Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
6,802
questions
5
votes
1
answer
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Is norm-continuous representation factored through a Lie quotient group?
I asked this 11 days ago at MSE, but there was no answer, I hope people here could help.
Let $G$ be a locally compact group, and $X$ a Hilbert space. A unitary representation $\varphi:G\to B(X)$ is ...
4
votes
0
answers
116
views
Tannakian reconstruction and the distribution algebra
$\DeclareMathOperator\Dist{Dist}\DeclareMathOperator\Lie{Lie}\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\End{End}$Let $G$ be an affine group scheme over a commutative ring $k$ (I am mainly ...
0
votes
0
answers
75
views
Is a Lagrangian subgroup of a metric group isomorphic to its quotient?
A metric group is a finite abelian group $G$ with a quadratic function
$$q:G\rightarrow \mathbb R/\mathbb Z\;,$$
that is,
$$M(a,b):= q(a+b)-q(a)-q(b)$$
is bilinear in $a$ and $b$ [edit: and non-...
6
votes
1
answer
277
views
Which finite simple groups are rational-relative-real?
A finite group $G$ is called rational if every element $g \in G$ is conjugate to all of its primitive powers $g^a, a \in (\mathbb{Z}/\operatorname{order}(g))^\times$.
Analogously, I'll call $G$ real ...
2
votes
0
answers
74
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Is that a local module the same as an $E_2$ module?
Let $A$ be an $E_2$ algebra in a braided monoidal category $C$, i.e. a commutative monoid in $C$. Denote the braiding by $c_{x,y}$ for objects $x,y\in C$. We define a local module over $A$ as follows:
...
7
votes
1
answer
312
views
Easy example of a non-symmetric braiding of $\operatorname{Rep}(G)$?
What is the smallest group $G$ such that $\operatorname{Rep}(G)$ has a non-symmetric braiding (or just an easy example)?
I seem to remember a result classifying all universal $R$-matrices of $\mathbb ...
-1
votes
1
answer
163
views
Orthogonality of irreducible and non-isomorphic representations [closed]
Let V and W be any two subspaces of $(\mathbb{C}^d)^{\otimes n}$ such that there exists two irreducible and non-isomorphic representations $\rho_V: G \to GL(V)$ and $\rho_W: G \to GL(W)$. Does this ...
7
votes
2
answers
265
views
Decomposition of tensors into symmetry classes according to Schur functors
I am mainly looking for references on this subject, as I was unable to find any, at least any that answers what I am looking for to a satisfactory degree.
As it is well-known and extremely easy to ...
6
votes
0
answers
222
views
Group homomorphism from $\mathrm{GL}_p$ to $\mathrm{SL}_p$ in characteristic $p$
If $k$ is a commutative field of characteristic $p>0$, then the map
$$ \theta \colon \mathrm{GL}_p(k) \to \mathrm{SL}_p(k) \colon A = (a_{ij}) \mapsto (\det A)^{-1} (a_{ij}^p) $$
is a group ...
8
votes
1
answer
332
views
Is the GL(2,R)-representation of smooth, odd and 0-homogeneous functions on the punctured plane irreducible?
Let me preface this by saying that I have next to no background in representation theory. I come from geometry but the following representations showed up naturally in my work.
We let $ V = C^\infty \...
1
vote
0
answers
105
views
Weyl group action on the Lie algebra [duplicate]
Let $W$ be the Weyl group of a complex semisimple Lie algebra $\frak{g}$. Certainly $W$ acts on the root system $R$ of $\frak{g}$ but can it be made to act on $\frak{g}$ or on the universal enveloping ...
2
votes
0
answers
63
views
A branching law involving 2-power exterior representations
Let $K=SU(n)$.
We take a maximal torus $T$ in $K$ and fixed a simple root system with fundamental weights $\eta_1,\dots,\eta_{n-1}$.
For $\mu$ a dominant weight of $K$, we denote by $(\tau_\mu,V_\mu)$ ...
