All Questions
75 questions
2
votes
1
answer
177
views
Action of $O(3,\mathbb{R})$ on the conic $\{x^2+y^2+z^2=0\}$
The action of the orthogonal group $O(3,\mathbb{R})$ on the conic
$C= \{ x^2+y^2+z^2=0 \}$ in $\mathbb{P}^2$ must be well-understood, but I could not find any reference.
Is it doubly transitive?
- 14k
5
votes
0
answers
140
views
Classification of visible actions for *reducible* representations?
Is there a classification of the pairs $(G,V)$ such that $G$ is reductive [and connected, if you like], and $G$ acts faithfully and visibly on $V$ - crucially, including all cases where $V$ is ...
- 3,230
5
votes
0
answers
293
views
On the deformation theory of associative algebras
Let us start by recalling the notion of a formal deformation:
Let $K$ be a field of characteristic zero and $A$ be an associative $K$-algebra. Consider a commutative augmented $K$-algebra $R$, with ...
- 541
4
votes
0
answers
177
views
Reference for Iwahori-Hecke algebras
I recently came across the notion of an Iwahori-Hecke algebra. I would like to learn the basics about this type of algebras (mainly to get an intuition about them, as they seem to be related to some ...
- 541
9
votes
1
answer
493
views
A compactification of the space of points on the affine line
I recently encountered an interesting space. It is a compactification of the space of $ n$ points in $ \mathbb A^1 $ modulo translation, $ (\mathbb A^1)^n / \mathbb G_a $.
Let $ n \in \mathbb N $ and ...
- 4,612
5
votes
0
answers
263
views
Reference/list of reductive subgroups of reductive groups?
Let $G$ be a (say, connected) reductive group over an algebraically closed field of characteristic zero (say, $\mathbb C$).
I am looking for simple examples of (ideally) complete characterizations of ...
- 3,799
12
votes
1
answer
567
views
Reference for character sheaves over $\mathrm{GL}_n(q)$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}$I know a little bit about complex representation theory of finite reductive groups as $\GL_n(q),\SO_n(q)$ etc via Deligne-Lusztig induction and ...
- 1,061
11
votes
2
answers
684
views
Invariants of $\mathrm{GL}_n$ representations
$\DeclareMathOperator\GL{GL}$Let $V=\mathbb C^n$ be the natural representation of $\GL_n(\mathbb C)$ and let $W=\operatorname{Sym}^2(V)$ be the symmetric square representation. Let $W^k$ denote the ...
- 673
11
votes
2
answers
977
views
Reference for combinatorics with view towards representation theory/algebraic geometry
I'm making this post to ask for a reference about combinatorics: I'm a PhD student in representation theory/algebraic geometry. My background is mostly in algebra and geometry (and also mostly ...
- 1,061
8
votes
3
answers
1k
views
Further reading in algebraic geometry
I recently finished reading W. Fulton's "Algebraic Curves" and also attended a lecture series on moduli spaces and am interested in learning about them as well. I looked for a few books to ...
4
votes
0
answers
163
views
External tensor product Calabi-Yau DG categories
Let $\mathcal{C}$ be a smooth proper DG-category such that the shift $[p]$ is a Serre functor for $D^{perf}(\mathcal{C})$ (we say that $D^{perf}(\mathcal{C})$ is $p$-Calabi-Yau). I am looking for a ...
- 7,300
8
votes
1
answer
264
views
Class group of hypersurfaces of finite representation type
Let $k$ be an algebraically closed field of characteristic different from $2,3$ and $5$, and let $R=k[[x,y,x_2,\dots,x_d]]/(f)$, where $f\in(x,y,x_2,\dots,x_d)^2$, $f\neq0$. By results of Buchweitz-...
- 411
10
votes
1
answer
299
views
Map from Bruhat stratification to Catalan stratification for the space of totally nonnegative upper-triangular matrices
$\DeclareMathOperator\SL{SL}$This question came up in a class "Total Positivity and Cluster Algebras" being taught by Chris Fraser.
