# Questions tagged [rt.representation-theory]

Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.

**3**

votes

**1**answer

137 views

### Generalization of Jordan's Lemma $A^2=B^2=I$ can be 2-block diagonalized

One of Jordan's lemma states that if two orthogonal matrices $A,B$ are such that $A^2=B^2=I$, then they can be co-diagonalized by block of size 2.
(the proof is easy, consider $x$ an eigenvector of $A+...

**1**

vote

**1**answer

81 views

### Relative position and change of torus

Let $G$ be a connected split reductive group over a field $k$ of characteristic $0$. Let $T$ and $T'$ be two split maximal tori of $G$ and $B \supset T, B' \supset T'$ be two Borel subgroups of $G$.
...

**8**

votes

**3**answers

422 views

### Tannaka duality for semisimple groups

Tannakian formalism tells us that for any rigid, symmetric monoidal, semisimple category $\mathcal{C}$ equipped with a fiber functor $F: \mathcal{C} \to Vect_k$ for a field $k$ (of characteristic $0$) ...

**3**

votes

**0**answers

61 views

### Weyl theorem - possible corollary - alternative characterization of projective representation of $Z_N\times Z_N$

For an integer $N$, let $\omega=e^{2i\pi/N}$ and $A$, $B$ be the clock and shift operators:
$A=\left(\begin{matrix}
1 & 0 & \cdots & 0 \\
0 & \omega & \cdots & 0 \\
\vdots &...

**2**

votes

**0**answers

61 views

### Centraliser of $\Delta U$ in $U\otimes U$

Let $U$ be a universal enveloping algebra of a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. What is a good reference for the centralizer of $\Delta (\mathfrak{g})$
in $U \otimes U$ ? Here $\...

**6**

votes

**2**answers

223 views

### Subalgebra of a group algebra

Let $k$ be a field, $G$ a finite group, and $k[G]$ the group algebra.
Let $A$ be a subalgebra of $k[G]$. In general, $A$ is not the group algebra of some subgroup $H$ of $G$.
Question: Is there any ...

**2**

votes

**0**answers

74 views

### When is the category of complexes of finite type?

For a ring $R$ define the category of complexes of length $n \geq 2$ as the category $C_n(R)$ with objects the complexes of the form $0 \rightarrow X_n \rightarrow \cdots X_1 \rightarrow 0$ with the ...

**5**

votes

**1**answer

349 views

### A binary hook-length formula?

This is purely exploratory and inspired by curiosity.
Setup: For an integer $k>0$, let $k=\sum_{j\geq0}k_j2^j$ be its binary expansion and denote the sum of its digits by $\eta(k):=\sum_jk_j$. ...

**1**

vote

**1**answer

99 views

### How to prove the following Whittaker formula

I am a theoretical physicist and
I need help in proving the alternate Whittaker formula
$W _ { k , m } ( z ) = \frac { \Gamma ( - 2 m ) } { \Gamma \left( \frac { 1 } { 2 } - m - k \right) } M _ { k , ...

**5**

votes

**0**answers

152 views

### Homeomorphisms of Springer fibers

Let $V$ be a complex $n$-dimensional vector space and denote by ${\cal F}$ its space of complete flags. Let $g \in Gl(V)$ be unipotent and consider the Springer fiber ${\cal F}_g$ of its fixed points ...

**4**

votes

**2**answers

210 views

### Non-faithful irreducible representations of simple Lie groups

For a complex simple Lie algebra $\frak{g}$, which of its finite dimensional irreducible representations give non-faithful representations of the corresponding simply-connected compact Lie group.
...

**2**

votes

**0**answers

78 views

### A theory of (or reference for) symmetric point arrangements

I wonder where I can find something written on symmetric point arrangements (see definition below). I am interested in general references, preferably books that introduce (or papers that use) some ...

