Quantum generalization of the Horn conjecture

Nonabelian theta functions of positive genus

“…Therefore the Knutson-Tao theorem leads to the expectation that whenever moduli spaces M (of type A) are nonempty, nonzero sections of suitable line bundles on them exist. For g = 0 this is established by works of Belkale on quantum horn and saturation conjectures [3]. We will verify this saturation principle in higher genus.…”

6j symbols for $$U_q (\mathfrak{s}\mathfrak{l}_2 )$$ and non-Euclidean tetrahedra

“…By Lemma 4.3.2 and Racah's formula(6), since the number of terms in the sum grows linearly with k.4.4. Euclidean limit of the asymptotic formulaIn this section we compare the quantities j 12 j 23 j 13 j 34 j 14 j…”

“…Given j 1 , j 2 ∈ [0, (r − 2)/2] ∩ Z/2, the tensor product V j 1 ⊗ V j 2 is isomorphic to the direct sum of objects V j 3 where the sum is over j 3 satisfying the quantum Clebsch-Gordan inequalities max(j 1 − j 2 , j 2 − j 1 ) ≤ j 3 ≤ min(j 1 + j 2 , r − 2 − j 1 − j 2 ) (1) and the parity condition j 1 + j 2 + j 3 ∈ Z. Geometrically, the condition (1) means that there exists a triangle in the unit sphere with edge lengths j a 2π/(r − 2), a = 1, 2, 3. Generalizations of these inequalities to Lie algebras of higher rank are described in [1], [7], [6], [37]. This paper concerns a generalization of this relationship in a different direction, namely from triangles to tetrahedra.…”

General sheaves over weighted projective lines

“…It would be interesting to explore the relationship between the theory developed in this paper and quantum Schubert calculus [3]. See also [2].…”