14
votes
Why is the catalecticant invariant under coordinate changes?
Dolgachev (2012, p. 57; pdf) observes that your matrix $\left( a_{i+j-2}\right) _{1\leq i\leq n+1,\ 1\leq j\leq n+1}$ (with determinant $\operatorname{Cat} f$) is the matrix of a symmetric bilinear ...
- 31.5k
11
votes
Embedding $\mathrm{SL}_n(3)$ into $\mathrm{SL}_n(p)$
$\DeclareMathOperator\SL{SL}$The idea I had in mind originally when I made the comment was more simple-minded. I'll deal with the case that $n$ is even for ease of exposition.
Each $3$-subgroup $S$ of ...
- 44.4k
11
votes
Accepted
Embedding $\mathrm{SL}_n(3)$ into $\mathrm{SL}_n(p)$
$\DeclareMathOperator\SL{SL}$Here is a quick and dirty argument. For $\SL(3,3)$ we can check directly that there is no representation of dimension less than $11$ over a field of characteristic $\ne 3$....
- 16.2k
11
votes
Why is the catalecticant invariant under coordinate changes?
Let $d=2n$ be the degree of your binary form $f$.
Let me introduce $n+1$ pairs of formal variables $\alpha^{(1)}=(\alpha^{(1)}_{1},\alpha^{(1)}_{2}),\ldots, \alpha^{(n+1)}=(\alpha^{(n+1)}_{1},\alpha^{(...
9
votes
Accepted
$N_{G}(E)/C_{G}(E)$ is the Weyl group of $G$?
$\newcommand{\ZZ}{{\mathcal Z}_G}
\newcommand{\NN}{{\mathcal N}_G}
\newcommand{\zz}{{\mathfrak z}_G}
\newcommand{\Lie}{{\rm Lie\,}}
\renewcommand{\tt}{{\mathfrak t}}
\renewcommand{\gg}{{\mathfrak g}}
\...
- 14.1k
9
votes
Accepted
is the embedding $\mathrm{Sp}_{2m}(p)\leqslant S_{p^m-1}$ possible?
No. The smallest degrees of the faithful permutation representations of the finite simple groups are listed in Table 4.5 of On the maximum orders of elements of finite almost simple groups and ...
- 37.4k
6
votes
Is the size of a conjugacy class in a finite classical group a polynomial?
As already noted in the comments, as worded, this question does not quite make sense. Here is an attempt to say something about it anyway, since the context the OP is asking about is interesting.
In ...
- 8,529
5
votes
Accepted
Is there any (specially Algebraic Geometrical) exposition of Koike Terada's Young-diagrammatic methods for the representation theory paper?
I discussed an approach to these ideas using algebraic geometry / classical invariant theory in a course I taught. You can find notes here:
https://mathweb.ucsd.edu/~ssam/old/20S-251C/notes.pdf
...
- 10.7k
5
votes
unipotent class in classical lie algebra bala-carter
I don't understand why you refer here to Bala-Carter, since their method provides a different approach to the traditional Dynkin classification and is formulated for semisimple groups (over ...
- 52.9k
5
votes
Connectedness of the linear algebraic group SO_n
This is a more elementary solution, which would make sense in the context of Springer's book. We consider
$SO(V,\langle,\rangle)$, or for short $SO(V)$, where $V$ is $n$-dimensional and $\langle,\...
- 1,388
4
votes
Accepted
Splitting of regular semisimple conjugacy classes in $SL_{n}(q)$
This appears to be well-studied, see e.g. http://www-bcf.usc.edu/~fulman/LAApaper.pdf
and references therein.
- 14k
4
votes
Accepted
Orbit sizes of $G=\operatorname{SO}^{+}_{2n}(2)$
As Mikko Korhonen said in the comments, the (non-zero) singular and non-singular vectors form single orbits by Witt's lemma.
So it remains to count the singular vectors, which are (row) vectors ${\...
- 37.4k
4
votes
Accepted
Normalisers and stabilisers in classical groups $\operatorname{PGL}_{4}$
The difference between the examples arises principally because $5 \equiv 1 \bmod 4$ and $3 \equiv 3 \bmod 4$.
For $q \equiv 1 \bmod 4$, $G := {\rm GL}(4,q)$ has centre $Z$ divisible by $4$, and ...
- 37.4k
4
votes
Accepted
Existence (or the number) of generating triple of involutions of $\operatorname{PGL}_2(p)$ with some conditions
Such triples exist, I think.
First, embed $PGL_2(p)$ in $S_{p+1}$ through its action on $1$-spaces from ${\mathbf F}_p^2$. This maps elements of order $p+1$ in $PGL_2(p)$ to $(p+1)$-cycles, and ...
