Triangular Diagonal Harmonics Conjectures

Abstract

The purpose of this paper is to present conjectures that extend to any “triangular” partitions (partitions “under any line” in the terminology of Blasiak-Haiman-Morse-Pun-Seelinger), properties of the Frobenius transform of multivariate diagonal harmonics modules. In their simplest version, these last modules correspond to the special case of “staircase” partitions, that is of the form $(n-1,n-2,\dots ,1)$. The conjectures are motivated and supported both by extensive experimental computer algebra calculations, as well as stability properties.

KEYWORDS:

• triangular partitions
• symmetric functions
• multivariate diagonal harmonics

Notes

1 In [9], the relevant object are denoted ${\mathcal{H}}_{\mathit{b}}{(x)}_{\text{pol}}$, with $\mathit{b}=\mathit{b}(\tau )$ the multiplicity vector of parts of τ.

2 Neither involving the θ_i.

3 The ${\mathcal{V}}_{\mu }$ are assumed to form a complete list of S_n-irreducible.

4 With a suitable indexing, corresponding to Specht-modules.

5 We here exploit the non-commutativity of tensors, as well as their bilinearity.

6 The Whittaker polynomial $W_{\mu }(q;\mathit{z})$ occurs as the highest t-degree component of the modified Macdonald polynomials ${\tilde{H}}_{\mu }(q,t;\mathit{z})$.

7 Calculated in part using Sagemath code made available by the authors of [9].

8 With positive integer coefficients for $s_{\lambda }\otimes s_{\mu }$ terms.

9 Recall that the plethysm $p_k(\mathit{q}+1)=p_k(\mathit{q})+1$ sends a Schur function $s_{\rho }$ to the sum

$s_{\rho }(\mathit{q}+1)=\sum _{\rho /\mu \mathrm{H}.\mathrm{S}.}{s}_{\mu }(\mathit{q}),$ (5.1)

where μ runs over all partitions such that $\rho /\mu$ is an horizontal strip (H.S.). This is to say that no two cells of the skew shape $\rho /\mu$ lie in the same column. In our framework, we have the inverse plethysm equivalence: $f(\mathit{q})=g(\mathit{q}-1)$ iff $g(\mathit{q})=f(\mathit{q}+1)$.

Additional information

Funding

This work was supported by NSERC.