Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
-1
votes
1answer
13 views
Is the decomposition of kronecker product $s_\mu \otimes s_\lambda $ always unique?
Let $s_\mu$ $ s_\lambda $ be two irreducible representations of $S_n$ over $\mathbb{C}.$
Is the decomposition of kronecker product $s_\mu \otimes s_\lambda $ always unique ?
That is, suppose ...
3
votes
0answers
96 views
In which finite groups is there a non-central g such that, for all irreducible characters, Chi(i)(g) <> zero?
What is the character of Pi(G), the tensor product of all inequivalent irreducible representations of G?
2
votes
0answers
145 views
The tallest possible lattice?
Let O be a complete discrete valuation ring and G a finite group. Recall that a finitely generated O-free OG-module $M$ such that the traces of the invertible endomorphisms of $M$ generate a strictly ...
7
votes
1answer
185 views
Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?)
Recently, I looked at articles that make use of Deligne's idea that "in characteristic 0 every deformation problem is governed by a differential graded
Lie algebra" as explained first in ...
7
votes
1answer
201 views
$\text{Rep}(D(G))$ as representation category of a vertex operator algebra
The category of representations $\text{Rep}(D(G))$ of the quantum double of a finite group is well-known to be a modular tensor category. Can these modular tensor categories also be obtained as ...
15
votes
1answer
261 views
Is there a natural notion of completion of a Coxeter system?
Let $(W,S)$ be a Coxeter system. Then any element of $W$ can be written as a finite products of elements of $S$. I want some notion of a "completion" of $W$, call it $\hat{W}$, whose elements are ...
7
votes
0answers
103 views
What happens to simple modules under Ringel duality?
If $A$ is a quasi-hereditary algebra then its Ringel dual $A'=End_A(T)$ is the endomorphism algebra of a (minimal) full tilting module $T$ for $A$. The algebra $A'$ is again quasi-hereditary, although ...
3
votes
1answer
73 views
Siegel domains and cuspidal functions
Let $F$ be a number field and $\mathbb{A}$ the ring of adeles over $F$. We consider $P_{n}$ the mirabolic subroup of $GL_{n}$.
Do we have a analog of Siegel subset for the quotient ...
3
votes
0answers
116 views
Is the formula for plethysm $S^n(S^3)$ known explicitely?
Is the formula for plethysm (in this the decomposition into irreducible GL representations of the composition of symmetric powers) $S^n(S^3)$ known explicitely? I know $S^n(S^2)$ e.g. in Macdonald's ...
2
votes
0answers
49 views
Isomorphic bound quiver algebras for different admissible Ideals
We know that for a path algebra KQ, whether or not KQ is finite dimensional (namely, Q may or may not have oriented cycles), there might be different admissible ideals I and J of KQ for which the ...
4
votes
0answers
62 views
Bushnell-Kutzko semi-simple types
Let $F$ be a non Archimedean local field with ring of integers $\mathcal{O}_F$. Let $P$ be a standard proper parabolic subgroup of $GL_n(F)$. Let $M$ be the standard Levi subgroup of $P$ and $\sigma$ ...
-1
votes
0answers
45 views
graded k-algebras [migrated]
Suppose we are given a positively graded $k$-algebra $A$ such that $A_i=0$, for $i\neq 0,1$ (i.e $A=A_0\oplus A_1$). Suppose furthermore all $A_i$ are finite dimensional as $k$ vector spaces and that ...
3
votes
0answers
176 views
A paper by Elashvili (translation request)
I would like to know if there is an English version of a paper by Elashvili called "Centralizers of nilpotent elements in semisimple Lie algebras".
If not, is there atleast an online version of the ...
2
votes
0answers
81 views
(Graded) Lie algebras with “nice” irreps
"(The only Lie algebras for which) all finite-dimensional representations are completely reducible (are the semisimple Lie algebras)" (Chapman, here on MO). Evidently this can't no longer hold when ...
5
votes
2answers
189 views
What can representations of affine Weyl groups do?
In Carter's Finite Groups of Lie type and Lusztig's Characters of Reductive Groups over a Finite Field, the representations of Weyl groups are helpful in finding the representations of algebraic ...
5
votes
0answers
53 views
Invariant theory of $SL_2$ over a field of positive characteristic
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $W$ be a finite dimensional $SL_2$-module over $k$. Let $V$ be the natural representation of $SL_2$.
What can be said - in ...
2
votes
1answer
100 views
Why is the semisimple quotient of a reductive group with semisimple rank 1 equal to PGL2?
Lets work over $\mathbb{C}$. Let $G$ be a reductive group with semisimple rank 1 (this means that a maximal torus in $G / R(G)$ has dimension 1). In section 25 of Humphreys book "linear algebraic ...
