Extended partial order and applications to tensor products

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 λ defined by Chari, Sagaki and the author [Posets, tensor products and Schur positivity, Algebra and Number Theory, to appear]. 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 (respectively, orthogonal) analogue of the row shuffle defined by Fomin et al. [Amer. J. Math. 127 (2005), 101–127]. The extension of the partial order reduces the number elements in the cover relation and may facilitate the proof of an analogue of Schur positivity along the partial order for symplectic and orthogonal types.