3
votes
0
answers
57
views
Locally finite positive energy modules generated by singular vectors at positive levels?
This is question is about whether or not certain modules for an affine Lie algebra are generated by their singular vectors. I begin with some background.
Backround on affine Lie algebras. Let $\...
3
votes
1
answer
164
views
Quantum group associated to a reductive group
In most of the classical references about quantum groups, these objects are defined as a one-parameter deformation of the universal enveloping algebra. However, I have read in several papers that it ...
0
votes
0
answers
47
views
A Multiplicative Average of Positive Operators
Let $G$ be a finite group. I have an action of $G$ on a matrix algebra of positive operators, $\mathcal{M}$. In particular, $\mathcal{M}$ has a $G$-module structure, yielding a linear representation ...
11
votes
1
answer
152
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Are all indecomposable $\mathbb{Z}_+$-modules over the character ring of a group, character rings of a subgroup?
A $\mathbb Z_+$-algebra is an algebra $A$ over $\mathbb C$ with given basis $\{v_i\}$ such that
$$v_iv_j=\sum_k n_{ijk}v_k,\hspace{10mm}n_{ijk}\in\mathbb Z_{\geq0}.$$
An example of such an object is ...
7
votes
2
answers
691
views
Tensor product of irreducible representations of an algebra
Let $A$ be an associative algebra over $\mathbb{C}$ with irreducible finite-dimensional representations on $V$ and $W$. Then is the tensor product of representations on $V \otimes W$ semi-simple?
The ...
1
vote
0
answers
71
views
Product of matrix entry and representation
Consider a semisimple lie algebra $\mathfrak{g}$ and the corresponding quantum group $\mathcal U_q(\mathfrak{g})$ over $\mathbf{Q}(q)$. Consider two dominant weights $\lambda,\mu\in P^+$, a matrix ...
6
votes
1
answer
321
views
Do doubly-transitive actions give rise to indecomposable representations for infinite groups?
This is a follow-up to this question.
Let $G$ be a group acting doubly-transitively on a set $X$. Then the vector space $V_X$ of functions $f\colon X\to\mathbb C$ with finite support such that $\sum_{...
4
votes
1
answer
254
views
Are polynomial algebras over fields (that are not algebraically closed) tame?
Let $A$ be an algebra over a field $K$. Loosely speaking, an algebra is said to be tame if for each $d \in \mathbb{Z}_{>0}$ all but finitely-many of the indecomposable $A$-modules of $K$-dimension $...
9
votes
1
answer
529
views
Do doubly-transitive actions give rise to irreducible representations for infinite groups?
Let $G$ be a group acting doubly-transitively on a set $X$. Then the vector space $V_X$ of functions $f\colon X\to\mathbb C$ with finite support such that $\sum_{x\in X}f(x)=0$ carries an action of $G$...
2
votes
1
answer
136
views
Solving the explicit isomorphism problem
Suppose $A \cong M_n(\mathbb{D})$ where $A$ is a simple algebra over division ring $D$. We want to find an explicit isomorphism between $A$ and $M_n(\mathbb{D})$. I read from Ivanyos et al. (2012) ...
3
votes
0
answers
113
views
Root space inner products and the partial order on roots
For a root system $R$ and a choice of positive roots $R^+$ it is a standard fact (see, e.g., Bourbaki, "Lie Groups and Lie Algebras," Theorem 1 of Section 1.3 of Chapter VI) that
if $(\...
3
votes
1
answer
266
views
Orbit of a parahoric subgroup on a flag variety
Let $G$ be a split reductive group over a nonarchimedean local field $F$ (I'm particularly interested in the case of $\operatorname{GSp}_{2n}$).
Given a parahoric subgroup $K \subset G(F)$, and a ...
3
votes
1
answer
306
views
A generalisation of induced representations
Let $G$ be a finite group, and $H\subseteq G$ a subgroup. Let $F$ be a field. Let $W$ be a finite-dimensional $F[H]$-module. Let $T$ be a left transversal of $H$ in $G$. Then we can define:
$W^G=\sum_{...
