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 ...
- 141
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-...
- 17k
10
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 ${\mathbb P}(U)$, ...
- 1,602
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 $U(n-r)$ ...
- 3,772
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. (...
- 148k
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 ...
- 148k
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),...
- 44.7k
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 ...
- 4,612
7
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 ...
- 1,219
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
$$
\lambda_1 = \dots = \lambda_{k_1} >
\lambda_{k_1 + 1} = \dots = \lambda_{k_1 + k_2} > \dots >
\lambda_{k_1 + \dots +...
- 39.3k
7
votes
Frobenius pushforward of an equivariant tautological bundle on the flag variety
EDIT. Corrected the statement ($\sigma$ should be $p-1$ times what I wrote) and answered the question in the comment.
In general, the push-forward of a line bundle on the flag variety $G/B$ will not ...
- 16.1k
6
votes
Schubert calculus expressed in terms of the cotangent space of the Grassmannians
For your first question: yes. See Stoll, Invariant forms on Grassman manifolds, p. 15. I think your second question is answered in the same book.
- 26.3k
6
votes
Accepted
Commuting matrices and cyclic modules
Let $A=\left(\begin{smallmatrix}1&0&0\\1&1&0\\0&0&1\end{smallmatrix}\right)$ and $B=\left(\begin{smallmatrix}1&0&0\\0&1&0\\1&0&1\end{smallmatrix}\right)$...
- 16.2k
6
votes
References for $K$-orbits in $G/B$
A good reference for this is "On Rationality Properties of Involutions of Reductive Groups" by Helminck and Wang (Advances in Math. 1993). The decomposition of $K\backslash G/B$ is studied ...
- 6,349
6
votes
Accepted
$\mathbb P^1$-bundle on a partial flag variety
If $\mathcal{U}_k \subset \mathcal{U}_{k+2} \subset V \otimes \mathcal{O}$ is the flag of tautological subbundles then
$$
Q = \mathcal{U}_{k+2} / \mathcal{U}_k
$$
is one of the possible choices.
- 39.3k
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).
- 31.5k
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 ...
- 24.2k
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.
- 8,597
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 ...
- 6,162
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 ...
- 954
4
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 ...
- 44.7k
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 ...
- 44.7k
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 ...
- 8,597
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,...
- 3,513
4
votes
Accepted
Embedding flag manifolds of real semisimple lie group
If $P$ is any parabolic subgroup in a semisimple real Lie group $G$, one can
construct a $G$-equivariant embedding of the partial flag manifold $G/P$ into some (high dimensional) real projective space ...
- 658
3
votes
Irreducibility of Gelfand-Serganova strata
Here is an explicit example. There are examples of realization spaces of matroids (which are, up to a torus quotient, Gelfand-Serganova strata in the Grassmannian) which are disconnected. I believe ...
- 1,046
3
votes
Accepted
Containment of Bruhat cells on flag variety
I'd say that the relevant fact here is as follows. For two Borels $B_1$ and $B_2$ with a common maximal torus $T$ let $x_1$ be the unique $T$-fixed point in the open $B_1$-orbit. Then the $B_2$-orbit $...
- 3,513
3
votes
Union of Schubert cells being affine
This is essentially an extension of my comment, just to answer the actual "is this the only case?" question. It is not, $Z$ will be affine whenever $S$ is an antichain in the Bruhat order. Indeed, ...
- 3,513
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 ...
- 765
2
votes
Positivity of coefficients of a polynomial derived from Schubert polynomials
You might want to try to prove Conjecture 1 in Kirillov's paper when the length difference between w and v is 2, starting from Liu's formula for skew divided difference operators in terms of /partial_{...
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