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Inazuma Eleven GO Strikers 2013 for Wii, Wii ISO.Q: An orthogonal representation of the Temperley-Lieb algebra An orthogonal representation of the Temperley-Lieb algebra is defined as follows. We first recall that the Temperley-Lieb algebra $TL_n$ is an algebra over the field $\mathbb{C}$ of complex numbers with $n$ generators $1,e_1,e_2,…,e_{n-1},$ satisfying the relations: $$e_ie_j=\delta_{ij}e_i, \;\;\; e_i^2=e_i, \;\;\; 1+e_i+e_ie_i=0, \;\;\; i=1,…,n-1.$$ We define the linear map $X\to End_\mathbb{C}(TL_n)$ with $X\in TL_n$ by $$X\mapsto \sum_{i,j}e_jXe_i.$$ This map is clearly a representation of $TL_n$ (in fact, an orthogonal representation), and it is easy to see that every representation of $TL_n$ is of this form. I am interested in the following question: Is there a characterization of the representations of $TL_n$ which are orthogonal? I.e., do orthogonal representations of $TL_n$ contain any information about the algebra $TL_n$ itself? Any help is appreciated. A: The representations you describe are trivial. The Temperley-Lieb algebra is the invariant algebra for $S_3$. So, a representation of the algebra determines a 3-dimensional space for $S_3$ acting on, and invariant for the action of the subgroup, the 3-cycle $\{1, (12), (123)\}$. The only invariant subspaces for this action are the trivial ones. So, a representation is trivial unless the two-dimensional space it is the restriction of the trivial representation is the invariant space for the three element cycle. If it is, there is no “internal” action on this subspace. Then, this two-dimensional space is the trivial representation. The representations are obviously not

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