2
votes
0
answers
202
views
Classification of generic representations of $\mathrm{GL}(n)$ over non-archimedean fields
Let $k$ be a local field and $n \in \mathbb{N}$.
Question:
I would like to know precisely, which irreducible (admissible) representations of $G := \textrm{GL}_n(k)$ are generic, i.e. which admit ...
1
vote
0
answers
61
views
The local structure theorem for spherical varieties under quasi-split group action
I want to understand a simplified version of the general $k$-local structure theorem proved in the paper "Reductive group actions":
For $k$ a characteristic zero algebraically closed field, $...
4
votes
0
answers
113
views
Two definitions of intertwining operators and Harish-Chandra's Plancherel measure
I guess this question is a well-known fact to experts, but I didn't find any explicit explanation in the literature.
So let $F$ be a $p$-adic field. (There're parallel definitions and results in the ...
4
votes
1
answer
198
views
Eigenvalue of Iwahori Hecke Algebra element for the Steinberg
In Iwahori-Matsumoto's paper the Iwahori Hecke Algebra for $G=GL_n(F)$ is generated by $X_{s_0}, X_{s_i},i\in\{0,...,n-1\}$ and $ X_{\rho}$ with the relations:
$
1) (X_{s_{i}}-q)(X_{s_{i}}+1)=0\:,\;\;...
3
votes
1
answer
255
views
Can any pair of associate parabolics be related by opposite parabolics?
Let $G$ be a reductive group, say over an algebraically closed field of characteristic zero.
We have the following definitions for a pair of parabolic subgroups $P_1$ and $P_2$ with Levi quotients $...
2
votes
0
answers
75
views
Almost split sequences for symmetric algebras
Let $k$ be an algebraically closed field and $A$ be a symmetric algebra.
I want to know how to compute almost split sequences ending at a non-projective indecomposable right $A$-module $X$.
Question: ...
5
votes
0
answers
98
views
Modern proof of a theorem of Dickson on finite representation type
In Theorem 3.1 the paper S. Dickson, On algebras of finite representation type
Trans. Amer. Math. Soc. 135 (1969), 127-141, Dickson gives a sufficient condition for an algebra to have infinite ...
2
votes
1
answer
95
views
Baur-Monk quantifier elimination (BG-invariants in 1-free variable)
$\DeclareMathOperator\Inv{Inv}$Baur-Monk quantifier elimination implies that a sentence in the language of modules is a combination of BG invariant statements.
A BG invariant sentence is a boolean ...
4
votes
2
answers
362
views
Splitting field for $\mathrm{GL}(2,p)$ - reference request
It seems to me from a quick glance at several sources describing the complex and modular irreducible representations of $\mathrm{GL}(2,p)$ that any field $K$ containing a primitive $(p-1)$-root of ...
1
vote
0
answers
160
views
Limit of groups with Kazhdan property (T)
Let $G_1 \le G_2 \le \cdots $ be countable groups with Kazhdan property (T). Let $G = \bigcup_i G_i$. Does it necessarily follow that $G$ has (T)?
This seems false but I cannot find a counterexample.
9
votes
1
answer
303
views
The convex hull of Schur polynomial evaluations
Let $r\leq n$ and $d$ be positive integers. A probability vector is a vector of non-negative entries that sum to 1. For each probability vector $\lambda$ of length $n$, let
$$s(\lambda)=(\dim[\pi] \...
1
vote
0
answers
158
views
Applications of hyperbolic polynomials?
The recently posted MO-Q "Positivity of the coefficients of Taylor series associated to the Riemann hypothesis" (see also this MO-Q) has re-kindled my interest in hyperbolic polynomials--...
2
votes
0
answers
164
views
Howe duality vs first fundamental theorem in invariant theory
I'm working on Howe duality, and R. Howe proved that the Howe duality of $\mathrm{GL}_n$ is equivalent to the first fundamental theorem (FFT) in invariant theory. So, Howe duality gives a ...
