Questions tagged [rt.representation-theory]

Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.

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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 ...
Sergei Akbarov's user avatar
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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 ...
Antoine Labelle's user avatar
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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-...
Andi Bauer's user avatar
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6 votes
1 answer
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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 ...
Theo Johnson-Freyd's user avatar
2 votes
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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: ...
H.Yang's user avatar
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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 ...
shin chan's user avatar
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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 ...
listener's user avatar
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 ...
Bence Racskó's user avatar
6 votes
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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 ...
Tom De Medts's user avatar
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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 \...
JaSch's user avatar
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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 ...
Lorenzo Del Vecchiopontopolos's user avatar
2 votes
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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 $(\tau_\mu,V_\mu)$ ...
emiliocba's user avatar
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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 $\...
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3 votes
1 answer
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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 ...
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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 ...
user82261's user avatar
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1 answer
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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 ...
shin chan's user avatar
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7 votes
2 answers
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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 ...
Nanoputian's user avatar
1 vote
0 answers
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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 ...
esteban's user avatar
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1 answer
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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_{...
Kenta Suzuki's user avatar
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4 votes
1 answer
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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 $...
Iteraf's user avatar
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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 $G$...
Kenta Suzuki's user avatar
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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) ...
KD9's user avatar
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0 answers
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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 $(\...
Fantas Anadolou's user avatar
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 ...
David Loeffler's user avatar
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_{...
semisimpleton's user avatar
2 votes
0 answers
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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 ...
Maty Mangoo's user avatar
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, $...
R. Chen's user avatar
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0 answers
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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 ...
youknowwho's user avatar
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\:,\;\;...
idocomb's user avatar
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3 votes
1 answer
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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 $...
Anthony Blanche's user avatar
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: ...
sola's user avatar
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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 ...
Benjamin Steinberg's user avatar
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 ...
Eladio Vuente's user avatar
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 ...
Benjamin Steinberg's user avatar
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.
jonan's user avatar
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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] \...
Ben's user avatar
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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--...
Tom Copeland's user avatar
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2 votes
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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 ...
zhichengzhang's user avatar
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|...
miniii's user avatar
  • 59
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 ...
Yellow Pig's user avatar
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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 = \...
Maty Mangoo's user avatar
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 ...
It'sMe's user avatar
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1 vote
0 answers
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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 ...
mick's user avatar
  • 703
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:\...
Q. Zhang's user avatar
  • 960
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 $\...
Isaac 's user avatar
  • 49
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 ...
Nanoputian's user avatar
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,...
zoran  Vicovic's user avatar
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 ...
Joshua Grochow's user avatar
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 ...
Maty Mangoo's user avatar
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(\...
T. Amdeberhan's user avatar

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