# Questions tagged [rt.representation-theory]

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

5,135
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### Multiplication formula in Grassmannian cluster categories

Grassmannian cluster categories are studied in A categorification of Grassmannian cluster algebras and Cluster categories from Grassmannians and root combinatorics. The category $CM(B_{k,n})$ of Cohen-...

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84 views

### The Jacobson radical as a bimodule

Let $A$ be a finite dimensional algebra with Jacobson radical $J$.
Question 1: In case $A$ is a Nakayama algebra with a linear quiver corresponding to a Dyck path $D$ (via its Auslander-Reiten ...

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31 views

### Real non-principal 2-blocks for finite groups of Lie type

Is it true that each finite group of Lie type has a non-principal real $2$-block? Here the principal block is the one containing the trivial character and a block is real if it contains the complex ...

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29 views

### Dual of smooth induced representation

Let $G$ be a locally profinite group with a closed subgroup $H$ and a smooth representation $(\pi,V)$ . Denote by $Ind_H^{\infty,G}(\pi)$ the smooth induced representation of $\pi$.
Is there a nice ...

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60 views

### Image of Frobenius element under irreducible representation is diagonalizable

Let $K/ \mathbb Q$ be a Galois extension, and $\rho$ be an irreducible representation of the Galois group $Gal(K/ \mathbb Q)$. Consider an integer prime $p$ which doesn't ramify in $K$, and let $\...

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### General construction of enveloping C*-algebra, left/right-regular representation, etc

In a number of contexts (e.g. groups, crossed products, groupoids, Fell bundles) there are similar constructions of enveloping C*-algebras and left/right-regular representations that incorporate ...

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104 views

### Question on the proper sub-representation of induced representation

$\DeclareMathOperator\Ind{Ind}$Let $G$ be a reductive group over a $p$-adic local field $F$, and $P=MN$ a parabolic subgroup.
Let $\sigma$ be an irreducible representation of $M(F)$ and consider its ...

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39 views

### Invariant theory of the indefinite orthogonal groups

I believe the following statements are true:
Let $V$ be a finite-dimensional real vector space with a positive-definite inner product $g$. Let $g_{\otimes n}$ denote the natural extension of $g$ to $...

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139 views

### Moments of Plücker coordinates on complex Grassmannian and log-concavity

Consider the Grassmannian $Gr(k,N)\simeq U(N)/(U(k)\times U(N-k))$ which parametrizes $k$-dimensional subspaces of $\mathbb{C}^N$. Let us put on it the $U(N)$-invariant probability measure. Let $\...

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79 views

### An integral Jacobson-Morozov theorem?

$\DeclareMathOperator\SL{SL}$I want to ask if there exists a version of the Jacobson–Morozov theorem for integer matrices. A first approximation would ask: given an integral unipotent matrix $m \in \...

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220 views

### Surjectivity in Deligne-Serre

Let $f$ be a newform of weight $k$ and level $N$ with integer coefficients. Deligne-Serre theorem theorem says there exist a nice associated representation $\rho_{f}^{(\ell)}:\text{Gal}(\overline{\...

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### Proof of parametrisation of $\hat{\mathfrak{g}}$-intertwiners of induced modules of affine Lie algebras

I am not 100% sure that this question belongs here, since I think the topic does but the specific problem I have might not. Feel free to tell me so if this is the case. It concerns the proof of ...

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34 views

### Irreducibility of the adjoint representation in positive characteristic

Let $G$ be a simple, simply connected, algebraic group over an algebraically closed field $k$ of characteristic $p>0$. Let $Ad$ be the adjoint representation of $G$ on $\mathrm{Lie}(G)$.
Given ...

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84 views

### Is the derived category of local systems equivalent to the derived category of sheaves of vector spaces with local system cohomology?

Let $k$ be a field and $X$ a topological space.
Write $\mathrm{Sh}(X)$ for the category of sheaves of vector spaces on $X$, and $\mathrm{Loc}(X)$ for the subcategory of local systems of finite ...

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146 views

### Confusion over spin representation and coordinate ring of orthogonal Grassmannian

This is a copy from MSE where the question did not attract much attention.
I'm working over $\mathbb{C}$ here. Let $G=\mathrm{SO}(2n+1)$ be the odd orthogonal group, and $P$ be the maximal parabolic ...

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114 views

### Level vs. conductor of a supercuspidal representation

What is the relation between level and conductor of a supercuspidal representation of $\operatorname{GL}_2(\mathbb{Q}_p)$ for some prime $p$?
Proposition 3.4 in Loeffler and Weinstein - On the ...

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164 views

### Finite group ${\rm Sp}_4({\Bbb F}_3)$: involutions coming from a 4-dimensional complex representation

I am interested in the finite unitary reflection group $G= G_{32}$, the group No. 32 in Table VII on page 301 of the paper:
Shephard, G. C.; Todd, J., A. Finite unitary reflection groups. Canad. J. ...

