A basis for the Diagonal Harmonic Alternants

TL;DR

This paper uses $sl[2]$ representation theory to predict a basis for Diagonal Harmonic Alternants, providing a formula for the number of 'string starters' and connecting algebraic and geometric approaches.

Contribution

It introduces an $sl[2]$-based framework to analyze Diagonal Harmonic Alternants and derives a formula for counting 'string starters,' advancing the understanding of their structure.

Findings

$sl[2]$ acts on Diagonal Harmonics, forming representations

DHA_n decomposes into a direct sum of $sl[2]$ strings

A formula for the number of string starters is provided

Abstract

It will be shown here that there are differential operators $E,F$ and $H=[E,F]$ for each $n\ge 1$, acting on Diagonal Harmonics, yielding that $DH_n$ is a representation of $sl[2]$ (see [3] Chapter 3). Our main effort here is to use $sl[2]$ theory to predict a basis for the Diagonal Harmonic Alternants, $DHA_n$. It can be shown that the irreducible representations $sl[2]$ are all of the form $P,EP,E^2P,\cdots,E^kP$, with $FP=0$ and $E^{k+1}P=0$. The polynomial $P$ is known to be called a "String Starter". From $sl[2]$ theory it follows that $DHA_n$ is a direct sum of strings. Our main result so far is a formula for the number of string starters. A recent paper by Carlsson and Oblomkov (see [2]) constructs a basis for the space of Diagonal Coinvariants by Algebraic Geometrical tools. It would be interesting to see if any our results can be derived from theirs.