bio | website | math.tifr.res.in/~venky |
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location | India | |
age | ||
visits | member for | 2 years, 5 months |
seen | 57 mins ago | |
stats | profile views | 2,388 |
interested in representations of groups, algebraic groups, arithmetic groups, rigidity, and related automorphic forms. Recently interested in braid groups and configuration spaces.
Oct 15 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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. |