All Questions
Tagged with rt.representation-theory reference-request
823 questions
6
votes
1
answer
291
views
Example of nice isomorphism between Cl$_{p,q}(\mathbb R)$ and matrix algebras over $\mathbb R,\mathbb C,\mathbb H,\mathbb R^2,\mathbb C^2,\mathbb H^2$
$\DeclareMathOperator\Cl{Cl}$It is known that every Clifford Algebra $\Cl(Q)$ over the real numbers where $Q: \mathbb R^n \to \mathbb R$ is a non-degenerate quadratic form is isomorphic to a matrix ...
- 63.9k
2
votes
0
answers
108
views
Invariants of Lie superalgebras
Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$
which is symmetric on $V_0$, skew symmetric on $V_1$, and ...
- 673
1
vote
0
answers
91
views
Explanation about Lapid-Rallis iductive argument (doubling method)
I am reading Lapid-Rallis "On the local factors of representations of classical groups" and I am completely stuck with the proof of Proposition 3.
In the case $\mathcal V$ is not anisotropic,...
- 11
10
votes
2
answers
594
views
Existence of a strongly continuous topologically irreducible representation of a compact group on an infinite dimensional Banach space?
Does there exists a triple $(G, X, \pi)$, where $G$ is a compact group, $X$ an infinite dimensional Banach space over $\mathbf{C}$, and $\pi : G \to B(X)$ a strongly continuous representation of $G$, ...
- 18.7k
2
votes
1
answer
257
views
Possible "algebraic" direction in hyperplane arrangements
I have been studying hyperplane arrangements from R. Stanley's notes and so far, I have read until lecture 5. But, Lecture 6 and end of Lecture 5 seems very combinatorial. I'm more interested in the &...
- 24.2k
8
votes
1
answer
516
views
Specht modules for symmetric group $S_{\infty}$
Specht modules of $S_n$, the symmetric group on n symbols is well-known.
Is there an analogue of these modules for $S_{\infty}$, the set of all permutations of $\mathbb N$?
Also, please share some ...
- 1,256
22
votes
2
answers
2k
views
A royal road to Coulomb branches of 3D $\mathcal{N}=4$ gauge theories
So, I've been very interested recently with the developements of the (now mathematically precise) theory of Coulomb branches - in particular because of its recent applications on representation theory ...
- 4,612
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
12
votes
2
answers
727
views
Suggestion for framing a course in Representation theory + Spectral graph theory
I am going to give a course in spectral graph theory to graduate students. I want to learn and teach the connection between the spectral graph theory and the representation theory of finite groups. I ...
- 13.6k
14
votes
0
answers
298
views
Representation theory of Kac-Moody algebras in positive characteristic
I have been trying to learn about the representation theory of Kac-Moody algebras in positive characteristic. However, my usual reference for the subject (Infinite dimensional Lie algebras by Kac) ...
- 1,389
9
votes
1
answer
434
views
Questions on the group $\mathrm{GL}(H)$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\U{U}$Let $H$ be an infinite dimensional complex Hilbert space. Consider the group $\GL(H)$ of bounded invertible operators on $H$.
Question 1. I've ...
- 18.7k
6
votes
5
answers
3k
views
Reference for quantum Schur-Weyl duality
I am trying to prove a version of quantum Schur-Weyl duality. I hope to be able to generalize the proof of the Schur-Weyl duality between $U_q(\mathfrak{gl}_n)$ and the Hecke algebra $H_r$. So I am ...
- 4,713
11
votes
2
answers
684
views
Invariants of $\mathrm{GL}_n$ representations
$\DeclareMathOperator\GL{GL}$Let $V=\mathbb C^n$ be the natural representation of $\GL_n(\mathbb C)$ and let $W=\operatorname{Sym}^2(V)$ be the symmetric square representation. Let $W^k$ denote the ...
- 63.9k
7
votes
1
answer
281
views
Question concerning the coefficients of block idempotents
Let $G$ be a finite group. Let $p$ be a prime number such that $p \mid |G|$.
Let Irr$(G)$ denote the set of ordinary irreducible characters of $G$.
For $\chi\in$ Irr$(G)$ define $e_{\chi} := \frac{\...
- 44.4k
7
votes
1
answer
292
views
Homotopy fixed points of involutive automorphisms of discrete groups
$\DeclareMathOperator\Map{Map}$Precis: I am looking for a reference/explanation of a (general) computation of homotopy fixed points $(BG)^{h\Gamma}$ of a finite group $\Gamma$ acting on a discrete (...
- 563
7
votes
0
answers
207
views
A reference for Bernstein's approach to KL conjectures
The exposition in the famous J. Bernstein's lecture notes on D-modules is rather sketchy.
Fortunately, there are many other sources for the information from this text (like Hotta's excellent "D-...
- 24.2k
3
votes
0
answers
205
views
Status of RFD groups and $C^*$-algebras
Motivated by this question and its great answers, I become very curious to know what do we know about RFD (residually finite dimensional) groups and $C^*$-algebras, e.g. do we know how these ...
