# Questions tagged [rt.representation-theory]

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

6,341
questions

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### Local component of cuspidal automorphic representation

Let $F$ be a number field and $\mathbb{A}$ its adele ring. $G$ be a classical group and $
\pi$ be a unitary cuspidal automorphic representation of $G(\mathbb{A})$.
Then I am wondering whether there is ...

4
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1
answer

289
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### Clebsch–Gordan decomposition formula for algebraic groups

$\DeclareMathOperator\SL{SL}$There is a well-known Clebsch–Gordan decomposition formula for irreducible representations of $\SL_2$. If $V_n$ denotes the unique $n+1$-dimensional irreducible ...

8
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2
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### Proofs of the Frobenius characteristic map

Let $\mathfrak{S}_n$ be the symmetric group on $n$ letters, $\mathsf{Rep}(\mathfrak{S}_n)$ be the abelian category of finite dimensional complex representations of $\mathfrak{S}_n$. A classical result ...

0
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96
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### Sum of weights of an irreducible representation of $U(N)$

Let $R$ be a finite-dimensional irreducible representation of $U(N)$, with the set of weights $W_R$. Each element of $W_R$ is a vector of length $N$ with integer entries.
Firstly, I would like to know ...

2
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### Young diagrams for the block matrices

Let S_n be the group of permutations of n elements. Consider map S_n -> S_mn of block permutations, and an irreducible representation of S_mn (over complex numbers), corresponding to Young diagram ...

4
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### Epstein zeta function of Barnes-Wall and related lattices

Sarnak and Strömbergsson studied Epstein zeta function $\zeta(L,s)=\sum\limits_{0\neq v\in L}\langle v,v\rangle^{-s}$ of a number of highly symmetric lattices in their Inventiones Math. paper.
In ...

2
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2
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### Complete representation theory of $\mathrm{SL}(2,\mathbb R)$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Is the complete representation theory of $\SL(2,\mathbb R)$, $\GL(2,\mathbb R)$, $\SL(2,\mathbb C)$, and $\GL(2,\mathbb C)$ known, in the sense ...

2
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69
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### Unitary dual of universal cover

The universal covering group $G$ of $\mathrm{SL}_2({\mathbb R})$ has infinite center. Is there an irreducible unitary representation $\pi$ of $G$, whose central character is injective? Or does every $\...

8
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### Alternate proof in Fulton–Harris of representation theoretic version of Littlewood–Richardson rule

$\DeclareMathOperator\Ind{Ind}$Let $d = d_1 + d_2$ with $d_1$, $d_2$ positive integers. Let $\lambda$ be a partition of $d_1$ and $\mu$ a partition of $d_2$, so that the Young symmetrizer construction ...

4
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### Equivalence of Yetter-Drinfeld modules to Drinfeld center: is there a purely categorical proof?

Let $H$ be an Hopf algebra over a field $k$, and let $\mathcal{C}$ be the monoidal category of left $H$-modules.
It is known that the Drinfeld center of $\mathcal{C}$ is equivalent (as a braided ...

1
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1
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136
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### Categories associated to digraphs

Let's take a directed graph, or a digraph, $G=(V,E)$ given by a finite set of vertices $V$ and a finite set of edges $E$. We can assume that pairs of vertices can have parallel edges between them and ...

2
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1
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### Proof of restrictableness of Lie algebra without basis

$\DeclareMathOperator\ad{ad}$According to the Strade and Farnsteiners' textbook, a Lie algebra $L$ is restrictable if and only if $\ad(L)$ is $p$-subalgebra of $\operatorname{Der}(L)$. Also, the ...

1
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0
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43
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### Is the Schofield semi invariant defined at $V/IV$?

Let $A=\mathbb{K}Q$ be the path algebra of an acyclic quiver $Q$ over an algebraically closed field $\mathbb{K}$, and $0\not=I\subset\mathbb{K}Q$ be an admissible ideal. Let $W$ be a left $A$-module ...

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### Asymptotic character theory of unitary groups via shifted Schur functions

In the paper "Shifted Schur Functions" http://arxiv.org/abs/q-alg/9605042 by Andrei Okounkov and Grigori Olshanski it is said that one of the motivations for that paper was the asymptotic ...

2
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149
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### A Vandermonde like determinant with exponentials

Let $n\geq m$ be non negative integers, and consider a list of $(n+m+1)$ distinct numbers (complex or real). I am interested in getting a closed form formula for the following determinant: $\det\left[\...

2
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### For locally profinite groups $H\lhd G$, is there a spectral sequence $\newcommand\@[2]{{\rm Ext}_#1^{#2}(\pi_1,\pi_2)}H^p(G/H,\@Hq)\implies\@G{p+q}$?

Let $G$ be a locally profinite group and let $H$ be a closed normal subgroup. Let $\pi_1$ and $\pi_2$ be two smooth complex representations of $G$. Is there always a spectral sequence as follows?
$$...

