$(G{L}_k\times S_n)$-Modules of Multivariate Diagonal Harmonics

Abstract

This is the first in a series of papers in which we describe explicit structural properties of spaces of diagonal rectangular harmonic polynomials in $k$ sets of $n$ variables, both as $G{L}_k$-modules and $S_n$-modules, as well as some of there relations to areas such as Algebraic Combinatorics, Representation Theory, Algebraic Geometry, Knot Theory, and Theoretical Physics. Our global aim is to develop a unifying point of view for several areas of research of the last two decades having to do with Macdonald Polynomials Operator Theory, Diagonal Coinvariant Spaces, Rectangular-Catalan Combinatorics, the Delta-Conjecture, Hilbert Scheme of Points in the Plane, Khovanov-Rozansky Homology of $(m,n)$-Torus links, etc.

arXiv comment26 pages, 1 figure

References in corpus (8)

• Toric braids and $(m,n)$-parking functions
• The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra
• Homology of torus knots
• Higher Trivariate Diagonal Harmonics via generalized Tamari Posets
• Rational parking functions and Catalan numbers
• Multivariate Diagonal Coinvariant Spaces for Complex Reflection Groups
• (GL_k x S_n)-modules and nabla of hook-indexed Schur functions
• On the Schur positivity of ${\Delta }_{e_2}{e}_n[X]$

Cited by in corpus (3)

• The Hilbert series of the superspace coinvariant ring
• Lefschetz theory for exterior algebras and fermionic diagonal coinvariants
• The Bosonic-Fermionic Diagonal Coinvariant Modules Conjecture