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There is an open locus $U$ of $GL_n/B$ such that if you take a point $p$ in $U$, then its torus orbit closure $\overline{T\cdot p}$ is isomorphic to the permutahedral variety. Equivalently, by Atiyah'...
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Let $$BS(\mathbf{i}) = P_{i_1} \times^B P_{i_2} \times \cdots \times^B P_{i_r}/B$$ be the Bott-Samelson variety associated to a word $\mathbf{i}$ in the vertices of the Dynkin diagram of a reductive ...
Antoine Labelle's user avatar
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Let $G$ be a complex simply-connected semisimple group. Then I know that the flag variety of $G$ can be described as the subscheme of $\prod_{\lambda} \mathbb{P}(V_\lambda)$ (with the product over all ...
Antoine Labelle's user avatar
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I have often heard two definitions of Richardson varieties. One is where you take the intersection of two open Schuberts $BwB$ and $B^-vB$ then take the closure. Another is where you take the ...
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Let $G$ be a semisimple simply connected algebraic group over $\mathbb C$ and let $B\subseteq G$ be a Borel subgroup. Furthermore let $T\subseteq B$ be a fixed maximal torus, $\lambda \in X^*(T)$ an ...
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Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$. Let $P$ be a parabolic subgroup of $G$, $\mathscr{F}:=G/P$ the partial flag variety associated to $P$. For a $G$-variety $X$, ...
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I want to know given a connected (maybe we can assume it to be simply connected or linear) real semisimple lie group $G$ and one of its maximal parabolic group $P$, how can we embed the flag variety $...
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In my research, one can find as a special case the following family of varieties. Fix integers $0<k<n$ and let $G=Gr(k,n)$ be the Grassmannian of $k$-planes in an $n$-dimensional vector space $V$...
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Let $X$ be the partial flag variety of flags $0 \subset V_k \subset V_{k+2} \subset V$ where $V$ is a fixed vector space of dimension $n$ and ${\rm dim} V_k = k$ and ${\rm dim} V_{k+2} = k+2$. Is it ...
Yellow Pig's user avatar
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If one considers the (complete) flag variety $F(\mathbb{C}^2)$ of $\mathbb{C}^2$, then this is biholomorphic to $\mathbb{P}^1_\mathbb{C}$, and thus the symmetric product of $d$ copies of $F(\mathbb{C}^...
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Let $G$ be a complex algebraic group with simple Lie algebra $\mathfrak{g} = \operatorname{Lie} G$. Let $\mathcal{B}$ be the flag variety consisting of all the Borel subalgebras of $\mathfrak{g}$. Let ...
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To fix (albeit standard) notation, let $G$ be a complex semisimple algebraic group, and $T \subset B \subset G$ choices of maximal torus and Borel subgroup, respectively. Let $X^\ast(T)$ be the ...
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Consider the flag manifold $M = U(n, \mathbb{C})/T^n$, where $T^n$ is the maximal torus of unitary diagonal matrices. Fixing a diagonal matrix $D$ with distinct reals on its diagonal, we can identify ...
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$\DeclareMathOperator\GL{GL}$Let $G$ be a linear algebraic group (probably reductive will be needed). Consider a faithful representation $G \to \GL(V)$. Given a parabolic subgroup $P < G$, we can ...
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Let $G$ be a reductive group, $K$ a symmetric subgroup of $G$ (e.g., fixed point of an involution), and $B$ a Borel subgroup of $G$. Then it is well known that $G/B$ has finitely many $K$-orbits. ...
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I have seen several different definitions of the so called "thick" affine flag variety associated to an affine Lie algebra, and I am having trouble seeing why they are the same. Some ...
Qixian Zhao's user avatar
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Consider the Plücker embedding of the variety of complete flags in $\mathbb C^n$: $$F_n\subset\mathbb P(\bigwedge\nolimits^1\mathbb C^n)\times\dots\times\mathbb P(\bigwedge\nolimits^{n-1}\mathbb C^n).$...
