We study defining inequalities of string cones via a potential function on a reduced double Bruha... more We study defining inequalities of string cones via a potential function on a reduced double Bruhat cell. We give a necessary criterion for the potential function to provide a minimal set of inequalities via tropicalization and conjecture an equivalence.
We obtain an explicit crystal isomorphism between two realizations of crystal bases of finite dim... more We obtain an explicit crystal isomorphism between two realizations of crystal bases of finite dimensional irreducible representations of simple Lie algebras of type A and D. The first realization we consider is a geometric construction in terms of irreducible components of certain Nakajima quiver varieties established by Saito and the second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke. We give a homological description of the irreducible components of Lusztig's quiver varieties which correspond to the crystal of a finite dimensional representation and describe the promotion operator in type A to obtain a geometric realization of Kirillov-Reshetikhin crystals.
For arbitrary reduced words we give formulas for the crystal structures on string and Lusztig dat... more For arbitrary reduced words we give formulas for the crystal structures on string and Lusztig data of type A as well as the defining inequalities of the corresponding polytopes revealing certain dualities between them.
We give a formula for the crystal structure on the integer points of the string polytopes and the... more We give a formula for the crystal structure on the integer points of the string polytopes and the *-crystal structure on the integer points of the string cones of type A for arbitrary reduced words. As a byproduct, we obtain defining inequalities for Nakashima–Zelevinsky string polytopes. Furthermore, we give an explicit description of the Kashiwara *-involution on string data for a special choice of reduced word.
We study defining inequalities of string cones via a potential function on a reduced double Bruha... more We study defining inequalities of string cones via a potential function on a reduced double Bruhat cell. We give a necessary criterion for the potential function to provide a minimal set of inequalities via tropicalization and conjecture an equivalence.
We obtain an explicit crystal isomorphism between two realizations of crystal bases of finite dim... more We obtain an explicit crystal isomorphism between two realizations of crystal bases of finite dimensional irreducible representations of simple Lie algebras of type A and D. The first realization we consider is a geometric construction in terms of irreducible components of certain Nakajima quiver varieties established by Saito and the second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke. We give a homological description of the irreducible components of Lusztig's quiver varieties which correspond to the crystal of a finite dimensional representation and describe the promotion operator in type A to obtain a geometric realization of Kirillov-Reshetikhin crystals.
For arbitrary reduced words we give formulas for the crystal structures on string and Lusztig dat... more For arbitrary reduced words we give formulas for the crystal structures on string and Lusztig data of type A as well as the defining inequalities of the corresponding polytopes revealing certain dualities between them.
We give a formula for the crystal structure on the integer points of the string polytopes and the... more We give a formula for the crystal structure on the integer points of the string polytopes and the *-crystal structure on the integer points of the string cones of type A for arbitrary reduced words. As a byproduct, we obtain defining inequalities for Nakashima–Zelevinsky string polytopes. Furthermore, we give an explicit description of the Kashiwara *-involution on string data for a special choice of reduced word.
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Papers by Bea Schumann