# Questions tagged [rt.representation-theory]

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

6,848
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### Irreducible tensor product representations in finite simple groups

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PSp{PSp}\DeclareMathOperator\PSU{PSU}$Background:
A representation $ \rho: G \to \GL(V) $ of a group $ G $ on a (complex) ...

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### A reference for this statement (representations of universal central extensions)

Let $G$ be a connected Lie group with universal cover $\tilde{G}$. I would really appreciate if someone could give me a reference for the proof of the following fact:
"Every projective unitary ...

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0
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### Young symmetrizers-like projections to the center of group algebra

Let $A:=\mathbb{C}S_n$ be the symmetric group aglebra.
Let $T$ be a standard Young tableaux of shape $\lambda$. Denote $R(T)$ and $C(T)$ as row and column stabilizers of $T$. For a set $S \subseteq ...

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### How to construct an explicit isomorphism of the split Quaternion Algebra $(a,b)_F$ over the field $F$ to $\mathrm{Mat}_2(F)$

$\DeclareMathOperator\Mat{Mat}$How to construct an explicit isomorphism of the split quaternion algebra $(a,b)_F$ over the field $F$ to $\Mat_2(F)$?
As it is known that the algebra of quaternions is ...

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0
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### Can every $\ast$-algebra be represented in this space of matrices?

Let $k$ be a field with characteristic $0$. For every set $X$, let $\mathcal{B}(X)$ be the set of (possibly infinite) matrices $T = (T_{x,y})_{x,y \in X}$ with coefficients in $k$ such that in each ...

4
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1
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### Extension of scalars for bounded chain complexes of $kG$-modules

I'm wondering if a generalization regarding a statement from Curtis-Reiner holds. The original statement is as follows:
(30.33) Theorem: Let $R$ and $S$ be complete discrete valuation rings, with $S$ ...

2
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0
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70
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### Is there an alternating power functor on braided monoidal categories?

Let $\mathsf{C}$ be a braided monoidal $\mathbb{C}$-linear category and $V\in \mathsf{C}$ any object. Then we can form the tensor power $V^{\otimes 2}$.
The braid group $B_2$ acts on $V^{\otimes 2}$, ...

0
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### When can a R matrix be brought to Hermitian form?

For this question, let "R matrix" denote a (preferrably invertible) solution of the (constant) Yang Baxter equation. Any R matrix (if you alternatively write it as tensor $R^{ab}_{cd}$) is ...

4
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### An algebraic group has how many representations?

Let $G$ be a connected, linear algebraic group over $\mathbb{C}$. Let $\rho:G\to \mathrm{GL}(V)$ be a rational representation. Does $\rho$ have at most countably many subrepresentations (up to ...

3
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### A question in $\operatorname{Spin}(7)$ geometry

$\DeclareMathOperator\Spin{Spin}$I am looking for a proof of a fact (I think it's true intuitively due to representation theory) in $\Spin(7)$ geometry. Let's take a closed $\Spin(7)$-manifold $(M^8,g)...

4
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2
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### Why the character of metaplectic (Weil) representation of symplectic group ${\rm Sp}(2n,{\mathbb C})$ is nonzero?

Recently I have just started to work on the trace formula of the metaplectic representation of ${\rm Sp}(2n,{\mathbb C})$ group. I have a naive question: is the trace (i.e., character of the ...

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0
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### Asymptotic behavior of Shalika germs near non-regular elements

Let $G$ be a connected reductive group over a $p$-adic field $F$. Let $T\subset G$ be a maximal torus. Fix a special maximal compact subgroup $K$ of $G(F)$ and for any closed subgroup $H\subset G(F)$ ...

2
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1
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### Question on the irreducibility of induced representations of $\mathrm{GL}_n$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\det{det}\DeclareMathOperator\Ind{Ind}$Let $F$ be a $p$-adic local field and $\chi_i$ be unramified characters of $F^{\times}$. For a integer $m$, ...

4
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### How does one compute the group action of the automorphism group on integral cohomology?

Suppose I have a curve $X$ (for concreteness, we can take $X$ to be a smooth, projective curve over a finite field $\mathbb F_q$, and even more concretely consider the family of curves described by ...

4
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### Which Lie groups are covers of matrix groups?

I would like to ask a variation on a question (not yet answered) I previously asked on math.SE, namely:
Which Lie groups are covers of matrix Lie groups? That is, which Lie groups $G$ admit discrete ...

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### Schubert varieties and cohomology vanishing

To fix (albeit standard) notation, let $G$ be a complex semisimple algebraic group, and $T \subset B \subset G$ choices of maximal torus and Borel subgroup, respectively. Let $X^\ast(T)$ be the ...

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1
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### Killing form that is not diagonalizable?

An example Lie algebra $L$ with non-diagonalizable Killing form would have to be non-semisimple, and the Killing form complex. (Otherwise diagonalizability is obvious.) I tried with a few $L$, but (in ...

