Schubert Calculus - Osaka 2012 (Collect)

Preface
Alain Lascoux (1944-2013)
Contents
Hiraku Abe and Sara Billey, Consequences of the Lakshmibai-Sandhya Theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry
§1. Introduction
§2. Preliminaries
2.1. The Flag Manifold
2.2. Flags and Permutations
2.3. Schubert Cells and Schubert Varieties in Flₙ(ℂ)
2.4. Combinatorics and Geometry
§3. Smooth Schubert varieties
3.1. Lie algebras and tangent spaces of Schubert varieties
3.2. Bruhat graphs
3.3. Lakshmibai-Sandhya Theorem
§4. 10 Pattern Avoidance Properties
Pattern Avoidance Property 1
Pattern Avoidance Property 2
Pattern Avoidance Property 3
Pattern Avoidance Property 4
Pattern Avoidance Property 5
Pattern Avoidance Property 6
Pattern Avoidance Property 7
Pattern Avoidance Property 8
Pattern Avoidance Property 9
Pattern Avoidance Property 10-1
Pattern Avoidance Property 10-2
Pattern Avoidance Property 10-3
§5. Pattern avoidance for Coxeter groups
5.1. A quick review on Coxeter groups
5.2. Coxeter patterns
5.3. Applications of Coxeter Patterns
§6. Computer tools for Schubert geometry
§7. Open Problems
References
[10]
[30]
[51]
[68]
[85]
Piotr Achinger and Nicolas Perrin, Spherical multiple flags
§1. Introduction
§2. Structure of G-orbits
§3. Minimal orbits for the weak order
3.1. Weak order
3.2. Minimal orbits: The case of opposite pairs
3.3. Minimal orbits: General case
§4. Distance and rank
4.1. Distance
4.2. Connection with the rank
§5. Proof of Theorem 1
§6. Example of non normal closures
References
[4]
Peter Fiebig, Moment graphs in representation theory and geometry
§1. Introduction
§2. Moment graphs
2.1. Moment graphs
2.2. Sheaves on moment graphs
2.3. Sections of sheaves
2.4. A topology on the moment graph
2.5. A restriction on the characteristic
2.6. The Braden-MacPherson sheaf
2.7. A generic decomposition of Z-modules
2.8. Z-modules admitting a Verma flag
§3. Representation theory
3.1. Simple highest weight modules
3.2. Characters of highest weight modules
3.3. Simple highest weight characters and the Kazhdan-Lusztig conjecture
3.4. Verma modules
3.5. Jordan-Hölder multiplicities
3.6. The category O
3.7. Verma flags
3.8. EGG-reciprocity
3.9. Deformed category O
3.10. Simple objects in O_A
3.11. Deformed projective objects
3.12. A functor into moment graph combinatorics
§4. Geometry
4.1. H-spaces
4.2. The equivariant derived category
4.3. Hypercohomology
4.4. The main example - flag varieties as T-spaces
4.5. The moment graph associated to a T-space X
4.6. Moment graph sheaves associated to equivariant sheaves
4.7. The localization theorem
4.8. (Equivariant) parity sheaves on stratified varieties
4.9. The Elias-Williamson work
References
[F2]
Takeshi Ikeda, Lectures on equivariant Schubert polynomials
§0. Introduction
§1. Grassmannians and Factorial Schur functions
1.1. Schubert varieties in Grassmannians
1.2. Weights of coordinate functions
1.3. GKM graph and equivariant Schubert classes
1.4. Factorial Schur functions
1.5. Localization map
1.6. Left divided difference operators
1.7. Double Schubert polynomials
1.8. Appendix to §1- Kempf-Laksov formula
§2. Schur's Q-functions and the Lagrangian Grassmannian
2.1. Lagrangian Grassmannian
2.2. Schubert varieties in LG(n)
2.3. Schur's Q-functions
2.4. GKM graph and Schubert classes for LG(∞)
2.5. Factorial Q-functions
2.6. Double Schubert polynomials for the symplectic flag variety
2.7. Factorial Q-functions and Kazarian's formula
§3. Equivariant multiplicity
3.1. Notation
3.2. Formal character
3.3. Restriction to one-parameter subgroups
3.4. G/P of cominuscule type
References
[17]
[35]
Bumsig Kim, Stable quasimaps to holomorphic symplectic quotients
§1. Introduction
§2. Holomorphic symplectic quotients
2.1. Symplectic quotients
2.2. Symmetry
§3. Stable Quasimaps
3.1. Quasimaps
3.2. Some results from [9]
3.3. Symmetric obstruction theory
§4. Twisted Quasimaps
4.1. Set-up
4.2. Twisted quasimaps
4.3. Obstruction Theory
4.4. T-action
§5. The Quiver Example
5.1. Nakajima's quiver varieties
