Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
6,802
questions
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Does the first fundamental representation of $\frak{sp}_n$ generates all the other fundamental representations
Let $\mathfrak{sp_n}$ be the symplectic Lie algebra, that is, the $C_n$ complex simple Lie algebra. Is it true that the first fundamental, which is to say the vector space, representation $V_1$ of $\...
1
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0
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87
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Homotopical interpretation of Langlands correspondence
Recently I began learning about homotopy theory, I am very far from being familiar with all the basic notions and constructions, however I heard of the notion of topological modular forms. I also ...
1
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0
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67
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Trivial representation of a maximal torus
Let G be a connected compact Lie group and $T\subset G$ a maximal torus. For an irreducible representation $V_\lambda$ of $G$, the multiplicity of the trivial representation of $T$ in $V_\lambda$ is ...
0
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76
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A criterion for when a symmetrized decomposable tensor is nonzero
In the problem I am interested in, one has a collection of vector spaces, all $2$-dimensional, denoted by $V_{i, j}$, where $1 \leq i \neq j \leq n$. In other words, the set of indices is
$$S = \{(i, ...
1
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0
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23
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Genericity of local representation with a non-generic local A-parameter
Let $\pi$ be an irreducible smooth representation of a classical $p$-adic group. Suppose that $\pi$ has a local L-parameter associated to some non-generic local A-parameter $\psi$. Then I am wondering ...
1
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0
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97
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Mulitplicity one property for $\mathcal{D}'$ and $L^2$ over a homogeneous space
Let $G$ and $G_0$ be Lie groups, and suppose that a homogeneous space $X=G/G_0$ have a $G$-invariant measure.
It is known (E.G.F. Thomas showed) that there is an admissible parametrization $\{\mathcal{...
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0
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117
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Extending the proof of Maschke's Theorem from finite groups to algebras
In the theory of representations of a finite group there is Maschke's Theorem that any finite-dimensional representation of a finite group $G$ can be decomposed into a direct sum of irreducible ...
3
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134
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What d.o. $\sum_i f_i(z)\partial_z^i$ correspond to subalgebras $M$ in polynoms $C[x_i]$ being Langlands dual to motive of $Spec(M) \to X$?
Briefly: The question is about presenting explicit examples of the construction discussed in the recent MO question "Relation between motives and geometric Langlands" and Will Sawin's asnwer ...
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96
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Representation of anti-commuting matrices in $\mathbb{C}^{2}$
This is a cross posting updated question from MSE. I have not got any answers there yet and I really want to understand this problem.
The basic question is the following. Let $V$ be a finite-...
20
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3
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Examples when quantum $q$ equals to arithmetic $q$
First, as a disclaimer, I should say that this post is not about any specific propositions, but is more of some philosophical flavor.
In the world of quantum mathematics, the letter $q$ is a standard ...
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Computing the induced homomorphisms of derived functors using acyclic resolutions
Let's suppose that $F\colon \mathcal A\to \mathcal B$ is a right exact additive functor between abelian categories such that $\mathcal A$ has enough projectives. Standard references shows that if $Q_\...
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139
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Efficient decomposition algorithm for characters of symmetric groups
Let $\chi$ be a rational character of $G:=S_n$, and we want to know whether it decomposes into irreducibles $\chi_\lambda$, for $\lambda\in\Lambda$, with $\Lambda$ given, as
$$
\chi=\sum_{\lambda\in\...
6
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Known posets of tilting modules for finite dimensional algebras
Question: For which classes of finite dimensional algebras $A$ is the poset of tilting $A$-modules known?
Here two famous examples:
-For the path algebra of a linear oriented quiver of Dynkin type $...
3
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0
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102
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In how many ways does a Lie algebra decompose as an orthogonal direct sum of Cartans?
For a prime $p$, the Lie algebra $\mathfrak{su}(p)$ can be decomposed into an orthogonal direct sum of $p+1$ Cartan subalgebras as follows. Consider the clock and shift matrices — these are a pair of ...
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Classification of complex irreducible representations of $\mathrm{GL}_n(\mathbb{F}_q)$ [duplicate]
Is there a classification of complex irreducible representations of the group $\operatorname{GL}_n(\mathbb{F}_q)$, where $\mathbb{F}_q$ is a finite field with $q$ elements?
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Is the irreducible $ \mathrm{SU}(3) $ subgroup of $ \mathrm{SU}(6) $ maximal?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\S{S}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PSL{PSL}\...
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1
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Irreducible subspaces in the space of functions on Grassmannian acted by $\mathrm{GL}_n(\mathbb{F}_q)$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gr{Gr}$Let $\mathbb{F}_q$ be a finite field with $q$ elements. Let $\Gr_{i,n}(\mathbb{F}_q)$ denote the Grassmannian of linear $i$-dimensional ...
