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A Demazure crystal is the basis at $q=0$ of a Demazure module. Demazure crystals play an important role in Schubert calculus because the character of a Demazure crystal in type A is identical to a key polynomial, which is closely related to Schubert polynomials. In this paper, we study tensor products of Demazure crystals. Each connected component of a tensor product of Demazure crystals need not be isomorphic to some Demazure crystal. We provide a necessary and sufficient condition for every connected component of a tensor product to be isomorphic to some Demazure crystal. Also, we obtain the explicit formula for connected components. As applications, we study the positivity for structure constants of products of key polynomials, and we obtain an equation of crystals, which is an analog of the Leibniz rule for Demazure operators.
We show that a tensor product of nonexceptional type Kirillov--Reshetikhin (KR) crystals is isomorphic to a direct sum of Demazure crystals; we do this in the mixed level case and without the perfectness assumption, thus generalizing a result of Naoi
A Cartesian decomposition of a coherent configuration $cal X$ is defined as a special set of its parabolics that form a Cartesian decomposition of the underlying set. It turns out that every tensor decomposition of $cal X$ comes from a certain Cartes
This paper studies the properties of Demazure atoms and characters using linear operators and also tableaux-combinatorics. It proves the atom-positivity property of the product of a dominating monomial and an atom, which was an open problem. Furtherm
We provide a new tableau model from which one can easily deduce the characters of irreducible polynomial representations of the orthogonal group $mathrm{O}_n(mathbb{C})$. This model originates from representation theory of the $imath$quantum group of
The Demazure character formula is applied to the Verlinde formula for affine fusion rules. We follow Littelmanns derivation of a generalized Littlewood-Richardson rule from Demazure characters. A combinatorial rule for affine fusions does not result,