7
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0
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119
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### Relation between Fourier series and Schur polynomials

Asked initially at MSE.
I would like to know how to express the Fourier series of a symmetric function, $f(\theta_1,...,\theta_N)$, in terms of Schur polynomials $s_\lambda(x_1,...,x_N)$ in the ...

2
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1
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### Explicit computation for the coefficients of the intertwining operator

In the following note by Casselman https://personal.math.ubc.ca/~cass/research/pdf/p-adic-book.pdf page 74, proposition 9.3.2, for the group $G=SL_{2}$ he computed explicitly the coefficients of the ...

3
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1
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### Computation of modulus character of the mirabolic subgroup of $\operatorname{GL}_n$ using roots

$\DeclareMathOperator\GL{GL}$For parabolic subgroup of a general linear group or classical group $G$, we can compute its modulus character using the positive roots associated to them.
But it seems ...

2
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0
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### Question on the geometric lemma in $p$-adic classical groups

$\DeclareMathOperator\GL{GL} \DeclareMathOperator\Sp{Sp} \DeclareMathOperator\Ind{Ind}\DeclareMathOperator\B{B} $My question is about the geometric lemma of $p$-adic classical groups not $\GL_n$. For $...

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### A rather strange algebra

Let $k$ be an algebraic closed field of zero characteristic and $X$ an affine smooth variety, with $A=\mathcal{O}(X)$ the algebra of regular functions and $\mathcal{V}$ the Lie algebra of vector ...

3
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### Factoring out an element of a root subgroup to make a conjugation integral

Fix a nonarchimedean local field $L/\mathbb{Q}_p$ with ring of integers $\mathcal{O}$, uniformizer $\varpi$, and residue field $k$. If I take a matrix
$$\begin{pmatrix} a & \varpi b \\ c & d \...

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### Question on the modulus character of not parabolic subgroup

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Sp{Sp}$
Last time, I asked a question on the computation about modulus character of parabolic subgroup of symplectic group and LSpice gave me a nice ...

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0
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### Representation theory of $\mathrm{GL_n}(\mathbb{F}_q)$

I am interested in learning about the classification of irreducible representations of $\mathrm{GL}_n(\mathbb{F}_q)$ as done by Green. One standard reference (besides Green's original work) is ...

4
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### Relation between exotic sheaves in Achar's notes and in Bezrukavnikov-Mirkovic

I am trying to calculate explicitly a certain simple exotic sheaf (a simple object of the heart of the exotic t-structure on the Springer resolution, which is defined in Theorem 1.5.1 of Bezrukavnikov-...

4
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1
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### Do parabolic inductions share a composition factor if and only if the inducing data are associate?

Let $F$ be a local field of characteristic zero and $G$ a connected reductive group over $F$. Let us call an inducing datum a triple $(P,M,\sigma)$, where $P$ is a parabolic subgroup of $G$, $M$ is a ...

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### $p$-adic counterpart of W-algebra

Representation theory and geometry over $k((t))$ and $\mathbb{Q}_p$ have many similarities, and there are many similar constructions, usually motivated from the other side (say the study of affine ...

4
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### Largest primitive subgroup of $\mathrm{GL}_8(\mathbb{C})$ of order $2^a 3^b 5^c$

The paper "Bounds for finite primitive complex linear groups" by M. Collins computes the largest possible value of $[G:Z(G)]$ for $G$ a primitive subgroup of $\mathrm{GL}_N(\mathbb{C})$, for ...

4
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1
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### Reference Request: Test vectors for local Rankin-Selberg L-factors in ramified cases

Let $F$ be a global number field, i.e. a finite extension of the field of rational numbers. Let $\sigma$, $\pi$ be automorphic representations of $\mathrm{GL}_n(F)$ and $\mathrm{GL}_{n+1}(F)$ ...

2
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### Theorem of highest weight of semisimple Lie algebras: what fails precisely for reductive case

Let $\mathfrak{g}$ a complex semisimple Lie algebra. It is well known that by the theorem of the highest weight, all finite-dimensional complex irreps of $\mathfrak{g}$ are (up to iso) classified by ...

3
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1
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### Involution of $\text{GL}_{m+n}(\mathbb{C})$ fixing Levi and exchanging parabolic subgroups

Is there any involution of $\text{GL}_{m+n}$ which is the identity on $\text{GL}_m\times\text{GL}_n\subset\text{GL}_{m+n}$ and that exchanges the positive and negative associated parabolic subgroups $...

10
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1
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### Braidings on Temperley-Lieb Category

Let $k$ be a field, and let $q\in k^{\times}$. We can then consider the Temperley-Lieb category $TL(q)$. The objects of $TL(q)$ are the non-negative integers, and morphisms are roughly isotopy classes ...