3
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1
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Cohomology of the partial flag variety associated with the minimal nilpotent orbit
Let $G$ be a semi-simple group over complex number; for simplicity let us assume that it is simply laced. Let $X$ be the orbit of the highest root line in the adjoint representation of $G$. This is a ...
4
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Representation theory of spinors - Understanding how $\mathrm{SO}_3$ acts in particle physics
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I have started to study particle physics, beginning with wikipedia and I am now reading David ...
7
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Chevalley-Solomon formula and Weyl character formula
Let $\Phi\subset V$ be a root system of rank $r$ with Weyl group $W$, a choice of positive roots $\Phi_+$ and exponents $d_1, \ldots, d_r$ (i.e. the invariant algebra $(\operatorname{Sym}^\bullet V)^W$...
3
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2-term silting objects in algebras of global dimension at most 2
Let $P$ be a 2-term silting complex (see defintion 2.1 in https://ntnuopen.ntnu.no/ntnu-xmlui/bitstream/handle/11250/2639675/survey_revised14042019.pdf?sequence=1 )in a finite dimensional algebra $A$ ...
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0
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The size of $\operatorname{Ext}^1(M,N)$ for $M$, $N$ modules of $\operatorname{SL}_n(\mathbb{F}_q)$
I was just wondering has there been any estimation or exact evaluation on the size (cardinality) of $\operatorname{Ext}^1(M,N)$, where $M$, $N$ are modules of $\operatorname{SL}_n(\mathbb{F}_q)$? What ...
1
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0
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82
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Is there a quaternionic analogue of Weyl's character formula?
I am pondering about this. The definition of a character makes sense for quaternionic matrices. Indeed, given a quaternionic representation of a quaternionic matrix group such as $GL(n, \mathbb{H})$ ...
2
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0
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42
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Depth and codepth of an algebra
Let $A$ be a finite dimensional $K$-algebra over a field $K$ and
$0 \rightarrow A \rightarrow I^0 \rightarrow I^1 \rightarrow \cdots$
a minimal injective coresolution of the regular module $A$.
The ...
4
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2
answers
214
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Order of abelian subgroup of the automorphism group of an abelian group
Suppose $A$ is a finite abelian group and $B\leq\operatorname{Aut}(A)$ is abelian. Is it possible for the order of $B$ to be strictly greater than the order of $A$? What if I additionally impose that ...
1
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0
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116
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A question related to Kirillov model
I am reading Jackson - The theory of admissible representations of $\operatorname{GL}(2, F)$ and am not able to understand the following map related to Kirillov model. This result appears on p. 54:
I ...
3
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0
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Question on the genericity of unramified representation
Let $F$ be a $p$-adic local field and $W$ be a 2n-dimensional symplectic space over $F$. Let $G_n$ be the isometry group of $W$ and $B_n$ be the Borel subgroup of $G_n$. Then the maximal torus $T_n$ ...
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0
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108
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What is the "weight" of an automorphic form for $\mathrm{PGL}_2$?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PGL{PGL}$I'm trying to understand what the notion of "weight" is for automorphic forms over $\GL_2(F)$ where $F$ is some number field, in ...
0
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Integral representation of completely alternating homogeneous functionals on semi-lattice of continuous functions
For a long time I've been interested in G. Choquet seminal work "Theory of capacities" (Annales de l’institut Fourier, tome 5 (1954), p. 131-295). More precisely part 53 about integral ...
4
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2
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References for $K$-orbits in $G/B$
Let $G$ be a reductive group, $K$ a symmetric subgroup of $G$ (e.g., fixed point of an involution), and $B$ a Borel subgroup of $G$. Then it is well known that $G/B$ has finitely many $K$-orbits. ...
6
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Does every Lie algebra appear as centralizer of an element in a semisimple Lie algebra?
Given a finite dimensional, complex, semisimple (fcss) Lie algebra $\mathfrak{g}$ and an element $x\in\mathfrak{g}$, denote by $\mathfrak{g}^x$ the centralizer of $x$ in $\mathfrak{g}$ i.e. the set $\{...
2
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About normal states in abstract von Neumann algebras
In the book "Fundamental of the theory of operator algebras" (KAdisong and Ringrose, Vol 2) we have the Corollary 7.1.16
but this was state only for concrete von Neumann algebras (because ...
2
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0
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91
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On the irreducible submodules of adjoint representations $\text{ad}^{0}$
Let $k$ be a finite field of characteristic $p$. Let $H$ be a subgroup of $\rm{GL}_{n}(k)$ of order prime to $p$ where $n\geq2$. Assume that the representation $H\hookrightarrow \rm{GL}_{n}(k)$ is ...
9
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Mappings of the sphere (to itself) defined by homogeneous polynomials
Preamble
$\DeclareMathOperator\SO{SO}$Let $\mathbb{S}^m\subset \mathbb{R}^{m+1}$ be the standard unit sphere. An observation of Do Carmo and Wallach states that
If $G$ is a subgroup of $\SO(m+1)$ ...
