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Search: MSC category 14M15 ( Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] )

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1. CJM Online first

Feigin, Evgeny; Finkelberg, Michael; Littelmann, Peter
Symplectic Degenerate Flag Varieties
A simple finite dimensional module $V_\lambda$ of a simple complex algebraic group $G$ is naturally endowed with a filtration induced by the PBW-filtration of $U(\mathrm{Lie}\, G)$. The associated graded space $V_\lambda^a$ is a module for the group $G^a$, which can be roughly described as a semi-direct product of a Borel subgroup of $G$ and a large commutative unipotent group $\mathbb{G}_a^M$. In analogy to the flag variety $\mathcal{F}_\lambda=G.[v_\lambda]\subset \mathbb{P}(V_\lambda)$, we call the closure $\overline{G^a.[v_\lambda]}\subset \mathbb{P}(V_\lambda^a)$ of the $G^a$-orbit through the highest weight line the degenerate flag variety $\mathcal{F}^a_\lambda$. In general this is a singular variety, but we conjecture that it has many nice properties similar to that of Schubert varieties. In this paper we consider the case of $G$ being the symplectic group. The symplectic case is important for the conjecture because it is the first known case where even for fundamental weights $\omega$ the varieties $\mathcal{F}^a_\omega$ differ from $\mathcal{F}_\omega$. We give an explicit construction of the varieties $Sp\mathcal{F}^a_\lambda$ and construct desingularizations, similar to the Bott-Samelson resolutions in the classical case. We prove that $Sp\mathcal{F}^a_\lambda$ are normal locally complete intersections with terminal and rational singularities. We also show that these varieties are Frobenius split. Using the above mentioned results, we prove an analogue of the Borel-Weil theorem and obtain a $q$-character formula for the characters of irreducible $Sp_{2n}$-modules via the Atiyah-Bott-Lefschetz fixed points formula.

Keywords:Lie algebras, flag varieties, symplectic groups, representations
Categories:14M15, 22E46

2. CJM 2011 (vol 64 pp. 1090)

Rosso, Daniele
Classic and Mirabolic Robinson-Schensted-Knuth Correspondence for Partial Flags
In this paper we first generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they lie, using the Robinson-Schensted-Knuth correspondence. Then we use this result to generalize the mirabolic Robinson-Schensted-Knuth correspondence defined by Travkin, to the case of two partial flags and a line.

Keywords:partial flag varieties, RSK correspondence
Categories:14M15, 05A05

3. CJM 2010 (vol 62 pp. 1246)

Chaput, P. E.; Manivel, L.; Perrin, N.
Quantum Cohomology of Minuscule Homogeneous Spaces III. Semi-Simplicity and Consequences
We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, specialized at $q=1$, is semisimple. This implies that complex conjugation defines an algebra automorphism of the quantum cohomology ring localized at the quantum parameter. We check that this involution coincides with the strange duality defined in our previous article. We deduce Vafa--Intriligator type formulas for the Gromov--Witten invariants.

Keywords:quantum cohomology, minuscule homogeneous spaces, Schubert calculus, quantum Euler class
Categories:14M15, 14N35

4. CJM 2010 (vol 62 pp. 870)

Valdimarsson, Stefán Ingi
The Brascamp-Lieb Polyhedron
A set of necessary and sufficient conditions for the Brascamp--Lieb inequality to hold has recently been found by Bennett, Carbery, Christ, and Tao. We present an analysis of these conditions. This analysis allows us to give a concise description of the set where the inequality holds in the case where each of the linear maps involved has co-rank $1$. This complements the result of Barthe concerning the case where the linear maps all have rank $1$. Pushing our analysis further, we describe the case where the maps have either rank $1$ or rank $2$. A separate but related problem is to give a list of the finite number of conditions necessary and sufficient for the Brascamp--Lieb inequality to hold. We present an algorithm which generates such a list.

Keywords:Brascamp-Lieb inequality, Loomis-Whitney inequality, lattice, flag
Categories:44A35, 14M15, 26D20

5. CJM 2008 (vol 60 pp. 875)

Mare, Augustin-Liviu
A Characterization of the Quantum Cohomology Ring of $G/B$ and Applications
We observe that the small quantum product of the generalized flag manifold $G/B$ is a product operation $\star$ on $H^*(G/B)\otimes \bR[q_1,\dots, q_l]$ uniquely determined by the facts that: it is a deformation of the cup product on $H^*(G/B)$; it is commutative, associative, and graded with respect to $\deg(q_i)=4$; it satisfies a certain relation (of degree two); and the corresponding Dubrovin connection is flat. Previously, we proved that these properties alone imply the presentation of the ring $(H^*(G/B)\otimes \bR[q_1,\dots, q_l],\star)$ in terms of generators and relations. In this paper we use the above observations to give conceptually new proofs of other fundamental results of the quantum Schubert calculus for $G/B$: the quantum Chevalley formula of D. Peterson (see also Fulton and Woodward ) and the ``quantization by standard monomials" formula of Fomin, Gelfand, and Postnikov for $G=\SL(n,\bC)$. The main idea of the proofs is the same as in Amarzaya--Guest: from the quantum $\D$-module of $G/B$ one can decode all information about the quantum cohomology of this space.

Categories:14M15, 14N35

6. CJM 2007 (vol 59 pp. 742)

Gil, Juan B.; Krainer, Thomas; Mendoza, Gerardo A.
Geometry and Spectra of Closed Extensions of Elliptic Cone Operators
We study the geometry of the set of closed extensions of index $0$ of an elliptic differential cone operator and its model operator in connection with the spectra of the extensions, and we give a necessary and sufficient condition for the existence of rays of minimal growth for such operators.

