Tensorial generalization of characters

Abstract

Abstract In rainbow tensor models, which generalize rectangular complex matrix model (RCM) and possess a huge gauge symmetry U(N 1) × … × 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: χ R 1 … R r ∼ C R 1 , … , R r N 1 … D R r N r $$ \left\langle \chi {R}_1,\dots, {R}_r\right\rangle \sim {C}_{R_1},\dots {,}_{R_r}\left({N}_1\right)\dots {D}_{R_r}\left({N}_r\right) $$ of representations R i of the linear group SL(N i ), with C R 1 , … , R r $$ {C}_{R_1},\dots {,}_{R_r} $$ , made of the Clebsch­Gordan coefficients of representations R i of the symmetric group. Moreover, not only the averages, but the operators χ R → $$ {\chi}_{\overrightarrow{R}} $$ themselves exist only when these C R → $$ {C}_{\overrightarrow{R}} $$ 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 χ R → $$ {\chi}_{\overrightarrow{R}} $$ are eigenfunctions of operators which generalize the usual cut-and­join operators W ̂ $$ \hat{W} $$ ; 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.

Keywords

• Matrix Models
• Random Systems