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 . The conjectures are motivated and supported both by extensive experimental computer algebra calculations, as well as stability properties.
Notes
1 In [9], the relevant object are denoted , with
the multiplicity vector of parts of τ.
2 Neither involving the θi.
3 The are assumed to form a complete list of Sn-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 occurs as the highest t-degree component of the modified Macdonald polynomials
.
7 Calculated in part using Sagemath code made available by the authors of [9].
8 With positive integer coefficients for terms.
9 Recall that the plethysm sends a Schur function
to the sum
(5.1)
where μ runs over all partitions such that is an horizontal strip (H.S.). This is to say that no two cells of the skew shape
lie in the same column. In our framework, we have the inverse plethysm equivalence:
iff
.