14 votes

Origin of terms "flag", "flag manifold", "flag variety"?

I think the concept may date back to René De Saussure (1868-1943). He was interested in the Euclidean geometry of 3-dimensional space and used the term "géometrie des feuillets". I think this may ...
J.M. Selig's user avatar
14 votes
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When the Littlewood-Richardson rule gives only irreducibles?

The answer is Yes, but this requires some elaboration. Knutson-Tao-Woodward prove Fulton's conjecture in $\S$6.1. In principle, you can follow the approach by De Loera-McAllister or Mulmuley-...
Igor Pak's user avatar
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10 votes
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Schubert calculus expressed in terms of the cotangent space of the Grassmannians

The tangent space to the Grassmanian corresponds to the following representation of $U(r)\times U(n-r)$, call it $\rho$: it is the $r\times (n-r)$ matrices, with $U(r)$ acting on the left and $U(n-r)$ ...
Anton Mellit's user avatar
  • 3,572
9 votes

Embeddings of flag manifolds

In general there is a more efficient way: $a_1,\ldots,a_k$ determines a Young diagram, and you can realize the flag variety as the stabilizer of a point in the unique closed orbit of ${\mathbb P}(U)$, ...
Victor Petrov's user avatar
9 votes
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Homology of the free loop space of generalized flag varieties

Serre proved that for any simply-connected $X$, if $X$ has finitely generated homology groups in each degree, then the loop space of $X$ has finitely generated homology groups in each degree. (...
Will Sawin's user avatar
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9 votes
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Parabolic subgroups of reductive group as stabilizers of flags

$\DeclareMathOperator\GL{GL}$Yes for reductive $G$, and this follows quickly from the dynamical characterization of parabolics: A subgroup $H$ of a reductive group $G$ is parabolic if and only if ...
Will Sawin's user avatar
  • 135k
8 votes

Real orbits of Complex Flag Varieties

Even though there is already a good answer to the question I would like to add that Aomoto's paper just marks the beginning of an extensive body of research on open $G_0$-orbits on $G/B$ or, more ...
Friedrich Knop's user avatar
8 votes
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Schubert cells in G/P for reductive G

You already answered your question: the center of any reductive group lies in any parabolic, so if $G$ is reductive, and $G_{\operatorname{ad}}$ its adjoint quotient (which is, of course, semi-simple),...
Ben Webster's user avatar
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7 votes

Closures of torus orbits in flag varieties

A point in the Grassmannian $ x \in G(k, n) $ defines a matroid $ M = M(x)$. Associated to this matroid is a matroid polytope $P(M)$. The torus orbit closure through $ x $ is the toric variety ...
Joel Kamnitzer's user avatar
7 votes
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Real orbits of Complex Flag Varieties

Aomoto (1966) gives a formula for the number of open orbits (page 15, between (46) and (47)), and specializes it to various special cases (just search the paper for "open").
Francois Ziegler's user avatar
7 votes

Varieties in the $\mathit{GL}(V)$-module $S_{\lambda}V$

If $\lambda = (\lambda_1,\dots,\lambda_{n+1})$ is a dominant weight and $$ \lambda_1 = \dots = \lambda_{k_1} > \lambda_{k_1 + 1} = \dots = \lambda_{k_1 + k_2} > \dots > \lambda_{k_1 + \dots +...
Sasha's user avatar
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7 votes

Frobenius pushforward of an equivariant tautological bundle on the flag variety

EDIT. Corrected the statement ($\sigma$ should be $p-1$ times what I wrote) and answered the question in the comment. In general, the push-forward of a line bundle on the flag variety $G/B$ will not ...
Piotr Achinger's user avatar
6 votes
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Lagrangian Grassmannian as a Spin Manifold

The complex Lagrangian Grassmannian $M=G/K=Sp(n)/U(n)$ is an isotropy irreducible Hermitian symmetric space, hence it admits a unique invariant complex structure. It occurs by painting black in ...
314159.'s user avatar
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6 votes

Schubert calculus expressed in terms of the cotangent space of the Grassmannians

For your first question: yes. See Stoll, Invariant forms on Grassman manifolds, p. 15. I think your second question is answered in the same book.
Ben McKay's user avatar
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6 votes
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Commuting matrices and cyclic modules

Let $A=\left(\begin{smallmatrix}1&0&0\\1&1&0\\0&0&1\end{smallmatrix}\right)$ and $B=\left(\begin{smallmatrix}1&0&0\\0&1&0\\1&0&1\end{smallmatrix}\right)$...
Dave Benson's user avatar
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6 votes

