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

3
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2answers
90 views

Generating Irreducible representations of a simple lie algebra with Schur functors

Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$. Let $Rep(\mathfrak{g})$ denote the category of finite dimensional $\mathfrak{g}$-modules. For every $V \in Rep(\mathfrak{g})$ define $...
1
vote
1answer
108 views

references on group representation over local fields / a question on an argument of a Ralph Greenberg's paper

I'm currently studying Iwasawa theory. 1) There are many $\mathbb{Z}_p$-modules on which some Galois groups act. So I often face some facts on the group representation over local fields or p-adic ...
3
votes
1answer
110 views

Clebsch–Gordan decomposition for $\mathrm{SU}(2)$, in indices

Let $\pi_m$, $m \geq 0$, be the unitary irreps of $\mathrm{SU}(2)$. The Clebsch–Gordan decomposition then gives that $$ \pi_m \otimes \pi_n = \bigoplus_{k=0}^{\min(m,n)}\pi_{m+n-2k}.$$ But suppose I ...
3
votes
1answer
151 views

Higher Extension Group Question

Suppose we have an associative unital ring $R$, and we have an $R$-module $M$ with a length 3 socle filtration, i.e. write $$soc(M) \text{ for the socle of } M,$$ $$soc^2(M) \text{ for the preimage ...
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vote
3answers
200 views

References request: representations of classical groups

Are there some good references about representations of classical groups? What are the fundamental representations of classical groups of type $B, D$? I would like to know explicit formulas of the ...
7
votes
1answer
172 views

Finite group with a character having one nonzero absolute value

Let $G$ be a finite group. Assume that $\chi$ is a complex irreducible character of $G$ of degree $n\geq 2$, with the property that for each element $g\in G$ either $\chi(g)=0$ or $|\chi(g)|=n$. ...
3
votes
1answer
89 views

Identity for $Ext^1$ for special algebras

Let $A$ be a finite dimensional algebra and assume all modules are also finite dimensional. A module $M$ is said to have dominant dimension at least $n$ in case the term $I_i$ for $i=0,1,...,n-1$ are ...
3
votes
0answers
73 views

Geometric interpretation of duality for representations of reductive groups

For a reductive group $G$ over a nonarchimedean local field, let $\Omega$ be a connected component of the variety of cuspidal data. Let $\Omega$ have dimension $d$. Let $\mathcal{M}^f(\Omega)$ be the ...
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vote
0answers
117 views

Reducibility of the Weil representation

Let $G$ be a finite abelian group, and let $G\longrightarrow \hat{G}$, $g\mapsto \chi_g$ be a group isomorphism from $G$ to its dual (i.e. the abelian group of group morphisms from $G$ to $\textbf{C}^{...
1
vote
0answers
61 views

Indecomposable modules for the big quantum group

I am study the representation theory of the big quantum group at a root of unity, and I am wonder if it is known a complete classification of the indecomposable modules for it. To be more specific, ...
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vote
0answers
44 views

Total number of extremal weights in irreps corresponding to fundamental weights equals total number of positive roots?

Let $\mathfrak g$ be a semisimple Lie algebra and $\mathfrak g = \mathfrak n \oplus \mathfrak h \oplus \mathfrak n^-$ be the triangular decomposition. A weight of a finite dimensional irreducible ...
4
votes
1answer
61 views

Multiplicities in Plancherel theorem for SL2(R)

The usual formulation of the Plancherel theorem one writes $f(1)$ as an integral over the dual $\widehat G$. The support of the measure is the set of representations which weakly occur in $L^2(G)$. ...
9
votes
0answers
155 views

cohomology of flag variety

I recently ran into a 30+ years old literature by Andersen and Jantzen on some calculations on cohomology of flag varieties (Cohomology of Induced Representations for Algebraic Groups). Here is the ...
1
vote
0answers
83 views

Some simple question of the base change of the unitary group to general linear group

Let $E/F$ be a quadratic extension of number fields and $\chi$ is a unitary automorphic character of $E^{\times}$. Let $\pi$ be an automorphic representation of $U(n)(F)$ associated to $E/F$, which ...
5
votes
3answers
353 views

Irreducible representations and invariant subspaces

Consider two invertible n-by-n matrices, $n>2$, $X$ and $Y$ over a finite field $k$ (say for simplicity $k=\mathbb Z/ \mathbb Z_2$). Is there any reasonable way to check that there is no proper ...
12
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2answers
684 views

What is a tensor category?