- 1,586
4
votes
1
answer
197
views
Moment integrals and determinants
Let $USp(2n)$ be the compact symplectic group of size $2n$, $dA$ its Haar measure
of total mass one, and $\det(1−A)$ being computed for the standard representation of
$A\in USp(2n)$ as a matrix of ...
- 85.6k
3
votes
1
answer
177
views
Quiver and relations for ADE singularities in dimension one
Let $A$ be an ADE-hypersurface singularity in dimension one.
For example in Dynkin type $A_n$, A is given by $K[[x,y]]/(x^2+y^{n+1})$.
Then $A$ is CM-finite and let $M$ be the direct sum of all ...
- 26.5k
11
votes
2
answers
977
views
Reference for combinatorics with view towards representation theory/algebraic geometry
I'm making this post to ask for a reference about combinatorics: I'm a PhD student in representation theory/algebraic geometry. My background is mostly in algebra and geometry (and also mostly ...
- 15.8k
0
votes
1
answer
187
views
Matrix representations of Lie groups of type $B_n$
For the Lie algebra $\mathfrak{so}(2n+1, \mathbb{C})$, there is a matrix representation given by the following matrices:
\begin{align}
\left( \begin{matrix} 0 & x & y \\ -y^T & A & B \\...
- 12.9k
10
votes
1
answer
396
views
Generalising the union-closed sets conjecture from lattice to a larger class of posets
(edit: I decided to simplify the question and only pose it for bounded posets first)
The Union-closed sets conjecture is equivalent for lattices P to:
There exists a join-irreducible element $a$ with ...
- 63.9k
7
votes
1
answer
778
views
Reference for the Brauer-Nesbitt theorem
In Wikipedia's article on the Brauer-Nesbitt theorem, they state that given a group $G$ and a field $E$, two semisimple representations $\rho_1,\rho_2 : G\longrightarrow \operatorname{GL}_n(E)$ are ...
- 7,527
4
votes
2
answers
299
views
Relation of the first Hochschild cohomology and the outer automorphism group
Let $R$ be a ring.
Qeustion: Is it true that the first Hochschild cohomology of $R$ is zero if and only if the outer automorphism group of $R$ is finite?
(It is not true, by the two answers. Is it ...
CommunityBot
- 1
12
votes
0
answers
402
views
Which abelian categories have homological dimension 1?
In this MSRI lecture Geometry of Quiver Varieties I, Victor Ginzburg describes all abelian categories of homological dimension $1$ as being either
a category of representations $\mathrm{Rep}_\mathbf{...
- 1,161
0
votes
0
answers
171
views
Total sum of characters over partitions with distinct parts
In my earlier quest, we looked at $\chi_{\mu}^{\lambda}=$value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu$ and $\lambda$ are (unrestricted) partitions of $n$. Then, ...
- 43.2k
7
votes
2
answers
713
views
Total sum of squares of characters of the symmetric group $\mathfrak{S}_n$
In my earlier MO post, I proposed the double sum $\sum_{\mu\vdash n}\sum_{\lambda\vdash n}\chi_{\mu}^{\lambda}$ regarding characters of the symmetric group $\mathfrak{S}_n$. Soon after, I started ...
- 43.2k
4
votes
1
answer
700
views
Total sum of characters of the symmetric group $\frak{S}_n$
Let $\chi_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that
$$\sum_{\lambda\vdash n}\...
- 250
2
votes
0
answers
218
views
Does the following corollary of Mackey's tensor product theorem hold for smooth representations?
Let $G$ be a locally profinite group, and let $H$ be a closed subgroup of $G$. Let $\sigma$ be a smooth representation of $G$, and let $\tau$ be a smooth representation of $H$ (henceforth, every ...
- 353
6
votes
2
answers
613
views
Counting $\pm 1$ and $0$'s in the character tables of $\frak{S}_n$
Let $\chi_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that
$$\sum_{\lambda\vdash n}\...
- 43.2k
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(...
- 32.1k
6
votes
1
answer
244
views
What is a Serre-smooth algebra?
Let $A$ be an $R$-algebra.
In the book "Noncommutative Geometry and Cayley-smooth Orders" by Le Bruyn one can find the notion of "Serre-smooth" in the introduction.
But no formal ...
- 4,277
8
votes
3
answers
1k
views
Further reading in algebraic geometry
I recently finished reading W. Fulton's "Algebraic Curves" and also attended a lecture series on moduli spaces and am interested in learning about them as well. I looked for a few books to ...
- 739
7
votes
0
answers
171
views
$\mathfrak{sl}_2(\mathbb{Z})$'s properties as a Lie algebra over a ring
I was wondering what was known about $\mathfrak{sl}_2(\mathbb{Z})$ as a Lie algebra; in particular, what is known about its representation theory? I know of some texts which treat Lie algebras ...
- 820
7
votes
1
answer
429
views
K-type in discrete series representation
The following result seems well known.