References for $K$-orbits in $G/B$

A good reference for this is "On Rationality Properties of Involutions of Reductive Groups" by Helminck and Wang (Advances in Math. 1993). The decomposition of $K\backslash G/B$ is studied ...
Paul Broussous's user avatar
5 votes
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Irreducibility of Gelfand-Serganova strata

The strata need not be irreducible. Quoting page 2 of Knutson, Lam, Speyer (https://arxiv.org/abs/0903.3694v1): "the strata can have essentially any singularity [Mn88]. In particular, the nonempty ...
Sam Hopkins's user avatar
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5 votes

Schubert calculus expressed in terms of the cotangent space of the Grassmannians

Regarding your second question, I think the answer is in the famous Kostant "Lie Algebra Cohomology and the Generalized Borel-Weil Theorem" or rather its second part.
Vít Tuček's user avatar
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5 votes
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Flag manifolds as homogeneous Kahler manifolds

Flag manifolds $G/C(S)$ even exhaust homogeneous symplectic manifolds of $G$: Borel-Weil (1954, Thm 1). Also restated with fewer details in (1954, Thm 1).
Francois Ziegler's user avatar
5 votes
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Tensor product of perverse sheaves on flag varieties

First a general comment: as Sasha alludes to, there are two tensor products of complexes of sheaves. Let $i : X \hookrightarrow X \times X$ denote the diagonal. We have, and given complexes of sheaves ...
Geordie Williamson's user avatar
5 votes

Description of the generalized grassmannians and flag varieties (parabolic quotients) associated to the exceptional groups

Edit 2: A good discussion of the lowest dimensional pieces of each of the flags below is found in Geometries, the principle of duality, and algebraic groups by Carr and Garibaldi. In particular for ...
Callum's user avatar
  • 877
4 votes
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Borel--Bott--Weil for the Grassmannians

Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. Math. 74 (1961), 329-387. W. Schmid, Homogeneous complex manifolds and representations of semisimple Lie groups, ...
Ben McKay's user avatar
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4 votes
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Combinatorics of the Cohomology Ring of the Lagrangian Grassmannians

I'm assuming you mean in $\mathbb{C}^{2n}$ (the answer for real Lagrangian Grassmannians is trickier). In this case, the answer is easy: The cohomology is isomorphic to the symmetric polynomials in ...
Ben Webster's user avatar
  • 43.9k
4 votes
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real orbits on flag varieties

Classic paper: Joseph A. Wolf (1969), The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components. Recent survey: Dmitri Akhiezer (2013), Real ...
Francois Ziegler's user avatar
4 votes
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Euler characteristic of a holomorphic homogeneous vector bundle

In the case $G$ is a complex semisimple Lie group and $P$ its parabolic subgroup, the answer is given by Kostant's version of Borel-Bott-Weil theorem [K]. Any homogeneous vector bundle is given by a ...
Vít Tuček's user avatar
  • 8,159
4 votes
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geometric meaning to pairs of SYT indexing for the basis of cohomology ring of full flag variety

Since, surprisingly, there are still no answers or even comments, let me note that the answer to the last question is well known to be "yes": the Schubert cell containing a flag $(E_1,\dots,...
Igor Makhlin's user avatar
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3 votes
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Explicit description of the Lagrangian Grassmannian as a homogeneous space

Like any normal person, I'm going to use the symplectic form where $\langle e_i,e_{j}\rangle =\pm \delta_{j,2n-i+1}$ with $1$ if $i\leq n$ and $-1$ if $i>n$. The compact symplectic group you have ...
Ben Webster's user avatar
  • 43.9k
3 votes

The automorphism group of a divisor in a complete flag variety

A very partial answer: The kernel of $r$ is precisely the center of $G$: let $V=H^0(L\otimes K_X)$. Then $X$ embeds into $\mathbf P(V)$ and $G/\mathrm{center}\to\mathrm{Aut}(\mathbf P(V))$ is ...
Friedrich Knop's user avatar
3 votes

Equivariant $K$-theory, singular vectors, and flag manifolds

The ideas were first developed by Givental and Lee in the context of quantum equivariant K-theory https://arxiv.org/abs/math/0108105, where they defined quantum K-theory as a certain lift of quantum ...
Peter Koroteev's user avatar
3 votes

Borel--Bott--Weil for the Grassmannians

Lars, search for a paper of Kostant ora paper of Griffiths-Schmid, you will find a complete answer to your question, even when \lambda is just an irreducible representation. best regards
jorge vargas's user avatar

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