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Questions tagged [rt.representation-theory]

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

3
votes
1answer
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
1answer
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
3answers
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
0answers
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
0answers
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
2answers
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
0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
2answers
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
3answers
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
8answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
3answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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}(...