1
vote
0answers
70 views
Real analytic diffeomorphisms of sphere
We know that for $n\ge 4$, $SL(n,\mathbb Z)$ acts real analytic diffeomorphism on the sphere $S^2$ through a finite image, what about $SL(3,\mathbb Z)$? That is to say, if $\Phi: SL(3,\mathbb Z)\to ...
4
votes
1answer
384 views
Topologie sur l'ensemble des sous-groupes de GL_n(R)
Bonjour,
Est ce qu'il existe une topologie naturelle sur l'ensemble des sous-groupes du groupe général linéaire ?
English translation: "is there a natural topology on the set of subgroups of the ...
7
votes
0answers
187 views
Characteristic Classes in Geometric Representation Theory
Geometric respectively topological methods are widely applied in representation theory. As far as I know mainly cohomological methods are used.
I wonder if there are concrete applications of the ...
6
votes
0answers
78 views
Joint representation of the semi-direct product of the metaplectic group and Heisenberg group
Given a symplectic space $W$ over a local field $F$ and a additive character $\psi$ of $F$, we can construct the Weil representation $\omega_\psi$, which can be viewed as a representation of the ...
6
votes
2answers
321 views
Representations of $SL(2)$ in characteristic 2
In characteristic zero one can use the Clebsch-Gordan rule to decompose tensor products of SL(2)-modules. In characteristic $p$ things are more complicated.
I am interested in the special case ...
3
votes
0answers
41 views
Closed form for 3j-symbol ratios
I am working on the spherical harmonic decomposition of cosmic microwave background maps, therefore I often deal with functions that are proportional to Wigner 3J symbols/Clebsch–Gordan ...
1
vote
0answers
54 views
What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE?
What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE? Is it possible to write set-theoretical solutions of Quantum ...
2
votes
0answers
90 views
On Eigenvalues of the symmetric linear transformation related to a lie algebra's representation?
Let $\mathfrak{g}$ be a quadratic (finite dimensional) lie algebra and $\rho:\mathfrak{g}\rightarrow \mathfrak{gl}(W)$ be an anti-symmetric representation of $\mathfrak{g}$ on a finite dimensional ...
1
vote
0answers
71 views
Finding density of Haar measure relative to the Liouville volume measure on $\frac{G^{\mathbb{C}}}{P}$
Let $G$ be a connected compact Lie group and $G^{\mathbb{C}}$ be the complexification of the Lie group $G$ then we know that by polar decomposition we can write $G^{\mathbb{C}}\cong G\times ...
4
votes
0answers
95 views
Projective representation of diffeomorphism group of $S^2$
We know that the projective representation of a group $G$ is classified by $H_{grp}^2(G,R/Z) = H^3(BG,Z)$, where $H^*_{grp}$ is the group-cohomology class.
Then do we have a classification of the ...
1
vote
0answers
134 views
Replacing the Lie commutator with something else [closed]
Take a vector space $V={A,B,C,...}$ (of matrices), and the commutator $[A,B]=AB-BA$, then a Lie algebra of $V$ is characterized by $[V,V]$ staying in $V$. (Loosely speaking.)
What happens ...
4
votes
1answer
135 views
Semidirect product of torus with cyclic group: representations/cohomology?
Let $p$ be prime and let $T^p$ be the $p$-torus and $\mathbb{Z}/p$ the cyclic group of order $p$ generated by $(12\ldots p)$. Consider the semidirect product $T^p\rtimes\mathbb{Z}/p$ with the natural ...
1
vote
1answer
157 views
Schur-Weyl duality for arbitrary tensor products of simple finite-dimensional $GL_n$-modules
Let $M$ and $N$ be two simple finite dimensional $GL_n$ modules. Is there a way of expressing the heighest weight vectors of the simple submodules of $M\otimes N$ in terms of the heighest weight ...
3
votes
0answers
83 views
Character of continuous series representation of GL(2)
It is wellknown that the character of an irreducible, unitary representation of $GL(n,\mathbb{C})$ uniquely determines the isomorphism classes. I fail to construct a function for $GL(2, \mathbb{C})$, ...
1
vote
1answer
130 views
labeling state vectors in representation space of a simple lie algebra
Given a simple lie algebra (over ${\mathbb C}$ or ${\mathbb R}$). What is the number
of operators such that their eigenvalues sufficiently label all state vectors in the algebra's representation ...
2
votes
0answers
54 views
Isometric representation semisimple?
The first lemma on p.35 of these notes states that unitary representations are semisimple. Could the same be said of isometries if the space doesn't have an inner product? This topic notes that the ...