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307
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### Geometric properties of the adjoint action of a reductive group

$\newcommand{\g}{\mathfrak{g}}$Let $G$ be a reductive algebraic group over field $k = \overline{k}$ and consider the characteristic polynomial $\g \to \g/\!/G := \operatorname{Spec} (k[\g]^G)$ induced ...

4
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105
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### Spectral projection of an eigenvalue associated to a generator of Hecke algebras

In his paper "Hecke Algebras of type $A_n$ (Inv. Math. 1988, EUDML link) and subfactors", in section 2 "Orthogonal representations...", Wenzl takes the usual third relation of a ...

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### Characters of alternating groups

I am studying certain properties of characters of alternating groups and I have found precisely three characters not satisfying it up to $A_{15}$. These are:
A character of dimension $3.696$ of $A_{...

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### A detail in I. Piatetski-Shapiro and S. Rallis's "Doubling paper": computing the integral on negligible orbits

I'm currently reading the paper "L-functions for the classical groups" by I. Piatetski-Shapiro and S. Rallis, where they introduced the doubling method over classical groups.
I'm confused at ...

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### Pure, residual, 2-dimensional, semisimple $G_{\mathbb{Q}}$ representations

Let $G_{\mathbb{Q}}$ be the absolute Galois group of the rationals. I want to know how many isomorphism classes of 2-dimensional, irreducible, pure Galois representations are there over a finite field ...

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### modules with the same associated variety

$S$ is a polynomial ring, $M$ is a finitely generated graded $S$-module, the associated variety of $M$ is
$$
\mathcal{V}\left( M\right)=V\left( Ann_S \left( M \right) \right),
$$
What is the ...

1
vote

1
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58
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### Non-negative integer matrix representation of a fusion ring

Context: I am a physics grad student working on topological lines in 2D CFTs.
Let $A$ be a unital based $\mathbb{Z}_{+}$ ring with finite rank (or a Fusion ring) with the basis $B = \{b_1, b_2, ... b_{...

7
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1
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### Average of polynomials over the real sphere

In quantum information, much can be done with the averaging formula
$$
\int_{\mathbb{C}P^{n-1}} (zz^*)^{\otimes t} dz = {n + t -1 \choose t}^{-1} \operatorname{\Pi}_{\mathrm{Sym}^t}$$
Here the ...

2
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1
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129
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### Orbits in the open set given by Rosenlicht's result

Let $G$ be a linearly reductive algebraic group, and let $X$ be an irreducible affine variety, over an algebraically closed field $\mathbb{K}$, with a regular action of $G$. By Rosenlicht's result, we ...

2
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0
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156
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### Relation between equivariant geometry and representation theory (of geometric objects)

Equivariant geometry studies "manifolds" with an extra structure $G\times M\rightarrow M$.
Representation theory studies "Lie algebroids" with an extra structure $\Gamma(M,A)\times ...

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161
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### Representability of the sheaf $\mathrm{Hom}(G,\mathrm{SL}_2)$

$\DeclareMathOperator\Spm{Spm}\DeclareMathOperator\SL{SL}\DeclareMathOperator\Hom{Hom}$Let $T$ be the topos of $\Spm\mathbb{Q}_p$-rigid analytic spaces, $G$ an abstract group, and $\Hom(G,\SL_2)$ the ...

4
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1
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184
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### Action of complex torus on a vector space

Consider a torus $T$ over $\mathbb{C}$. Let $\rho: T\rightarrow \operatorname{GL}_{n}(\mathbb C)$ be a finite dimensional complex representation.
Is there an elementary way (undergrad level) to see ...

0
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0
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40
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### Homomorphism group of the additive group of rational numbers $\mathbb{Q}$ into quasi-cyclic group $\mathbb{Z}(p^\infty)$ [duplicate]

I read somewhere that the group of homomorphisms from the additive group of rational numbers $\mathbb{Q}$ into the quasi-cyclic group $\mathbb{Z}(p^\infty)$ is isomorphic to the additive group of ...

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0
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41
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### Functional equation Archimedean exterior square

Is there any reference where the functional equation of the Jacquet-Shalika integral representation of the Archimedean local exterior-square L-function is established? Any help is appreciated.

4
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### On decomposition of the space of automorphic forms (via central characters)

Let $F$ be a number field and $\mathbb{A}$ be its adele. For simplicity, we assume $G$ is a connected reductive group.
Given a unitary central character $\chi: Z_{G}(F)\backslash Z_{G}(\mathbb{A}) \to ...

3
votes

1
answer

332
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### What is the name for algebras generated by elements, all of whose cubes vanish?

Given a ring $R$ with identity $1$, we can define the exterior algebra of order $k$ over $R$ to be the algebra over $R$, generated by elements $x_1, \dots, x_k$ satisfying $x_i^2 = 0$ for each index $...

9
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2
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366
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### Using Schur-Weyl duality

I am trying to gain a better understanding of Schur-Weyl duality specifically applied to symmetric functions. My motivating example is trying to understand the Frobenius character of the multilinear ...

