Questions tagged [rt.representation-theory]

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

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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 ...
Theo Johnson-Freyd's user avatar
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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?
asv's user avatar
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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}\...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
62 views

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 ...
asv's user avatar
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3 votes
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277 views

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 ...
Alexander Braverman's user avatar
4 votes
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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 ...
Andrea's user avatar
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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 $(\operatorname{Sym}^\bullet V)^W$...
Antoine Labelle's user avatar
3 votes
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38 views

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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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 ...
user236626's user avatar
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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 $GL(n, \mathbb{H})$ ...
Malkoun's user avatar
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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 ...
Mare's user avatar
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4 votes
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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 ...
tomasz's user avatar
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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 ...
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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 $T_n$ ...
Andrew's user avatar
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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 ...
HASouza's user avatar
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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 ...
Vladimir B.'s user avatar
4 votes
2 answers
168 views

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. ...
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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 $\{...
Hugo MTV's user avatar
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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 ...
Gabriel Palau's user avatar
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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 ...
stupid boy's user avatar
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303 views

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 $\SO(m+1)$ ...
Willie Wong's user avatar
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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 ...
Isle of sand's user avatar
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61 views

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, ...
Dibyendu's user avatar
2 votes
1 answer
146 views

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 ...
asv's user avatar
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13 votes
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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 ...
John Baez's user avatar
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6 votes
1 answer
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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]. ...
Jacob FG's user avatar
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0 answers
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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 ...
211's user avatar
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6 votes
1 answer
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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 ...
Mario's user avatar
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4 votes
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98 views

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 ...
asv's user avatar
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5 votes
0 answers
77 views

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 $k$-...
asv's user avatar
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4 votes
1 answer
158 views

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 ...
Ji Woong Park's user avatar
3 votes
0 answers
89 views

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 ...
Marty's user avatar
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2 votes
0 answers
62 views

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 ...
Mare's user avatar
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2 votes
2 answers
200 views

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 ...
stupid_question_bot's user avatar
1 vote
0 answers
125 views

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 ...
total dependent random choice's user avatar
1 vote
0 answers
76 views

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 $2n$-...
Andrew's user avatar
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2 votes
2 answers
123 views

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 ...
kevkev1695's user avatar
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4 votes
1 answer
143 views

Representations of $\mathrm{GL}_n(\mathcal{O})$ in functions on Grassmannians

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gr{Gr}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers. The natural representation of the group $\GL_n(\...
asv's user avatar
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3 votes
1 answer
262 views

Can we relate the character of the permutation representation of $G$ on the cosets $G/\langle g_i\rangle$ to the number of cycles of $g_i$?

Let $G$ be a finite group generated by permutations $g_1,\dots,g_s$ such that $g_1g_2\cdots g_s$ is the identity permutation. The corresponding Hurwitz representation $V_{\text{Hur}}$ has character $$\...
utx7563yu's user avatar
1 vote
0 answers
54 views

Simplest way to classify reducibility of principal series for $p$-adic $\mathrm{SL}_2$

Let $F$ be a $p$-adic field and $G_1=\mathrm{SL}_2(F)\subset \mathrm{GL}_2(F)=G$. For simplicity, we assume $p>2$. Denote by $|\cdot|$ the normalized absolute value on $F$. Here I shall focus on ...
youknowwho's user avatar
8 votes
1 answer
260 views

Some fusion rings/categories I don't recognize

Recently (what I believe are) all multiplicity-free fusion categories up to rank 7 have been posted on the AnyonWiki. Most of the fusion rings belonging to these categories belong to one of the ...
Gert's user avatar
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3 votes
0 answers
121 views

Composition of Frobenius $n$-homomorphisms, characteristic-free?

This question is, as so often, a crossbreed of curiosity and laziness. The former has led me to an interesting, but somewhat unsatisfactory paper by Khudaverdian and Voronov (arXiv:2002.02395v2) and, ...
darij grinberg's user avatar
9 votes
3 answers
302 views

$G$-module structure of the relation module for a presentation of a finite group $G$

Let $F_n$ be a free group of rank $n\ge 2$, and $F_n\rightarrow G$ a surjection with $G$ finite. Let $R$ be the kernel. From this, we get an action of $G$ on the abelianization $R/R'$ (a free abelian ...
stupid_question_bot's user avatar
2 votes
0 answers
68 views

Simple question on the genericity of induced representation

$\DeclareMathOperator\GL{GL} \DeclareMathOperator\Sp{Sp} \DeclareMathOperator\Ind{Ind}$ Let $F$ be a $p$-adic field and $\Sp(2n)$ symplectic group over 2n dimensional symplectic space over $F$. Let $B=...
Andrew's user avatar
  • 865
5 votes
2 answers
196 views

What are the Schur indices of irreducible representations of $\operatorname{SL}(2,p)$?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$What are the Schur indices of the irreps of $\SL(2,p)$? ($p$ an odd prime.) Presumably this is in a book somewhere? Section 6 of the paper &...
stupid_question_bot's user avatar
1 vote
0 answers
155 views

Applications of Artin's theorem on induced representations

Let $G$ be a finite group and let $R(G)$ be the (complex) representation ring of $G$. As stated in Serre's book on representation theory, Artin's theorem says the following: Theorem: Let $X$ be a ...
Jean-Pierre's user avatar
1 vote
0 answers
90 views

Question on the unramified representation

$\DeclareMathOperator\GL{GL}$Let $F$ be a $p$-adic field and $\chi$ be an unramified character of $\GL_1(F)$. Consider an induced representation $\pi$ of $\GL_2(F)$ induced from the character $\chi|\...
Andrew's user avatar
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0 votes
0 answers
46 views

A question on projective unitary representation of a Lie group

$\DeclareMathOperator\GL{GL}$Let $\mathcal{H}$ be a Hilbert space and $\GL(\mathcal{H})$ denote the group of invertible linear transformations of $\mathcal{H}$. Assume that $G=\{ f:\mathbb{P}\mathcal{...
Mahtab's user avatar
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6 votes
1 answer
371 views
+50

Symmetric power lift of modular forms

Let $f_1$ and $f_2$ be two cuspforms of weights $k_1$ and $k_2$ and nebentypus $\epsilon_1$ and $\epsilon_2$ respectively such that $f_1 \neq f_2 \otimes \chi$ for some Dirichlet character $\chi$ of ...
user15243's user avatar
  • 474
14 votes
0 answers
469 views

Is the monster group maximal in SO(196883)?

$\DeclareMathOperator\SO{SO}$The smallest degree of a nontrivial complex representation of the monster group $ M $ is $ 196883 $. This irrep has Schur indicator $ 1 $, so the image must lie in the ...
Ian Gershon Teixeira's user avatar

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