73 research outputs found
Comparing skew Schur functions: a quasisymmetric perspective
Reiner, Shaw and van Willigenburg showed that if two skew Schur functions s_A
and s_B are equal, then the skew shapes A and B must have the same "row overlap
partitions." Here we show that these row overlap equalities are also implied by
a much weaker condition than skew Schur equality: that s_A and s_B have the
same support when expanded in the fundamental quasisymmetric basis F.
Surprisingly, there is significant evidence supporting a conjecture that the
converse is also true.
In fact, we work in terms of inequalities, showing that if the F-support of
s_A contains that of s_B, then the row overlap partitions of A are dominated by
those of B, and again conjecture that the converse also holds. Our evidence in
favor of these conjectures includes their consistency with a complete
determination of all F-support containment relations for F-multiplicity-free
skew Schur functions. We conclude with a consideration of how some other
quasisymmetric bases fit into our framework.Comment: 26 pages, 7 figures. J. Combin., to appear. Version 2 includes a new
subsection (5.3) on a possible skew version of the Saturation Theore
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 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.Comment: 16 pages, final version, to appear in AJo
Inequalities between Littlewood–Richardson coefficients
We prove that a conjecture of Fomin, Fulton, Li, and Poon, associated to
ordered pairs of partitions, holds for many infinite families of such pairs. We also show that the bounded height case can be reduced to checking that the conjecture holds for a finite number of pairs, for any given height. Moreover, we propose a natural generalization of the conjecture to the case of skew shapes.Natural Sciences and Engineering Research Council of CanadaFonds Québécois de la Recherche sur la Nature et les Technologie
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