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For Grassmannians, the Schubert cells can be indexed by certain Young Tableaux, whose partition determines the dimensions of intersections of the chosen subspace with the standard complete flag. For ...
staedtlerr's user avatar
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$\DeclareMathOperator\Fl{Fl}$It is known that $H^*(\Fl(m)) \cong R^{\mathbb Z}(m)$, where $\Fl(m)$ denotes the variety of complete flags in $\mathbb C^m$, and $R^{\mathbb Z}(m)$ is the coinvariant ...
staedtlerr's user avatar
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Background Let $n$ be a positive integer, let $W$ be the Weyl group of $\text{GL}_n$. Its set of Borel subgroups is isomorphic to the full flag variety $\mathcal{F}_n$. In this question, I studied ...
EJB's user avatar
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Let $A, B\in M_n(\mathbb{C})$ be matrices that commute. We suppose that there exists a vector $v\in\mathbb{C}^{n}$ such that $(\mathbb{C}[A,B]).v$ generates $\mathbb{C}^{n}$. We call such a pair a ...
prochet's user avatar
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Dale Peterson famously gave a series of lectures on the quantum cohomology of flag varieties $G/P$ at MIT in 1997. These lectures are often cited in subsequent papers by other authors on the subject (...
SamJeralds's user avatar
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Let ${\mathcal B}=Fl(V)$ be the variety of complete flags in an $(m+2n)$-dimensional vector space $V$ over $\mathbb C$. Then we have the standard line bundles $\mathcal{O}(\lambda)$ on $\mathcal B$ ...
IntegrableSystemsEnthusiast's user avatar
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Let $m,n$ be nonnegative integers. Let $k$ be a field of positive characteristic. Let the vector space $V$ over the field $k$ have basis $e_1,...,e_{m+n}, f_1,..., f_n$. Let $k^*$ act on $V$ by $t \...
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Given the famous Littlewood-Richardson rule, in terms of Schur polynomials: $$s_\mu s_\nu=\sum_\lambda c^{\lambda}_{\mu\nu} s_\lambda,$$ is there a classification of the cases where the LR ...
Nicolas Medina Sanchez's user avatar
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I want to apologize in advance if my question is too elementary as I am not an expert in Lie theory. I have posted it before on stackexchange without receiving an answer. Let $G$ be a compact Lie ...
Lennart Meier's user avatar
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$ \newcommand\Fl{\mathcal{F}\!\ell} \newcommand\numC{\mathbb{C}} \newcommand\numZ{\mathbb{Z}} \newcommand\ringO{\mathbb{O}} \newcommand\ringK{\mathbb{K}} \newcommand\power{\...
Gaussler's user avatar
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Let $F_n$ denote the full flag manifold consisting of all complete flags in $\mathbb{R}^n$. For the standard basis $\{ e_1,\dots,e_n\}$ of $\mathbb{R}^n$, denote the standard opposite flags: $\sigma_- ...
sunny's user avatar
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1 answer
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If $G$ is a classical semisimple algebraic/Lie group over an algebraically closed field (maybe just say $\mathbb{C}$), viꝫ. $\mathit{SL}_n$, $\mathit{SO}_n$, $\mathit{Sp}_n$ (isogenies irrelevant here)...
Gro-Tsen's user avatar
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There is a really nice proof of Borel–Weil–Bott, essentially using parabolic induction (see Proof of Borel-Weil-Bott Theorem, Lurie - A proof of the Borel–Weil–Bott theorem or Demazure - A very simple ...
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Pulcinella
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Let $G=GL_n(q)$ be the general linear group over $\mathbb{F}_q$ and $T$ be the torus of diagonal matrices. We also pick a Levi subgroup of the form $L=GL_{n_1}(q)\times GL_{n_2}(q) \times \cdots \...
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Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$, Borel $B \supset T$ and Weyl group $W$. Set $X:=G/B$ and $C_w:=BwB/B \subset X$ for $w \in W$ the ...
KKD's user avatar
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Let $C_n(\mathbb{R}^d)$ denote the configuration space of $n$ distinct points in $\mathbb{R}^d$, say $\mathbf{x}_1, \ldots, \mathbf{x}_n$. The symmetric group $\Sigma_n$ acts on $C_n(\mathbb{R}^d)$ ...
Malkoun's user avatar
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3 votes
2 answers
626 views

Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ and Borel $B \supset T$ defining a set of simple roots $\Delta$. Additionally let $\rho$ be the half ...
