All Questions
Tagged with reference-request rt.representation-theory
823 questions
2
votes
2
answers
267
views
Irreducible representations of $G_4 = \langle a,b \mid a^{16}, b^{2}, baba^{-7}\rangle$ and other Semidihedral groups
I would like to know the irreducible representations of the group $G_4 = \langle a,b \mid a^{16}, b^2, baba^{-7}\rangle$ and its character table.
More than that, I would like to know the irreducible ...
5
votes
1
answer
321
views
The ¨irreducible¨ representation variety of surface group
Let $S$ be a closed surface of genus larger than 1, $G$ be a compact, simply connected simple Lie group with finite center.
Consider the representation variety $M(S,G)=Rep(\pi_1(S), G)$. Witten´s ...
- 325
1
vote
1
answer
204
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 ...
- 1,013
1
vote
0
answers
148
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}^{...
- 1,479
4
votes
1
answer
183
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 $L^2(G)$. ...
user1688
17
votes
2
answers
742
views
Equivariant Möbius inversion
I'll first explain what Möbius 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 ...
- 156k
5
votes
3
answers
849
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 ...
- 835
4
votes
0
answers
218
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 ...
- 286
6
votes
1
answer
334
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 ...
- 6,180
1
vote
0
answers
53
views
A normalized embedding $\mathbb C \rightarrow \mathfrak a_M^{\ast}$ via $\tilde{\alpha}$
Let $G$ be a connected, reductive group over a field $k$. Let $S$ be a maximal $k$-split torus of $G$ with Weyl group $W$, $\Delta$ a set of simple roots of $S$ in $G$, and $P = MN$ a maximal ...
- 6,180
8
votes
1
answer
203
views
Reference request: Coxeter length and irreducible characters
Let $S_n$ be the symmetric group on $\{1,2,\ldots, n\}$ and $\ell$ the Coxeter length on $S_n$. There is a well-known formula to compute this length, namely for a $\pi \in S_n$ we have
$$\ell(\pi) = |\...
- 809
5
votes
1
answer
192
views
Given a representation-infinite algebra, when is every AR component infinite?
Let $A$ be a finite dimensional algebra over an algebraically closed field $K$. The Auslander-Reiten quiver $\Gamma_A$ of $A$ is a means of presenting the category of finitely generated right $A$-...
- 482
6
votes
0
answers
236
views
When is an irreducible unramified principal series representation $K$-spherical?
Let $G = \operatorname{GL}_n(\mathbb Q_p)$, $T$ the usual maximal torus of $G$, and $K = \operatorname{GL}_n(\mathbb Z_p)$.
Let $\chi$ be an unramified character of $T$, with $\chi(t_1, ... , t_n) =...
- 6,180
5
votes
0
answers
99
views
Does there always exist an irreducible representation occurring with multiplicity one when inducing from $M=Z_K(A)$ to $K$?
This question is a more specific version of Does there always exist an irreducible representation occurring with multiplicity one when inducing from a closed subgroup to a compact Lie group? .
Since ...
- 1,942
5
votes
1
answer
305
views
Schur covers of affine 2-transitive groups
I am interested in Schur covers of minimal 2-transitive groups. A theorem of Burnside gives that every finite 2-transitive group is either almost simple or affine. In the time since, these groups have ...
- 7,633
18
votes
2
answers
1k
views
What is the archimedean Hecke algebra?
Let $\mathbf G$ be a connected, reductive group over $\mathbb Q$. For each nonarchimedean place $v$, let $K_v$ be a maximal compact subgroup of $\mathbf G(\mathbb Q_v)$. The space $\mathscr H(\mathbf ...
- 6,180
5
votes
0
answers
92
views
Question concerning the representation dimension of a special algebra
I would like to know, if the following problem is still open:
Let $k$ denote an algebraically closed field of characteristic 3.
Determine the representation dimension of $k(C_3\times C_3)$, where $...
- 1,755
3
votes
2
answers
214
views
History of an open problem on partial tilting modules
The following is an open problem:
Given a partial tilting module $T$ over a finite dimensional algebra $A$ (that is $Ext_A^i(T,T)=0$ for all $i \geq 1$ and $pd(T) < \infty$), then $T$ is a tilting ...
- 26.5k
6
votes
2
answers
429
views
Natural bijection between Dyck paths and tilting modules
Let $A_n=kQ_n$ be the path algebra of linear oriented Dynkin graph $Q_n$ of Dynkin type $\mathcal{A_n}$ (so $A_n$ is the unique hereditary Nakayama algebra given by quiver and relations).
The number ...
