The MacMahon R-matrix

Abstract

Abstract We introduce an R-matrix acting on the tensor product of MacMahon representations of Ding-Iohara-Miki (DIM) algebra $$ {U}_{q,t}\left({\widehat{\widehat{\mathfrak{gl}}}}_1\right) $$ U q , t g l ^ ^ 1 . This R-matrix acts on pairs of 3d Young diagrams and retains the nice symmetry of the DIM algebra under the permutation of three deformation parameters q, t −1 and $$ \frac{t}{q} $$ t q . We construct the intertwining operator for a tensor product of the horizontal Fock representation and the vertical MacMahon representation and show that the intertwiners are permuted using the MacMahon R-matrix.