4
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1
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148
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### Complexification of a Lie subalgebra of a compact real form

I'm currently reading the paper Lie algebra Cohomology and the Generalized Borel–Weil theorem written by B. Kostant, and I have a question about Remark 3.9 he made.
In this paper, $\mathfrak{g}$ is a ...

3
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0
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87
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### Finding a bigger Frobenius algebra for a given local algebra

Let $A=K\langle x_1,\ldots,x_n\rangle/I$ be a local finite dimensional algebra with admissible relations $I$.
Question: Is there a canonical way to check whether $A$ is isomorphic to $B/\operatorname{...

3
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1
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98
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### Does every nilpotent orbit have an element supported on the simple root spaces?

Let $G$ be a connected reductive algebraic group (over $\mathbb{C}$) and $\mathfrak{g}$ its Lie algebra. Let $O \subset \mathfrak{g}$ be an orbit of a nilpotent element. Let $\Pi = \{\alpha_1, \dots ,\...

2
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0
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### Quiver and relations for Hopf algebras associated to quiver algebras

Let $A=KQ/I$ be a finite dimensional quiver algebra with admissible relations $I$.
$A$ can be made into a restricted Lie algebra over a field of characteristic $p$ via
$[x,y]=xy-yx$ and $x^{p}=x^p$. ...

3
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0
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### References on coefficient quivers

I would like to study about coefficient quivers, but I cannot find a good reference, as book for example. I could find many papers working with coefficient quivers, but none of them give a book or a &...

9
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### Rigorous proof of the pentagon identity

I briefly recall the statement of the pentagon identity in quantum dilogarithm and cluster algebra.
For $b\in\mathbb{C}$ with $\operatorname{Re}(b)>0,\operatorname{Im}(b)\geq0$, Faddeev, Kashaev ...

1
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0
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### Line bundles on toric varieties associated to Weyl chamber

I am interested in studying toric varieties associated to the fan of Weyl chambers. General information would be best but I am also interested in the specific case of the Weyl chamber of $\mathfrak{sl}...

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votes

1
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### Pseudo commutativity

Operator commutativity is the basis for things like homomorphisms and linearity, e.g., $f(x+y) = f(x) + f(y)$.
Is there any meaning or development on a more general nature of this property? E.g., $f(x+...

4
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0
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### Conjugacy of cocharacters from conjugacy of labelled diagrams

Everything to follow is over some fixed algebraically closed field $k$. Although all the definitions make sense regardless of characteristic, the meat of the question is about small positive ...

3
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### Intertwining operators and induced representation

Let $G=GL(n, F)$, $B$ be a Borel subgroup and let $B=AN$ be the Langlands decomposition. Let $\nu \in \mathfrak{a}^*_{\mathbb{C}}$ be in the positive Weyl chamber. Consider the normalized induced ...

3
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153
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### Representability of $\operatorname{Hom}(G_{\mathbb{Q}}, \operatorname{GL}_2)$

Let $G_{\mathbb{Q}}$ be the absolute Galois group of the rationals, and let $F: \mathrm{Aff}/\textbf{Q}_p\longrightarrow \mathrm{Sets}$ be the functor which associates to every affine $\mathbb{Q}_p$ ...

3
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1
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### Does $\mathrm{Ext}^i_G(\pi,\pi')$ vanish if $\pi$ and $\pi'$ are smooth irreducible representations of $G$ with different central characters?

Let $G$ be a $p$-adic group, ie. the group of $F$-points of some connected reductive group over $F$, where $F$ is a $p$-adic field. We consider complex smooth representations of $G$. Any irreducible ...

0
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1
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93
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### Centralizer of a reductive subgroup

Let $G$ be a reductive group over $\mathbb{C}$ and $H\subseteq G$ a reductive subgroup. Let $\rho$ be a faithful irreducible finite dimensional representation of $G$ over $\mathbb{C}$. Assume that $\...

2
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102
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### Construction of a certain long exact sequence

Let $A$ be a noetherian ring (not necessarily commutative) or for simplicity even a finite dimensional algebra over a field.
Let $X$ and $U$ be finitely generated $A$-modules and let $add(U)$ be the ...

0
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0
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56
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### How to canonically induce a morphism between module categories of path algebras when given a morphism of quivers?

Let $\pi:P\to Q$ be a morphism of quivers, i.e., $s(\pi(a))=\pi(s(a)),t(\pi(a))=\pi(t(a)) $ for arrows $a$, start $s$ and tail $t$. Is there a canonical way to induce from $\pi$ a morphism between ...

3
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0
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77
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### Example of an irreducible component with an open set of infinitely many codimension 2 (codimension 3) orbits

Let $\mathbb{K}$ be an algebraically closed field of characteristics $0$. Let $A$ be a finite dimensional (associative and unital) algebra over $\mathbb{K}$. Assume there is a quiver $Q=(Q_0,Q_1)$, ...

3
votes

1
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214
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### Projective dimension of group ring

Assume that $G$ is a group and $R$ is a p.i.d. What can we say about the projective dimension of $R[G]$? For example can we say that this dimension is at most $1$ for reductive groups? (I think if $...