Skip to main content

All Questions

Filter by
Sorted by
Tagged with
19 votes
2 answers
2k views

Dual versions of "folding" symmetric ADE Dynkin diagrams?

Start with the Dynkin diagram of an irreducible root system, typically associated with a simple Lie algebra over $\mathbb{C}$ or a simple algebraic group. Most of the simply-laced ADE diagrams ...
Jim Humphreys's user avatar
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. ...
user1258240's user avatar
13 votes
3 answers
1k views

Characterization of Frobenius complements

I have learned that Frobenius complements are characterized (among finite groups) by having a fixed point free complex representation. That is, a finite group $G$ is a Frobenius complement if and only ...
Joonas Ilmavirta's user avatar
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 ...
Abdelmalek Abdesselam's user avatar
44 votes
10 answers
11k views

The finite subgroups of SL(2,C)

Books can be written about the finite subgroups of $\mathrm{SL}(2,\mathbb C)$ (and their immediate family, like the polyhedral groups...) I am about to start writing notes for a short course about ...
Mariano Suárez-Álvarez's user avatar
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 ...
23 votes
2 answers
859 views

Moments of Plücker coordinates on complex Grassmannian

Consider the Grassmannian $Gr(k,N)\simeq U(N)/(U(k)\times U(N-k))$ which parametrizes $k$-dimensional subspaces of $\mathbb{C}^N$. Let us put on it the $U(N)$-invariant probability measure. Let $\...
Abdelmalek Abdesselam's user avatar
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 ...
David Corwin's user avatar
  • 15.4k
18 votes
3 answers
1k views

What happened to the fourth paper in the series "On the classification of primitive ideals for complex classical Lie algebras" by Garfinkle?

In a series of papers in Compositio Math. entitled On the classification of primitive ideals for complex classical Lie algebras I, II and III, Garfinkle describes an algorithm that allows one to ...
Tobias Kildetoft's user avatar
16 votes
3 answers
1k views

Reference for representation theory of SL_2(Z/n)

There are many references for the representation theory (say over $\mathbf C$) of $\operatorname{SL}_2(\mathbf{F}_q)$ and $\operatorname{GL}_2(\mathbf{F}_q)$, for instance lecture 5 in Fulton--Harris &...
Dan Petersen's user avatar
  • 40.2k
14 votes
1 answer
544 views

Are there nice isomorphisms $\operatorname{S}^2(k^n)\cong\Lambda^2(k^{n+1})$?

This might be forced to migrate to math.SE but let me still risk it. The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism ...
მამუკა ჯიბლაძე's user avatar
13 votes
3 answers
3k views

How to Compute the coordinate ring of flag variety?

Let $G$ be algebraic group over $\mathbb(C)$(semisimple), $B$ be Borel subgroup. consider flag variety $G/B$. It is known that $G/B$ is projective variety. So one consider the homogeneous coordinate ...
Shizhuo Zhang's user avatar
13 votes
0 answers
523 views

Euler Subgroups and Automorphic L-functions

Recently, I have read about the Whittaker expansion for $\mathrm{GL}_n$ and was struck by the utility of the mirabolic subgroup, $\mathrm{P}_n\subset \mathrm{GL}_n$ of matrices with bottom row $(0\; 0 ...
Spencer Leslie's user avatar
11 votes
1 answer
627 views

Representations of the automorphism group of graphs via spectral graphs theory

Given a (simple) graph $G=(V,E)$ with $V=\{1,...,n\}$ and let $A$ be its adjacency matrix. I am interested in the representation theory (over $\Bbb R$) of the automorphism group $\def\Aut{\mathrm{Aut}...
M. Winter's user avatar
  • 13.6k
10 votes
1 answer
2k views

Representation theory over Z

In his answer to my question here, Victor Protsak quoted the following result: Let $C_2$ be a finite cyclic group of order $2$. Then every $\mathbb{Z}[C_2]$ structure on $\mathbb{Z}^n$ is isomorphic ...
New to this's user avatar
10 votes
0 answers
1k views

Complexes of representations with complementary central charges

This is another question asking for references. There is an important phenomenon of correspondence between (complexes of) representations of infinite-dimensional Lie algebras with the complementary ...
Leonid Positselski's user avatar
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 (...
Igor Khavkine's user avatar
7 votes
1 answer
672 views

The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group

I asked this question on Math.Stack but have not had any answers. Question What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group, $A$? The trivial ...
JP McCarthy's user avatar
  • 1,037
6 votes
2 answers
788 views

Reference request: Projective representations of a simply connected real semisimple Lie group lift to unitary representations

I recently got interested in representation theory in quantum mechanics and I read the following theorem: Let $G$ be a simply-connected Lie group with $H^2(\mathfrak{g},\mathbb{R})=0$ and let $\...
Lucas Smits's user avatar
6 votes
1 answer
227 views

Quantum group representations from (convolution) matrix units?

