All Questions
2,543 questions
11
votes
5
answers
2k
views
How do you switch between representations of an algebraic group and its Lie algebra?
I'm interested in the structures of categories like $Rep(GL_n), Rep(SL_n)$, etc. of algebraic representations of an algebraic group. I understand that there should be some relation between these and ...
- 7,273
1
vote
1
answer
434
views
Tori acting on vector spaces
Let $T$ be a torus defined over a field $K$ of characteristic $p>0$. Suppose that $T$ acts (algebraically) on some vector space $V$ (over the same field $K$). Let $W$ be a subspace of $V$. Now ...
- 7,800
4
votes
1
answer
1k
views
Group Cohomology for Reductive Groups
Can anyone provide a reference to proofs of statements of the following type: The higher algebric group cohomology of a reductive group $G$ over $\mathbb{C}$ vanishes.
I am interested not just in ...
- 7,108
3
votes
4
answers
2k
views
Simplicity of (complex) orthogonal groups
I need a reference for the proof that the complex orthogonal group
$SO_{2n+1}($ℂ$) = \{A\in SL_{2n+1}($ℂ$): A^TA = Id\}$ is simple in a group theoretical sense (if it is true).
How about ...
- 2,178
10
votes
2
answers
1k
views
Is there a way to see a topological group as the "Cayley graph" of its "infinitesimal generators"?
At the time of writing, the most recent blog post over at What's new by Terrence Tao is Cayley graphs and the geometry of groups, and that (excellent, as with most of Tao's writing) post most ...
- 316
15
votes
2
answers
4k
views
Hopf algebra duality and algebraic groups
Background:
Let $G$ be a linear algebraic group over an algebraically closed field $k$ and let $I \subseteq k[G]$ be the ideal of the identity element. The hyperalgebra $U(G)$ of $G$ is defined to be ...
CommunityBot
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6
votes
3
answers
2k
views
What's the classification of the algebraic subgroups of Sp(4,R)?
Hi! I would like to know if there is an explicit classification of the algebraic (i.e., Zariski closed) subgroups of the symplectic group Sp(4,R) and/or more generally Sp(2n,R) somewhere in the ...
- 9,399
10
votes
1
answer
412
views
Reference for Pic(G) and central extensions.
Let $G$ be a connected reductive group over a (perfect, why not) field $F$. Let $m$, $pr_1$, $pr_2$ denote the multiplication, first, and second projection maps from $G \times G$ to $G$.
Then I'm ...
- 14.2k
22
votes
3
answers
5k
views
Do semisimple algebraic groups always have faithful irreducible representations?
For simplicity, I will be talking only about connected groups over an algebraically closed field of characteristic zero.
The basic theorem of affine algebraic groups is that they all admit faithful, ...
CommunityBot
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7
votes
3
answers
3k
views
congruent to 1 mod p
This is a somewhat vague question: for a prime number p, we often see that various counts come out to be 1 modulo p. What are the possible reasons for this?
Here are some I've encountered:
For some ...
- 39.9k
16
votes
3
answers
2k
views
On Category O in positive characteristic
Let $G$ be a semisimple algebraic group over an algebraically closed field $k$. In the case that $k$ has characteristic 0, there has been intensive study of the BGG category O of representations of ...
- 52.9k
9
votes
1
answer
777
views
Geometric (or intuitive) interpretation of additional derivatives in characteristic p > 0
In characteristic $p > 0$ there are "extra" differential operators, i.e., ones that are outside the algebra generated by first-order derivations.
Is there any interpretation of these operators in ...
- 12.4k
4
votes
2
answers
923
views
What is the correct formulation of the CDE triangle?
The CDE triangle with C Cartan matrix and D decomposition matrix is well-known for finite groups. I have not seen a full account of this for finite dimensional algebras which I find surprising as ...
- 44.7k
5
votes
2
answers
1k
views
Conjugate cocharacters in a maximal torus
Let $G$ be a linear algebraic group over an algebraically closed field $k$, and $T$ a maximal torus of $G$.
Suppose we have two cocharacter $\mu, \mu' : \mathbb{G}_m \to T$, which are conjugate under ...
- 7,108
2
votes
1
answer
278
views
What is the family derived from the absolute Frobenius on the Hilbert scheme?
Let $f$ be a Hilbert polynomial, and $X := Hilb_h(P^d_{F_p})$ a Hilbert scheme defined over $F_p$. Then there is an absolute Frobenius map $F: X \to X$. I'm even interested in the case $f \equiv 1$, ...
- 1,411
9
votes
4
answers
1k
views
Compact simple simply connected algebraic groups over $Q_p$ or other local non-archimedean fields
My motivation is to understand the following situation: Given absolutely and almost simple algebraic group $G$ defined over a number field $k$ and a finite valuation $v$ on $k$, when $G(k_v)$ can be ...
