has basis }\left\{\left(m_{P^{\left(1\right)}}\otimes \dots \otimes m_{P^{\left(n\right)}}\right)\otimes v_T|T\in {\hat{S}}^{\lambda },P^{\left(i\right)}\in {\hat{H}}^{s\left(T\left(i\right)\right)}\right\}$$ with $G_{H,1,n}$ action given by $$t_i\left(h\right)\left(m_{P^{\left(1\right)}}\otimes \dots \otimes m_{P^{\left(n\right)}}\otimes v_T\right)=m_{P^{\left(1\right)}}\otimes \dots \otimes h{m}_{P^{\left(i\right)}}\otimes \dots \otimes m_{P^{\left(n\right)}}\otimes v_T,$$$$s_i\left(m_{P^{\left(1\right)}}\otimes \dots \otimes m_{P^{\left(n\right)}}\otimes v_T\right)=s_i\left(m_{P^{\left(1\right)}}\otimes \dots \otimes m_{P^{\left(n\right)}}\right)\otimes s_i{v}_T,$$ where $$s_i{v}_T={\left(s_i\right)}_{TT}{v}_T+\left(1+{\left(s_i\right)}_{TT}\right)v_{s_{i}T},$$ with
   1. ${\left(s_i\right)}_{TT}=\left\{\begin{array}{rcl}\frac{1}{c\left(T\left(i\right)\right)-c\left(T\left(i-1\right)\right)},& \text{if}& s\left(T\left(i\right)\right)=s\left(T\left(i-1\right)\right),\\ 0,& \text{if}& s\left(T\left(i\right)\right)\ne s\left(T\left(i-1\right)\right),\end{array}\right\}$
   2. $c\left(T\left(i\right)\right)$ is the content of the box containing $i$ in $T,$
   3. $s_{i}T$ is the same as $T$ except that $i$ and $i-1$ are switched,
   4. $v_{s_{i}T}=0$ if $s_{i}T$ is not standard.

		Proof.
	The following argument determining the simple $G_{H,1,n}$ modules is often called Clifford theory. Let $G^{\lambda }$ be a simple $G_{H,1,n}=H^n⋊S_n$ module. Let $H^{\gamma }$ be a simple $H^n$ submodule of $G^{\lambda }.$ Then $w{H}^{\gamma }$ is another simple $H^n$ submodule of $G^{\lambda }$ and $$G^{\lambda }=\sum _{w\in S_n}w{H}^{\gamma },$$ since the right hand side is an $H^n⋊S_n$ submodule of $G^{\lambda }.$ Let $$S_{\gamma }=\left\{w\in S_n|w{H}^{\gamma }\cong H^{\gamma }\right\}=\left\{w\in S_n|w\gamma =\gamma \right\}.$$ Thus $$G^{\lambda }=\sum _{w_i\in S_n/S_{\gamma }}{w}_{i}N={\mathrm{Ind}}_{H^n⋊S_{\gamma }}^{H^n⋊H_n}\left(N\right),\text{ where }N=\sum _{w\in S_{\gamma }}w{H}^{\gamma }.$$ Then $$N\cong H^{\gamma }\otimes S_{\gamma }^{\lambda }\text{ with action }hw\left(m\otimes v\right)=hwm\otimes wm,$$ where $S_{\gamma }^{\lambda }$ is a simple $S_{\gamma }$ module. Since we are free to choose $\gamma$ in its $S_n$ orbit we may assume that $\gamma$ is of the form $$\gamma =\left({\gamma }_1,\dots ,{\gamma }_1\left({\mu }_1\text{ times}\right),{\gamma }_2,\dots ,{\gamma }_2\left({\mu }_2\text{ times}\right),\dots ,{\gamma }_l,\dots ,{\gamma }_l\left({\mu }_l\text{ times}\right)\right)\text{ so that }{S}_{\gamma }=S_{{\mu }_1}\times S_{{\mu }_2}\times \dots \times S_{{\mu }_l},$$ where $\mu =\left({\mu }_1,\dots ,{\mu }_n\right)$ is a partition of $n.$ An irreducible representation of $S_n$ is indexed by a tuple of partitions, one partition for each ${\gamma }_i$ that appears in $\gamma ,$ so that the total number of boxes in the tuple of partitions is $n.$ Let us make this construction more explicit. Using the notation in (???), the simple $H^n$ modules are indexed by the set ${\hat{H}}^n$ and a simple $\hat{H}$ module $$H^{\left({\gamma }_1,\dots ,{\gamma }_n\right)}\text{has basis }\left\{m_{P^{\left(1\right)}}\otimes \dots m_{P^{\left(n\right)}}|P^{\left(i\right)}\in {\hat{H}}^{{\gamma }_i}\right\}.$$ Then the action of $S_n$ on $H^n$ modules in (???) is given by $$w\left(m_{P^{\left(1\right)}}\otimes \dots \otimes m_{P^{\left(n\right)}}\right)=m_{P^{w\left(1\right)}}\otimes \dots m_{P^{w\left(n\right)}},\text{for }w\in S_n,$$ defines an action of $S_n$ on the irreducible $H^n$ modules. The resulting action of $S_n$ on ${\hat{H}}^n$ is given by $$w\left({\gamma }_1,\dots ,{\gamma }_n\right)=\left({\gamma }_{w\left(1\right)},\dots ,{\gamma }_{w\left(n\right)}\right).$$ Returning to the setup in question (???), $$w{H}^{\gamma }=H^{w\gamma },\text{ for }w\in S_n.$$ (The fact that $w{H}^{\gamma }=H^{w\gamma }$ means that, in this case, the cocycles (factor sets) that appear in Clifford theory are trivial.) $\square$