1
vote
0
answers
67
views
Orbit projection geometry
Background:
As shown in [1] and [2], for a closed smooth submanifold $M$ of $\mathbb R^d$, the domain $D_M$ of the projection map $P_M:D_M\rightarrow M$ has a dense interior $\Omega_M$ over which $P_M|...
6
votes
1
answer
455
views
Kazhdan-Lusztig theory for quantizations of symplectic resolutions/the rational Cherednik algebra?
In Kazhdan-Lusztig theory Beilinson-Bernstein localization plays an important role. There are results about localization for quantizations of sympectic resolutions by Losev. There are also results ...
2
votes
0
answers
110
views
Understanding segments in Bernstein-Zelevinsky Classification
All reps shall be admissible in what follows.
Let $k$ be a non-arch. field, $n = a\cdot b$ natural numbers and $P = M \cdot N \subset \mathrm{GL}_n(k)$ the standard parabolic subgroup with
$$
M = \...
0
votes
0
answers
86
views
Connected components of $Q(\mathrm{s\tau-tilt}A)$
I'm reading about support $\tau$-tilting modules and their mutations. I'm trying to understand the mutation quiver.
Let $A$ be a finite dimensional algebra over an algebraically closed field, which is ...
1
vote
0
answers
40
views
About nilpotent Jordan algebras, matrix representations and formally real algebras
Given an non-commutative associative unital algebra A of characteristic $0$, one can construct a Jordan algebra $A+$ using the same underlying addition vector space.
Notice first that an associative ...
7
votes
1
answer
321
views
Decomposition of a tensor product of representations of $\mathrm{GL}_l(\mathbb{C})$ and decomposition of Littlewood-Richardson numbers?
For a positive integer $m$, denote $T(m)=\{(\lambda_1,\dots,\lambda_m)\in \mathbb{Z}^m:\lambda_1\ge \lambda_2\ge\dots \ge\lambda_m\}$ and $T^+(m)=\{ (\lambda_1,\dots,\lambda_m)\in \mathbb{Z}^m:\...
1
vote
1
answer
87
views
The automorphism group of $2^{2n}{:}Sp_{2n}(2)$
Let $G=2^{2n}{:}Sp_{2n}(2)$ be the split extension, where the symplectic group $Sp_{2n}(2)$ acts naturally on the vector space $2^{2n}$. With the aid of GAP it turns out that the automorphism group $\...
5
votes
0
answers
154
views
Properties of semisimple monoidal category
In my work, I have constructed a semisimple category which has two monoidal structures: the usual direct sum; and a new "tensor product". This "tensor product" have several nice ...
0
votes
0
answers
149
views
Heisenberg group
Let
$X_{j}=\frac{\partial}{\partial x_{j}}-\frac{1}{2}y_{j}\frac{\partial}{\partial t}$, $j=1,2,\dots,n$
$Y_{j}=\frac{\partial}{\partial y_{j}}+\frac{1}{2}x_{j}\frac{\partial}{\partial t},j=1,2,\dots,...
2
votes
1
answer
213
views
Tame/wild classification of *cyclic* quivers?
There is a famous classification of the path algebras of finite acyclic quivers into finite, tame, and wild representation types. For quivers with cycles, it is standard that the 2-loop quiver (with ...
2
votes
0
answers
105
views
Admissible representations of an $\ell$-group are a (neutral) Tannakian category?
Let $G$ be an $\ell$-group in the sense of Bernstein/Zelevinsky (sometimes also called td-group), i.e. $G$ is a
Hausdorff
locally compact
totally disconnected
topological group. Prominent examples ...
2
votes
0
answers
223
views
Ramanujan's theta functions and hook lengths?
Given an integer partition $\lambda\vdash n$ of $n$, one may associate a Young diagram $Y(\lambda)$ to it followed by a computation of hook length $h_{\square}$ for each cell $\square=(i,j)$ in $Y(\...