Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
3
votes
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 ...
1
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 ...
1
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, ...
1
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
votes
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 ...
0
votes
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
votes
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}}.$$
A Gaussian ...
1
vote
0answers
69 views
Existence of a certain direct summand
Let $P$ be a cyclic group of order $p^n$. Let $M$ be a $\mathbb{Z}_p[P]$-module and $M_p :=\mathbb{Q}_p\otimes M$ be the associated $\mathbb{Q}_p[P]$-module. When will $M_p$ have a direct summand ...
1
vote
0answers
55 views
Is the root cone is contained in the weight cone?
Originally posted on math.stackexchange. Let $G$ be a semisimple algebraic group over a field $k$. Let $A_0$ be a maximal split torus of $G$, and $\mathfrak a_0 = \operatorname{Hom}(X(A_0), \mathbb ...
1
vote
0answers
104 views
Errata for Casselman's unpublished notes
In the first chapter of W. Casselman's unpublished notes on representation theory, there is at least one stated result which is not true:
A counterexample to this last result is given in the question ...
6
votes
1answer
87 views
Relationship between irreducible representations of the Schur covering group and elements of $H^2(G,U(1))$
Let $G$ be a finite group and let $D(g)$ be a projective representation of $G$ i.e.
\begin{equation}
D(g) D(h) = e^{i \omega(g,h)} D(gh)
\end{equation}
These can be classified by the equivalence ...
14
votes
1answer
340 views
Equivariant Mobius inversion
I'll first explain what Mobius inversion says, and then state what I am fairly sure the equivariant version is. I can write out a proof, but I also can't believe this hasn't been done already; this is ...
5
votes
0answers
75 views
Restricting projective representations of Lie groups to lattices
Let $G$ be a simple Lie group with trivial center, and let $\Gamma$ be a lattice in $G$. Is it true that an infinite-dimensional projective representation of $G$ restricted to $\Gamma$ can be de-...
2
votes
0answers
90 views
Definition 2.3 in Lusztig's “Character Sheaves I”
About Definition 2.3 in Character Sheaves I: We have the usual $G$, $B$, $T$, Weyl group $W$ and roots $R$. Lusztig wrote down a definition of $R_{\mathcal{L}}$ as:
$$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 ...
1
vote
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
votes
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
votes
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
votes
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 ...