The Casimir element is the sum of the elements in the conjugacy class of $s_{12},$$${\kappa }_n=\sum _{1\le i<j\le n}\sum _{h\in H}{t}_i\left(h\right)t_j\left(h^{-1}\right)s_{ij},$$ with notations as in (???).

1. The Casimir element ${\kappa }_n$ for $G_{H,1,n}$ is a central element of the group algebra of $G_{H,1,n}$ such that $${\kappa }_n\text{ acts on }{G}_n^{\lambda }\text{ by the constant }\sum _{b\in \lambda }c\left(b\right).$$
2. Let $H^{*}$ be an index set for the conjugacy classes of $H$ and let $\mu \in H^{*}.$ Let $$z\left(\mu \right)=\sum _{h\in {𝒞}_{\mu }}\sum _{i=1}^n{t}_i\left(h\right),$$ where the sum is over all elements of $H$ in the conjugacy class $\mu .$ Then $z\left(\mu \right)$ is an element of the center of the group algebra of $G_{H,1,n}$ and $$z\left(\mu \right)\text{ acts on }{G}_n^{\lambda }\text{ by the constant }\sum _{\alpha \in \hat{H}}\frac{{\chi }_H^{\alpha }\left(\mu \right)}{\dim \left(H^{\alpha }\right)}.$$

