All Questions
Tagged with reference-request rt.representation-theory
823 questions
13
votes
1
answer
1k
views
Outer automorphism action on representations of $S_6$
Let $S_6$ be the symmetric group on 6 letters and let $\alpha \colon S_6 \to S_6$ be an outer automorphism (note that $S_6$ is the only permutation group that has an outer automorphism and that $\...
9
votes
1
answer
497
views
Highest weight representations of Kac—Moody algebras: what is inside the weight spaces?
Let $V(\lambda)$ be the unique irreducible representation of a Kac—Moody algebra $\mathfrak{g}$ with the highest weight $\lambda$. If $\mathfrak{g}$ is not of finite type, then even for $\lambda$ one ...
7
votes
0
answers
597
views
Reference for shtuka and trace formula
I really want to learn the work of Laurent Lafforgue and the joint work of Zhiwei Yun and Wei Zhang. They both involve shtuka and trace formula, which I only know the basic idea. So I would like to ...
4
votes
1
answer
615
views
Representation of Heisenberg-Weyl elements and their exponentials
There is possibly a huge literature on the subject but I am a newcomer on analytic representations and my need is rather specific. I simplify it below.
Let $A,B$ be two symbols (standing for ...
6
votes
1
answer
366
views
Group of order $5p^aq^b$
In Lectures by Dan Bump on Modular representation theory,
Theorem 13.14 states that whenever $G$ is a non-abelian simple group of order $|G|=p^aq^br$ for distinct primes $p$,$q$, and $r$, every $r$-...
2
votes
1
answer
340
views
Exterior powers of $Sym^p T$ over Gr(k,n)
Let G=Gr(k,n) the Grassmannian of $k$-dimensional subspaces of $\mathbb{C}^n$ and denote by $T$ the (rank $k$) tautological bundle over $G$, and by $Sym^p T$ its $p$-th symmetric power. Is there any ...
18
votes
5
answers
2k
views
Good source for representation of GL(n) over finite fields?
I'd like to gain some understanding of unitary representations of GL(n) over finite fields. Any good source would be appreciated.
======== edit =========
My original question was ambiguous. ...
2
votes
0
answers
72
views
"Spectral gaps" of commutativity measure
There's a notion of commutativity measure $P(G)$ of a finite group $G$ which is probably folklore: count commuting pairs in $G \times G$ and divide by $|G \times G|$. There are some results:
$P(...
3
votes
0
answers
184
views
Mackey Obstruction Class with Integral Coefficients
Consider an exact sequence of groups
\begin{equation}
1\rightarrow H\rightarrow K\rightarrow G \rightarrow1~.
\end{equation}
Mackey theory enables us to understand representations of $K$ in terms of ...
5
votes
0
answers
213
views
If an irreducible admissible representation is generic, so is its contragredient?
Let $G$ be a $p$-adic reductive group, and $\pi$ be an irreducible admissible representation of $G$ that is generic, do we know that the contragredient representation of $\pi$ is also generic?
If $G$ ...
8
votes
0
answers
134
views
Rational homotopy type of Hilbert scheme components
What is known about rational homotopy type of irreducible component of $Hilb^n(\mathbb C^k)$ containing configuration space? I've searched arXiv for a while and found nothing but surmise that Betti ...
4
votes
0
answers
76
views
Comparing parametrizations of unipotent radical
Let ${G}$ be a simple algebraic group over $\mathbb{C}$ with maximal torus $T$ and set of simple roots $\{\alpha_i\}_{i\in \Delta}$. We then have a Borel supgroup $B=TU$ with unipotent radical $U$. ...
5
votes
1
answer
1k
views
The Casimir invariant of an irreducible representation of a compact Lie group
Let $G$ be a compact Lie group (not necessarily connected) and $\rho:G\to \mathrm{End}(V)$ an irreducible (hence finite-dimensional) unitary representation of $G$. Let $\mathfrak{g}$ be the Lie ...
4
votes
2
answers
288
views
The explicit indecomposable representations of (any) Euclidean quiver of type E
It is known that for any quiver $Q$ that is an orientation of $\tilde{\mathbb{E}}_8$, the hereditary path algebra $KQ$ ($K$ being an algebraically closed field) is tame (but not finite). That is, in ...
6
votes
0
answers
1k
views
Definition of Admissible Representation
Let $G$ be a connected, reductive group over a number field $k$. Let $v$ be a place of $k$.
If $v$ is finite, an admissible representation of $G(k_v)$ is defined to be an abstract representation of $...
2
votes
0
answers
571
views
Clebsch–Gordan(CG) coefficients for SO(N) and Sp(N) group
I know how to calculate the CG coefficients for $SU(N)$, but there are other simple Lie group like $SO(N)$ and $Sp(N)$. But up to now I can't find any textbook tells me how to calculate these and I ...
