Extended partial order and applications to tensor products

TL;DR

This paper extends a partial order on dominant weights of classical Lie algebras, analyzes its structure, and explores implications for tensor product dimensions and potential Schur positivity analogs.

Contribution

It introduces an extended partial order on k-tuples of weights, characterizes its extremal elements, and links the order to tensor product dimensions in classical Lie algebra representations.

Findings

The extended partial order has a unique minimal and maximal element.

Tensor product dimensions increase along the order for k=2.

The maximal element's tensor product has the largest dimension for any k.

Abstract

We extend the preorder on k-tuples of dominant weights of a simple complex Lie algebra g of classical type adding up to a fixed weight $\lambda$ defined by V. Chari, D. Sagaki and the author. We show that the induced extended partial order on the equivalence classes has a unique minimal and a unique maximal element. For k=2 we compute its size and determine the cover relation. To each k-tuple we associate a tensor product of simple g-modules and we show that for k=2 the dimension increases also along with the extended partial order, generalizing a theorem proved in the aforementioned paper. We also show that the tensor product associated to the maximal element has the biggest dimension among all tuples for arbitrary k, indicating that this might be a symplectic (resp. orthogonal) analogon of the row shuffle defined by Fomin et al. The extension of the partial order reduces the number of elements in the cover relation and may facilitate the proof of an analogon of Schur positivity along the partial order for symplectic and orthogonal types.