**5**

votes

**0**answers

92 views

### Group with Character Degrees {1,pq,pr,qr}, where p,q and r are distinct primes

I am currently trying to bound the derived length of certain solvable groups assuming that they have only two irreducible monomial complex character degrees. Using induction, it often suffices to ...

**6**

votes

**0**answers

100 views

### Motivation and Difference of Category O Definition for Kac-Moody Algebras

My first encounter with Category $\mathcal{O}$ was (perhaps unusually) learning about Kac-Moody algebras from Kac's book. Kac takes the following definition:
The Category $\mathcal{O}$ has objects $\...

**3**

votes

**0**answers

51 views

### Expressing $\sum_{g\in [G/H]}ge_Hg^{-1}\in Z(\mathbb{C}[G])$ in terms of primitive central idempotents?

Suppose $G$ is a finite group, and $H$ a subgroup. For an irreducible character $\chi$ of $G$, there is a central idempotent in the group algebra $\mathbb{C}[G]$:
$$
e_\chi=\frac{\chi(1)}{|G|}\sum_{g\...

**2**

votes

**1**answer

96 views

### Characterisation of even nilpotent elements in $\mathfrak{sl}_n$

Is there a ''nice'' classification of even nilpotent elements in $\mathfrak{sl}_n,$ using the correspondence between nilpotent elements and partitions of n? By an even element, I mean an element $e$, ...

**8**

votes

**1**answer

163 views

### Unitary representations of finite groups over finite fields

I would like to learn the basic theory of unitary representations of finite groups over finite fields.
Here, the unitary group $\operatorname{GU}(n,\mathbb{F}_{q^2})$ consists of all invertible ...

**3**

votes

**0**answers

29 views

### Bounds for the number of edges in an Alperin diagram

If $A$ is an algebra over a field $k$ and $M$ is a finite-dimensional $A$-module, then Alperin showed in a paper [Diagrams for modules, JPAA, 1980] how to associate a diagram to $M$ with the vertices ...

**1**

vote

**0**answers

58 views

### Continuity of theta correspondence

If theta correspondence established a map from unitary dual of G to unitary dual of H,then is the map continous w.r.t Fell topology of both unitary duals?

**5**

votes

**0**answers

99 views

### Tensoring Harish-Chandra bimodules with Verma modules

The question is about the functor $T_\lambda$ defined by Bernstein and Gelfand in the paper Tensor Products of Finite and Infinite Dimensional
Representations of Semisimple Lie Algebras.
Setup: Let $\...

**3**

votes

**1**answer

156 views

### Distinguished dominant integral weight related to a branching problem

Let $G$ be a simple compact connected Lie group and let $K$ be a connected closed subgroup of $G$.
Let $\widehat G$ and $\widehat K$ denote the corresponding unitary duals, that is, the (equivalence ...

**2**

votes

**1**answer

41 views

### Equivalence of definition of category $\mathcal{O}^\mathfrak{p}$

Denote by $\mathfrak{g}$ a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$.
Let $\Phi$
be the root system of $(\mathfrak{g},\mathfrak{h})$, write $W$ ...

**7**

votes

**0**answers

119 views

### Geometric Interpretations of Nil-Hecke Ring and Affine Hecke Algebra

I am interested in two related constructions which give us either the cohomology or the $T \times \mathbb{C}^*$-equivariant $K$-theory of flag varieties.
Let $G$ be a semisimple, simply connected ...

**4**

votes

**0**answers

69 views

### $q$-Kostant partition function and flow polytopes?

The Kostant partition function is known to be related to volumes and Ehrhart polynomials of flow polytopes of graphs (see e.g. https://link.springer.com/article/10.1007/s00031-008-9019-8 or https://...

**3**

votes

**0**answers

61 views

### Compatibility of $\mathrm{SL}_2$ representations, bilinear forms and isotropic flags

Let $V$ be a finite dimensional $\mathbf{C}$-vector space with a symplectic (non-degenerate anti-symmetric bilinear) form $\omega: V\times V \to \mathbf{C}$ and a symplectic $\mathrm{SL}_2$-...