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0
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### Question on two types of Frobenius theorem in $p$-adic groups

Let $G$ be a $p$-adic classical group and let $P_0$ be a minimal parabolic subgroup of $G$. Let $P=MN$ be a
standard parabolic subgroup containing $P_0$. Let $\text{Ind}$ and $\text{Jac}$ be the ...

0
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0
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### Matrix of the minimal projective presentation of a $\tau$-rigid module

Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbb{K}$ of characteristics zero. Suppose $A$ is given by the bound quiver $(Q,I)$. We will use $P_l$ to denote the ...

2
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1
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### Isomorphisms $S^d(S^m(V)^*) \cong \Lambda^d(S^{m+d-1}(V)^*)$

$\DeclareMathOperator\SL{SL}$Let $K$ be an algebraically closed field. Let $V$ be a 2-dimensional vector space over $K$. The group $G=\SL_2(K)$ acts naturally on $V$ by left-multiplication on column ...

2
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### How to multiply dots with Young idempotents in the degenerate affine Hecke algebra (type A)

Let $\widehat{\cal H}_n$ be the type A degenerate affine Hecke algebra on $n$ strands, and let $x_1,\cdots,x_n$ be the dots. Inside of this algebra lies the algebra $\mathbb C S_n$, and the Young ...

4
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2
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### Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$

Let $G$ be a connected algebraic group over an algebraically closed field $k$ of characteristic zero (I'm mostly interested in the case of a reductive group).
By the Tannakian formalism, $G(k)$ can be ...

2
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0
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### Have analytic continuations of holomorphic discrete series representations a formal dimension?

Let $G$ be a connected real semisimple Lie group, for example, $SU(p,q)$. I am interested in unitarizable highest weight representations for $G$. If $K$ denotes a maximal compact subgroup of $G$ and $\...

5
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1
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### Characters with all higher exterior powers irreducible

Let $G$ be a finite group and we take for the field the complex numbers.
Call an irreducible character $\xi$ with degree $m$ of $G$ perfect, if all exterior powers $\bigwedge\nolimits^k \xi$ are ...

4
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### Irreducible representation of $\mathrm{GL}(2,\mathbb{R})$ that is not admissible

It is a basic theorem of Harish-Chandra that an irreducible unitary representation $\pi$ of a reductive group $G$ over $\mathbb{R}$ on a Hilbert space is admissible, meaning that every irreducible ...

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### An interesting identity involving skew-Schur functions

Denote $\rho=(-\frac12,-\frac32,-\frac52,\dots)$. I was reading this interesting paper, where in particular, the authors claim that one can get the expression in (2.9)
\begin{align*}
\prod_{k\geq1}(1+...

3
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### Connection between certain finite groups and Frobenius algebras

This question is maybe not so precise yet, but contains some ideas that maybe just search for the right definition.
Let $G=F/U$ be a finite group with presentation $F/U$ (here the presentation really ...

8
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### Quiver and relations for a monoid related to Catalan numbers

Let $C_n$ be the monoid consisting of monotone maps $\{1,...,n\} \rightarrow \{1,...,n\}$ with $f(i) \leq i$ for all $i$.
The cardinality of $C_n$ is given by the Catalan numbers.
Consider $A_n= \...

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### $K_0$-basis modules with a unique extension related to parking functions

Let $B=B_n$ be the quiver algebra of type $A_n$ (with some orietnation) with $n$ points.
A $B$-module $M$ is a basis of the Grothendieck group $K_0$ if $M$ has $n$ indecomposable direct summands and ...

3
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1
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### Does the reproducing property of the unitary group Poisson kernel require a multiple of the identity?

The Poisson kernel of the unitary group is
$$ P(Z,U)=\frac{\det(1-ZZ^\dagger)^N}{\det(1-ZU^\dagger)^N\det(1-UZ^\dagger)^N}.$$
It has a reproducing property, $\int dU P(Z,U)f(U)=f(Z)$, where $dU$ is ...

5
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### Decomposing an endomorphism as a tensor product

$\DeclareMathOperator\End{End}$Let $f$ be an endomorphism of the finite-dimensional vector space $V$, over the field $K$. The question of whether $f$ is decomposable, that is, whether $V$ can be ...

4
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1
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### The number of irreducible characters of simple groups of Lie type

Let $S$ be a simple group of Lie type defining over $\mathbb{F}_q$ where $q$ is a power of a prime $p$ and let $\sigma$ a nontrivial field automorphism of $S$.
Set $\mathrm{C}_{S}(\sigma)$ the ...

16
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3
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### Conjectures in the representation theory of the symmetric group

Question: What are current open conjectures about the representation theory of the symmetric group?
I am interested mostly in the characteristic 0 case, but conjectures for the modular case can also ...

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0
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### Absolute irreducibility implies free action on framed universal deformation ring

Let $\overline{\rho}: G\longrightarrow \text{GL}_n(\mathbb{F}_p)$ be a residual representation of the Galois group of a number field. Let $R_{\overline{\rho}}^{\square,\text{univ}}$ be the universal ...