		Proof.
	$$\begin{array}{rcl}\frac{1}{\left|H\right|}\sum _{h\in H}h{m}_Q\otimes h^{-1}{m}_P& =& \sum _{h,R,S}{A}_{RQ}^{\alpha }\left(h\right)A_{SP}^{\beta }\left(h^{-1}\right)\left(m_R\otimes m_S\right)\\ & =& \sum _{R,S}\left(m_R\otimes m_S\right)\sum _{h\in H}{A}_{RQ}^{\alpha }\left(h\right)A_{SP}^{\beta }\left(h^{-1}\right).\end{array}$$ define an element of $\mathrm{Hom}\left(H^{\alpha },H^{\beta }\right),$$${\phi }_{QS}^{\alpha \beta }:H^{\alpha }\to H^{\beta },\text{by }{\phi }_{QS}^{\alpha \beta }\left(m_P\right)={\delta }_{QP}{m}_S.$$ Then, as elements of $\mathrm{Hom}\left(H^{\alpha },H^{\beta }\right),$ $$g\sum _{h\in H}h{\phi }_{QS}^{\alpha \beta }{h}^{-1}=\sum _{h\in H}gh{\phi }_{QS}^{\alpha \beta }{\left(gh\right)}^{-1}g,$$ and $$\mathrm{Tr}\left(\sum _{h\in H}h{\phi }_{QS}^{\alpha \alpha }{h}^{-1}\right)=\mathrm{Tr}\left(\sum _{h\in H}{\phi }_{QS}^{\alpha \alpha }\right)=\left|H\right|{\delta }_{QS},$$ and thus, by Schur's lemma, $$\sum _{h\in H}h{\phi }_{QS}^{\alpha \beta }{h}^{-1}=\frac{\left|H\right|}{d_{\lambda }}{\delta }_{\alpha \beta }{\delta }_{QS}\mathrm{id}$$ where $d_{\alpha }=\dim \left(H^{\alpha }\right).$ Thus $$\sum _{h\in H}{A}_{RQ}^{\alpha }\left(h\right)A_{SP}^{\beta }\left(h^{-1}\right)={\left(\sum _{h\in H}{A}^{\alpha }\left(h\right){\phi }_{QS}{A}^{\beta }\left(h^{-1}\right)\right)}_{RP}=\frac{\left|H\right|}{d_{\alpha }}{\delta }_{\alpha \beta }{\delta }_{QS}{\delta }_{RP}.$$ Hence $$\frac{1}{\left|H\right|}\sum _{h\in H}h{m}_Q\otimes h^{-1}{m}_P=\frac{1}{d_{\alpha }}{\delta }_{\alpha \beta }\left(m_P\otimes m_Q\right).$$ Let $x_1=0$ and, for $2\le k\le n$ let $$x_k=\sum _{h\in H}\sum _{1\le i<k}{t}_i\left(h\right)t_k\left(h^{-1}\right)s_{ik}\text{so that }{x}_1+x_2+\dots +x_n={\kappa }_n.$$ Then $x_k=s_k{x}_k{s}_k+\sum _{h\in H}{t}_{k-1}\left(h\right)s_k{t}_{k-1}\left(h^{-1}\right)$ and $$\begin{array}{rcl}{x}_k\left(m_P\otimes v_T\right)& =& \left(s_k{x}_{k-1}{s}_k+\sum _{h\in H}{s}_k{t}_k\left(h\right)t_{k-1}\left(h^{-1}\right)\right)\left(m_P\otimes v_T\right)\\ & =& s_k\left(c\left(T\left(k-1\right)\right){\left(s_k\right)}_{TT}\left(s_k{m}_P\otimes v_T\right)+c\left(T\left(k\right)\right)\left(1+{\left(s_k\right)}_{TT}\right)\left(s_k{m}_P\otimes v_{s_{k}T}\right)+\sum _{h\in H}{t}_{k-1}\left(h^{-1}\right)t_k\left(h\right)\left(m_P\otimes v_T\right)\right)\\ & =& s_k\left(c\left(T\left(k\right)\right)s_k\left(m_P\otimes v_T\right)+\left(c\left(T\left(k-1\right)\right)-c\left(T\left(k\right)\right)\right){\left(s_k\right)}_{TT}\left(m_P\otimes v_T\right)+\sum _{h\in H}{t}_{k-1}\left(h^{-1}\right)t_k\left(h\right)\left(m_P\otimes v_T\right)\right)\\ & =& \left\{\begin{array}{rcl}{s}_k\left(c\left(T\left(k\right)\right)s_k{v}_T+\left(\left(-1\right)+1\right)\right)v_T,& \text{if}& s\left(T\left(k\right)\right)=s\left(T\left(k-1\right)\right)\\ s_k\left(c\left(T\left(k\right)\right)s_k{v}_T+\left(\left(0\right)+0\right)\right)v_T,& \text{if}& s\left(T\left(k\right)\right)\ne s\left(T\left(k-1\right)\right)\end{array}\right.\\ & =& c\left(T\left(k\right)\right)v_T.\end{array}$$ Since ${\sum }_{h\in 𝒞\mu }$ ????????????????? acts on $H^{\alpha }$ by the constant $$\frac{{\chi }_H^{\alpha }\left(\mu \right)}{\dim \left(H^{\alpha }\right)}$$ it follows that $$\left(\sum _{h\in {𝒞}_{\mu }}{t}_i\left(h\right)\right)\left(m_P\otimes v_T\right)=\sum _{h\in {𝒞}_{\mu }}\left(t_i\left(h\right)m_P\otimes v_T\right)=\frac{{\chi }_H^{\left({\alpha }^i\right)}\left(\mu \right)}{\dim \left(H^{\left({\alpha }_i\right)}\right)}\left(m_P\otimes v_T\right).$$ Part (b) of the theorem now follows by summing over $i.\square$

