The MacMahon R-matrix

TL;DR

This paper introduces a new R-matrix for MacMahon representations of the DIM algebra, acting on 3D Young diagrams and preserving symmetry among deformation parameters, with applications to intertwiners.

Contribution

It constructs an R-matrix for MacMahon representations of DIM algebra and demonstrates its role in permuting intertwiners involving 3D Young diagrams.

Findings

Defined an R-matrix acting on 3D Young diagrams

Preserves symmetry among deformation parameters q, t^{-1}, t/q

Shows intertwiners are permuted by the MacMahon R-matrix

Abstract

We introduce an $R$-matrix acting on the tensor product of MacMahon representations of Ding-Iohara-Miki (DIM) algebra $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$. 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 $\frac{t}{q}$. 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.