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92 views

### How is Harish-Chandra restriction compatible with Harish-Chandra series?

Suppose $G$ is a connected reductive group over $\overline{\mathbb{F}}_p$, with Frobenius $F$. Let $(L_0,\Lambda_0)$ be a cuspidal pair with $L_0$ a Levi subgroup of a Levi subgroup $L$, and let ${^{\...

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261 views

### Combinatorial proof of invertibility of a symmetric matrix associated to the ring of matrices over a finite field

Let $F$ be a finite field of $q$ elements with characteristic $p$. Let $M_n(F)$ be the ring of $n\times n$ matrices over $F$. We define a $q^{n^2}\times q^{n^2}$ symmetric matrix $L$ over the ...

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60 views

### Coefficient ring of Satake isomorphism

Let $G$ be a split reductive group over local field $F$, $G^L$ be the (complex) Langlands dual group of $G$. Denote $H$ to be the $\mathbb{Z}$-Hecke algebra of $G$, that is the ring of $G(\mathcal{O}...

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166 views

### A question related to newform and irreducible cuspidal representation of $\operatorname{GL}_n$

I was reading adelization of classical automorphic forms and learnt that each cusp form corresponds to an automorphic representation of $\operatorname{GL}_n(\mathbb{A}_\mathbb{Q})$. I understood the ...

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### Decomposition of the homogeneous polynomial ring $\{\mathbb R[x_{ij}]_{1\le i,j\le n}\}$ of degree 2 into Specht modules

I have tried to decompose this as following spans over the real field.
$V_1=\operatorname{span} \langle x_{ij}^2\rangle$
$V_2=\operatorname{span} \langle x_{ij}x_{jk}\rangle$
$V_3=\operatorname{...

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122 views

### Coinvariants of tensor products of Hopf algebras

Let $G$ be a Hopf algebra, considered as a right $G$-comodule in the obvious way.
The axioms of Hopf algebras imply that
$$
G^{coinv(G)} == \{g \in G : \Delta(g) = g \otimes 1\} = \mathbb{C}1.
$$
...

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249 views

### Decomposing a polynomial ring into Specht Modules

Let $S_{\pi}$ where $\pi$ is an integer partition of $n$, denote the Specht module corresponding to $\pi$.
I am trying to decompose the set of all homogeneous polynomials in $x_1,x_2,...,x_n$ ...

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309 views

### Representation of central extension

Let $G$ be a finite abelian group of rank $n$ and $H\rightarrow G$ a central extension with cyclic finite kernel.
Is it true that we can find a faithful representation $H\rightarrow {\rm GL}_{k(n)}(\...

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164 views

### A transversal for the $\operatorname{Ad}(K)$ action on a sphere in $\mathfrak{p}$

This exercise level question has been unanswered on MSE for a few years. I hope you can answer it either there or here.
$G$ is a semisimple Lie group with a choice of Cartan decomposition on its Lie ...

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### Branching from $GL(a+b)$ to $GL(a)\times GL(b)$ using Gel'fand-Cetlin patterns

If one iterates the multiplicity-free branching rule from $GL(n)$ representations (finite-dim, over $\mathbb C$) to $GL(n-1)$ all the way down to $GL(0)$, one obtains triangular "Gel'fand-Cetlin (or ...

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### Do you recognise this setup of structure on a poset?

The setup is that we have a finite poset $P$, with a multiplicative rank function $r_{xy}:P\times P\rightarrow \mathbb{N}$, and a symmetric pairing $\langle\ ,\ \rangle:P\times P\rightarrow\mathbb{N}$....

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250 views

### Closed form for Jacobi sum $\sum_{a\in \mathbb{F}_{p^2}}\chi(a)\chi(1-a)$

Let $\mathbb{F}_{p^2}$ be a field with $p^2$ elements and $\chi:\mathbb{F}_{p^2}^*\to\mathbb{C}^{*}$
be a multiplicative and non-trivial character on the multiplicative group $\mathbb{F}_{p^2}^*$ (...

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116 views

### Almost split sequences coming from bimodules

Let $A$ be a finite dimensional algebra with enveloping algebra $A^e$.
Auslander and Reiten proved in "On a theorem of E. Green on the dual of the transpose" that
$Hom_A(Tr_{A^e}(A),M) \cong \tau(M)$ ...

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### Bimodule isomorphism for representation-finite blocks of the Schur algebra

Let $A$ be a representation-finite block of a schur algebra with $n \geq 2$ simple modules. Then the global and dominant dimension of $A$ are equal to $g=2n-2$. You can find quiver and relations for ...

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### Complete list of indecomposable representations of Temperley-Lieb algebras at roots of unity?

The Temperley Lieb algebra $TL_n$ at roots of unity is not semisimple. The standard representations $V_{n,p}$ are indecomposable but, in general, not irreducible. If $K_{n,p}$ is the sub-...

