3,724 reputation
1921
bio website math.tifr.res.in/~venky
location India
age
visits member for 2 years, 4 months
seen 7 hours ago

interested in representations of groups, algebraic groups, arithmetic groups, rigidity, and related automorphic forms. Recently interested in braid groups and configuration spaces.


7h
comment Least collaborative mathematician
+1 for "Betti number"
11h
comment Conceptual algebraic proof that Grassmannian is closed in Plucker embedding
I checked; this is standard terminology.
13h
comment Conceptual algebraic proof that Grassmannian is closed in Plucker embedding
Sorry: I may have used a non-standard terminology. I do mean that the image of a connected $X$ under a dominant map contains a dense open set.
14h
comment A Karrass-Solitar theorem for surface groups
Does not a comparison of "Euler characteristics indicate that $N$ must be free, and if $n$ is the number of generators of $N$ , then $1-n=q(2-2g)$ which shows that $q$ the index must be finite?
19h
revised Conceptual algebraic proof that Grassmannian is closed in Plucker embedding
added 48 characters in body
19h
comment Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book
Moreover, the quotient $\wedge ^s V /\Omega ^* \wedge ^{s-2}V$ is irreducible, with highest weight $x_1+\cdots+x_s$ in the obvious notation
20h
revised Conceptual algebraic proof that Grassmannian is closed in Plucker embedding
edited body
20h
comment Conceptual algebraic proof that Grassmannian is closed in Plucker embedding
@David: if $Gv$ is an orbit, then the complement of $Gv$ in its closure has smaller dimension by the dominant mapping theorem; of course, in a beginning course on algebraic geometry, I do not know if this is proved before one gets to Grassmannians
21h
answered Conceptual algebraic proof that Grassmannian is closed in Plucker embedding
22h
comment Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book
Clearly there is a map from $\wedge ^{s-2}V \otimes \wedge ^2 V$ into $\wedge ^s V$. Since $\wedge ^2 V$ contains a vector invariant under the symplectic group (namely the invariant symplectic form in the dual $V^*$, we have a map from $\wedge ^{s-2}V$ into $\wedge ^s V$.
2d
revised Go I Know Not Whither and Fetch I Know Not What
fixed grammar
Sep
15
comment A p-Sylow of a group $G$ is characteristic in its normalizer $N_G(S)$?
this is a beginner course exercise and not meant for MO.
Sep
14
comment A function canonically associated to an irreducible representation in L^2(M) for a Riemannian G-manifold M. Who has seen it?
The map is equivariant for the action of $G$
Sep
14
comment A function canonically associated to an irreducible representation in L^2(M) for a Riemannian G-manifold M. Who has seen it?
The evaluation at $x\in M$ gives a map $\phi$ from $M$ into the dual $V^*$ of $V$; the function $v$ is simply (up to a fixed scalar multiple) the $G$ invariant norm on $V^*$ applied to $\phi (x)$ for $x\in M$; you can define this for any set $X$ on which $G$ acts, in place of $M$.
Sep
13
comment Are irreducible matrix algebra neccesarily simple?
Part of the Wedderburn theory is that if an algebra has a faithful irreducible representation, then it is semi-simple.
Sep
13
comment quasi-split algebraic group
A ternary quadratic form $F$ over a number field has quasisplit $SO(F)$ if and only if $F$ is equivalent to the form $xy+z^2$.
Sep
13
comment Are irreducible matrix algebra neccesarily simple?
This is a consequence of Wedderburn theory; the answer is yes.
Sep
13
comment quasi-split algebraic group
Given a non-degenerate quadratic form $F$ in 2n variables, the group $SO(F)$ is quasi-split if and only if $F$ may be written a direct sum of $n$ hyperbolic forms> If $F$ is a form in $2n+1$ variables, $F$ is a direct sum of $n$ hyperbolic forms plus the form $x^2$. There is a similar description for unitary groups.
Sep
13
revised Kernel of the character of congruence groups
fixed grammar
Sep
8
reviewed Edit suggested edit on Hilbert Class fields and Pure cubic fields