All Questions
Tagged with reference-request rt.representation-theory
823 questions
8
votes
1
answer
1k
views
Irreducible decomposition of tensor product of irreducible $S_n$ representations
Are there well known results on the irreducibles in the decomposition of tensor products of irreducible $S_n$ representations? I would also like to know of some references where I can find formulas (...
- 596
11
votes
3
answers
3k
views
Reference request for projective representations of finite groups over a non-problematic field
I would like to get a reference where I can learn about the theory of projective representations of finite groups over the complex numbers (or over any field K such that the order of the given group ...
- 2,406
18
votes
6
answers
2k
views
Explicit formula for the trace of an unramified principal series representation of $GL(n,K)$, $K$ $p$-adic.
Let $K$ be a non-arch local field (I'm only interested in the char 0 case), let $\mathbb{G}$ be a connected reductive group over $K$ and let $G=\mathbb{G}(K)$. If $V$ is a smooth irreducible complex ...
- 41.4k
35
votes
6
answers
5k
views
Character-free proof that Frobenius kernel is a normal subgroup?
The question is in the title, but here is some background/reminders:
A subgroup $H\neq\{1\}$ of a finite group $G$ is called a Frobenius complement if $H\cap H^g = \{1\}$ for all $g\in G\backslash H$....
- 13k
2
votes
1
answer
938
views
S. Agnihotri, "Quantum cohomology and the Verlinde algebra"
I am looking for the Oxford PhD thesis of S. Agnihotri, "Quantum cohomology and the Verlinde algebra". I can't seem to find it online. Does anyone know how / where I can find this? Thank you!
- 21k
9
votes
3
answers
3k
views
About representation theory of Heisenberg group
Actually I am an undergraduate student, but I want to study Heisenberg groups over arbitrary field.
Firstly, why is this group important? I know that the Heisenberg group is important in the field ...
12
votes
5
answers
2k
views
How does the group algebra look as a Lie algebra
It's probably a well known question, so it is just a reference question.
Let $G$ be a finite group and let $C[G]$ be a group algebra. Then we can define a bracket on $C[G]$ by $[f,h]=f*h-h*f$. What ...
- 2,179
5
votes
1
answer
264
views
Group not leaving subset invariant
Let $Y,X$ be two sets of size n,m. Let $Y\subset X$.
What is the maximal group(in size) $G< Sym(X)$ such that gY=Y imply that $g=1$?
Here I mean that the only permutation which permutes elements of ...
- 2,179
8
votes
2
answers
572
views
reference containing the list of irreducible finite dimensional representation of real general linear group
It seems that it is not easy to find a reference containing a classification and construction of finite dimensional irreducible representations of $GL_n(\mathbb{R})$. One way to look at it is via $(\...
- 2,709
4
votes
1
answer
348
views
Irreducible representation flipping two elements
Does there are exist simple proof for the following statement?
Let $\rho,V$ be an irreducible representation of group $G$ of dimention $n$.
Assume that there are exist $g \in G$ such that $\rho(g)$ ...
- 2,179
3
votes
1
answer
312
views
Representations and support.
I am interested in the question: Does there are exist concept of support in representation theory?
When I say support I mean number of non-zero values of $f \in C[G]$.
Do you know theorems which ...
- 2,179
10
votes
3
answers
1k
views
p-adic representations of a quaternion algebra over a local field
How to determine a complete set of isomorphism class representatives of the irreducible algebraic representations of $D^{\times}/F$ (where $D$ is a quaternion algebra over a local field $F/\mathbb{Q} ...
10
votes
1
answer
1k
views
Representations of central extensions
Let $G$ be central extension of an abelian group $A$ by some group $H$.
Is it possible to characterize all irreducible representions of $G$
in terms of irreducible representations of $A$ and $H$?
- 2,179
10
votes
2
answers
955
views
Semisimplicity of étale cohomology representations
Let $K$ be a number field and $G=Gal(\overline{K}/K)$ the absolute Galois group of $K$. Let $\ell$ be a prime number.
Let $A/K$ be an abelian variety. Then the representation of $G$ on $V_\ell(A)$ is ...
- 1,673
5
votes
2
answers
507
views
Symmetric matrices as a module over the skewsymmetric ones
I'm trying to understand the Cartan decomposition of a semisimple Lie algebra, $\mathfrak g=\mathfrak k \oplus \mathfrak p$, where $[\mathfrak k,\mathfrak p] \subseteq \mathfrak p$, cf. the wikipedia ...
- 4,280
7
votes
1
answer
818
views
Uncertainty principle for non-commutative groups
Is it true that for every group $G$ and $f\in \mathbb C[G]$ it holds that $$\dim(\mathbb C[G]*f)\mathop{supp}(f)\geq |G| ?$$
Here, $\mathbb C[G]$ is the group algebra, and by $\mathbb C[G]*f$ I ...
- 2,179
12
votes
5
answers
4k
views
Good books in Modular Representation Theory
Hi every one!