Keywords:resolvents, manifolds with conical singularities, spectral theor, boundary value problems, Grassmannians
Categories:58J50, 35J70, 14M15

7. CJM 2007 (vol 59 pp. 36)

Develin, Mike; Martin, Jeremy L.; Reiner, Victor
Classification of Ding's Schubert Varieties: Finer Rook Equivalence
K.~Ding studied a class of Schubert varieties $X_\lambda$ in type A partial flag manifolds, indexed by integer partitions $\lambda$ and in bijection with dominant permutations. He observed that the Schubert cell structure of $X_\lambda$ is indexed by maximal rook placements on the Ferrers board $B_\lambda$, and that the integral cohomology groups $H^*(X_\lambda;\:\Zz)$, $H^*(X_\mu;\:\Zz)$ are additively isomorphic exactly when the Ferrers boards $B_\lambda, B_\mu$ satisfy the combinatorial condition of \emph{rook-equivalence}. We classify the varieties $X_\lambda$ up to isomorphism, distinguishing them by their graded cohomology rings with integer coefficients. The crux of our approach is studying the nilpotence orders of linear forms in the cohomology ring.

Keywords:Schubert variety, rook placement, Ferrers board, flag manifold, cohomology ring, nilpotence
Categories:14M15, 05E05

8. CJM 2003 (vol 55 pp. 157)

Shimada, Ichiro
Zariski Hyperplane Section Theorem for Grassmannian Varieties
Let $\phi \colon X\to M$ be a morphism from a smooth irreducible complex quasi-projective variety $X$ to a Grassmannian variety $M$ such that the image is of dimension $\ge 2$. Let $D$ be a reduced hypersurface in $M$, and $\gamma$ a general linear automorphism of $M$. We show that, under a certain differential-geometric condition on $\phi(X)$ and $D$, the fundamental group $\pi_1 \bigl( (\gamma \circ \phi)^{-1} (M\setminus D) \bigr)$ is isomorphic to a central extension of $\pi_1 (M\setminus D) \times \pi_1 (X)$ by the cokernel of $\pi_2 (\phi) \colon \pi_2 (X) \to \pi_2 (M)$.

Categories:14F35, 14M15

9. CJM 2000 (vol 52 pp. 265)

Brion, Michel; Helminck, Aloysius G.
On Orbit Closures of Symmetric Subgroups in Flag Varieties
We study $K$-orbits in $G/P$ where $G$ is a complex connected reductive group, $P \subseteq G$ is a parabolic subgroup, and $K \subseteq G$ is the fixed point subgroup of an involutive automorphism $\theta$. Generalizing work of Springer, we parametrize the (finite) orbit set $K \setminus G \slash P$ and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of $\theta$-stable (resp. $\theta$-split) parabolic subgroups. We also describe the decomposition of any $(K,P)$-double coset in $G$ into $(K,B)$-double cosets, where $B \subseteq P$ is a Borel subgroup. Finally, for certain $K$-orbit closures $X \subseteq G/B$, and for any homogeneous line bundle $\mathcal{L}$ on $G/B$ having nonzero global sections, we show that the restriction map $\res_X \colon H^0 (G/B, \mathcal{L}) \to H^0 (X, \mathcal{L})$ is surjective and that $H^i (X, \mathcal{L}) = 0$ for $i \geq 1$. Moreover, we describe the $K$-module $H^0 (X, \mathcal{L})$. This gives information on the restriction to $K$ of the simple $G$-module $H^0 (G/B, \mathcal{L})$. Our construction is a geometric analogue of Vogan and Sepanski's approach to extremal $K$-types.

Keywords:flag variety, symmetric subgroup
Categories:14M15, 20G05

10. CJM 1997 (vol 49 pp. 1281)

Sottile, Frank
Pieri's formula via explicit rational equivalence
Pieri's formula describes the intersection product of a Schubert cycle by a special Schubert cycle on a Grassmannian. We present a new geometric proof, exhibiting an explicit chain of rational equivalences from a suitable sum of distinct Schubert cycles to the intersection of a Schubert cycle with a special Schubert cycle. The geometry of these rational equivalences indicates a link to a combinatorial proof of Pieri's formula using Schensted insertion.

Keywords:Pieri's formula, rational equivalence, Grassmannian, Schensted insertion
Categories:14M15, 05E10

11. CJM 1997 (vol 49 pp. 417)

Boe, Brian D.; Fu, Joseph H. G.
Characteristic cycles in Hermitian symmetric spaces
We give explicit combinatorial expresssions for the characteristic cycles associated to certain canonical sheaves on Schubert varieties $X$ in the classical Hermitian symmetric spaces: namely the intersection homology sheaves $IH_X$ and the constant sheaves $\Bbb C_X$. The three main cases of interest are the Hermitian symmetric spaces for groups of type $A_n$ (the standard Grassmannian), $C_n$ (the Lagrangian Grassmannian) and $D_n$. In particular we find that $CC(IH_X)$ is irreducible for all Schubert varieties $X$ if and only if the associated Dynkin diagram is simply laced. The result for Schubert varieties in the standard Grassmannian had been established earlier by Bressler, Finkelberg and Lunts, while the computations in the $C_n$ and $D_n$ cases are new. Our approach is to compute $CC(\Bbb C_X)$ by a direct geometric method, then to use the combinatorics of the Kazhdan-Lusztig polynomials (simplified for Hermitian symmetric spaces) to compute $CC(IH_X)$. The geometric method is based on the fundamental formula $$CC(\Bbb C_X) = \lim_{r\downarrow 0} CC(\Bbb C_{X_r}),$$ where the $X_r \downarrow X$ constitute a family of tubes around the variety $X$. This formula leads at once to an expression for the coefficients of $CC(\Bbb C_X)$ as the degrees of certain singular maps between spheres.

Categories:14M15, 22E47, 53C65

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