Let $N^+$ denote the space of uni-upper-triangular ...
- 24.2k
3
votes
1
answer
473
views
Borel–Weil–Bott for partial flag varieties
Is there a generalization of Borel-Weil-Bott for partial flag varieties, i.e. homogeneous spaces of the form $G/P$ with $P$ parabolic and $G$ semisimple? If so, I would like a reference.
- 3,079
4
votes
1
answer
343
views
Subgroups of algebraic groups containing regular unipotent elements
Let G be a simple algebraic group. Let H be a reductive subgroup of G which contains a regular unipotent element of G. Such subgroups were classified by Saxl and Seitz in all good characteristics. I'...
- 2,751
0
votes
0
answers
294
views
Reference: Irreducible components of the Steinberg variety are conormal bundles
The (partial) Springer resolution is defined as a map $\mu: T^*\mathcal{F} \to \mathcal{N}$, where $\mathcal{F}$ is the partial flag variety consisting of $n$-step partial flags of $\mathbb{C}^d$, and ...
- 159
5
votes
0
answers
118
views
Where to read about the toric variety coming from a principal nilpotent element of a (semi)simple algebraic group?
Given a principal (regular) nilpotent element $e$ in the Lie algebra $\mathfrak g$ of a complex semisimple algebraic group $G$, let $\mathfrak s=(e,f,h)$ be an $\mathfrak{sl}_2$-triple for $e$. Then ...
- 18.4k
4
votes
0
answers
82
views
Support of the fundamental class of irreducible components of partial Springer fibres corresponding to flag signatures
The (partial) Springer resolution is defined as a map $\mu: T^*\mathcal{F} \to \mathcal{N}$, where $\mathcal{F}$ is the partial flag variety consisting of $n$-step partial flags of $\mathbb{C}^d$, and ...
- 159
3
votes
0
answers
102
views
Is there a source in which Demazure's function $p$ defined in SGA3, exp. XXI, is calculated?
Suppose that $\mathcal{R}=(M,R,M^*,R^*)$ is a root datum. In section 1.2 of SGA3, exp. XXI, Demazure defines the $\mathbb Z$-linear map $p:M\to M^*$ by
$$p(x)=\sum_{u\in R^*}(u,x)u$$
and proves many ...
- 3,137
22
votes
2
answers
2k
views
A royal road to Coulomb branches of 3D $\mathcal{N}=4$ gauge theories
So, I've been very interested recently with the developements of the (now mathematically precise) theory of Coulomb branches - in particular because of its recent applications on representation theory ...
- 3,318
18
votes
1
answer
566
views
Subgroup $\mathrm{E}_6$ generated by $\mathrm{Spin_7}$ and $\mathrm{SL}_3$
Let $\mathbb{O}$ be the octonion algebra (say over $\mathbb{R}$) and let $J_{3}(\mathbb{O})$ be the set of $3 \times 3$ hermitian matrices with octonion coefficients, that is:
$$ J_3(\mathbb{O}) = \...
- 7,300
9
votes
3
answers
589
views
Subgroup of $\mathrm{GL}_n$ stabilizing linear subspace skew-symmetric matrices
I am currently reading "Schiffer variations and the generic Torelli theorem for hypersurfaces" by Voisin, where it is claimed that the subgroup of $\mathrm{SL}_{2m}$ ($m \geq 3$) which preserves a ...
- 7,300
2
votes
0
answers
640
views
Areas of algebraic geometry useful for geometric representation theory
What topics/areas of algebraic geometry (aside from perverse sheaves/D-modules, etale cohomology, and possibly derived algebraic geometry) is it useful to learn/master if one is interested in doing ...
- 2,964
5
votes
2
answers
2k
views
Canonical reference for Chern characteristic classes
I'm a little uncertain about the definitions for
Chern roots
Chern classes
Chern characters
From perusing several discussions, I gather that if one correlates the nomenclature with that of ...
