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Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
1
vote
1
answer
232
views
Transfer for the group of coinvariants: a reference request
Let $G$ be a group and $M$ be a $G$-module,
that is, an abelian group written additively on which $G$ acts:
$$ (g,m)\mapsto g m.$$
We consider the group of coinvariants
$$ M_G:=G/\langle g m -m\ |\ g\ …
4
votes
Accepted
What is the minimum possible k-rank of a quasi-split reductive group over a field?
$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\im{im}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\SL{SL}$I add details to Friedrich's comment:
The question easily reduces to the case of a sem …
8
votes
When is the normalizer of the maximal torus maximal?
$
\newcommand{\g}{{\mathfrak g}}
\newcommand{\h}{{\mathfrak h}}
\newcommand{\t}{{\mathfrak t}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\Ad}{{\rm Ad}}
\newcommand{\ad}{{\rm ad}}
$Theorem. Let $G$ be …
3
votes
Accepted
Does the F-unitary group isomorphism arises from a conformal isometry?
The answer to Question 1 is No. Indeed, take $V_1=V_2=V:=K^n$ and write
$$ h_1(z)=h_2(z)=h(z):=\lambda_1 z_1 \bar z_1+\dots+\lambda_n z_n\bar z_n\quad\text{with}\
\lambda_i\in F^\times.$$
Write $\wide …
3
votes
1
answer
291
views
The torsion subgroup of the coinvariants for a $G$-module
Let $G$ be a finite group and $M$ be a finitely generated $G$-module,
that is, a finitely generated abelian group on which $G$ acts.
Consider the functor
$$ (G,M)\rightsquigarrow F(G,M):= (M_G)_{\rm T …
1
vote
Indecomposable integral representations of a group of order 2 "by hand"
See Appendix A in M. Borovoi and D. A. Timashev, Galois cohomology and component group of a real reductive group.
2
votes
Accepted
Regular embeddings of a reductive groups with induced center
The answer is Yes.
Let $G\hookrightarrow G'$ be a smooth regular embedding. We write $Z'=Z(G')$ for the center of $G'$, which is an $F$-torus where $F={\Bbb F}_q$.
We construct a regular embedding $G' …
7
votes
Accepted
Root system of fixed point Lie sub-algebra
Let ${\frak g}$ be a simple Lie algebra over $\Bbb C$, and let $\theta$ be an inner involution of ${\frak g}$,
that is, an inner automorphism of ${\frak g}$ of order dividing 2.
Such automorphisms are …
8
votes
2
answers
464
views
Parabolics and simple roots for a special unitary group: reference request
I am looking for a reference where the relative root system, the relative system of simple roots, and parabolic $\Bbb R$-subgroups for the real algebraic group ${\rm SU}(p,q)$ are explicitly computed. …
5
votes
1
answer
198
views
Coordinate-free description of an alternating trilinear form on pure octonions
Let $O$ denote the division algebra of octonions over $\Bbb R$, and write $V$ for the 7-dimensional quotient space $O/{\Bbb R}$.
The compact group $G_2:={\rm Aut}(O)$ naturally acts on $V$,
and clearl …
5
votes
Accepted
Symmetric and alternating powers of defining representation of classical groups
You can find the decomposition of $S^kV$ and $\Lambda^k V$ into a direct sum of irreducible representations in Table 5 in the book: Onishchik and Vinberg, Lie Groups and Algebraic Groups, Springer-Ver …
11
votes
2
answers
589
views
To describe an invariant trivector in dimension 8 geometrically
$\newcommand\Alt{\bigwedge\nolimits}$Let $G=\operatorname{SL}(2,\Bbb C)$, and let $R$ denote the natural 2-dimensional representation of $G$ in ${\Bbb C}^2$.
For an integer $p\ge 0$, write $R_p=S^p R$ …
3
votes
2
answers
395
views
Indecomposable integral representations of a group of order 2 "by hand"
This question is a duplicate of
that 2010 MO question.
I am interested in classifying isomorphism classes of $n$-dimensional integral representations of the cyclic group $C_2$ of order $2$.
Clearly, a …
6
votes
1
answer
209
views
Finite group ${\rm Sp}_4({\Bbb F}_3)$: involutions coming from a 4-dimensional complex repre...
I am interested in the finite unitary reflection group $G= G_{32}$, the group No. 32 in Table VII on page 301 of the paper:
Shephard, G. C.; Todd, J., A. Finite unitary reflection groups. Canad. J. M …
6
votes
Accepted
Connectedness of the stabilizer in a semisimple group of a semisimple element in the Lie alg...
A reference: Steinberg, Torsion in reductive groups,
Advances in Math. 15 (1975), 63–92, Corollary 3.11.
In positive characteristic $p$: see loc. cit., Theorem 3.14. It says that if (and only if) $ …