2,352 reputation
1020
bio website tau.ac.il/~borovoi
location Tel Aviv, Israel
age
visits member for 4 years, 6 months
seen 17 hours ago

I am a professor at Tel Aviv University, interested in linear algebraic groups, homogeneous spaces, number theory and nonabelian cohomology.


Jul
26
comment Rational structures on the flag variety over a finite field
@user148212: I continue. Since the action of the Frobenius $F$ on any point $x\in X$ is uniquely determined, and our variety $X$ is reduced, the action of $F$ on the structure sheaf of $X$ is uniquely determined, thus the rational structure on $X$ is uniquely determined - if it exists. The existence of a rational structure can be proved by an explicit construction, see my previous comment.
Jul
26
comment Rational structures on the flag variety over a finite field
@user148212: I add details. Write $x_0=eB\in X$. Since it is a rational point, we have $F(x_0)=x_0$, where $F$ denotes the Frobenius. Now consider any point $x=gB\in G/B$ where $g\in G$, then $x=g x_0$. Since we want the action of $G$ on $X$ to be rational, we have $F(x)=F(g x_0)=F(g) F(x_0)=F(g) x_0$. Since the rational structure on $G$ is given, the element $F(g)\in G$ is uniquely determined. Since the $G$-action on $X$ is given, the product $F(g) x_0$ is uniquely determined. Thus $F(x)$ is uniquely determined.
Jul
26
comment Rational structures on the flag variety over a finite field
@user148212: Yes, the choice of a rational point and of a G-structure on a homogeneous space implies the choice of a rational structure.
Jul
25
comment Rational structures on the flag variety over a finite field
@user148212: Note that the reference to Theorem 12.2.1 in Springer's book works when the Frobenius in $G$ comes from a structure of an algebraic $\mathbb{F}_q$-groups (not in the case when you deal with Suzuki groups or Ree groups).
Jul
24
comment Rational structures on the flag variety over a finite field
@user148212: On your variety $X=G/B$ you already have a canonical structure of $G$-variety (over $\overline{F}_q$) and a canonical rational point $eB=B$, the image of the unit element $e$ of $G$. Therefore, the $F_q$-rational structure on $X$ is unique, if exists. What is not evident is the existence of an $F_q$-rational structure. This can be proved by an explicit construction of a closed (in your case) subvariety in a projective space, see the proof of Corollary 5.5.4 in Springer's book mentioned by Daniel Loughran.
Jul
4
awarded  Inquisitive
Jul
3
comment What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?
@user52824: Thank you! As I expected, my questions reduce to linear algebra. Now what is the induced quadratic (resp., Hermitian) form on $\det(V)$? Could you please either explain or give references? I would be grateful if you could write an answer based on your comment. I will accept it immediately.
Jul
3
revised What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?
edited title
Jul
3
asked What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?
Jul
2
awarded  Curious
May
25
answered Real Lie groups versus real linear algebraic groups: differences in connexity and fundamental group
Apr
22
comment $SL(n) \times SL(n)$-invariants of $m$-tuples of matrices
The determinants of your matrices $M_1$, ... $M_m$ are polynomial invariants in the $SL$ case, and a natural guess would be that the algebra of polynomial invariants is generated by these determinants. Is this correct?
Apr
22
comment $SL(n) \times SL(n)$-invariants of $m$-tuples of matrices
I think you can find the answer to your question in the book "The Classical Groups: Their Invariants and Representations" by Hermann Weyl (I don't have this book on my table).
Apr
19
comment $SL(n) \times SL(n)$-invariants of $m$-tuples of matrices
What precisely is the (trivial) answer in the $GL$ case?
Mar
18
awarded  Good Answer
Feb
25
awarded  Necromancer
Feb
25
revised Projective arrows
added 179 characters in body
Feb
22
revised Projective arrows
A proposition and a corollary were added.
Feb
22
revised Projective arrows
added 106 characters in body
Feb
22
answered Projective arrows