Tensorial generalization of characters

Abstract

In rainbow tensor models, which generalize rectangular complex matrix model (RCM) and possess a huge gauge symmetry $U(N_1)\times\ldots\times U(N_r)$, 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: $\Big<\chi_{R_1,\ldots,R_r}\Big> \sim C_{R_1,\ldots, R_r}D_{R_1}(N_1)\ldots D_{R_r}(N_r)$ of representations $R_i$ of the linear group $SL(N_i)$, with $C_{R_1,\ldots, R_r}$ made of the Clebsch-Gordan coefficients of representations $R_i$ of the symmetric group. Moreover, not only the averages but the operators $\chi_{\vec R}$ themselves exist only when these $C_{\vec R}$ 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 $\Big<{\rm character}\Big> \sim { character}$, which opens a way to lift the notion and the theory of characters (Schur functions) from matrices to tensors. In particular, operators $\chi_{\vec R}$ are eigenfunctions of operators which generalize the usual cut-and-join operators $\hat W$; they satisfy orthogonality conditions similar to the standard characters, but they do not form a {\it full} linear basis for all gauge-invariant operators, only for those which have non-vanishing Gaussian averages.