- 10.5k
9
votes
3
answers
2k
views
Borel's presentation for the cohomology of a Flag Variety
If $G$ is a simple complex Lie group, $T\subset B\subset G$ is a choice of Borel and maximal torus, and $W$ is the Weyl group, then
1) $H^{*}(G/B,K)=K[T^{\vee}]/(K[T^\vee]^W_{+})$
and
2) $K[T^\vee]^...
- 1,537
5
votes
0
answers
92
views
Question concerning the representation dimension of a special algebra
I would like to know, if the following problem is still open:
Let $k$ denote an algebraically closed field of characteristic 3.
Determine the representation dimension of $k(C_3\times C_3)$, where $...
- 1,755
3
votes
1
answer
203
views
Symplectic representation of modular group
The modular group $\Gamma_{g}$ of isotopy classes of diffeomorphisms of a genus $g$ surface $S$ acts on $H^1(S,\mathbb{Q})$ (or $H^1(S,\mathbb{Z})$) respecting the intersection pairing. This gives a ...
- 33
7
votes
0
answers
597
views
Reference for shtuka and trace formula
I really want to learn the work of Laurent Lafforgue and the joint work of Zhiwei Yun and Wei Zhang. They both involve shtuka and trace formula, which I only know the basic idea. So I would like to ...
- 409
2
votes
1
answer
340
views
Exterior powers of $Sym^p T$ over Gr(k,n)
Let G=Gr(k,n) the Grassmannian of $k$-dimensional subspaces of $\mathbb{C}^n$ and denote by $T$ the (rank $k$) tautological bundle over $G$, and by $Sym^p T$ its $p$-th symmetric power. Is there any ...
- 776
4
votes
0
answers
76
views
Comparing parametrizations of unipotent radical
Let ${G}$ be a simple algebraic group over $\mathbb{C}$ with maximal torus $T$ and set of simple roots $\{\alpha_i\}_{i\in \Delta}$. We then have a Borel supgroup $B=TU$ with unipotent radical $U$. ...
- 2,516
5
votes
1
answer
261
views
A "prequestion" about meromorphic representations of algebraic groups
In a comment exchange around an answer to Is a group scheme determined by its category of representations? there arose the issue of Tannakian reconstruction for non-affine algebraic groups (e. g. ...
- 18.4k
6
votes
0
answers
268
views
duality between quiver variety and affine Grassmannian
Let $\frak{g}$ be a ADE type simple lie algebra. There are (at least) two geometric ways to get highest weight irreducible representations of $\frak{g}$. One is by considering constructible functions ...
- 849
0
votes
0
answers
197
views
'Adelic torus' not arising from a rational torus
Let $G$ be a reductive group over a global field $F$, and $\gamma$ a strongly regular semi-simple element of $G(F)$. Then the centralizer $G_\gamma$ is defines an $F$-torus $T$, and hence by base ...
- 3,799
3
votes
2
answers
1k
views
Rational Characters of a reductive group have the same rank as split component
Let $G$ be a connected reductive group defined over a perfect field $F$. The split component $A$ of $G$ is the unique maximal $F$-split subtorus of the radical of $G$. For an algebraic group $H$ ...
- 6,180
4
votes
1
answer
783
views
Three dimensional representations of Alternating group
The alternating group $A_5$ has $2$ irreducible representation of degree $3$. The characters for these representations have irrational values. I guess the ring of invariants of these representations ...
- 125
3
votes
0
answers
188
views
Bott-type vanishing results for the weighted Grassmannian wGr(2,5)
If $G=Gr(k,n)$ denotes the Grassmannian of k-dimensional subspaces in $V= \mathbb C^n$, representation theory gives us a Bott-type result for the cohomology groups $H^q(G, \Omega^p(k))$ of the twisted ...
- 776
3
votes
1
answer
195
views
How to embed $S^2\mathbb{C}^2$ into $S^2S^3\mathbb{C}^2$ and get the ideal of the twisted cubic?
