Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants

Abstract

The set of conjugacy classes appearing in a product of conjugacy classes in a compact, 1-connected Lie group K can be identified with a convex polytope in the Weyl alcove [37]. In this paper we identify linear inequalities defining this polytope. Each inequality corresponds to a non-vanishing Gromov-Witten invariant for a generalized flag variety G/P , where G is the complexification of K and P is a maximal parabolic subgroup. This generalizes the results for SU (n) of Agnihotri and the second author [1] and Belkale [7] on the eigenvalues of a product of unitary matrices and quantum cohomology of Grassmannians.

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

“…Hence so is (θ ab − θ 0 ab ) 2 for θ ab sufficiently close to 0, π, which proves (37). Then (38) follows since (θ ab − θ 0 ab ) 2 has a simple zero at l ab = l 0 ab .…”

“…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.…”

The Deligne–Simpson problem—a survey

“…The DSP for an arbitrary compact connected simple simply-connected Lie group is considered in [67]. In most papers cited in this subsection the results are related to GromovWitten invariants of Grassmanians.…”

Moduli of parabolic Higgs bundles and Atiyah algebroids

“…Though most of the statements we make can at least formally be translated into this setting, we have refrained from working in this generality as it seems that the dust has not settled on the notion of stability for parabolic principal bundles, cf. [6,54,3,2]. By working in the context of stacks rather than moduli schemes these problems would of course be avoided, and we intend to take up this matter in the future.…”