Let $A=F(\mathbb{G})$ be the algebra of functions on a finite quantum group with a Haar state $$h=:\int_\mathbb{G}:F(\mathbb{G})\rightarrow \mathbb{C}.$$ There is a convolution product on $A=F(\...
JP McCarthy's user avatar
  • 1,037
5 votes
2 answers
2k views

Canonical reference for Chern characteristic classes

I'm a little uncertain about the definitions for Chern roots Chern classes Chern characters From perusing several discussions, I gather that if one correlates the nomenclature with that of ...
Tom Copeland's user avatar
  • 10.5k
4 votes
1 answer
215 views

Does a $G$-structure on $M$ automatically descend to a contact $G$-structure on $\mathbb{P}T^*M$?

If $M$ is a real smooth manifold of dimension $n+1$, by $D\in\mathbb{P}T^*M$ I mean a tangent hyperplane at some point of $M$. I denote by $b$ the canonical projection of the $(2n+1)$-dimensional ...
Giovanni Moreno's user avatar
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 ...
Yousef's user avatar
  • 43
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}\...
T. Amdeberhan's user avatar
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 ...
Mare's user avatar
  • 26.5k
35 votes
4 answers
2k views

Why there is a relation among the second-order minors of a symmetric $4\times 4$ matrix?

A $4\times 4$ symmetric matrix $$ \left( \begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{12} & a_{22} & a_{23} & a_{24} \\ a_{13} & a_{23} & a_{33} & ...
Giovanni Moreno's user avatar
33 votes
2 answers
2k views

What do cluster algebras tell us about Grassmannians?

One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. ...
Matthew Pressland's user avatar
30 votes
1 answer
2k views

Is there an accessible exposition of Gelfand-Tsetlin theory?

I'm hoping to start an undergraduate on a project that involves understanding a bit of Gelfand-Tsetlin theory, and have been tearing my hair out looking for a good reference for them to look at. ...
Ben Webster's user avatar
  • 44.7k
26 votes
1 answer
2k views

Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?

This question was motivated by a more general question raised by Jan Weidner here. In general one starts with a variety $X$ (say smooth) over an algebraic closure of a finite field $\mathbb{F}_q$ of ...
Jim Humphreys's user avatar
25 votes
3 answers
6k views

Introductory References for Geometric Representation Theory

Would anyone be able to recommend text books that give an introduction to Geometric Representation Theory and survey papers that give an outline of the work that has been done in the field? I'm ...
Siddharth Venkatesh's user avatar
21 votes
2 answers
2k views

A new combinatorial property for the character table of a finite group?

Let $G$ be a finite group and $\Lambda = (\lambda_{i,j})$ its character table with $\lambda_{i,1}$ the degree of the ith character. Consider the following combinatorial property of $\Lambda$: for ...
Sebastien Palcoux's user avatar
18 votes
2 answers
1k views

Has anyone seen this construction of the Weil representation of $\mathrm{Sp}_{2k}(\mathbb{F}_p)$?

$\def\FF{\mathbb{F}}\def\CC{\mathbb{C}}\def\QQ{\mathbb{Q}}\def\Sp{\text{Sp}}\def\SL{\text{SL}}\def\GL{\text{GL}}\def\PGL{\text{PGL}}$Let $p$ be an odd prime. The Weil representation is a $p^k$-...
David E Speyer's user avatar
18 votes
5 answers
3k views

Moments of the trace of orthogonal matrices

Let $O_n$ be the (real) orthogonal group of $n$ by $n$ matrices. I am interested in the following sequence which showed up in a calculation I was doing $$a_k = \int_{O_n} (\text{Tr } X)^k dX$$ where ...
J. E. Pascoe's user avatar
  • 1,429
17 votes
5 answers
3k views

Reference for this theorem in representation theory?