KKD's user avatar
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1 answer
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I am interested in computing tensor products of perverse sheaves on (partial) flag varieties. For a specific example - consider the product of the big projective on $\mathbb{P}^1$ with itself (This is ...
Adam Gal's user avatar
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0 answers
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Let $G$ be a simple simpy connected complex algebraic group. I was wondering if there is a clear relationship between the generalized minors (defined by Berenstein, Fomin and Zelevinsky) and bases of ...
F.H.A's user avatar
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Let $G$ be a reductive algebraic group over $\mathbb C$ and let $G^\vee$ be its Langlands dual group. What are the precise relationships between the complete flag variety $X$ of $G$ and the flag ...
Qixian Zhao's user avatar
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0 answers
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The lattice of linear subspaces in a vector space V can be provided with a structure of monoid by considering the subspace generated by the union of two subspaces as the monoid operation. When looking ...
FreddyG's user avatar
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3 votes
1 answer
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All literature on the Schubert cells of the generalized flag varieties $G/P$ ("generalized" here means that $P$ is an arbitrary parabolic) assumes that $G$ is a semisimple complex group. I ...
Filip's user avatar
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Is there a notion of holonomic D module which admits the six functor formalism in the world of twisted D modules? Recall that twisted D modules on $X$ are well-defined for any $T$ torsor $\tilde{X}\...
Pulcinella's user avatar
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5 votes
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Is it known whether for a generalized complex flag variety $X$ (that is, $G/P$ for a complex semisimple Lie group $G$ and a parabolic $P$), the homology of the free loop space $H_*(\Lambda X, \mathbb{...
Filip's user avatar
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3 votes
1 answer
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It is known that, given a complex vector space $V$ of dimension $\mathit{dim}(V)=n+1$, all irreducible representations for the group $\mathit{GL}(V)$ are parametrized by Young tableaux $\lambda$. For ...
gigi's user avatar
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2 votes
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This may be a stupid question. I'm reading the paper "Automorphisms of moduli spaces of vector bundles over a curve" of Indranil Biswas, Tomas L. Gomez, V. Munoz (arXiv link). I have a ...
Aoki's user avatar
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This question was posted at MSE, but it did not receive any answer there. Let $G$ be a connected semisimple algebraic group over $\mathbb C$, $X$ the flag variety of $G$, $B_0$ a Borel subgroup, $\...
Qixian Zhao's user avatar
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6 votes
0 answers
189 views

For $U(n)$ the Lie group of $n \times n$ unitary matrices, and $T^n$ its maximal torus subgroup, the homogeneous space $$ U(n)/T^n $$ is called the complete flag variety of order $n$. For the special ...
Quin Appleby's user avatar
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0 answers
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The Weyl group $W$ of $G_2$, is a group of order $12$ which is generated by the subgroup of permutations of $e_1$, $e_2$ and $e_3$ and by by the element $\tau$ which maps $(e_1,e_2,e_3)$ to $(-e_1,-...
Malkoun's user avatar
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8 votes
1 answer
503 views

Let $k$ be a field of characteristic zero, $G$ be a reductive group with a Borel $B$ and $\mathcal{F}:=G/B$ the associated flag variety. Let $W$ be the Weyl-group of G. Then let $S \subset W$ and $Z=\...
KKD's user avatar
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3 votes
0 answers
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Let $k$ be a field, $d,n \in \mathbb{N}$ and denote by $Gr(d,n)$ the Grassmannian, which parameterizes the $d$-dimensional linear subspaces of $n$-dimensional $k$-vector space, considered as a ...
KKD's user avatar
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1 vote
1 answer
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Let $F,F'$ be two locally free sheaves on a smooth complex algebraic variety. There is a cup-product $H^i(X, F) \otimes H^j(X,F') \to H^{i+j}(X,F \otimes F')$. In particular if $F$ is the sheaf of ...
Nicolas Hemelsoet's user avatar
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3 votes
1 answer
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Let $G/B$ be a compact homogeneous complex manifold, and let $E = G \times_{\rho} V$ be a hololmorphic homogeneous vector bundle over $G/B$. Does there exist a presentation of the Euler characteristic ...
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