5.2. Twisted quasimap to a Nakajima quiver variety
5.3. Obstruction theory
§6. Stabilities on Quiver bundles
6.1. King's stability
6.2. Quiver sheaves
References
[11]
Valentina Kiritchenko, Divided difference operators on polytopes
§1. Introduction
§2. Main construction
2.1. String spaces and parapolytopes
2.2. Polytopes and convex chains
2.3. Divided difference operators on parachains
2.4. Examples
§3. Polytopes and Demazure characters
3.1. Characters of polytopes
3.2. Gelfand-Zetlin polytopes for SLₙ
3.3. Applications to arbitrary reductive groups
3.4. Examples
§4. Bott towers and Bott-Samelson resolutions
4.1. Bott towers
4.2. Bott-Samelson varieties
4.3. Degenerations of string spaces
References
Allen Knutson, Schubert calculus and puzzles
§1. Schubert varieties and interval positroid varieties
1.1. Schubert varieties
1.2. Schubert calculus
1.3. First positivity result
1.4. Interval rank varieties
§2. Vakil's Littlewood-Richardson rule, from shifting
2.1. Combinatorial shifting
2.2. Geometric shifting
2.3. Vakil's degeneration order
2.4. Partial puzzles
§3. Equivariant and K-extensions
3.1. K-homology
3.2. K-cohomology
3.3. Equivariant K-theory
3.4. Equivariant cohomology
§4. Puzzles for the Belkale-Kumar Schubert calculus of other partial flag manifolds
References
[Kn1]
Thomas Lam, Whittaker functions, geometric crystals, and quantum Schubert calculus
§1. Introduction
1.1. Whittaker functions
1.2. Mirror symmetry for flag varieties
1.3. Geometric crystals
1.4. Whittaker functions as integrals over geometric crystals
1.5. Geometric analogues of Schur functions and geometric RSK
1.6. Organization
§2. Background on semisimple groups
2.1. Notations
2.2. Relations for X_i and Y_i
2.3. Torie charts
2.4. Canonical form
2.5. Direct geometric interpretation of ω_U and ω_x
2.6. Example
§3. Geometric crystals
3.1. Decorated geometric crystals
3.2. The geometric crystal with highest weight t
3.3. Weight map in coordinates
3.4. The positive decorated geometric crystal
3.5. Combinatorial crystals from geometric crystals
§4. Whittaker functions and Whittaker modules
4.1. Quantum Toda lattice
4.2. Center and Harish-Chandra homomorphism
4.3. Whittaker modules
4.4. Principal series representations
§5. Whittaker functions as integrals over geometric crystals
5.1. Definition
5.2. Convergence
5.3. Whittaker vectors in W_μ
5.4. Pairing
5.5. Proof of Theorem 5.1
§6. Whittaker functions as geometric analogues of Schur functions
6.1. Integrals over Gelfand-Tsetlin patterns
6.2. Identities
§7. Mirror symmetry for flag varieties
7.1. Toda lattice
7.2. Cohomology of flag varieties
7.3. Quantum cohomology of flag varieties
7.4. Mirror conjecture and quantum D-module
7.5. Quantum equals affine and Schubert bases
References
[Giv]
[OSZ]
Alain Lascoux, Tableaux and Eulerian properties of the symmetric group
§1. Eulerian structures
§2. Keys
§3. Tensor product
§4. Tableauhedron
§5. Postulation
§6. Eulerian polynomials
References
Cristian Lenart, Satoshi Naito, Daisuke Sagaki,Anne Schilling and Mark Shimozono, Quantum Lakshmibai-Seshadri paths and root operators
§1. Introduction
§2. Lakshmibai-Seshadri paths
2.1. Basic notation
2.2. Paths and root operators
2.3. Lakshmibai-Seshadri paths
2.4. Characterization of the set ?(λ)_{cl} of paths
§3. Quantum Lakshmibai-Seshadri paths
3.1. Quantum Bruhat graph
3.2. Definition of quantum Lakshmibai-Seshadri paths
§4. Main result
4.1. Statement of the main result and some technical lemmas
4.2. Explicit description of the image of a quantum LS path under the action of root operators
References
[NS4]
Abraham Martín del Campo and Frank Sottile, Experimentation in the Schubert Calculus
§1. Introduction
§2. Background
2.1. The Shapiro Conjecture
2.2. Schubert calculus of enumerative geometry
2.3. Experimentation on a supercomputer
§3. History and generalizations of the Shapiro Conjecture
3.1. The Shapiro Conjecture for other flag manifolds
§4. Lower bounds and gaps on the number of real solutions
4.1. Topological lower bounds