Let $G$ be a finite group and $\chi$ be an irreducible character of $G$ (characteristic zero algebraically closed base field). If $H$ is the kernel of $\chi$ then the irreducible representations of $G/...
Sebastian Burciu's user avatar
16 votes
2 answers
818 views

Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$

$V=\mathbb C^n$ is a $\mathbb CS_n$-module, where $S_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power $V^...
Benjamin Steinberg's user avatar
16 votes
2 answers
992 views

Maximal number of maximal subgroups

Let $G$ be a finite group. I want to find an upper bound on the number of the maximal subgroups. My questions is does it possible to prove that the number of maximal subgroups of any finite group $G$ ...
Klim Efremenko's user avatar
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 ...
Steven Sam's user avatar
  • 10.7k
15 votes
2 answers
2k views

Isomorphism between Spin(3,2) and Sp(4, R)

I've been using the fact that Spin(3,2) is isomorphic to Sp(4, R) for a while, but I've never seen a proof. Can anyone point me in the direction of a good reference?
MRD1729's user avatar
  • 393
14 votes
4 answers
1k views

actions of the hyperoctahedral group

I am looking for actions (i.e., permutation representations) of the hyperoctahedral group $H_n$ (also known as the group of signed permutations) studied in the literature, i.e., homomorphisms from $...
Martin Rubey's user avatar
  • 5,822
14 votes
2 answers
3k views

What is the non-motivic motivation behind automorphic representations?

In one of my last questions: What is the "reason" for modularity results? it was pointed out to me that "the notion of automorphic representation developed independently of any concern with ...
James D. Taylor's user avatar
14 votes
2 answers
3k views

How many ways are there to prove flag variety is a projective variety?

I am looking for references talking about different ways to prove flag variety $G/B$ is projective variety. Now I have some in mind: There is a proof in Humphreys Linear algebraic groups, he first ...
Shizhuo Zhang's user avatar
12 votes
1 answer
729 views

Schur's Theorem about immanants

$\DeclareMathOperator\Imm{Imm}$I am looking for a proof in English or French of Schur's theorem that, for every $H$ in the space $\mathbb H_n^+$ of positive semi-definite Hermitian matrices, and every ...
Denis Serre's user avatar
  • 52.3k
12 votes
1 answer
744 views

Is the following construction of the 0-Hecke monoid (well) known?

Let W be a Coxeter group with Coxeter generators S. The corresponding 0-Hecke monoid H(W) has generating set S, the braid relations of W and the relations that each element of S is an idempotent. If ...
Benjamin Steinberg's user avatar
12 votes
3 answers
2k views

What is a good introduction to branching rules in representation theory?

I'm looking for a book or introductory article, that explains branching rules in representation theory of real Lie groups. When a Lie group has a set of irreducible representations, I'd like to know ...
Manuel Bärenz's user avatar
12 votes
1 answer
1k views

Smallest dimension of nontrivial representation of a simple Lie algebra over `$\mathbb{C}$`

The question involved here is natural and very classical, but I'm unsure what has been formally stated and proved in the literature. The only approach I know involves assembling facts that apparently ...
Jim Humphreys's user avatar
11 votes
4 answers
2k views

Textbook source for finite group properties deducible from character table?

Various questions have been posted on MO (some answered, some not) involving the character table of a finite group $G$ over a splitting field such as $\mathbb{C}$ of characteristic 0. My basic ...
Jim Humphreys's user avatar
10 votes
1 answer
389 views

Operads and the Stable Module Category

I am seeking references to places where operads and their algebras have been studied for the stable module category. Colored operads are fine too. Let $k$ be a field and $R$ a $k$-algebra. The stable ...
David White's user avatar
  • 30.3k
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 ...
Rick Sternbach's user avatar
9 votes
1 answer
460 views

Connections between linear representations and permutation representations

A finite group $\Gamma$ might be represented by a linear transformation $$\rho : \Gamma\to\mathrm{GL}(\Bbb R^d),$$ or by permutations $$\phi :\Gamma\to\mathrm{Sym}(n).$$ Of course, latter ones can ...
M. Winter's user avatar
  • 13.6k
9 votes
3 answers
2k views

Borel's presentation for the cohomology of a Flag Variety

If $G$ is a simple complex Lie group, $T\subset B\subset G$ is a choice of Borel and maximal torus, and $W$ is the Weyl group, then 1) $H^{*}(G/B,K)=K[T^{\vee}]/(K[T^\vee]^W_{+})$ and 2) $K[T^\vee]^...
DCT's user avatar
  • 1,537