**5**

votes

**0**answers

100 views

### When does the canonical $t$-structure restrict to perfect complexes?

I am interested in non-Noetherian(!) rings such that the canonical $t$-structure on $D(R)$ (the derived category of left $R$-modules) restricts to perfect complexes i.e. to the subcategory of ...

**6**

votes

**0**answers

90 views

### Orthogonality relations for characters of VOAs?

If $G$ is a finite group, the characters of its irreps satisfy
$$
\langle \chi_1,\chi_2\rangle := \frac{1}{|G|}\sum_{g\in G} \chi_1(g)\; \overline{\chi_2(g)} = \delta_{\chi_1,\chi_2}.
$$
Alexei ...

**5**

votes

**2**answers

291 views

### Is the Perron-Frobenius dimension of a G-Set given by its cardinality?

Given a ring $R$ with finite additive basis $\{e_i\}_{i=1}^{n}$, such that $e_i e_j=\sum c_{ijk}e_k$ with $c_{ijk}\in \mathbb{N}$, we define the Perron-Frobenius dimension $FPDim(e_i)$ of a basis ...

**12**

votes

**3**answers

544 views

### Reference request: Grassmannian and Plucker coordinates in type B, C, D

Grassmannian $Gr(k,n)$ is the set of $k$-dimensional subspace of an $n$-dimensional vector space. What are the Grassmannian in types B, C, D? What are the analog of Plucker coordinates and Plucker ...

**6**

votes

**1**answer

217 views

### Is a finite group given by its character table if its Sylow subgroups are so?

As pointed out by Mikko Korhonen in this answer, Özdem Çelik proved (in 1976 here) that a finite group whose Sylow subgroups are cyclic (called a Z-group) is determined by its character table.
...

**2**

votes

**0**answers

27 views

### Standard name for rational Levi subgroups of rational parabolic groups

I am looking for a standard name for the groups above. They appear in Harish-Chandra theory. It would be convenient to have a shorter name for them. I tried looking online but I did not find useful ...

**4**

votes

**0**answers

66 views

### What are the zonal spherical functions for a finite unitary group acting on a unit sphere?

Given a prime power $q$ and a dimension $d$, consider the Hermitian form $(\cdot,\cdot) \colon \mathbb{F}_{q^2}^d \times \mathbb{F}_{q^2}^d \to \mathbb{F}_{q^2}$ given by
$$
(x,y) = \sum_{i\in [d]} ...

**4**

votes

**1**answer

186 views

### Finite groups with the same character table *including* class types, and square-free order

There are non-isomorphic finite groups with the same (complex) character table, as $D_4$ and $Q_8$.
$$\scriptsize\begin{array}{c|c}
\text{class}&1&2A&2B&2C&4 \newline
\text{...

**2**

votes

**0**answers

83 views

### Quiver algebra not derived equivalent to its opposite algebra

Let $A=KQ/I$ be a quiver algebra with the following two properties:
a)Q is an acyclic quiver
b) the injective envelope of $A$ is projective.
Question 1: Is there an algebra $A$ with properties a)...

**14**

votes

**8**answers

1k views

### Applications of the idea of deformation in algebraic geometry and other areas?

The idea of proving something by deforming the general case to some special cases is very powerful. For example, one can prove certain equalities by regarding both sides as functions/sheaves, and show ...

**2**

votes

**0**answers

63 views

### Minimal Embedding for flags varieties

I would like to understand how to construct a parametization of a flag
variety $F(V,n_1,\ldots,n_r)\subseteq \mathbb{P}^N$ in its minimal embedding.
First, I would like to know if there is a closed ...

**2**

votes

**0**answers

72 views

### Generic representation of PGL(3)

Let $G$ be the group $PGL(3,F)$, where $F$ is non-archimedean locally compact field, and $(\widetilde{H}_{n})_{n\in\mathbb{N}}$ the decreasing sequence of open and compact subgroups given by (image in ...