2
votes
0answers
81 views
Weyl Character Formula to find $M_\lambda(\mu)$ [migrated]
In Introductory Lie algebra book by Humphreys, he has used Weyl Character Formula to find the dimension of $V(\lambda)$ in the examples followed by the proof of this formula. But how to find the ...
8
votes
1answer
472 views
Why are they called Specht Modules?
I know that the simple modules of $\mathbb{C}S_n$ are called $\it{Specht}$ $ \it{Modules}$, and they are named after the German Mathematician Wilhelm Specht
because he studied them, but I think these ...
1
vote
0answers
36 views
Analytic vectors of elliptic elements of the enveloping algebra in a strongly continuous representation of a Lie group
Let $G$ be a semi-simple real Lie group, and $\rho$ a strongly continuous unitary representation of $G$ in a Hilbert space $H$. Let $\mathfrak{g}$ be its Lie algebra and $d\rho$ be the infinitesimal ...
5
votes
2answers
151 views
Heisenberg subalgebras of affine Lie algebras
It seems to be "well-known" that (infinite-dimensional) Heisenberg subalgebras of an affine Lie algebra $\hat{\mathfrak{g}}$ corresponding to a finite-dimensional simple Lie algebra $\mathfrak{g}$ of ...
4
votes
0answers
114 views
What are global sections of the determinant bundle on the Beilinson-Drinfeld Grassmannian?
Let $X$ be a smooth proper algebraic curve over $\mathbb{C}$, and let $G$ be a reductive group over $\mathbb{C}$. Let $Gr_{X,n}$ be the Beilinson-Drinfeld Grassmannian (for n points in $X$), which ...
0
votes
1answer
201 views
When one can expect $\widehat{(fg)} = \hat{f} \ast \hat{g}$; $f, g\in L^{1} (G)$?
Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows:
$$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$
It is ...
6
votes
0answers
291 views
Iwahori-Hecke algebras as endomorphism (or convolution) algebra?
Let $H_n(q,k)$ be the Iwahori Hecke algebra of symmetric group $S_n$ over an algebraically closed field $k$ of characteristic $p>0$, where $q$ is an invertable element in $k$. Assume that $q$ is a ...
0
votes
0answers
41 views
How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?
We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and
...
3
votes
1answer
302 views
What sort of ring-theoretic properties does the representation ring of a compact Lie group possess?
Recall the definition of the representation ring $R(G)$ of a compact Lie group $G$. I'd like a reference that gives me basic ring-theoretic properties that $R(G)$ always has, or enough info that I can ...
1
vote
1answer
256 views
finitely many algebraic representations
Is there a comlete discription of algebraic groups having only finitely many irreducible representations? ( giving some references would be very kind )
0
votes
0answers
68 views
global dimension III
Let $A$ be the followig matrix
$A=\begin{pmatrix}
R & M \\
0 & S
\end{pmatrix}$, where $R$ and $S$ are semisimple. Now suppose an arbitrary algebraic group acting on $A$. What can one say ...
0
votes
0answers
203 views
generators for derived category
Let $G$ be a algabraic group $G$ over a field $k$. We denote by $D^b(\mathrm{Repr}(G))$ the derived category of finite dimensional representations. Under what kind of assmumptions one has a generating ...
2
votes
2answers
250 views
semisimple category with finite number of simple objects
Since I am not an expert in algebraic groups; Is there a description of algebraic groups with semisimple category of finite dimensional representations such that they have only finitely many ...
2
votes
1answer
106 views
Is there any way to determine the “effeciency” of Jantzen's sum formula?
Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a reductive algebraic group over $k$.
In order to determine the structure of the Weyl modules $V(\lambda)$, ...
0
votes
0answers
62 views
Bounding global matrix coefficient for PGL_2
I'm trying to find a reference that gives a bound for the adelic matrix coefficient for $\text{PGL}_2$ using the bound towards Ramanujan conjecture. More specifically:
Let $G=\text{PGL}_2$. Let $F$ ...
15
votes
1answer
206 views
Is the representation category of quantum groups at root of unity visibly unitary?
Let $\mathfrak g$ be a simple Lie algebra.
By taking the specialization at $q^\ell=1$ of a certain integral version¹ of the quantum group $U_q(\mathfrak g)$,
and by considering a certain quotient ...
1
vote
2answers
191 views
When representation of two different coadjoint orbits are equivalent?
Let $G$ be a compact connected Lie group and $\mu:T\to S^1$ be a representation of a maximal torus $T \subset G$ and $\lambda=d\mu$ be a weight for some $\lambda\in\mathfrak{t}^*$ (where ...
