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Questions tagged [rt.representation-theory]

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

7
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0answers
88 views

The $\frak{sl}_2$-representation on a symplectic manifold

Any symplectic manifold $(M,\omega)$ carries a representation of $\frak{sl}_2$: Define the maps $$ L: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~~~~ \Lambda: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~...
2
votes
0answers
82 views

Quadratic algebras and Koszul algebras

Let $A$ be a quadratic algebra and $B$ the Ext-algebra of $A$. In case $A$ is a Koszul algebra, we should have that the global dimension of $A$ plus one is equal to the Loewy length of $B$ (is there a ...
1
vote
1answer
36 views

Definition of the weight lattice for a nonreduced root system

Let $(V,\Phi)$ be a root system with dual root system $(V^{\ast},\Phi^{\vee})$. Let $\Delta = \{\alpha_1, ... , \alpha_n\}$ be a set of simple roots for $V$, and let $\Delta^{\vee} = \{\alpha_1^{\vee}...
2
votes
0answers
22 views

About Extension group and weights in $\mathcal{O}^\mathfrak{p}$

Denote $M_I(\lambda)$ be the generalized Verma module with highest weight $\lambda$ and $L(\mu)$ is the simple highest weight module with highest weight $\mu$. Suppose $\text{Ext}_{\mathcal{O}^\...
1
vote
0answers
45 views

Finite-dimensional graded Lie algebras with $2$ generators

Does anyone know of a classification of those (complex) Lie algebras which are: generated by two elements $\mathbb{Z}$-graded Lie algebras finite dimensional
6
votes
1answer
91 views

P-adic representations corresponding to the same cuspidal pair

Let $G(F)$ be a reductive $p$-adic group. A result of Bernstein says that we can correspond each smooth irreducible representation to a “cuspidal pair” where it is embedded, and at most finitely many ...
6
votes
1answer
131 views

Irreducibility of Gelfand-Serganova strata

To keep the notations simple I'll restrict my attention to the complete flag variety although the question should be equally valid for partial flag varieties. Consider $G=SL_n(\mathbb C)$ with Borel $...
1
vote
0answers
26 views

Rigid $Hom$-orthogonal modules in wild hereditary algebras

Let $Q$ be a simply-laced wild quiver with at least one multiple edge, $k$ be an algebraically closed field, $1$ be the source of and $2$ be the sink of one set of such edges. Can we find rigid ...
2
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0answers
40 views

Properties of extendable irreducible characters to a normal Sylow subgroup

Let $G$ be a finite group with a nilpotent commutator subgroup, and denote by $mcd(G)$ the set of degrees of the irreducible monomial characters. Suppose $|mcd(g)|=2$. Furthermore, we call a monomial ...
2
votes
0answers
71 views

Length 2 modules over finite dimensional algebras

Given a finite dimensional algebra $A$ over an infinite field and two simple modules $S,T$. Question 1: Is there a useful (homological/computational) crtierion to decide when there are infinitely ...
12
votes
1answer
213 views

Tilting Objects in BGG Categories $\mathcal{O}$

Let $\mathcal{O}$ be the BGG category $\mathcal{O}$ with respect to a finite dimensional semisimple Lie algebra $\mathfrak{g}$ and its Borel subalgebra $\mathfrak{b}$ (as define in this book by ...
2
votes
0answers
54 views

Uniserial modules for group algebras

Recall that a module is uniserial in case it has a unique composition series. Let $G$ be a finite group and $kG$ its group algebra, that we assume is not semi-simple. Questions: Can uniserial ...
7
votes
1answer
384 views

Stabilizer of Sp(n) and U(n) in GL(n)

I would be very grateful for a reference to the following results (which are, I think, true, though I never saw it in the literature). Let $G\subset GL(n,{\Bbb C})$ be $U(n)$, abd $A\in GL(2n,{\Bbb ...
1
vote
0answers
110 views

About Hom and weight space of nilpotent Lie algebra cohomology

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Denote by $\Phi$ the root system of $(\mathfrak{g},\mathfrak{h})$ and denote by $\mathfrak{g}_\alpha$ the root subspace of $\mathfrak{g}$ ...
3
votes
1answer
119 views

Question on $\operatorname{Ext}$ in a local Frobenius algebra

Let $A$ be a finite dimensional local Frobenius algebra with simple module $k$ and an indecomposable non-projective module $M$ (that is also finite dimensional). Question: Is there an example of ...
3
votes
0answers
37 views

About isomorphism of Extension groups between Category $\mathcal{O}^\mathfrak{p}$ and $\mathcal{O}$

In the paper: Kostant modules in blocks of category $\mathcal{O}^\mathfrak{p}$ In section 8.2 (p.22), I use the notation in Humphrey's Category $\mathcal{O}$ book. Then the passage said the ...
5
votes
2answers
216 views

Basic theorem on induction for representations of $p$-adic groups

I know a lot of places where the following is sparsely proved, but I remember there was some paper where I read it in basically the same form I write it, but unfortunately I can't remember where it ...
12
votes
1answer
241 views

Is there a Giambelli identity with dual representations?