- 1,411
10
votes
4
answers
1k
views
Algebraicity of holomorphic representations of a semisimple complex linear algebraic group
Let $G$ be a complex linear algebraic group, given to us as a closed subgroup of some $\mathrm{GL}(n,\mathbb{C})$. Suppose moreover that $G$ is semisimple. Then it's a fact that every finite-...
- 4,540
7
votes
0
answers
491
views
Alterations of regular varieties
Let $X$ be a regular quasi-projective variety over a perfect field $k$. The existence of a "good compactification" of $X$, i.e. a regular projective variety $\bar{X}$ with an embedding $X\...
- 4,450
0
votes
0
answers
700
views
Questions on orbit properties of group action on varieties
Let $F$ be a p-adic field or $\mathbb{R},\mathbb{C}$, $G$ a group(not necessarily reductive) over $F$, $X$ an algebraic variety defined over $F$, and $G$ acts on $X$. Now we have several questions ...
- 18.7k
5
votes
1
answer
499
views
software for computations on flag varieties in arbitrary characteristic
Is there any software that will compute cohomology of vector bundles (or just line bundles) on flag manifolds?
The only one I know of is Macaulay2, via the Schubert2 package, but it works with what ...
- 5,846
6
votes
2
answers
597
views
Points of reductive groups
Let $G$ be a (connected) reductive group over a field $k$. Then there is a natural functor from the category of representations of $G$ to the category of representations of $G(k)$.
Under which ...
- 4,015
2
votes
1
answer
474
views
Automorphism of algebraic group preserving a hyperspecial maximal compact
Suppose that $K/\mathbb{Q}_l$ is a finite extension, with ring of integers $\mathcal{O}_K$. Suppose $\mathcal{G}/K$ is a (linear) algebraic group (connected+reductive), and $\Gamma\subset \mathcal{G}(...
- 8,296
2
votes
0
answers
165
views
Connected components of the set of Hodge structures
Let $\mathbb S$ be the torus $\mathbb C^\times$ viewed as an algebraic group over $\mathbb R$. Let $G$ be any affine algebraic group over $\mathbb R$. The set $Hom(\mathbb S,G)$ of morphisms of real ...
- 21
8
votes
4
answers
3k
views
method of finding roots of polynominal equations with arithmetic operations and roots and other functions
Lets recall Platonic construction in plane geometry. It is impossible to square a circle using only ruler and callipers. But is also known that it is possible to do it with ruler which has a mark on ...
- 41.1k
0
votes
1
answer
470
views
Natural embedding GL_n(C) -> C^{n^2} \ {0} induces zero on cohomology
The question looks like an exercise in elementary algebraic topology, but I didn't manage to solve it. I am considering this question because it is a toy example in a problem I'm thinking about.
Let'...
CommunityBot
- 1
33
votes
1
answer
1k
views
Is the group of integer points on a finite-type group scheme over Z finitely presented?
Let $G$ be a group scheme of finite type over $\mathbf{Z}$. Must $G(\mathbf{Z})$ be finitely presented?
(The question is inspired by a not yet successful attempt to answer a question of Brian Conrad....
CommunityBot
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8
votes
1
answer
696
views
Are groups in (Var/k, rational maps) necessarily algebraic groups?
Let $RVar$ be the category which has complex algebraic verieties as objects and rational maps as morphisms [Edit: for $RVar$ to be a category, the rational maps have to be dominant].
Let's consider a ...
- 23.4k
15
votes
2
answers
814
views
Can the failure of the multiplicativity of Euler factors at bad primes be corrected?
Warning: This one of those does-anyone-know-how-to-fix-this-vague-problem questions, and not an actual mathematics question at all.
If $X$ is a scheme of finite type over a finite field, then the ...
- 9,418
14
votes
2
answers
3k
views
How many ways are there to prove flag variety is a projective variety?
I am looking for references talking about different ways to prove flag variety $G/B$ is projective variety. Now I have some in mind:
There is a proof in Humphreys Linear algebraic groups, he first ...
- 5,515
9
votes
1
answer
820
views
Quantum equivariant $K$-theory and DAHA.
Theorem 3.2 of the paper "Quantum cohomology of the Springer resolution" by Braverman, Maulik and Okounkov relates equivariant quantum cohomology of the cotangent bundle of $G/B$ to the trigonometric ...
CommunityBot
- 1
7
votes
1
answer
799
views
Liftability of Enriques Surfaces (from char. p to zero)
Let $k$ be an algebraically closed field of characteristic $p > 0$, $X$ a variety over $k$.
We say $X$ lifts to characteristic zero, if there exists a local ring $R$ containing $\mathbb Z$ with ...
- 22.6k
2
votes
2
answers
1k
views
Automorphism group of bi-elliptic surface
$X$ = bi-elliptic surface (smooth and over $\mathbb{C}$),
Aut($X$) = the group of automorphisms of $X$,
Aut$^0(X)$ = connected component of the identity in Aut($X$).
Is Aut$^0(X)$ always an affine ...