For example, if $$\begin{array}{rcl}H=\mathbb{Z}/r\mathbb{Z}& \text{if}& \hat{H}=\left\{0,1,2,\dots ,r-1\right\},\\ H=\mathbb{Z},& \text{if}& \hat{H}={ℂ}^{*},\\ H={ℂ}^{*},& \text{if}& \hat{H}=\mathbb{Z},\end{array}$$ where in the last case $\hat{H}$ indexes the rational representations of $H={ℂ}^{*}.$

Characters of $G_{H,1,n}$

Let $\mu \in G_n^{*}.$ Then $$\left|Z_{G_n}\left(\mu \right)\right|=\sum _{\beta \in H^{*}}\left(\left|Z_{S_n}\left({\mu }^{\left(\beta \right)}\right)\right|{\left|Z_H\left(\beta \right)\right|}^{l\left({\mu }^{\left(\beta \right)}\right)}\right).$$ Let $H^{*}$ be an index set for the conjugacy classes of $H$ and, for each $\beta \in H^{*},$ let $$x^{\left(\beta \right)}=\left\{x_1^{\left(\beta \right)},x_2^{\left(\beta \right)},\dots \right\}\text{be a set of variables indexed by }\beta ,$$ and let $$p_{\mu }\left(x\right)=\prod _{\beta \in H^{*}}{p}_{{\mu }^{\left(\beta \right)}}\left(x^{\left(\beta \right)}\right),\text{for }\mu \in G_n^{*},$$ so that $p_{\mu }\left(x\right)$ is the product of power symmetric functions from each of the variable sets $x^{\left(\beta \right)}.$ Define a "change of variables" from the $x^{\left(\beta \right)}$ variables, which are indexed by $\beta \in H^{*},$ to $y^{\left(\alpha \right)}$ variables indexed by $\alpha \in \hat{H},$ by setting $$p_r\left(y^{\left(\alpha \right)}\right)=\sum _{\beta \in H^{*}}\frac{{\chi }_H^{\alpha }\left(\beta \right)p_r\left(x^{\left(\beta \right)}\right)}{\left|Z_H\left(\beta \right)\right|},\text{for each }\alpha \in \hat{H},\text{ and each }r\in {\mathbb{Z}}_{>0}.$$ Define $$s_{\lambda }\left(y\right)=\prod _{\alpha \in \hat{H}}{s}_{{\lambda }^{\left(\alpha \right)}}\left(y^{\left(\alpha \right)}\right),\text{for each }\lambda \in {\hat{G}}_n.$$ Then $$s_{\lambda }\left(y\right)=\sum _{\mu \in {\hat{G}}_n^{*}}\frac{{\chi }_{G_n}^{\lambda }\left(\mu \right)p_{\mu }\left(x\right)}{\left|Z_{G_n}\left(\mu \right)\right|}\text{ and }{p}_{\mu }\left(x\right)=\sum _{\lambda \in {\hat{G}}_n}$$ χ Gn λ μ sλ y , for $\lambda \in {\hat{G}}_n$ and $\mu \in G_n^{*}.$

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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