Let $X:=x^3$, $Y:=x^2y$, $Z:=xy^2$ and $W:=y^3$ be the 4 independent generators of $S^3\mathbb{C}^2$, and observe that the kernel of the natural epimorphism (total symmetrisation)
$$
p:S^2S^3\mathbb{C}...
- 2,527
4
votes
1
answer
2k
views
Reference request for an introduction to deformation theory in algebraic geometry
I'd like some introductory references for deformation theory in algebraic geometry. I'm interested in survey articles too but I primarily want references which give all the definitions and go through ...
8
votes
1
answer
549
views
Ring of invariants for the regular representation
The symmetric group $S_n$ acts on $\mathbb C^n$ by permuting the coordinates. In this case the ring of invariants is generated by elementary symmetric polynomials in n-variables. Now consider the ...
- 195
5
votes
1
answer
436
views
Is the Veronese variety "enough" to describe all the $SL(V)$-orbits in $\mathbb{P}(\textrm{Sym}^dV)$?
I apologise in advance if the question will look ridicolous to experienced eyes: in this case a good reference will be enough to clarify my doubts.
Let $V$ be a complex vector space of dimension $n$, ...
- 2,527
2
votes
0
answers
156
views
Extension of the Hilbert-Mumford Criterion
Let $X$ be a smooth variety, $L$ a line bundle on $X$ and $G$ a reductive group actin on $X$ with a linearization of the action to $L$. Say we are over the complex numbers.
Both the concept of GIT ...
- 2,384
3
votes
0
answers
168
views
Invariant Theory over finite adeles
Classical invariant theory, among the other things, classifies polynomial functions over a vector space $V$ endowed with a quadratic form $Q$ which are invariant under the action of $SO(V,Q)$.
I am ...
- 2,384
6
votes
0
answers
455
views
Cohomology of Bott-Samelson varieties?
How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here.
Is there ...
- 1,719
25
votes
3
answers
6k
views
Introductory References for Geometric Representation Theory
Would anyone be able to recommend text books that give an introduction to Geometric Representation Theory and survey papers that give an outline of the work that has been done in the field? I'm ...
8
votes
1
answer
1k
views
The Bialynicki-Birula Stratification of the Affine Grassmannian
Let $G$ be a connected, simply-connected complex semisimple group with affine Grassmannian $\mathcal{G}r$. Fix a maximal torus and Borel $T\subseteq B\subseteq G$. I am reading "Loop Grassmannian ...
- 4,920
5
votes
1
answer
514
views
Reference for the Natural Ample Line Bundle on the Affine Grassmannian
Let $G$ be a connected, simply-connected complex semisimple group. Let $$\mathcal{G}r:=G((t))/G[[t]]$$ be its affine Grassmannian. I have read that $\mathcal{G}r$ possesses a natural very ample line ...
- 4,920
6
votes
1
answer
401
views
Stratifications and Filtrations of the Affine Grassmannian
Let $G$ be a connected, simply-connected complex semisimple group. Let $$\mathcal{G}r=G(\mathcal{\mathbb{C}((t))})/G(\mathcal{\mathbb{C}[[t]]})$$ be the affine Grassmannian of $G$. We know that $\...
- 4,920
7
votes
2
answers
1k
views
Strata of the Affine Grassmannian
Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and denote by $\mathcal{G}$ its affine Grassmannian. Fix a maximal torus $T\subseteq G$. We know that $\mathcal{G}$ ...
- 4,920
8
votes
1
answer
382
views
Action of the endomorphism monoid on an irreducible GL-module
Let $G=\mathrm{Gl}_n(\mathbb C)$ and $V$ an irreducible $G$-module on which $G$ acts polynomially. In other words, the algebraic group action of $G$ on the affine space $V$ extends to an algebraic ...
- 4,902
1
vote
1
answer
308
views
Holomorphic representations of complex reductive Lie groups and the boundary of orbits (Reference request)
I have difficulties finding an appropriate reference for the following question (which I hope that it to be true).
Let $U$ be a compact Lie group, $G:=U^{\mathbb{C}}$ its complexification and $\tau: U^...