- 26.5k
3
votes
0
answers
97
views
Reference Request: Branching Rules of $\mathfrak{s}\mathfrak{l}_n$ in $\mathfrak{s}\mathfrak{l}_{n+1}$
I have heard that the branching rules are well-known for the simple Lie algebra $\mathfrak{s}\mathfrak{l}_n$ in $\mathfrak{s}\mathfrak{l}_{n+1}$ over fields of characteristic zero. Where can I find a ...
user117370
6
votes
0
answers
239
views
Direct sum decomposition of the space of cuspidal automorphic forms
$\newcommand{\G}{\mathbb{G}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\A}{\mathbb{A}} \newcommand{\Autom}{\mathcal{A}} \newcommand{\cen}{\mathcal{Z}} \newcommand{\lieg}{\...
- 5,382
8
votes
0
answers
129
views
Is there a splitting rule for the restriction of a $GL(23, \mathbb{Q})$-representation to $O(23, \mathbb{Q})$?
I am interested in a $23$-dimensional $\mathbb{Q}$-vector space $V$ which I am viewing as a GL$_{23}(\mathbb{Q})$ representation. Schur functors can be defined over $\mathbb{Q}$, so we get ...
- 233
5
votes
2
answers
439
views
Difference of adjacent dominant weights is a root?
The basic set-up here makes sense in the theory of abstract root systems if one brings (integral) weights into the picture, but it may be more natural to think about the classical characteristic 0 ...
- 52.9k
5
votes
1
answer
152
views
Group rings over central products
I have a proof of the following result but I was wondering if anyone had a reference for it. I have asked on math.stackexchange here but didn't receive any replies.
Let $G$ a finite group given by ...
- 79
2
votes
1
answer
294
views
Decomposition into Weyl modules
Let $G$ be a split reductive group over an arbitrary field $k$. By definition, see Jantzen (*), an ascending chain $$0 = V_0 \subset V_1 \subset V_2 \subset \dots$$ of submodules of a $G$-module $V$ ...
13
votes
1
answer
358
views
Cartography of the duals of GL, PGL, SL, etc
A short version of this question could be
What are the duals of $PGL(2,\mathbf{Q}_p)$, $PGL(2,\mathbf{R})$ and $PGL(2,\mathbf{C})$?
I should obviously add some precisions.
there are different ...
- 5,424
5
votes
0
answers
383
views
Kac's theorem for quiver representations over an arbitrary ground field
Let $Q$ be a quiver without loops (cycles of length 1). Kac proved that if $K$ is algebraically closed, the dimension vectors of indecomposable representations of $Q$ over $K$ are exactly the ...
- 6,292
8
votes
1
answer
849
views
Representations of groups with the same derived group, how much control do we have over the central character?
Let $G_1 \subset G$ be the rational points of $p$-adic reductive groups sharing the same derived group. There are some well known results relating representations of $G_1$ to representations of $G$, ...
- 6,180
8
votes
1
answer
320
views
Connections between representations of $\operatorname{SL}_n$ and $\operatorname{GL}_n$
Let $G = \operatorname{GL}_n(F)$ for a $p$-adic field $F$, and let $G_D = \operatorname{SL}_n(F)$. I am wondering if there is a connection between irreducible, admissible representations of $G$ and ...
- 6,180
2
votes
0
answers
81
views
Fourier transform in the complex motion group
I am looking for a reference that deals with the unitary dual of the complex motion group $\mathbb C^2 \rtimes SU(2)$ i.e., the semi-direct product of $\mathbb C^2$ with the special unitary group $K=...
- 650
4
votes
0
answers
288
views
Meaning of a highly ramified character for reductive groups
Let $F$ be a $p$-adic local field, and $G$ a connected reductive group over $F$. What is the meaning of a "highly ramified character" of $G(F)$? I have seen this terminology in many places in ...
- 6,180
5
votes
1
answer
433
views
Generic supercuspidal representations of $\operatorname{GL}_n$ can be defined by integrals over $U$
Let $(V,\pi)$ be an irreducible, admissible, supercuspidal representation of $G = \operatorname{GL}_n(F)$ for $F$ a $p$-adic field. Let $B = TU$ be the usual Borel subgroup, maximal torus, and ...
- 6,180
3
votes
1
answer
203
views
Symplectic representation of modular group
The modular group $\Gamma_{g}$ of isotopy classes of diffeomorphisms of a genus $g$ surface $S$ acts on $H^1(S,\mathbb{Q})$ (or $H^1(S,\mathbb{Z})$) respecting the intersection pairing. This gives a ...
- 33
7
votes
0
answers
202
views
What is the kernel of the action of the Iwahori-Hecke algebra?
The Iwahori-Hecke algebra $H_n(q)$ acts on the $n$th tensor power of the standard representation of $U_q(\mathfrak{sl}_m)$. What is the kernel of this action? Does anyone know a reference?
I'm happy ...
- 425
3
votes
1
answer
145
views
Does the Leclerc-Thibon involution exchange vertex operators of the first and second type?
