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Questions tagged [rt.representation-theory]

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

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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) ...
Sebastien Palcoux's user avatar
0 votes
0 answers
46 views

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 ...
Mahtab's user avatar
  • 267
1 vote
0 answers
41 views

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 ...
Alimzhan's user avatar
2 votes
0 answers
160 views

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 ...
Samuil Lee's user avatar
1 vote
0 answers
70 views

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 ...
Luiz Felipe Garcia's user avatar
4 votes
1 answer
86 views

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 $S$ ...
Sam K's user avatar
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2 votes
0 answers
70 views

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 $V^{\otimes 2}$, ...
Gutiérrez's user avatar
0 votes
0 answers
72 views

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 ...
Hauke Reddmann's user avatar
4 votes
1 answer
568 views

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 ...
Doug Liu's user avatar
  • 525
3 votes
0 answers
260 views

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)...
Partha's user avatar
  • 883
4 votes
2 answers
320 views

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 ...
Aztec's user avatar
  • 41
1 vote
0 answers
58 views

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 $H\subset G(F)$ ...
youknowwho's user avatar
2 votes
1 answer
191 views

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 $m$, ...
Andrew's user avatar
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4 votes
0 answers
106 views

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 ...
Asvin's user avatar
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4 votes
0 answers
146 views

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 ...
Iian Smythe's user avatar
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6 votes
0 answers
138 views

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 ...
SamJeralds's user avatar
-1 votes
1 answer
105 views

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 ...
Hauke Reddmann's user avatar
7 votes
0 answers
119 views

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 ...
thedude's user avatar
  • 1,499
2 votes
1 answer
74 views

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 ...
R. Chen's user avatar
  • 121
3 votes
1 answer
78 views

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 ...
Andrew's user avatar
  • 939
2 votes
0 answers
96 views

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 $...
Andrew's user avatar
  • 939
11 votes
0 answers
410 views

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 ...
jg1896's user avatar
  • 3,063
3 votes
1 answer
124 views

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 \...
Ashwin Iyengar's user avatar
1 vote
0 answers
38 views

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 ...
Andrew's user avatar
  • 939
1 vote
0 answers
158 views

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 ...
MO B's user avatar
  • 545
4 votes
0 answers
168 views

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-...
Yellow Pig's user avatar
  • 2,540
4 votes
1 answer
121 views

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 ...
user449595's user avatar
12 votes
1 answer
192 views

$p$-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 ...
Estwald's user avatar
  • 1,321
4 votes
0 answers
153 views

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 ...
Irwin's user avatar
  • 123
4 votes
1 answer
124 views

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 $\mathrm{GL}_{n+1}(F)$ ...
Hetong Xu's user avatar
  • 619
2 votes
0 answers
172 views

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 ...
user267839's user avatar
  • 5,984
3 votes
1 answer
109 views

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 $...
jrg's user avatar
  • 33
10 votes
1 answer
327 views

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 ...
JeCl's user avatar
  • 1,001
1 vote
0 answers
123 views

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 ...
Andrew's user avatar
  • 939
0 votes
0 answers
91 views

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 ...
It'sMe's user avatar
  • 767
2 votes
1 answer
166 views

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 ...
Jon Elmer's user avatar
  • 175
2 votes
0 answers
62 views

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 ...
Fan Zhou's user avatar
  • 301
4 votes
2 answers
138 views

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 ...
Antoine Labelle's user avatar
2 votes
0 answers
56 views

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 $\...
d'Alembert's user avatar
5 votes
1 answer
202 views

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 ...
Mare's user avatar
  • 26.2k
4 votes
0 answers
87 views

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 ...
Stefan  Dawydiak's user avatar
0 votes
0 answers
101 views

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+...
T. Amdeberhan's user avatar
3 votes
0 answers
71 views

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 ...
Mare's user avatar
  • 26.2k
8 votes
1 answer
220 views

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= \...
Mare's user avatar
  • 26.2k
2 votes
0 answers
43 views

$K_0$-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 ...
Mare's user avatar
  • 26.2k
3 votes
1 answer
89 views

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 ...
Marcel's user avatar
  • 2,542
5 votes
0 answers
165 views

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 ...
Pierre's user avatar
  • 2,195
4 votes
1 answer
237 views

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 ...
user44312's user avatar
  • 603
16 votes
3 answers
1k views

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 ...
Mare's user avatar
  • 26.2k
1 vote
0 answers
43 views

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 ...
kindasorta's user avatar
  • 1,995

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