- 13.1k
12
votes
1
answer
1k
views
isomorphism of abelian varieties
Let $A, B, C$ and $D$ be abelian varieties (over $\mathbb{C}$) such that $A \times B \cong C \times D$, and $A \cong C$. From the irreducibility of abelian varieties, we can say that $B$ and $D$ are ...
- 27k
4
votes
2
answers
1k
views
Are all connected solvable affine algebraic groups supersolvable?
The basic question is whether there is a notion of chief factor of a connected solvable algebraic group that matches my intuition. A few smaller assertions are sprinkled through the explanation, and ...
- 10.7k
15
votes
2
answers
2k
views
How Does a Borel Subgroup Know Which Weights Are Dominant
Let $G$ be a simple group (say $SL_n$) and let $B$ be a Borel subgroup (say upper triangular matrices). Then all irreducible representations of $G$ are induced from one-dimensional representations of $...
- 44.7k
6
votes
3
answers
3k
views
Whenever I read "centraliser of maximal split torus", I think of...
Inspired by this question
I'd like to ask something more specific:
In the theory of connected reductive groups over fields, one often reads about the centraliser of a maximal split torus. Here is ...
CommunityBot
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48
votes
5
answers
15k
views
Algebraically closed fields of positive characteristic
I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far ...
CommunityBot
- 1
23
votes
3
answers
5k
views
Relation between Hecke Operator and Hecke Algebra
In the study of number theory (and in other branches of mathematics) presence of Hecke Algebra and Hecke Operator is very prominent.
One of the many ways to define the Hecke Operator $T(p)$ is in ...
- 45.7k
5
votes
1
answer
4k
views
Simply connectedness of algebraic group
$G$ is a semisimple algebraic group over $k$, if $G_{\bar k}$ is simply connected when we do base change to $\bar k$, can we descent the simply connectedness to $G$?
Here, simply connectedness means ...
- 52.9k
3
votes
3
answers
3k
views
Proof of Steinberg's tensor product theorem
Let $G$ be a simply connected semi-simple algebraic group over an algebraically closed field of positive characteristic. The Steinberg tensor product theorem gives a tensor product decomposition of an ...
- 22.6k
12
votes
0
answers
716
views
Lifting abelian varieties in (the closed fiber of) a fixed Neron model
Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth ...
- 1,609
8
votes
3
answers
1k
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References for theorem about unipotent algebraic groups in char=0?
There is a textbook theorem that the categories of unipotent algebraic groups and nilpotent finite-dimensional Lie algebras are equivalent in characteristic zero. Indeed, the exponential map is an ...
- 52.9k
7
votes
2
answers
2k
views
Reductive subgroup corresponding to a subdiagram of the Dynkin diagram of a simple group
I am looking for a reference for the following well-known fact: For any subdiagram $\Delta_0$ of of the Dynkin diagram $\Delta=D(G)$
of an adjoint simple group $G$ over an algebraically closed field $...
- 778
1
vote
1
answer
435
views
Definition of congruence subgroup for non-matrix groups
For an algebraic group that cannot be embedded into $GL_n$, is there a nice definition for congruence subgroup? Do we just define it as the compact open subgroup of $G(A_f)$, where $A_f$ is the finite ...
- 7,108
12
votes
3
answers
4k
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Books on reductive groups using scheme theory
Prof. Conrad mentioned in a recent answer that most of the (introductory?) books on reductive groups do not make use of scheme theory. Do any books using scheme theory actually exist? Further, are ...
CommunityBot
- 1
11
votes
1
answer
2k
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Realizations and pinnings (épinglages) of reductive groups
Let $G$ be a reductive group over an (say, algebraically closed) field $k$. Springer (in his book on algebraic groups) calls for a chosen maximal torus $T$ in $G$ a family $(u_\alpha) _{\alpha \in \...
- 7,108
3
votes
2
answers
733
views
If the morphism of root data induced by an isogeny of a reductive group is a Frobenius, is then the isogeny itself a Frobenius?
Let $G$ be a reductive (or just semisimple) algebraic group over an algebraically closed field $k$ of characteristic $p > 0$, let $T$ be a maximal Torus and let $f:G \rightarrow G$ be an isogeny. ...
- 7,108
2
votes
1
answer
414
views
generators of the ideal of an unipotent-generated algebraic group
Given any affine algebraic group $G$ over an algebraically closed field $\mathbb{F}$ of characteristic $0$ with a faithfull representation in $GL_n(\mathbb{F})$ . If one knows the generators of the ...
- 53
5
votes
3
answers
739
views
Smoothness of hyperplane sections
Suppose $X\subset \mathbb{P}^n$ is a smooth hypersurface defined over $\mathbb{Q}$. For a "generic" prime $p$, what can be said about the set of hyperplanes $H$ in $\mathbb{P}^n(\mathbb{F}_p)$ for ...
- 16.9k
9
votes
2
answers
656
views
How does the order of a pole of a zeta function indicate any geometric information?
Here, I'm primarily concerced about zeta functions of hypersurfaces over fields of finite characteristic.
Assume $F_q$ to be a finite field with q elements. Consider the zeta function of the ...
- 65.4k