This question is about $U_q ( \hat{\mathfrak{sl}}_2 )$ representation theory. There is a notion of vertex operators $\Phi_{\pm }(z)$ of first and $\Psi_{\pm}(z)$ of the second type. They are defined ...
- 385
8
votes
2
answers
617
views
Reference request: Models of cuspidal representations of GL(n,k) where k is a finite field
Let $k=\mathbb{F}_q$ where $q$ is a prime power of odd cardinality.
Where could I find explicit models of all irreducible cuspidal (complex) representations of $GL_n(k)$ for $n\ge 3$?
I understand ...
- 960
9
votes
0
answers
470
views
Branching rules for compact Lie groups
Let $G$ be a compact connected Lie group, and let $H\subset G$ be a closed subgroup. For an irreducible representation $\pi:G\to\mathrm{End}_\mathbb{C}(V)$ of $G$ ($\dim\pi<\infty$) I want to know ...
- 1,959
1
vote
1
answer
174
views
Reference for a result of Auslander about the global dimension
One of Auslanders famous theorems is that he proved that the global dimension of a semiprimary ring is equal to the maximum of the projective dimensions of the simple modules of the ring. This result ...
- 26.5k
5
votes
1
answer
198
views
Murnaghan-Nakayama rule when all cycles have same size
Let $\lambda \vdash nk$. Let $n^k$ denote the partiton with $k$ parts of size $n$. We can compute $\chi^\lambda(n^k)$ by using the Murnaghan-Nakayama rule, as a signed sum over border-strip tableaux, (...
- 15.8k
2
votes
1
answer
208
views
Dimension of restricted root spaces of split Lie algebras
Let $\mathfrak g$ be a real simple split Lie algebra. Let $\mathfrak g = \mathfrak k \oplus \mathfrak p$ be the Cartan decomposition. Let $\mathfrak a\subseteq \mathfrak p$ be a maximal abelian ...
- 1,782
4
votes
2
answers
418
views
Indecomposable representations of euclidean quivers
The classification of indecomposable representations of a Euclidean quiver is well-known over an algebraically closed field. I am interested in an analogous classification, but over an arbitrary field....
- 61
17
votes
1
answer
1k
views
References for Langlands classification
I kindly ask about some references concerning the representation theory of the Langlands dual of a compact Lie group, and how it relates to things related to the original compact Lie group.
My ...
- 5,215
6
votes
1
answer
428
views
Reference on Highest Weight Module of Kac-Moody Algebra
I am trying to understand this paper. The construction requires the understanding of the following concepts in the representation theory of simple and affine Lie algebras:
The construction of Verma ...
- 989
9
votes
2
answers
2k
views
alternating and symmetric powers of the standard representation of the symmetric group
Let $n \geq 7$ and $V = \mathbb{C}^n$ be the standard representation for $S_{n+1}$, the symmetric group of cardinal $(n+1)!$
Let $k$ be an integer such that $2 \leq k \leq n$. Is it true or false ...
- 7,300
2
votes
0
answers
61
views
Question on outer Ext-products
For group algebras $A=KG$ over a field $K$ with finite group $G$ there exists an outer product on Ext:
$Ext_A^i(M,N) \otimes_K Ext_A^j(M',N') \rightarrow Ext_A^{i+j}(M \otimes_K M',N \otimes_K N')$.
...
- 26.5k
0
votes
0
answers
195
views
Automorphisms of weighted quiver
I am reading this paper strongly primitiv species with potentials I : mutations.
In page 6, they give the definition of weighted quiver: a weighted quiver is a pair $(Q,d)$, where $Q$ is a loop-...
- 775
6
votes
1
answer
938
views
Does the orthogonal group act irreducibly on totally symmetric tensors?
Consider the action of the orthogonal group $\operatorname{O}(d)$ on $k$-way tensors $(\mathbb{R}^d)^{\otimes k}$ defined by
$$Q(x_1\otimes\cdots\otimes x_k)=Qx_1\otimes\cdots \otimes Qx_k$$
and ...
- 7,633
1
vote
0
answers
63
views
Reference request for formula on global dimension
Given a finite dimensional algebra $A$ over an algebraically closed field $K$.
Let $A^e=A^{op} \otimes_K A$ be the enveloping algebra of $A$.
Who noted first that the global dimension of $A$ is equal ...
- 26.5k
15
votes
1
answer
961
views
Who conjectured the Cartan determinant conjecture
The Cartan determinant conjecture states that every finite dimensional algebra of finite global dimension has the property that the determinant of its Cartan matrix is equal to one. Who stated this ...
- 26.5k
8
votes
1
answer
402
views
Separating closed $SO(p,q)$ orbits by invariant polynomials
Consider the real Lie group $SO(p,q)$ (I believe that it happens to be a linearly reductive algebraic group over $\mathbb{R}$, if that's relevant). Also, if relevant, I'm mostly interested in the (...
- 21.5k