For natural numbers $a,b$ with $b\leq n-1$, let $V_{ (a|b)}$ be the irreducible representation of $GL_n$ with highest weight vector $(a+1, 1^b, 0^{n-b-1})$ where the exponentiation denotes repetition. ...
9
votes
1answer
148 views

Representations of degenerate Clifford algebras

Given a real finite-dimensional vector space $V$ with a symmetric bilinear form $b$, we define the Clifford algebra $Cl(V,b)$ as the quotient of the tensor algebra $\bigotimes V$ by the two sided ...
6
votes
0answers
123 views

Which representations of the Lie algebra of a Lie group come from representations of the group itself?

I think this is very classic mathematics, but I can't find a complete answer in the literature. Let $G$ be a Lie group, $\mathfrak{g}$ the Lie algebra of $\mathfrak{g}$. Suppose $\rho : \mathfrak{g} \...
7
votes
2answers
204 views

“Closed bicategories”

I am interested in the following property that a bicategory may or may not have. Let $\mathbf{B}$ be a bicategory. Every one-morphism $f\colon x\rightarrow y$ defines a functor $\mathbf{B}(y,z)\...
7
votes
1answer
140 views

Divisors of the regular character of a finite group

Recall that the regular character $\rho=\hspace{-.2cm}\sum\limits_{\chi\in\operatorname{Irr}(G)}\hspace{-.2cm}\chi(1)\chi$ of a finite group $G$ takes values $$ \rho(g)= \left\{\begin{array}{cl} ...
2
votes
0answers
110 views

Maximal number of $S_n$-conjugates living in a hyperplane

Let $v=(a_1,\dots,a_n)\in\mathbb{R}^n$ where the $a_i$ are distinct and positive. For $\sigma\in S_n$, let $\sigma(v)=(a_{\sigma(1)},\dots,a_{\sigma(n)})$. For any hyperplane $H$ through the origin, ...
8
votes
1answer
204 views

Integral of product of Schur functions

Schur functions are irreducible characters of the unitary group $\mathcal{U}(N)$. This implies the integration formulae $$ \int_{\mathcal{U}(N)}s_\lambda(AUA^\dagger U^\dagger)dU=\frac{|s_\lambda(A)|^...
2
votes
0answers
80 views

On a conjecture about tilting modules

There is the following conjecture on tilting modules (see also History of an open problem on partial tilting modules): Given a partial tilting module $T$ over a finite dimensional algebra $A$ (that ...
2
votes
0answers
28 views

About Extension group in Category $\mathcal{O}^\mathfrak{p}$

Let $M_I(\lambda)$ be the generalized Verma module with highest weight $\lambda$, $L(\lambda)$ be the simple highest weight module with highest weight $\lambda$. Suppose $\mu\le \lambda\le \nu$, does ...
4
votes
0answers
98 views

Tensor product of bimodules

Im not sure whether this question is appropriate for MO, but I do not have much experience with bimodules (or I forgot many things). Let $A$ be a finite dimensional (connected) algebra over a field $...
8
votes
0answers
108 views

$n$-fold tensor products of $D(A)$ for finite dimensional algebras

Let $A$ be a finite dimensional quiver algebra over a field $K$ and let $D(-):=Hom_K(-,K)$ denote the natural duality (assume algebras are connected). Define $\psi_A:=sup \{ n \geq 1 | D(A)^{\otimes ...
4
votes
0answers
76 views

How to determine the unramified character corresponding to an unramified Langlands parameter?

Let $F$ be a p-adic field with ring of integers $\mathcal{O}$. Let $\textbf{G}$ be a connected split reductive algebraic group over $F$. For simplicity, we assume that $\textbf{G}$ is a Chevalley ...
3
votes
1answer
90 views

Are there non-trivial automorphisms of stable framed quiver representations?

Let $Q=(Q_0,Q_1)$ be a quiver and $q\in Q_0$ a chosen vertex. Let $d$ be a dimension vector with $d_q=1$ and let $\theta\in \mathbb R^{Q_0}$ be a $d$-generic stability parameter. Let $M$ be a $\theta$-...
19
votes
1answer
923 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. ...
5
votes
0answers
279 views

Zeta function of the affine Grassmanian and Weil conjecture

Let $G$ be a split connective reductive group over $k=\mathbb F_q$, then the affine Grassmannian $X=Gr_G$ is representable by an ind-projective strict ind-scheme over $k$. (That is, there exists an ...
2
votes
0answers
36 views

If $f \in \operatorname{c-Ind}_Q^P \sigma$, then $f|_N$ is compactly supported modulo $Q \cap N$?