A monoidal category is a well-defined categorical object abstracting products to the categorical setting. The term tensor category is also used, and seems to mean a monoidal category with more ...
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0answers
45 views

Comments/references on an obscure category of “rudimentary representations”

Consider a finite toset (cool word) $A=\{a_1<\ldots<a_N\}$. Let an ordered monomial be a finite formal product $b_1\ldots b_M$ with $b_i\in A$ and $b_i\le b_{i+1}$. Consider the following ...
3
votes
1answer
123 views

Invertible bimodules which are isomorphic in the stable module category

I'm in the following situation. I have a self-injective finite-dimensional basic algebra $\Lambda$ (hence Frobenius) over a perfect field and two finite-dimensional invertible $\Lambda$-bimodules $M$ ...
2
votes
0answers
55 views

local gamma factors and their compatibility with the local langlands correspondence

Let $F$ be a p-adic field and $(\rho,V)$ be an n-dimensional indecomposable representation of the local Weil-Deligne group $W_F'$. Then we know that $\rho\simeq \rho'\otimes Sp(m)$, where $\rho'$ is ...
5
votes
1answer
191 views

Fell's trick for Lie groups

Let $\Gamma$ be a countable discrete group and $\lambda$ the left regular representation of $\Gamma$ on $l^2(\Gamma)$. Let $\rho:\Gamma\rightarrow U(H)$ be a unitary representation of $\Gamma$ on some ...
5
votes
0answers
90 views

Are the integral forms of quantized coordinate algebras always Noetherian?

Let $U_q(\mathfrak{g})$ be the quantized enveloping algebra of a complex semisimple Lie algebra and let $\mathcal{O}_q(G)$ be the quantized coordinate algebra of the corresponding simply-connected ...
19
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1answer
1k views

How do mathematicians and physicists think of SL(2,R) acting on Gaussian functions?

Let $\mathcal{N}(\mu,\sigma^2)$ denote the Gaussian distribution on $\mathbb{R}$: $$ \mathcal{N}(\mu,\sigma^2)(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}.$$ A Gaussian ...
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vote
0answers
69 views

Existence of a certain direct summand

Let $P$ be a cyclic group of order $p^n$. Let $M$ be a $\mathbb{Z}_p[P]$-module and $M_p :=\mathbb{Q}_p\otimes M$ be the associated $\mathbb{Q}_p[P]$-module. When will $M_p$ have a direct summand ...
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0answers
55 views

Is the root cone is contained in the weight cone?

Originally posted on math.stackexchange. Let $G$ be a semisimple algebraic group over a field $k$. Let $A_0$ be a maximal split torus of $G$, and $\mathfrak a_0 = \operatorname{Hom}(X(A_0), \mathbb ...
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0answers
104 views

Errata for Casselman's unpublished notes

In the first chapter of W. Casselman's unpublished notes on representation theory, there is at least one stated result which is not true: A counterexample to this last result is given in the question ...
6
votes
1answer
87 views

Relationship between irreducible representations of the Schur covering group and elements of $H^2(G,U(1))$

Let $G$ be a finite group and let $D(g)$ be a projective representation of $G$ i.e. \begin{equation} D(g) D(h) = e^{i \omega(g,h)} D(gh) \end{equation} These can be classified by the equivalence ...
14
votes
1answer
340 views

Equivariant Mobius inversion

I'll first explain what Mobius inversion says, and then state what I am fairly sure the equivariant version is. I can write out a proof, but I also can't believe this hasn't been done already; this is ...
5
votes
0answers
75 views

Restricting projective representations of Lie groups to lattices

Let $G$ be a simple Lie group with trivial center, and let $\Gamma$ be a lattice in $G$. Is it true that an infinite-dimensional projective representation of $G$ restricted to $\Gamma$ can be de-...
2
votes
0answers
90 views

Definition 2.3 in Lusztig's “Character Sheaves I”

About Definition 2.3 in Character Sheaves I: We have the usual $G$, $B$, $T$, Weyl group $W$ and roots $R$. Lusztig wrote down a definition of $R_{\mathcal{L}}$ as: $$R_{\mathcal{L}}=\{\alpha\in R\;|\;...
4
votes
2answers
300 views

$G\cong C_4\times A_5$ or $C_2\times C_2\times A_5$?

Let $G$ be a finite group of order $240$. If $G\cong C_4\times A_5$ or $C_2\times C_2\times A_5$, then the all degrees of irreducible $\mathbb{C}$-characters of $G$ are $ [1,1,1,1,~3,3,3,3,3,3,3,3, ~...
2
votes
1answer
114 views

$Ext_A^1(J,J)$ for the Jacobson radical $J$ of an algebra $A$

Let algebras be finite dimensional over a field $K$ and let $J$ denote the Jacobson radical (this is the intersection of all maximal right ideals) of an algebra. Being hereditary means that the ...
6
votes
0answers
156 views

Relation between Lie group characters and spherical functions on symmetric spaces

Setup: Let $G/K$ be an irreducible compact Riemannian symmetric space, where $G$ is a simply connected compact real Lie group, and $K$ is the maximal compact connected subgroup of some noncompact real ...
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0answers
80 views