3
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0
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A class of bipartite graphs appearing in higher Auslander--Reiten theory
Let $G = (V,E)$ be a simple undirected bipartite graph with vertices $V$, edges $E$, and a chosen partition $V = X \cup Y$.
Recall that the bipartite complement of $G$ is the graph on the same vertex ...
2
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0
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Is there any (specially Algebraic Geometrical) exposition of Koike Terada's Young-diagrammatic methods for the representation theory paper?
I am talking about the paper by Koike, Kazuhiko and Terada, Itaru, Young-diagrammatic methods for the representation theory of the classical groups of type ($B_n$), ($C_n$), ($D_n$), J. Algebra 107, ...
2
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1
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157
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Integral over the space of $p$-adic matrices
$\DeclareMathOperator\Mat{Mat}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers with a uniformizer $\pi$. Let $|\cdot|\colon \mathbb{F}\to \mathbb{R}$ be ...
13
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Is there a simple proof that representations of GL(n,k) are determined by their restriction to diagonal matrices?
Let $k$ be a field of characteristic zero. The general linear group $\mathrm{GL}(n,k)$ has a subgroup $\mathrm{D}(n,k)$ consisting of invertible diagonal matrices. These are linear algebraic groups ...
6
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1
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Tame-Wild dichotomy; why can't tame algebras be wild?
I would like to understand the Tame-Wild dichotomy, and in particular why an algebra cannot be tame and (semi-)wild at the same time. I've looked in the papers by Drozd and Crawley-Boevey [D80, CB88].
...
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0
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Affine Springer fibers for symmetric spaces
Springer fibers are defined to be the varieties of "isotropic" full flags which are fixed by a certain element in the symmetric space. In a similar manner, affine Springer fibers can be ...
6
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1
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Trying to understand the topology of the Weil group for the local Langlands conjecture
I am trying to study the representation of the Weil group from the book "The Local Langlands Conjecture for $GL(2)$". I have some problem with the topology of this group.
Let $F$ be a non ...
4
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0
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103
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Hecke algebra $\mathcal{H}(K_1\backslash \mathrm{GL}_n(\mathbb{F})/K_1)$
$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers, and let $\frak{m}$ be its maximal ideal Let $\GL_n(\mathcal{O})$ be the group ...
5
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0
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Spherical functions in the space of functions on real Grassmannians
Let $G=O(n)$ be the orthogonal group. Let $K=S(O(k)\times O(n-k))$ be the subgroup of $O(n)$.
Then the pair $(G,K)$ is symmetric, and the homomogeneous space $G/K$ is the Grassmannian of $k$-...
4
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1
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Symplectic structure of Higgs branch
I've been reading Kamnitzer's survey Symplectic resolutions, symplectic duality, and Coulomb branches. Here the Higgs branch is defined as a projective GIT quotient, but I couldn't figure out how the ...
3
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0
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89
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Auslander-Reiten sequences where irreducible morphisms are all epi/mono
Let's work in the setting of modules over an Artin algebra $A$, or a finite-dimensional $k$-algebra $A$, or if you like, modules over a connected quiver $Q$ without oriented cycles.
Let $M$ be such a ...
2
votes
0
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62
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Bounds for sum of the homological dimensions in the incidence algebra of a Boolean lattice
Let $A$ be a finite dimensional algebra.
Define $\varphi_A:= \sup \{ \operatorname{pd} M + \operatorname{id} M \mid M \in \operatorname{ind}(A) \}$, where $\operatorname{pd} M$ denotes the projective ...
2
votes
2
answers
203
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Factorizations of an $n$-cycle in $S_n$ into a product $xy$ where $|x| = 2, |y| = 3$
Let $S_n$ be the symmetric group on $n$ letters. When (and how) can an $n$-cycle in $S_n$ be factored into a product $xy$, where $x,y$ have orders 2,3 respectively?
More precisely, I'd like to ...
1
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0
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126
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Exploring the Intersection of Expander Graphs, Number Theory, Representation Theory and Recent Computer Science Developments [closed]
I have a solid understanding of the basics of expander graphs and their properties and the recent development of High-Dimensional Expanders and their application to Random Walks, along with other ...
1
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0
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81
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Question on the geometric lemma in $p$-adic representation theory
$\DeclareMathOperator\GL{GL} \DeclareMathOperator\Sp{Sp} \DeclareMathOperator\Ind{Ind}\DeclareMathOperator\B{B} $
Let $F$ be a $p$-adic field and $\Sp_{2n}$ the symplectic group over a $2n$-...
2
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2
answers
124
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Infinite radical ideal cubed equals zero for tame hereditary Artin algebras
Let $A$ be a tame hereditary Artin algbera and mod$A$ the category of finitely generated (left) $A$-modules. Further, let rad$_A$ be the radical ideal of mod$A$, which is the smallest ideal containing ...