- 3,571
4
votes
is the embedding $\mathrm{Sp}_{2m}(p)\leqslant S_{p^m-1}$ possible?
$\DeclareMathOperator\Sp{Sp}$Derek's answer is definitive. I think, at least for $p> 3$ odd, we can also see that the answer is no by considering the irreducible characters of $\Sp(2m,p)$. The ...
- 44.4k
3
votes
Universal character ring for classical groups
I suggest the following:
M. J. Newell, Modification Rules for the Orthogonal and Symplectic Groups. Proc. Roy. Irish Acad. 54, 153 (1951),
R. C. King, Modification Rules and Products of Irreducible ...
- 2,552
3
votes
On $(2,3)$-generation of finite simple classical groups
This is not a definitive answer (I doubt there is one), but too long for a comment.
Indeed, it is known that among the finite simple groups there are, apart from $\operatorname{PSp}_4(2^k)$, $\...
- 3,186
3
votes
Accepted
An upper bound on the dimension of a subalgebra of $\mathfrak{so}(p,q)$ with non-trivial centre
[Answer completely rewritten.]
Indeed we have:
$\DeclareMathOperator\so{\mathfrak{so}}$Fix $n\ge 4$ and $p,q\ge 0$ with $p+q=n$; write $r=\min(p,q)$. Then the minimal codimension for the centralizer ...
- 63.9k
2
votes
Is the size of a conjugacy class in a finite classical group a polynomial?
I agree that the question needs a better formulation. In any case, an approach by Demetris Deriziotis might be useful because it's based on a different kind of analysis: see here (freely ...
- 52.9k
2
votes
Accepted
Holomorphic map to Möbius group
Condition $U\subseteq \mathbb{C}^2$ can be replaced by $U\subseteq \mathbb{C}^1$ (just restrict your map
on a little disk in a complex line in $U$. Moreover, the restricted map lifts to a map into $SU(...
- 91.8k
2
votes
Accepted
Subgroups of $\operatorname{PGL}_n$
$\renewcommand{\O}{{\rm O}}
\newcommand{\GO}{{\rm GO}}
\newcommand{\PO}{{\rm PO}}
\newcommand{\PGO}{{\rm PGO}}
\newcommand{\GL}{{\rm GL}}
\newcommand{\PGL}{{\rm PGL}}
\newcommand{\SL}{{\rm SL}}
\...
- 14.1k
1
vote
Accepted
An explicit matrix form
It looks like you are working with respect to the orthogonal form with matrix $\begin{pmatrix} & w_0 \\ w_0 \end{pmatrix}$, where $w_0 = \operatorname{antidiag}(1, \dotsc, 1)$. That's the one ...
- 12.9k
1
vote
Accepted
Intersection of identity components
$\DeclareMathOperator\Cent{C}\newcommand\oG{\overline G}\newcommand\oe{\overline e}$Put $\oG = \operatorname{PGL}_n(\mathbb C)$. I hope you will permit me to denote the involutions by $\oe_i$ instead ...
- 12.9k
1
vote
Existence (or the number) of generating triple of involutions of $\operatorname{PGL}_2(p)$ with some conditions
You might find helpful the paper of Liebeck and Shalev "Classical Groups, Probabilistic Methods, and the (2,3)-Generation Problem". There, they show that there are three involutions that generate all ...
- 3,407
1
vote
Accepted
Is the Singer cycle preserved by field automorphisms and graph automorphisms?
This is true by Proposition 4.3.6.(I) of Kleidman and Liebeck's book "The Subgroup Structure of the Finite Classical Groups", which says that, in all cases for the linear and unitary groups, there is ...
- 37.4k
1
vote
Can elements in the orthogonal group of a non-split Azumaya algebra with an orthogonal involution have reduced norm -1?
In the end it turns out that:
The answer to Question 1 is "Yes" (i.e., if $A$ is non-split and $R$ is semilocal, then all elements in $O(A,\sigma)$ have reduced norm $1$).
The answer to Questions 3 ...
- 2,928
1
vote
The double cover of $[W(E_7),W(E_7)] \cong Sp_6(\mathbb F_2)$ as a Galois group over $\mathbb Q$
More details on my comment:
I mean that there is an induced map $H^2( E_7^o , \mathbb Z/2) \mathbb Z/2 = H^2( W(E_7), \mathbb Z/2) \to H^2( \pi_1(E_7^o , \mathbb Z/2)) \to H^2( E_7^o , \mathbb Z/2)$...
- 148k
1
vote
When does an irreducible representation remain irreducible after restriction to a semi-simple subgroup?
If you
(a) work on the level of Lie algebras, with restriction to a standard Levi, and
(b) want to find out precisely which irreps remain irreducible under this restriction,
then I believe the ...
- 51
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