I am reading some paper and it uses Modular Representation Theory. I even dont really know about Representation Theory and I am looking for a good book for beginner. Could you please ...
- 233
18
votes
2
answers
2k
views
Virasoro action on the elliptic cohomology
I'm trying to understand better the mathematical notion of elliptic cohomology. Note that I only know the physics definition of the elliptic genus given in Witten's paper.
Let $X$ be a Calabi-Yau ...
- 6,123
8
votes
3
answers
716
views
Generators and relations for irreps of the Brauer algebra
The field of definition will be the complex numbers, $V$ is a vector space of dimension $m$, and $O(V)$ is the orthogonal group preserving some nondegenerate bilinear form on $V$. The centralizer ...
- 10.7k
4
votes
0
answers
420
views
Etale cohomology analogue for the semistable reduction theorem
Let $K$ be a field, $X/K$ a smooth projective variety, $l\neq char(K)$ a prime number and $q\ge 0$. Then we define $\overline{X}:=X_{K_{sep}}$ and denote by
$\rho_{X, l}^{(q)}$ the representation of $...
- 1,673
2
votes
1
answer
524
views
Character formulas for non-integrable modules?
Let $\mathfrak{g}$ be a Kac-Moody Lie algebra (actually, I'd already be happy with an answer addressing the case where $\mathfrak{g}$ is a simple Lie algebra over $\mathbb{C}$).
1st ?: I'm wondering ...
- 1,335
15
votes
2
answers
1k
views
Branching rule from symmetric group $S_{2n}$ to hyperoctahedral group $H_n$
Embed the hyperoctahedral group $H_n$ into the symmetric group $S_{2n}$ as the centralizer of the involution $(1, 2) (3, 4) \cdots (2n-1, 2n)$ (cycle notation). Label representations of $S_{2n}$ by ...
- 10.7k
3
votes
0
answers
264
views
Is it possible to construct category $\mathcal{O}^{\mathfrak{p}}$ with non-standard parabolic subalgebra
The usual definition of the parabolic category $\mathcal{O}^{\mathfrak{p}}$ is the following. We consider Lie algebra $\mathfrak{g}$ of the rank $r$ with the root system $\Delta$ and the set of ...
- 214
13
votes
3
answers
1k
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Non-vanishing cohomology of line bundles on projective varieties in prime characteristic?
This is a somewhat naive question about the expected non-vanishing behavior of sheaf cohomology groups $H^i(X, \mathcal{L})$, where $X$ is a smooth projective variety of dimension $d$ over an ...
- 52.9k
4
votes
2
answers
824
views
decomposition into irreducible unitary representations: references for explicit formulas?
I'm looking for references of the decomposition of $L^2(\Gamma\backslash G)$, where $G$ is a connected Lie group, and $\Gamma\subset G$ a discrete lattice; for simplicity one may assume that $G$ is ...
- 313
4
votes
1
answer
358
views
Is there an account of the algebra of highest weight tensors?
The context for this question is Schur-Weyl duality. Let $V$ be a vector space. For $r>0$ consider $\otimes^rV$. This has commuting actions of the symmetric group $S(r)$ and $G=GL(V)$.
My question ...
- 9,132
8
votes
1
answer
1k
views
Character table for the affine group of Z/p^nZ
Initial caveat: the following question could probably be answered by Google, MathSciNet or my library, if I could find the right search terms or book... but I've not had any luck today, so I hope ...
- 25.8k
10
votes
0
answers
881
views
Invariance of Euler characteristic under base change for sheaf cohomology of flag varieties
BACKGROUND:
Over an algebraically closed field of arbitrary characteristic, most of the basic structure theory of affine (= linear) algebraic groups can be developed concretely without quoting ...
- 52.9k
20
votes
3
answers
4k
views
Harmonic analysis on semisimple groups - modern treatment
For my finals, I am digging through the book by Varadarajan An introduction to harmonic analysis on semisimple Lie groups. I find it a rather hard read and I feel it's a bit outdated now. Any ...
2
votes
0
answers
223
views
Explicit formulas for the action of the Hall algebra of the cyclic quiver on q-Fock space?
In their paper on the decomposition numbers of Schur algebra, Vasserot and Varagnolo introduce an action of the (twisted) Hall algebra of a cyclic quiver $\Gamma$ on q-Fock space.
Without q-shifts, ...
- 44.7k
25
votes
1
answer
4k
views
What is a special parahoric subgroup?
Let me take this question again from the top.
I would like to know what a special parahoric subgroup is.
I think this is a "real" question, though not an especially good one -- it indicates my ...
20
votes
7
answers
9k
views
Elementary reference for algebraic groups
I'm looking for a reference on algebraic groups which requires only knowledge of basic material on the theory of varieties which you could find in, for example, Basic Algebraic Geometry 1 by ...
- 15.4k
5
votes
0
answers
219
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Character tables of the p-core of the binary modular congruence group of p-power level
Let $p \geq 5$ be a prime and let $n$ be positive integer. In his Ph.D thesis (See The characters of binary modular congruence group, Bulletin of the
American Mathematical Society. 79 (1973), no. 4.), ...
