Abstract
In rainbow tensor models, which generalize rectangular complex matrix model (RCM) and possess a huge gauge symmetry U(N1) × … × U(Nr), we introduce a new sub-basis in the linear space of gauge invariant operators, which is a redundant basis in the space of operators with non-zero Gaussian averages. Its elements are labeled by r-tuples of Young diagrams of a given size equal to the power of tensor field. Their tensor model averages are just products of dimensions: \( \left\langle \chi {R}_1,\dots, {R}_r\right\rangle \sim {C}_{R_1},\dots {,}_{R_r}\left({N}_1\right)\dots {D}_{R_r}\left({N}_r\right) \) of representations Ri of the linear group SL(Ni), with\( {C}_{R_1},\dots {,}_{R_r} \), made of the ClebschGordan coefficients of representations Ri of the symmetric group. Moreover, not only the averages, but the operators \( {\chi}_{\overrightarrow{R}} \) themselves exist only when these \( {C}_{\overrightarrow{R}} \) are non-vanishing. This sub-basis is much similar to the basis of characters (Schur functions) in matrix models, which is distinguished by the property \character) ~ character, which opens a way to lift the notion and the theory of characters (Schur functions) from matrices to tensors. In particular, operators \( {\chi}_{\overrightarrow{R}} \) are eigenfunctions of operators which generalize the usual cut-andjoin operators \( \hat{W} \); they satisfy orthogonality conditions similar to the standard characters, but they do not form a full linear basis for all gauge-invariant operators, only for those which have non-vanishing Gaussian averages.
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Itoyama, H., Mironov, A. & Morozov, A. Tensorial generalization of characters. J. High Energ. Phys. 2019, 127 (2019). https://doi.org/10.1007/JHEP12(2019)127
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DOI: https://doi.org/10.1007/JHEP12(2019)127