Eigenvalues of products of unitary matrices and quantum Schubert calculus

Agnihotri, Sharad; Woodward, Chris

doi:10.4310/mrl.1998.v5.n6.a10

Cited by 87 publications

(173 citation statements)

References 23 publications

Supporting

Mentioning

165

Contrasting

Unclassified

Order By: Relevance

“…We note that the above argument comes from [AW98]. For example, SU(2)/SU(2) ∼ = [−2, 2] and SU(3)/SU(3) is homeomorphic to a disc (see below).…”

Section: 1mentioning

confidence: 99%

The topology of moduli spaces of free group representations

Florentino¹,

Lawton²

2009

Math. Ann.

View full text Add to dashboard Cite

Abstract. For any complex affine reductive group G and a fixed choice of maximal compact subgroup K, we show that the Gcharacter variety of a free group strongly deformation retracts to the corresponding K-character space, which is a real semi-algebraic set. Combining this with constructive invariant theory and classical topological methods, we show that the SL(3, )-character variety of a rank 2 free group is homotopic to an 8 sphere and the SL(2, )-character variety of a rank 3 free group is homotopic to a 6 sphere.

show abstract

“…We note that the above argument comes from [AW98]. For example, SU(2)/SU(2) ∼ = [−2, 2] and SU(3)/SU(3) is homeomorphic to a disc (see below).…”

Section: 1mentioning

confidence: 99%

The topology of moduli spaces of free group representations

Florentino¹,

Lawton²

2009

Math. Ann.

View full text Add to dashboard Cite

show abstract

“…Now, as in the proof of theorem 5.10, we start with a path (x t ) t∈ [0,1] going from x 0 ∈ M (the same x 0 used to define M 0 in proposition 5.13) to some fixed point x 1 of β satisfying µ(x 1 ) ∈ Q 0 ⊂ U . We can lift the path u t = µ(x t ) ∈ U to a path u t ∈ U starting at (1 S , 1 G ) ∈ U .…”

Section: 3mentioning

confidence: 99%

“…In all of the following, we will assume that the conjugacy classes C 1 , ... , C l of U are chosen in a way that Hom C (π g,l , U ) = ∅. When g = 0 (that is, for the case of the punctured sphere group), giving necessary and sufficient conditions for this to be true is a difficult problem (see for instance [1]). When g ≥ 1, however, and if U is semi-simple, one always has Hom C (π g,l , U ) = ∅ (see [17]).…”

Section: Introductionmentioning

confidence: 99%

“…Observe that β (1) is an involution that reverses the 2-form defining the quasi-Hamiltonian structure on a single conjugacy class of C of U (see lemma 4.5) and is compatible with the conjugation action of U on C and the momentum map µ (1) : C ֒→ U . Likewise, β (1) is an involution that reverses the 2-form defining the quasi-Hamiltonian structure on the double U × U (see lemma 4.6) and is compatible with the conjugation action of U on U × U and the momentum map µ (1) : (a, b) ∈ U × U → aba −1 b −1 ∈ U . Lemmas 4.7 and 4.8 show that we can apply proposition 3.13 and therefore prove that β (l) and β (g) are form-reversing involutions on the quasi-Hamiltonian spaces C 1 × · · · × C l and (U × U ) g respectively, compatible with the respective actions of U and their respective momentum maps.…”

Section: The Involution β We Begin By Introducing the Following Map βmentioning

confidence: 99%

See 1 more Smart Citation

Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups

Schaffhauser

2008

Math. Ann.

View full text Add to dashboard Cite

Abstract. The importance of explicit examples of Lagrangian submanifolds of moduli spaces is revealed by papers such as [9,25]: given a 3-manifold M with boundary ∂M = Σ, Dostoglou and Salamon use such examples to obtain a proof of the Atiyah-Floer conjecture relating the symplectic Floer homology of the representation space Hom(π 1 (Σ = ∂M ), U )/U (associated to an explicit pair of Lagrangian submanifolds of this representation space) and the instanton homology of the 3-manifold M . In the present paper, we construct a Lagrangian submanifold of the space of representations M g,l := Hom C (π g,l , U )/U of the fundamental group π g,l of a punctured Riemann surface Σ g,l into an arbitrary compact connected Lie group U . This Lagrangian submanifold is obtained as the fixed-point set of an anti-symplectic involutionβ defined on M g,l . We show that the involutionβ is induced by a form-reversing involution β defined on the quasi-Hamiltonian space (The fact thatβ has a non-empty fixed-point set is a consequence of the real convexity theorem for group-valued momentum maps proved in [28]. The notion of decomposable representation provides a geometric interpretation of the Lagrangian submanifold thus obtained.

show abstract

“…Biswas [11], Agnihotri-Woodward [1] and Belkale [7] [37], Corollary 4.13, Ab is a convex polytope of maximal dimension in ~b . We wish to find the defining inequalities for Ab.…”

mentioning

confidence: 99%

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

Teleman¹,

Woodward²

2003

Annales De L’institut Fourier

View full text Add to dashboard Cite

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.

show abstract

Eigenvalues of products of unitary matrices and quantum Schubert calculus

Cited by 87 publications

References 23 publications

The topology of moduli spaces of free group representations

The topology of moduli spaces of free group representations

Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups

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

Contact Info

Product

Resources

About