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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\ …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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.
Mikhail Borovoi's user avatar
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' …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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. …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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 $R_p=S^p R$ …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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) $ …
Mikhail Borovoi's user avatar

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