4
votes
0answers
62 views

Closed orbits of complete flags in $\mathbb{C}^n$

Let $B$ be a symmetric (or antisymmetric) non-degenerate bilinear form on $\mathbb{C}^n$ and let $G$ be the associated group of automorphisms $O(n)$ (resp. $Sp(n)$). What can we say about the ...
2
votes
1answer
33 views

Representation-preserving isomorphism

Suppose we have two algebraic groups $G,H$ over a field $K$ (maybe reductive) and we know in advance that $G(A)\cong H(A)$ for some $K$-algebra A. If we attach algebraic representations $\rho_{G(A)}, ...
5
votes
1answer
59 views

Schur functors as spaces of “flag tensors”?

Consider the following construction: for a vector space $V$, define $W \subseteq \bigwedge^2 V \otimes V$ by $W = \langle\ \alpha \otimes v : v \in \text{Span}(\alpha) \ \rangle$, that is, $W$ is ...
1
vote
0answers
29 views

Exactness of Hom functor for torus representations?

Given a reductive algebraic group $G$ and a maximal torus $T$. Is it true that the functors $$ Hom_T(-,\lambda) $$ are exact, where $\lambda$ denotes one of the the simple one-dimensional ...
1
vote
0answers
19 views

A question about Cayley-Chow forms

I'm reading some papers about $k$-stable theory and I have a question about Cayley-Chow forms. Maybe this question looks silly. Let X be a variety of $\mathbb{P}^N$ with dimension n and degree d. ...
1
vote
0answers
106 views

Construction of line bundles on the flag variety

Edit: The following is phrased in terms of algebraic geometry, but can be thought of analytically as well. Hence I added some tags... I am a bit confused about the subject in the title. For ...
9
votes
1answer
93 views

The unique closed orbit in GIT quotient fibers for polynomial actions of Gl

The following reasoning must contain a flaw somewhere because I end up with something absurd, and I cannot figure out where the mistake is. I hope that someone can point it out to me. Let $M$ be the ...
4
votes
1answer
82 views

Proof that ideal of Plücker relations is a prime ideal

I am reading section 8.4 of Fulton's Young tableaux where he defines a certain ring as follows. Fix a complex vector space $E$ of dimension $m$ and integers $d_1,\ldots d_s$ such that $m \geq d_1 > ...
0
votes
0answers
89 views

Hamiltonian reduction for unit sphere

Let $(M, \omega)$, be Symplectic Vector space and $N\subset M$ be unit sphere. Then why $N/ker\omega\mid _N$ is naturally $\mathbb{P}^{n-1}(\mathbb{C})$ Here ker$\omega$=$\{ y\in M: \omega(x,y)=0, ...
8
votes
1answer
273 views

Learning Roadmap for Borel - Weil - Bott Theorem

Next semester I may study a course where the ultimate goal is to get to the Borel - Weil - Bott (BWB) Theorem, if not at least try to understand it in the case that we have $G = \text{SL}_n$. I have ...
5
votes
1answer
89 views

The regular representation for affine group schemes

I want to understand the regular representation of an affine algebraic group. An affine algebraic group as I know it, is a functor from the category of $k $ -algebras to groups that is representable ...
12
votes
1answer
181 views

Introduction to the trace formula for people outside number theory

I am looking for references on the trace formula, by which I mean the Selberg trace formula and its successor the Arthur-Selberg trace formula. I am aware that there are "standard references" on the ...
7
votes
2answers
245 views

Algebraic geometry in representation theory?

I heard that today algebraic geometry plays some significant role in representation theory, which is a little surprising because when I learnt representation theory it is basically algebra, topology, ...
3
votes
0answers
85 views

Linearizations induced by the trivial one?

Let $k$ be a field, algebraically closed for simplicity. $G=GL_{n+1}(k),$ $X=\mathbb{P}_k^n$ and consider the action $G\times X\rightarrow X$ given by $(g,x)\mapsto gx$ (thus induced by usual left ...
7
votes
1answer
124 views

Trivial summand of a representation's symmetric power

The following comes from Exercise 13.17 of Fulton and Harris's book, Representation Theory: A First Course. Let $V$ denote the standard representation of $\mathfrak{sl}_3\mathbb{C}$, with weights ...
0
votes
1answer
75 views

Eigenvectors of algebraic group representation

In a paper of Kollar and Szabo there is a lemma (Lemma $1$) in which the following terminology is used: "Every representation $H\rightarrow GL(n,K)$ has an $H$-eigenvector" ($H$ is an algebraic ...
3
votes
0answers
64 views