**1**

vote

**0**answers

58 views

### Artin’s theorem on induced representations and the kernel

Let $G$ be a finite group, and let $X$ be a family of subgroups of $G$ closed under conjugation and under passage to subgroups. Suppose further that $G$ is the union of the elements of $X,$ and denote ...

**2**

votes

**0**answers

98 views

### Satake correspondence for groups over finite field

I asked the same question in MSE, but I didn't get any answer. So I decided to post it here, too.
In Langlands' program, Satake correspondence gives a correspondence between unramified ...

**8**

votes

**0**answers

294 views

### What are the character tables of the finite unitary groups?

I need to know the (complex) character table of the finite unitary group $U_n(q)$. Lusztig and Srinivasan (1977) provide an abstract description, but parsing it requires a stronger background in ...

**4**

votes

**3**answers

190 views

### Real points of reductive groups and connected components

Let $\mathbf G$ be a connected reductive group over $\mathbb R$, and let $G = \mathbf G(\mathbb R)$. Then $G$ is not necessarily connected as a Lie group, e.g. $\mathbf G = \operatorname{GL}_n$. ...

**6**

votes

**1**answer

370 views

### On a problem for determinants associated to Cartan matrices of certain algebras

This is a continuation of Classification of algebras of finite global dimension via determinants of certain 0-1-matrices but this time with a concrete conjecture and using the simplification suggested ...

**6**

votes

**1**answer

98 views

### Bijection from $S^2$ to itself interchanging actions of $A_5$

Let $X$ and $Y$ be two copies of $S^2$, and let $A_5$ act on each of them (as a group of rotations). Call these actions $\theta_X$ and $\theta_Y$.
Moreover, let $g \in A_5$ be a fixed element of ...

**4**

votes

**1**answer

172 views

### On definitions and explicit examples of pure-injective modules

I am interested in the following assumption on left $R$-modules: for a module $I$ and all injective homomorphisms $A\to B$ of finitely generated (or possibly finitely presented) modules I want the ...

**3**

votes

**0**answers

87 views

### Recognizing a restriction from $SL_2(\mathbb{C})$ to $SL_2(\mathbb{Z})$

I am aware that classifying all $SL_2(\mathbb{Z})$ representations is more or less completely intractable, but I was wondering what is known about the following simpler question: How do I recognize ...

**2**

votes

**0**answers

65 views

### Do the values of the global dimension constitute an interval?

Let $Q$ be a fixed finite connected quiver and $k$ a fixed field. Set $Z_Q:= \{ gldim(kQ/I) < \infty | I $ an admissible ideal $\}$.
Question: Is $Z_Q$ an intervall?
This is true for example in ...

**0**

votes

**0**answers

75 views

### Find representation set of orbits when group acts on a set

Let group $G$ acts on a set $S$. Burnside's lemma gives as how to count numbers of orbits. I am interested how to find the orbits. By finding orbits I mean how to find a representative from each orbit....

**1**

vote

**0**answers

24 views

### Formal character and unit

Denote by $\mathfrak{g}$ a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. Let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$.
...

**3**

votes

**0**answers

76 views

### Relative position on flag variety

Let $G$ be a semisimple algebraic group over $\mathbb{C}$. Consider the $G$ diagonal action on $G/B \times G/B$, the orbit is indexed by $W$, the Weyl group of $G$ by Bruhat decomposition. There is a ...

**1**

vote

**0**answers

35 views

### Highest-$\ell$-weight tensor products and diagram subalgebras

Let $U_q(\mathcal{L}({\mathfrak{g}}))$ be a quantum loop algebra and $I$ the set of indexes of Dynking diagram of $\mathfrak{g}$. Consider $J\subset I$ a connected subdiagram, so that $U_q(\mathcal{L}(...