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Description

I will first describe explicit (GL_k x S_n)-modules, in k sets on n variables, whose graded Frobenius correspond (conjecturally) to the symmetric functions that occur in the rectangular shuffle theorem. I will then discuss many properties of the associated character, and show how the k-variate version (in fact one can assume that k goes to infinity) sheds new light and simplifies many aspect of the problems that have been considered in the last 25 years in relation to spaces of diagonal harmonic polynomials. I will also show how some of the properties alluded to are entirely natural in view of the natural ties that the subject seems to have with the study of (m,n)-links on the torus. I will also explain how to directly relate this to the Delta-conjecture, opening a clear path to its generalization to the rectangular context.