We relate the n! conjecture (by Garsia and Haiman) to the geometry of
principal nilpotent pairs, and state a conjecture generalizing the n!
conjecture to arbitrary semisimple algebraic groups. We also show, using
Borel's fixed point theorem, how to reduce the n! conjecture to staircase
partitions. Finally we study the interplay between characteristic p and the n!
conjecture for box partitions.Comment: Main conjectures has changed, 28 page

Kumar, Shrawan

Thomsen, Jesper Funch

English

arXiv

The aim of this paper is to formulate a conjecture for an arbitrary simple Lie algebra g in terms of the geometry of principal nilpotent pairs. When g is specialized to sln, this conjecture readily implies the n! result and it is very likely that, in fact, it is equivalent to the n! result in this case. In addition, this conjecture can be thought of as generalizing an old result of Kostant. In another direction, we show that to prove the validity of the n! result for an arbitrary n and an arbitrary partition of n, it suffices to show its validity only for the staircase partitions

A conjectural generalization of n! result to arbitrary groups

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