2
votes
0
answers
169
views
integrable modules reference request
Let $\mathfrak{g} = \displaystyle\varinjlim_{n} \mathfrak{g}_n$ be a locally simple Lie algebra. Let $M$ be an integrable module over $\mathfrak{g}$, that is, for every $n$, as a $\mathfrak{g}_n$ - ...
6
votes
1
answer
242
views
Imbedding of a representation of a compact subgroup
Let $G$ be a compact subgroup of $O(n)$. Let $\rho$ be a continuous finite dimensional representation of $G$.
Question Is it true that there exists a continuous finite dimensional representation $\...
4
votes
2
answers
829
views
Reference Request: Definition of Induced Representation for reductive groups over a local field
Let $G$ be a connected, reductive group over a local field $F$ of characteristic zero, and $H$ a closed subgroup of $G$ which is defined over $F$. Let $\mu_H, \mu_G$ be right Haar measures on $H(F), ...
6
votes
1
answer
148
views
Equivalent orthogonal representations are orthogonaly equivalent
Could anyone point me to a reference for the following fact :
Let $G$ denote the orthogonal or symplectic complex group and $H$ be a complex Lie group, then if $\rho_1\,:H\, \rightarrow G$, $\rho_2\,...
11
votes
0
answers
818
views
How to compute Ext-groups for categories without enough injectives/projectives?
I am studying a category of representations of an infinite-dimensional Lie algebra, which is an abelian category. Unfortunately, the category does not have enough injectives/projectives. I would ...
11
votes
1
answer
620
views
Gauss, Jacobi, Kloosterman sums and representation theory in the $\mathbb F_1$-world
This question is inspired by Why are Bessel function and Kloosterman sum similar? - it developed in me desire to understand Kloosterman sums better.
There seems to be common knowledge that Gauss, ...
5
votes
0
answers
303
views
Recovering an A-infinity structure on an Ext-algebra from a quiver presentation
Let $A=KQ/I$ be a basic finite dimensional algebra given by a quiver with relations. Let $S$ denote the direct sum of the corresponding simple modules.
According to [Keller: A-infinity algebras in ...
3
votes
1
answer
225
views
$f^{\lambda}$: asymptotics and analytic continuations
Let $\mathbb{Y}_n$ denote the set of all partitions of $n\in\mathbb{N}$ and $\mathbb{Y}$ Young's lattice of all partitions. The partition function $g_0(n)=\sum_{\lambda\in\mathbb{Y}_n}1$ has an ...
7
votes
3
answers
1k
views
the character tables of irreducible representations of $SL(3,Z_q)$
The following paper gives a classification of the character tables of irreducible representations of $SL(3,GF(q))$ where $q$ is a power of a prime number, and $ GF(q)$ a finite field of $q$ elements.
...
3
votes
0
answers
116
views
Extension of representations of certain compact Lie groups
Let $G$ be a real, connected, non-compact, semisimple Lie group with finite center and real rank $1$. Let $G=KAN$ be an Iwasawa decomposition, then $K\subset G$ is a maximal compact subgroup, and $\...
2
votes
1
answer
431
views
Lagrange interpolation vs homogeneous symmetric polynomials?
This question is a follow-up on another MO query here.
Question. For $r\geq$ an integer, is it true that there exists homogeneous symmetric polynomial $P_r(x_1,\dots,x_n)$ with positive ...
3
votes
1
answer
372
views
Reference for real and complex projective representation of finite group
I'm not a mathematician. I've only learnt about irreducible representation of finite group, symmetric group and simple Lie group. In fact, I don't know projective representation belong to which part ...
4
votes
1
answer
628
views
Closed orbits for the action of general linear groups
Let $G=GL_n(K)$ where $K$ is an algebraically closed field of characteristic zero. Let $V$ be a finite dimensional rational representation of $V$.
Assume that $v\in V$ has a reductive stabilizer $H\...
5
votes
1
answer
250
views
About generalized Springer theory for spin groups
I am interested in the detailed computation of the generalized Springer theory for spin groups (type B or D). In the last sentence in Section 14 of Lusztig's Intersection cohomology complex on a ...
1
vote
0
answers
75
views
Representation-finite trivial extension algebras
Given a quiver algebra A such that its trivial extension T(A) is representation-finite. Is T(A) automatically stable equivalent to a trivial extension algebra of a hereditary representation-finite ...
9
votes
0
answers
409
views
The proof of Kazhdan's density theorem (And does it hold over positive characteristic?)
When proving identities about traces of functions on representations of $p$-adic groups, Kazhdan's density theorem indicates one only has to check equalities of traces on tempered representations. ...
5
votes
1
answer
261
views
A "prequestion" about meromorphic representations of algebraic groups
In a comment exchange around an answer to Is a group scheme determined by its category of representations? there arose the issue of Tannakian reconstruction for non-affine algebraic groups (e. g. ...