- 2,396
4
votes
1
answer
254
views
Embedding into Permutation Representation
Let $\rho$ be irreducible representation of group $G$.
How one can characterize all subgroups $H< G$ such that $\rho$ can be embedded into permutation representation $F^X$, where $X=G/H$.
- 2,179
10
votes
1
answer
842
views
Is there a "correct" general setting for the principle: "tensoring any object with a projective object yields another projective"?
Apparently this principle was first formulated for left modules over the group algebra $A=kG$ of a finite group, where $k$ is a field of characteristic $p>0$ dividing $|G|$. (See Exercise 2 on p. ...
- 52.9k
8
votes
1
answer
593
views
Representability of polymatroids over $GF(2)$
A polymatroid is a finite set $X$ and a rank function $d : P(X) \to {\mathbb N}$ such that
1) $d(\varnothing)=0$,
2) $A \subset B$ implies $d(A) \leq d(B)$, and
3) $d(A \cap B) + d(A \cup B) \leq d(...
- 25.5k
5
votes
2
answers
3k
views
irreducible representations of O(2) - reference?
I am looking for a list of the irreducible representations of O(2). Could someone please provide a reference?
EDIT: I am particularly interested in the representations on IR^2 (irreducible or not)
- 2,935
51
votes
7
answers
11k
views
How is representation theory used in modular/automorphic forms?
There is certainly an abundance of advanced books on Galois representations and automorphic forms. What I'm wondering is more simple: What is the basic connection between modular forms and ...
- 15.4k
10
votes
1
answer
334
views
What is known about higher-categorical reconstruction theorems? (reference request)
The answer to my question is almost certainly "not much" — at least, I've asked a few people, and that was their answer. But I'd like to refine this answer, and MathOverflow seems like the best ...
- 54.6k
18
votes
4
answers
5k
views
The only great book that Bourbaki ever wrote?
OK, the title is opinionated and contentious, but I have a definite
question. I know that the title refers to the Bourbaki volume
Groupes et Algèbres de Lie (Chapters 4-6), published in 1968, but
...
- 12.4k
12
votes
1
answer
1k
views
Reference Request for Drinfeld and Laumon Compactifications
Background
Let $X$ denote a smooth projective curve over $\mathbb{C}$ and let $G$ denote a semi-simple simply connected algebraic group over $\mathbb{C},$ which has associated flag variety $G/B.$
...
- 2,706
8
votes
1
answer
920
views
Looking for references talking about category of topological vector spaces
It's known that category of topological vector spaces is not abelian but quasi-abelian or exact category. I am looking for the references playing with this category(category theory). All the related ...
- 5,515
10
votes
2
answers
897
views
Applications of classifying thick subcategories
So, relatively recently, Balmer introduced this notion of a spectrum for a tensor triangulated category and used it to prove a generalization of a classification theorem done in several areas of ...
- 13.5k
12
votes
3
answers
3k
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Why do Physicists need unitary representation of Kac-Moody algebra?
My advisor mentioned to me that he talked to Witten last summer on representation theory, and Witten told him that unitary representations of Kac-Moody algebra are important to working physicists. But ...
- 5,515
18
votes
1
answer
2k
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Does the Tannaka-Krein theorem come from an equivalence of 2-categories?
Possibly the correct answer to this question is simply a pointer towards some recent literature on Tannaka-Krein-type theorems. The best article I know on the subject is the excellent
André Joyal ...
- 54.6k
6
votes
1
answer
2k
views
How to calculate partition function of a QFT by summing over irreducible representations of the symmetry group?
By definition computing the partition function of a QFT amounts to doing a Feynman Path Integral exactly. At a schematic level I can see why this can become a question of summing/integrating over ...
- 3,541
2
votes
1
answer
940
views
Is simple non-holonomic D-module a local concept?
It is well known that we can use the Riemann-Hilbert correspondence to describe holonomic D-modules in terms of a category of perverse sheaves on some variety $X$. And $U|\rightarrow Perv(U)$ is a ...
- 5,515
8
votes
3
answers
784
views
Characterisation of parabolic subalgebras: reference sought
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $\mathfrak{p}$ a subalgebra. As we all know, $\mathfrak{p}$ is parabolic if it contains a Borel (thus maximal solvable) subalgebra. In this ...
- 1,051
3
votes
1
answer
152
views
Defining a family of rotations with certain properties
Let $d \ge 2$, and consider the sphere $S^{d-1}$ embedded in $\mathbb R^d$. Does there exist a family of rotations $\{\mathcal O_v\}_{v \in S^{d-1}}$ which satisfies:
$\mathcal O_v e_1 = v$, and
$\...
- 8,512
4
votes
1
answer
4k
views
Irreducible representations of Heisenberg algebra
I need some references for irreducible representations of the Heisenberg algebra with three generators or the category of its finite length modules.
Here, the Heisenberg algebra with three generators ...
- 43