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### Effect of extending scalars on maps of modules

Let $k$ be a field and $R$ be a $k$-algebra. Let $M$ and $N$ be left $R$-modules. Finally, let $\ell$ be a field extension of $k$. We thus have an $\ell$-algebra $\ell \otimes R$, and both $\ell \...

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### Invariance under derived equivalence of a Gorenstein projective bimodule

A module $M$ over an finite dimensional algebra $A$ is called Gorenstein projective in case there exists an exact complex $(P_i)$ of projective $A$-modules such that the complex stays exact after ...

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102 views

### Global splitting field for algebras

Let $A$ be a finite dimensional algebra.
A field $K$ is a splitting field for an indecomposable $A$-module $M$ in case the local algebra $End_A(M)/(rad(End_A(M))$ is 1-dimensional.
$K$ is called a ...

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### Higher analogue of the Auslander-Bridger transpose

Let $A$ be an Artin algebra and $M$ a module with $Ext^i(M,A)=0$ for $i=1,...,n-2$.
Then in case $P_{n-1} \rightarrow ... \rightarrow P_0 \rightarrow M \rightarrow 0$ is the beginning of a minimal ...

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161 views

### Jack function in power symmetric basis

In Macdonald's book, the Jack symmetric function $J_{\lambda}(x_1,\ldots, x_n)$ for a partition $\lambda$
is defined by three properties (orthogonality, triangularity, and normalization). In the ...

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### Preprojective algebra of finite dimensional algebras

The preprojective algebra of a module $M$ over a finite dimensional algebra $A$ is
defined as $P_M:= \bigoplus\limits_{n=0}^{\infty}{Hom_A(M, \tau^{-n}(M))}$ with the canonical multiplication.
...

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### What is the name of the real form corresponding to the quaternionic symmetric space?

Let $G$ be a compact simple Lie group. Choose a system of positive roots, and let $\mathrm{SU}(2) \subset G$ correspond to the highest root, and $\mathbb{Z}/2 \subset \mathrm{SU}(2)$ the centre. The ...

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### Question on a subcategory being extension-closed

In the article "Homological theory of noetherian rings" by Idun Reiten from 1996, it was stated that it seems to be not known whether the subcategory $\operatorname{Tr}(\Omega^i(\mathrm{mod}\text{-}A))...

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### Isomorphism of certain irreducible representations over finite fields

We are given a faithful representation of a cyclic group of order 5 $\rho: C_5=G \rightarrow End_{\mathbb{F}_3}(V) $ with $dim_{\mathbb{F}_3}V=8$ as vector space. It is also known that $V=U\oplus W$ ...

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### Realization of limit of discrete series using Dirac operators

I wonder if there is a geometric realization of limit of discrete series in the flavor of Atiyah-Schmid or Parthasarathy realizing discrete series using Dirac operators on G/K. I know you can see ...

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505 views

### History of the notion of irreducible representation

I am looking for the earliest references where the study of irreducible representations appears. There has been many articles and books on the history of representation theory. A fundamental feature ...

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### Reference for equivariant derived Künneth formula

I'm looking for a reference for the following statement in as much generality as possible, assuming it is correct.
Let's $X$ and $Y$ be "spaces" with a $G$-action. We can take the $G$-product defined ...

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### Dimension bound on invertible bimodules for blocks

A co-author and I have recently proved the following result:
Theorem.
Let $B$ be a block of a finite group defined over some algebraically closed field $k$ of characteristic $p>0$. If $M$ is a $B$-...

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### Mysterious inequality for the homological dimension of modules

Let $A$ be an Artin algebra and $M$ an indecomposable $A$-module.
Let $pd(M)$ denote the projective dimension of $M$, $id(M)$ the injective dimension of $M$, $domdim(M)$ the dominant dimension of $M$ ...

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282 views

### “Simple” proof of irreducible characters of finite groups being non-zero

A search brought up this, with reference to a book by I. M. Isaacs. However, the proof in the book leverages on a lot of field theory knowledge. I am wondering, is there a simpler proof (or a proof ...

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### Freeness of completed homology over universal deformation ring

In Theorem 7.4 of the paper "patching and the $p$-adic Langlands program for $\mathrm{GL}2(\mathbf{Q}_p)$" (arXiv link), it is proved that in the minimally ramified case, the completed homology of ...

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236 views

### Isomorphism for Ext spaces for finite dimensional algebras

Let $A$ be an Artin algebra with enveloping algebra $A^e$.
Then we have $Hom_{A^e}(X,A^e) \cong Hom_A(D(A) \otimes_A X,A)$ for a bimodule $X$. (see for example in the article "A theorem of Green on ...

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165 views

### On properties of an algebra as a bimodule

Let $A$ be a two-sided artinian ring.
Recall that a module $M$ is said to have dominant dimension at least $n$ in case the terms $I_i$ in the minimal injective coresolution of $M$ are projective for $...