2
votes
1
answer
75
views
Contravariant finiteness of subcategories
Let $A$ be a finite dimensional algebra . Let $P_{\inf}$ be the full subcategory of modules having finite projective dimension and $P_r$ the subcategory of modules having projective dimension bounded ...
3
votes
0
answers
205
views
Finitistic dimension via tilting modules
is the following true (all algebras and modules are assumed to be finite dimensional):
The finitistic dimension of an algebra is equal to the supremum of projective dimensions of tilting modules?
It ...
1
vote
0
answers
136
views
Representations of finite groups over commutative rings-question and reference request
In a textbook of representation theory I have encountered the following statement without proof:
Let $R$ be a commutative ring and $G$ a finite group. If $M$ is a simple $RG$-module then the ...
6
votes
0
answers
184
views
Reference request: fusion rules for unitary dual of SL(3,R)?
By the fusion rules, I mean: given two unitary irreps of the group, what unitary irreps occur in their tensor product and with what "multiplicity"? (I am guessing that direct integrals ...
6
votes
0
answers
233
views
Tracking down a copy of "Mixed categories, Ext-duality and representations (results and conjectures)"
I am trying to find a copy, ideally digital, of the following preprint:
Title: Mixed categories, Ext-duality and representations
(results and conjectures)
Authors: A. Beilinson and V. Ginzburg
Year:...
0
votes
1
answer
337
views
Homological dimensions of tensor products of algebras
Given two finite dimensional algebras $A$ and $B$ over a field. The Gorenstein dimension of an algebra A is defined as the injective dimension of the module A. The finitistic dimension of an algebra A ...
1
vote
0
answers
98
views
Ireducible representations in characteristic p [duplicate]
It is known that the intersection of kernels of all ireducible representations of a finite group in characteristic zero is the trivial group.
In characteristic $p>0$ I have understood that this ...
7
votes
1
answer
168
views
Region of convergence of Eisenstein series is a union of Weyl chambers when groups have discrete series?
Let $G$ be a reductive algebraic group, and let $M$ be a Levi subgroup of $G$. In Urban's paper Eigenvarieties for Reductive Groups, it seems to be assumed that if $G(\mathbb{R})$ and $M(\mathbb{R})$ ...
8
votes
1
answer
476
views
How does Jacquet's "Generic Representations" classify tempered representations?
Let $L$ be a $p$-adic field $G = GL_n(L)$. Let $P$ be a standard parabolic subgroup with Levi decomposition $P = MU$, where $M \cong G_1\times \ldots \times G_r$, for $G_i \cong GL_{n_i}(L)$.
The ...
15
votes
2
answers
762
views
Gelfand-Tsetlin algebras and "Jucys-Murphy elements" for $\mathfrak{gl}_n$
I'm trying to figure out/find in literature the details concerning Gelfand-Tsetlin algebras for $\mathfrak{gl}_n(\mathbb C)$ (Okounkov-Vershik style, if you wish).
Consider the chain $$\mathcal U(\...
25
votes
3
answers
1k
views
what else is in $\prod_{j=1}^n(1+q^j)$?
From time to time, I run into the finite product $\prod_{j=1}^n(1+q^j)$. And, the more it happens, the more fascinated I've become. So, herein, I wish to get help in collecting such results. To give ...
6
votes
0
answers
268
views
duality between quiver variety and affine Grassmannian
Let $\frak{g}$ be a ADE type simple lie algebra. There are (at least) two geometric ways to get highest weight irreducible representations of $\frak{g}$. One is by considering constructible functions ...
7
votes
1
answer
222
views
Some intuition on the $SL_n$-module $V_{[1,1,...,1]}$
(This question highly overlaps with this and also this.)
The irreducible ${\sf SL}_{n-1}$-module $V_{[1,1,\ldots,1]}$ is the one providing the minimal projective embedding $\mathbb{P}(V_{[1,1,\ldots,...
5
votes
1
answer
242
views
Jack polynomials and the Witt algebra
The symmetric Jack polynomials $J_n^{\alpha}(x_1,x_2,..,x_{n+1})$, a special subset of the symmetric Jack functions presented in Stanley's paper in equation a) on page 80, can be represented by the ...
11
votes
1
answer
626
views
Formula for $U(N)$ integration wanted
Before you jump on the "duplicate" buttom, let me say that I do not want to hear about Weingarten calculus and I do not want to see a character of the symmetric group.
What I would like is a formula ...
4
votes
1
answer
345
views
Irreducible representations of the reductive quotient
For a linear algebraic group over an algebraically closed field of characteristic zero $G$, with unipotent radical $U$, we have that $G/U$ is reductive.
When $G$ is solvable, then Lie's theorem says ...
6
votes
2
answers
2k
views
Alternative or reprint of Carter's "Finite Groups of Lie Type: Conjugacy Classes and Complex Characters"
I would like to learn about character theory of finite groups of Lie type and some Deligne-Lusztig theory. The classic textbook on the subject seems to be Roger W. Carter's Finite Groups of Lie Type: ...