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 ...
14
votes
Accepted
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-...
10
votes
Accepted
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 $U(n-r)$ ...
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 ${\mathbb P}(U)$, ...
9
votes
Accepted
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. (...
9
votes
Accepted
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 ...
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 ...
8
votes
Accepted
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),...
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 ...
7
votes
Accepted
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").
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 +...
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 ...
6
votes
Accepted
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 ...
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.
6
votes
Accepted
Commuting matrices and cyclic modules
Let $A=\left(\begin{smallmatrix}1&0&0\\1&1&0\\0&0&1\end{smallmatrix}\right)$ and $B=\left(\begin{smallmatrix}1&0&0\\0&1&0\\1&0&1\end{smallmatrix}\right)$...
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 ...
5
votes
Accepted
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 ...
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.
5
votes
Accepted
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).
5
votes
Accepted
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 ...
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 ...
4
votes
Accepted
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, ...
4
votes
Accepted
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 ...
4
votes
Accepted
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 ...
4
votes
Accepted
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 ...
4
votes
Accepted
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,...
3
votes
Accepted
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 ...
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 ...
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 ...
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
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