4.2. Real osculating instances of Schubert problems
4.3. Experiment
4.4. Symmetric Schubert problems
4.5. Lower bounds and gaps
§5. Galois groups of Schubert problems
5.1. Galois groups
5.2. Vakil's combinatorial criterion
5.3. Homotopy continuation
5.4. Frobenius method and elimination theory
5.5. Galois groups for Gr(4,8) and Gr(4,9)
References
[3]
[20]
[39]
[61]
Masaki Nakagawa and Hiroshi Naruse, Generalized (co)homology of the loop spaces of classical groups and the universal factorial Schur P- and Q-functions
§1. Introduction
§2. E-(co)homology of ΩSp and Ω₀SO
2.1. Generalized (co)homology theory
2.2. E-(co)homology of the loop space of SU
2.3. E-homology of the loop space of Sp
2.4. E-homology of the loop space of SO
2.5. E-cohomology of the loop space of Sp
2.6. E-cohomology of the loop space of SO
§3. Rings of E-(co)homology Schur P- and Q-functions
3.1. Ring of symmetric functions
3.2. Rings of E-homology Schur P- and Q-functions
3.3. Rings of E-cohomology Schur P- and Q-functions
3.4. Identifications with symmetric functions
§4. Universal factorial Schur P- and Q-functions
4.1. Lazard ring ? and the universal formal group law
4.2. Definition of P^?_λ (xₙ|b), Q^?_λ(xₙ|b)
4.3. ?-supersymmetric series
4.4. Stability Property
4.5. Universal factorial Schur functions
4.6. Factorization Formula
4.7. Basis Theorem for P^?_λ (x), Q^?_λ(x)
4.8. Vanishing Property
4.9. Algebraic localization map of types B_∞, C_∞, or D_∞
§5. Dual universal (factorial) Schur P- and Q-functions
5.1. One row case
5.2. Definition of \hat{p}^?_λ (y) and \hat{q}^?_λ (y)
5.3. Basis Theorem for \hat{p}^?_λ (y) and \hat{q}^?_λ (y)
5.4. Hopf algebra structure
5.5. Concluding remarks
§6. Appendix
6.1. Universal factorial Schur functions
6.2. Root data
References
[14]
[33]
[53]
Piotr Pragacz, Positivity of Thom polynomials and Schubert calculus
§1. Introduction
§2. Preliminaries
§3. Schubert varieties and Schubert classes
§4. Ample vector bundles and positive polynomials
§5. Thom polynomials for singularities of maps
§6. Thom polynomials for invariant cones
§7. Lagrangian Thom polynomials
§8. Legendrian Thom polynomials
References
[5]
[24]
[43]
Toshiaki Shoji, Character sheaves on exotic symmetric spaces and Kostka polynomials
§1. Introduction
§2. Geometric realization of Kostka polynomials
§3. Springer correspondence for GLₙ and Kostka polynomials
§4. The enhanced nilpotent cone and Kostka polynomials
§5. Symmetric space GL_{2n}/Sp_{2n}
§6. Exotic symmetric space GL_{2n}/Sp_{2n}×V
§7. Springer correspondence for exotic symmetric space
§8. F_q-structure on character sheaves
§9. The exotic symmetric space and Kostka polynomials
§10. H^F-invariant functions on Χ^F
References
[SS1]
Hugh Thomas and Alexander Yong, Cominuscule tableau combinatorics
§1. Introduction
1.1. Lie-theoretic data and jeu de taquin
1.2. Dual equivalence
1.3. Growth diagrams and their applications
§2. Growth diagrams and dual equivalence
2.1. Cominuscule growth diagrams
2.2. Proof of Theorem 1
2.3. Proof of Corollary 2
2.4. Proof of Theorem 3
§3. Further discussion of dual equivalence
3.1. Computing c^ν_{λ,μ} (G/P)
3.2. The Haiman table and the generalized Robinson-Schensted correspondence
3.3. Reading word order?
§4. Schützenberger's evacuation involution
§5. Cartons
5.1. Statement of the rule
5.2. The proof
§ Acknowledgements
References
[Sa87]
Julianna Tymoczko, Billey's formula in combinatorics, geometry, and topology
§1. Introduction
§2. Billey's Formula in Combinatorics
§3. What Billey's formula means
3.1. Representations of quantum groups
3.2. Orbit values of Kostant polynomials
3.3. Kumar's criterion for Schubert varieties
3.4. Restriction to fixed points and GKM theory
§4. Billey's formula for the Grassmannian G(k,n)
§5. Billey's formula for subvarieties
5.1. Examples of poset pinball
5.2. Schubert calculus and poset pinball
§6. Conjectures and open questions
References
[10]
[29]
Outline of the Conference
Program of the 5th MSJ-SI Schubert Calculus Summer School
Poster Session