3,889 reputation
11023
bio website math.tifr.res.in/~venky
location India
age
visits member for 2 years, 6 months
seen 34 mins ago

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


Oct
15
comment Computing the Zariski closure of a subgroup of SL(n,Z)
I liked your paper! It is just that I did not immediately see this dealt with arbitrary Zariski closure case. Now you assure that this does not.
Oct
15
comment Computing the Zariski closure of a subgroup of SL(n,Z)
this seems to give a criterion for Zarski density in SL(n), whereas the OP seems to ask for a procedure to compute the Zariski closure of any f.g. subgroup of $SL(n,Z)$.
Oct
14
answered Computing the Zariski closure of a subgroup of SL(n,Z)
Oct
14
comment Compact elements in $G(K)$ for a reductive group $G$ over a nonarchimedean local field $K$
@user52484: Yes, thank you for pointing this out; otherwise you may have a product of anisotropic and isotropic simply connected groups, for which KT cannot hold.
Oct
13
comment Compact elements in $G(K)$ for a reductive group $G$ over a nonarchimedean local field $K$
This is called the "Kneser-Tits conjecture" for local fields and has been proved by various people; there is a proof (by Raghunathan and Prasad) by reducing this to groups of $K$-rank one. Note also that Kneser-Tits is false for many fields; it is a relatively recent result due to Gille (I think) that KT is true for number fields.
Oct
13
answered Compact elements in $G(K)$ for a reductive group $G$ over a nonarchimedean local field $K$
Oct
13
reviewed Approve suggested edit on Compact elements in $G(K)$ for a reductive group $G$ over a nonarchimedean local field $K$
Oct
13
comment Characterisation of Q-rank 1
In Kazhdan's paper, however, he does not prove the result for which the OP has asked; he seems to prove property T only for simple groups of real rank at least three. It is only in Kirillov-Delaroche paper that I find property (T) proved for all simple lie groups of real rank at least two
Oct
12
revised Characterisation of Q-rank 1
added 7 characters in body
Oct
4
revised Location of the zeros set of holomorphic function
added 6 characters in body
Oct
3
reviewed Approve suggested edit on Why do roots of polynomials tend to have absolute value close to 1?
Oct
3
revised Faithful representation of the projective unitary group with the lowest dimension?
fixed typos
Sep
27
comment Subgroups of $SL_3(\mathbb{Z})$ that are finitely generated, Zariski-dense, infinite index, and torsion-free
@Rivin: ha! ha! more like a rat!
Sep
27
comment Subgroups of $SL_3(\mathbb{Z})$ that are finitely generated, Zariski-dense, infinite index, and torsion-free
@Rivin: thanks!! How did you find out it was I? (I had used a "false name").
Sep
26
comment Can any reductive $k$-group be written as a semidirect product of $k$-linear groups?
As to the original question, take $G$ to be a finite group (e.g the group of upper triangular unipotent matrices matrices with entries in $Z/3Z$. Then the commutator is the centre and the extension above does not split. If it did, the group would be abelian.
Sep
26
comment Can any reductive $k$-group be written as a semidirect product of $k$-linear groups?
@anon: what you say is not right: take the torus inside the diagonals in $GL_n$ all of whose entries except the first one are $1$. This maps iso to $GL_n/SL_n=G_m$.
Sep
25
comment Irreducible action of an algebraic group
Take the standard representation $V$ of $G=N(T)$ the normalizer of $T$, the group of diagonals in $GL(V)$. Then $V$ is irreducible for $G$; as a rep of $G^0=T$, $V$ decomposes as a direct sum of one dimensional distinct characters.
Sep
25
comment Irreducible action of an algebraic group
Itis easy to see that if $V$ is any (say finite dimensional ) module over a group $H$ which is a sum $\sum W_i$ of irreducible $H$ modules $W_i$, then $V$ is a direct sum of irreducibles. To see this, let $W$ be an $H$ subspace of $V$ which is of the largest dimension, which is a direct sum of irreducibles; then $W\neq 0$ since $W$ contains one of the $W_i$. If some $W_j$ does not lie in $W$, then $W+W_j$ is the direct sum of $W$ and $W_j$, since $W_j$ is irreducible; this contradicts maximality of $W$ and hence $W=V$.
Sep
25
comment Irreducible action of an algebraic group
of course; as a rep of the trivial group, $V$ is a direct sum of one dimensional trivial representations of the trivial group
Sep
25
comment Irreducible action of an algebraic group
It is true: take any irreducible $G^0$ submodule $W$ of the irreducible $G$-module $V$; then $V$ is the sum of $g(W)$ for $g\in G$. Hence we have a surjection from the direct sum of $g(W)$ onto $V$. This means that $V$ is a direct sum of irreps of the same dimension. Of course, isotypical is false.