There is a claim in a proof in Casselman's notes on representation theory which I have not been able to verify. I have asked several people and nobody knows why it is true. The thing I can't figure ...
1
vote
0answers
83 views

Question on vanishing Hochschild cohomology

Recall that for an $K$-algebra $A$ with $A^e:=A^{op} \otimes_K A$ the Hochschild cohomology is defined as $HH^n(A,M):=Ext_{A^e}^n(A,M)$. Question: Is there a finite dimensional selfinjective ...
1
vote
1answer
69 views

Gorenstein projective modules of a certain triangular matrix algebra

Let $B$ be a finite dimensional selfinjective algebra over a field $k$ with a finite dimensional non-projective $B$-module $M$ and $$A=\pmatrix{k&M\\0&B}.$$ A module $N$ over an algebra $C$ ...
5
votes
0answers
64 views

Spinor representation for $\operatorname{Spin}(V \oplus V^*)$

I'm studding Hitchin's Generalized Calabi-Yau Manifolds https://arxiv.org/abs/math/0209099 and I've stuck here: Suppose that $V$ is a vector space and denote its dual by $V^*$. Now we know that the $\...
17
votes
2answers
902 views

A character identity

This is related to my question, but it concerns a specific point of the proof of Schur's Theorem. Let $G$ be a finite group and $\chi$ an irreducible character of $G$. Is it true that $$\forall g\in ...
12
votes
1answer
414 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 ...
8
votes
1answer
120 views

Finite multiplicities

Let $G$ be a locally compact group and $\Gamma$ a lattice in $G$. Is it known whether the space $$ \mathrm{Hom}_G\left(\pi,L^2(\Gamma\backslash G)\right) $$ is finite dimensional for $\pi\in\widehat ...
2
votes
0answers
38 views

$c$-matrix reduction in hereditary algebras

Let $k$ be an algebraically closed field, $Q$ be a finite connected quiver and $Q'$ a subquiver of $Q$. Let $C$ be the $c$-matrix of a chamber of the scattering diagram/semi-invariant picture of $kQ$. ...
1
vote
1answer
91 views

finite dimensional modules are highest weight modules [closed]

Let $\mathfrak{g}$ be a basic classical simple Lie super algebra. I want to prove that every finite dimensional module over $\mathfrak{g}$ has a highest weight vector. My feeling is, since $e_i$'s ...
5
votes
1answer
204 views

Calculating the Ext-algebra with a computer

Given a finite dimensional quiver algebra $A$ over an arbitrary field and a module $M$ of finite injective dimension or finite projective dimension. Let $B$ be the Ext algebra of $M$, that is $B:=\...
4
votes
2answers
241 views

Fundamental representations and weight space dimension

For the Lie algebra $\frak{sl}_n$, its fundamental representations can be realised as the exterior powers of the first fundamental representation. From this we can see that their weight spaces are all ...
2
votes
0answers
95 views

About relation between Kostka numbers and Littlewood-Richardson coefficient

The fact that Kostka numbers equals to Littlewood-Richardson coefficients for some partitions is already known $\colon$ \begin{align} K_{\lambda \mu} = c_{\sigma \lambda}^\tau \end{align} where $\...
5
votes
0answers
80 views

Derived invariant acyclic algebras

Call a connected quiver algebra $A=KQ/I$ (finite quiver Q and admissible ideal I) derived-invariant in case every quiver algebra derived equivalent to $A$ is even isomorphic to $A$. For example local ...
3
votes
1answer
114 views

If the sum of a right (principal) ideal with a left one contains an invertible element and the product is zero then do they contain idempotents?

I am trying to solve a problem on additive categories, that gives the following question on (non-commutative unital associative) rings: if for elements $a$ and $b$ of a ring $R$ we have $ab=0$ and $a+...
6
votes
0answers
73 views

Numbers where there is a unique group with integral character table

Call a number $n$ special in case there is a unique group of order n whose character table consists of only integers. This should be equivalent to the condition that the number of conjugacy classes ...
29
votes
1answer
675 views

Number of irreducible representations of a finite group over a field of characteristic 0

Let $G$ be a finite group and $K$ a field with $\mathbb{Q} \subseteq K \subseteq \mathbb{C}$. For $K=\mathbb{C}$ the number of irreducible representations of $KG$ is equal to the number of conjugacy ...
12
votes
3answers
485 views

Which partitions realise group algebras of finite groups?

Fix an algebraically closed field $K$ (maybe of characteristic zero first for simplicity, like $\mathbb{C}$). Given a partition $p=[a_1,...,a_m]$ of an integer $n$. We can identify $p$ with the ...
6
votes
2answers
303 views

How should I think about the Grothendieck-Springer alteration?

Given a simple complex Lie algebra $\mathfrak{g}$, recall the Springer resolution of its nilpotent cone $\widetilde{\mathcal{N}}\to \mathcal{N}$. Several times I have seen someone explaining Springer ...