Let $G$ be a reductive Lie group with a maximal compact subgroup $K$. If $\mu$ is an irreducible unitary representation of $K$, then there exist only finitely ...
CommunityBot
- 1
4
votes
1
answer
321
views
Reference for fact about reduction mod $p$ of a representation of a finite group
Let $G$ be a finite group, let $M$ be a $\mathbb{Z}[G]$-module whose underlying abelian group is finite-rank free abelian, and let $\mathbb{F}$ be a field of characteristic $0$. For some prime $p$ ...
- 13k
2
votes
0
answers
91
views
When does a stable endomorphism ring have injective dimension at most one?
tLet $A$ be a Frobenius algebra (we can assume that $A$ is given by quiver and relations) and let $M$ be a basic $A$-module without projective direct summands (we can assume we know the decomposition ...
- 26.5k
11
votes
1
answer
617
views
Is there a unique "natural" action of $\mathsf{SL}_{n+1}$ on $\mathbb{R}^n$?
Context
By acting naturally via $\mathsf{SL}_3$ on $\mathbb{RP}^2=\{[x:y:z]\}$ and by taking the induced action on the affine hyperplane $z=1$ (which we identify with $\mathbb{R}^2$), one can realize ...
- 108k
6
votes
1
answer
226
views
Tensor representations of the quantum algebra $U_q(\mathfrak{sl}(2))$ at the roots of unity
I'm trying to understand how the representation theory of $U_q(\mathfrak{sl}(2))$
works and I had a look to some books and lecture notes available on the internet. The case of $q^m\neq1$ is discussed ...
- 28.1k
2
votes
1
answer
75
views
Reference for the action of the Mullineux involution on a partition with an added good node
Could you help me to locate a reference in the literature to the following fact: if you act by the Mullineux involution $M$ on a partition $Q$ which is a union of a regular partition $P$ with an added ...
- 1,570
7
votes
3
answers
599
views
Root system of fixed point Lie sub-algebra
It is known that a non-simply laced simple root system can be constructed from the simply-laced root system by folding the Dynkin diagram and hence the corresponding non-simply-laced Lie algebra can ...
- 14.1k
6
votes
1
answer
203
views
Question on a subcategory being extension-closed
In the article "Homological theory of noetherian rings" by Idun Reiten from 1996, it was stated that it seems to be not known whether the subcategory $\operatorname{Tr}(\Omega^i(\mathrm{mod}\...
- 26.5k
4
votes
0
answers
163
views
External tensor product Calabi-Yau DG categories
Let $\mathcal{C}$ be a smooth proper DG-category such that the shift $[p]$ is a Serre functor for $D^{perf}(\mathcal{C})$ (we say that $D^{perf}(\mathcal{C})$ is $p$-Calabi-Yau). I am looking for a ...
- 7,300
2
votes
0
answers
382
views
Is there a roadmap to learning representation theory of finite group over finite field?
I've been wanted to learn some basic theories of the (non-semisimple) representation of the finite group over a finite field.
I have been guessing that the materials might be contained in the books on ...
- 1,013
5
votes
3
answers
578
views
Non-symmetric quiver varieties
Given a symmetric Cartan datum $(I,\cdot)$, H. Nakajima has defined a family of varieties - known as quiver varieties - and has used them to give geometric constructions of the representation theory ...
- 63.9k
8
votes
1
answer
452
views
Characterization of automorphic discrete spectra
I recently learned about automorphic spectral decomposition from the book "Spectral decomposition and Eisenstein series" by Moeglin and Waldspurger. (Let me call it M-W)
I have a question ...
CommunityBot
- 1
1
vote
1
answer
275
views
The norm of the principal series intertwining operator for $\operatorname{GL}_2$
Is there a known bound on the norm of the standard intertwining operator for the principal series of $G = \operatorname{GL}_2(\mathbb Q_p)$?
Background:
For a character $\chi = (\chi_1,\chi_2)$ of the ...
- 11.6k
3
votes
2
answers
529
views
What does the regular representation of the coinvariant ring of a unitary reflection group look like?
Let $V$ be a complex vector space of finite dimension $n$ and let $W$ be a finite unitary reflection group. This is, $W$ is a subgroup of $GL(V)$ generated by reflections, i.e., elements $r \in GL(V)$ ...
6
votes
2
answers
331
views
Lie powers of a graded vector space and Klyachko's theorem
Let $V$ be a $\mathbb{Z}_2$-graded vector space (aka super vector space) and $L(V)$ be the free $\mathbb{Z}_2$-graded Lie algebra (aka super Lie algebra). The free super Lie algebra is also graded by ...
- 21.5k
2
votes
1
answer
317
views
English translation of Emmy Noether's Hyperkomplexe Grössen und Darstellungstheorie
I'm wondering if anybody knows where one can find an English translation of Emmy Noether's classical paper E. NOETHER, Hyperkomplexe Grössen und Darstellungstheorie, Math. Zeit. 30(1929), 641–692 ?...
- 188k