What is the Induced Representation in Geometric Terms

As is well known, for $G$ a Lie group, and $H$ a subgroup of $G$ such that $G/H$ is homogeneous space (or maybe this is always a homogeneous space?), we have a correspondence between representations ...
3
votes
0answers
65 views

Decomposing Semisimple Perverse Sheaves

Assume $\mathbf{G}$ is an algebraic group over an algebraic closure $\overline{\mathbb{F}_p}$ for some prime $p>0$. Let $\mathscr{M}\mathbf{G}$ be the category of all ...
11
votes
1answer
838 views

Undergraduate roadmap for Langlands program and its geometric counterpart

What are the topics which an undergraduate with knowledge of algebra, galois theory and analysis learn to understand Langlands program and its goemetric counterpart? I would also like to know what are ...
3
votes
1answer
72 views

Dimension of the GL-orbit of d-forms in one less variable

Let $V:=k[x_0,\ldots,x_n]_d$ be the $k$-vector space of homogeneous polynomials of degree $d$. Let $G:=\mathrm{Gl}(n+1,k)$ act on $V$ induced by the canonical action on the linear forms: For ...
3
votes
1answer
218 views

Left adjoint and right adjoint/ Nakayama isomorphism

I am reading a paper "2-vector spaces and groupoid" by Jeffrey Morton and I need a help to understand the following. Let $X$ and $Y$ be finite groupoids. Let $[X, \mathbb{Vect}]$ be a functor ...
19
votes
0answers
276 views

Tate conjecture for Fermat varieties

I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here) and have a question about his work on (conjecturing!) the Tate conjecture ...
3
votes
1answer
276 views

Motivation for studying quadratic algebras, Koszul algebras, Koszul duality

I'm trying to gain a practical understanding of Koszul duality in different areas of mathematics. Searching the internet, there's lots of homological characterisations and explanations one finds, but ...
2
votes
1answer
103 views

Question about Tamafumi Kaneyama's Paper: “On Equivariant Vector Bundles On An Almost Homogeneous Variety”

My reference: http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.nmj/1118795362&page=record I have two question about Proposition 3.3.: Proposition3.3. ...
1
vote
0answers
51 views

How to characterize the image of $PGL(V)$ in $\mathbb{P}(W)$ for an irreducible $GL(V)$-representation $W$

I'd like to ask two versions of my question, one a very specific case that I suspect may have a fairly easy and classical answer, the other the general case which I would like to find references on. ...
3
votes
1answer
194 views

Intuition on the definition of “rational maps”

I'm studying some representation theory on $S_n$ and $GL(V)$ and tensor spaces, and have come across a lot of material involving rational representations. I'm not really an algebraic geometer by ...
13
votes
0answers
253 views

Application of Hilbert's basis theorem in representation theory

In Smalo: Degenerations of Representations of Associative Algebras, Milan J. Math., 2008 there is an application of Hilbert's basis theorem that I don't understand: Two orders are defined on the set ...
10
votes
2answers
481 views

Proof $\mathbb{A}^n$ is irreducible, without Nullstellensatz

As the title suggests, could anyone either provide me with or direct me to a proof that affine n-space $\mathbb{A}^n$ is irreducible, without using the Nullstellensatz? This is an exercise in a ...
6
votes
1answer
271 views

Does the $p$-torsion of an elliptic curve with good reduction over a local field always determine whether the reduction is ordinary or supersingular?

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $E/K$ an elliptic curve with good reduction. Does the $\mathbb{F}_p[\mathrm{Gal}(\overline{K})]$-module $E[p](\overline{K})$ determine whether the ...
5
votes
2answers
520 views

What is the differences between scheme-theoretical intersection and set-theoretical intersection?

In the paper representations of general linear groups, there are two concepts: scheme-theoretical intersection and set-theoretical intersection. What are their differences and relations? Thank you ...
0
votes
1answer
89 views

Questions about Wronskian

I am reading a paper Bispectral and ($\mathfrak{gl}_n, \mathfrak{gl}_m$) dualities. I have some questions about some computations with Wronskian and dimensions of some vector spaces. On page 9 (line ...
4
votes
2answers
235 views

what are good references for learning about vector bundles and their sheaves of sections?

I am a beginner in representation theory and algebraic geometry, so that references giving clear explanations of things like the tautological line bundle on $\mathbb P^n$, its dual, and the associated ...
3
votes
1answer
265 views

Line bundles, line bundles on a homogeneous space, and sections of line bundles

I have some difficulty in understanding the concepts: line bundles, line bundles on a homogeneous space, and sections of line bundles. These concepts are on page 140 (the first paragraph of section ...
0
votes
1answer
212 views

questions about coroot

I am reading the lecture notes of geometric representation theory: http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf. I have a question on coroot. In general, if we have a root $\alpha$, then the ...