Group schemes and Hyperspecial maximal compact subgroups

Let $F$ be a number field. For each non-archimedean place $v$ let $O_v$ denote the ring of integers. Let $G$ be a connected linear algebraic group defined over $F$. Consider the set of sequences $(K_v)...
7
votes
0answers
71 views

On constructible Hall algebra and instantons

I heard in a talk by Yan Soibelman that by starting with a quiver $Q$ with a set of vertices $I$ we can either symmetrize or anti-symmetrize its Euler-Ringel form $\chi_Q$. He claims that anti-...
5
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0answers
118 views

Restriction of discrete series

QUESTION Let $G$ be a simple Lie group with equal rank; namely, the rank of $G$ equals the rank of its maximal compact subgroup. Suppose that $G'$ is a reductive subgroup of $G$ with equal rank. If $\...
2
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0answers
114 views

Two basic question on parabolic induction

I want to ask some basic two questions on the parabolic induction. Let $F$ be a local fields. Let $\chi_1,\chi_2$ be two characters of $GL_1(F)$ and $GL_1 \times GL_1$ be the Levi part of the ...
3
votes
1answer
167 views

Testing whether a quiver algebra is cellular with a computer

With a friend I made a program in the GAP-package QPA to check whether a given finite dimensional quiver algebra is quasi-hereditary. It is very slow since it has to go through all permutations of ...
3
votes
0answers
93 views

Some basic question on the parabolic induction

I would like to ask some basic question about parabolic induction. Let $F$ be a local field and $G=GL_n(F)$ and $P=MN$ its parabolic subgroup whose Levis subgroup $M=GL_{n_1}(F) \times GL_{n_2}(F)$ ...
7
votes
0answers
232 views

Higher traces in Hochschild cohomology

Let $A$ be an associative algebra over a field $k$. Let $\rho:A \rightarrow \mathrm{End}(M)$ a left module, finite dimensional over $k$. Then the map $a \mapsto \mathrm{tr}_M \rho(a)$ is a well ...
4
votes
2answers
220 views

Weyl's Branching Rule for $SU(N)$-Setting

On the Wikipedia page for restricted representations https://en.wikipedia.org/wiki/Restricted_representation there is presented a number of explicit "branching rules". In particular, there is the ...
4
votes
0answers
124 views

Integral structures via lattices

I am looking at the paper "p-adic Groups" by Bruhat (in the Boulder Proceedings, 1965). I have a question about one of the statements. Let $k$ be the quotient field of a complete discrete valuation ...
2
votes
1answer
130 views

semisimple support of character sheaves

So the essential question is: How should we think about, or if possible compute, the semisimple support of a cuspidal character sheaf? For example, let $G=SL_2$. We have the cuspidal character ...
9
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0answers
167 views

Some question on the induced representation of tensor product

I would like to some question concering the induced representation of tensor product. Let $F$ be a local fields of charateristic 0 and $0<a<m$ are two positive integers. For $1 \le i \le m$, ...
4
votes
0answers
171 views

Conjugacy class representatives for the automorphism group of a finite abelian group

Given a finite abelian group $A$, I'd like a list of conjugacy class representatives for its automorphism group ${\rm Aut}(A)$. In fact, it's not important that I have exactly one representative from ...
4
votes
0answers
67 views

Good range and fair range

Let $G$ be a noncompact simple Lie group with complexified Lie algebra $\mathfrak{g}$. Fix a Cartan involution $\theta$, which defines a maximal compact subgroup $K$ of $G$. Take a $\theta$-stable ...
10
votes
0answers
173 views

Bernstein-Zelevinsky classification: viewing a representation as a subrepresentation or a quotient

$\DeclareMathOperator{\GL}{GL}$ $\DeclareMathOperator{\Ind}{Ind}$I have a question on the details of the Bernstein-Zelevinsky classification. This classification allows us to obtain irreducible ...
4
votes
1answer
124 views

Minimum dimension of faithful representation of mapping class groups?

Let $\Sigma_{g}$ be a closed orientable surface of genus $g$. Let $d_g$ denote the minimum dimension of a faithful representation of the mapping class group of $\Sigma_g$. For $g=1$, the mapping class ...
7
votes
0answers
146 views

A robust version of Schur's lemma?

Does a robust version of Schur's lemma exist? Specifically, I was wondering about something like this: Let $B$ be a bounded operator over a vector space $V$, with underlying field $\mathbb{C}$ and ...
4
votes
0answers
66 views

Modular $S$-matrix for an extended affine Lie algebra

This is a refinement of this old question of mine. In order to find an answer, I've been working my way through q-alg/9511026, which contains all the information I need. In this paper, the authors ...
6
votes
1answer
136 views

Irreducibility of the unramified principal series

Let $G = \operatorname{GL}_n(F)$ with the usual Borel subgroup $P = TU$. Let $\chi = \chi_1 \otimes \cdots \otimes \chi_n$ be an unramified character of $T$. Suppose that $\chi$ is regular, which is ...