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Questions tagged [rt.representation-theory]

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

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Question on two types of Frobenius theorem in $p$-adic groups

Let $G$ be a $p$-adic classical group and let $P_0$ be a minimal parabolic subgroup of $G$. Let $P=MN$ be a standard parabolic subgroup containing $P_0$. Let $\text{Ind}$ and $ |_M$ be the normalized ...
Andrew's user avatar
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Matrix of the minimal projective presentation of a $\tau$-rigid module

Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbb{K}$ of characteristics zero. Suppose $A$ is given by the bound quiver $(Q,I)$. We will use $P_l$ to denote the ...
It'sMe's user avatar
  • 767
2 votes
1 answer
149 views

Isomorphisms $S^d(S^m(V)^*) \cong \Lambda^d(S^{m+d-1}(V)^*)$

$\DeclareMathOperator\SL{SL}$Let $K$ be an algebraically closed field. Let $V$ be a 2-dimensional vector space over $K$. The group $G=\SL_2(K)$ acts naturally on $V$ by left-multiplication on column ...
Jon Elmer's user avatar
  • 155
2 votes
0 answers
44 views

How to multiply dots with Young idempotents in the degenerate affine Hecke algebra (type A)

Let $\widehat{\cal H}_n$ be the type A degenerate affine Hecke algebra on $n$ strands, and let $x_1,\cdots,x_n$ be the dots. Inside of this algebra lies the algebra $\mathbb C S_n$, and the Young ...
Fan Zhou's user avatar
  • 301
3 votes
2 answers
123 views

Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$

Let $G$ be a connected algebraic group over an algebraically closed field $k$ of characteristic zero (I'm mostly interested in the case of a reductive group). By the Tannakian formalism, $G(k)$ can be ...
Antoine Labelle's user avatar
2 votes
0 answers
47 views

Have analytic continuations of holomorphic discrete series representations a formal dimension?

Let $G$ be a connected real semisimple Lie group, for example, $SU(p,q)$. I am interested in unitarizable highest weight representations for $G$. If $K$ denotes a maximal compact subgroup of $G$ and $\...
d'Alembert's user avatar
5 votes
1 answer
191 views

Characters with all higher exterior powers irreducible

Let $G$ be a finite group and we take for the field the complex numbers. Call an irreducible character $\xi$ with degree $m$ of $G$ perfect, if all exterior powers $\bigwedge\nolimits^k \xi$ are ...
Mare's user avatar
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4 votes
0 answers
80 views

Irreducible representation of $\mathrm{GL}(2,\mathbb{R})$ that is not admissible

It is a basic theorem of Harish-Chandra that an irreducible unitary representation $\pi$ of a reductive group $G$ over $\mathbb{R}$ on a Hilbert space is admissible, meaning that every irreducible ...
Stefan  Dawydiak's user avatar
1 vote
0 answers
88 views

An interesting identity involving skew-Schur functions

Denote $\rho=(-\frac12,-\frac32,-\frac52,\dots)$. I was reading this interesting paper, where in particular, the authors claim that one can get the expression in (2.9) \begin{align*} \prod_{k\geq1}(1+...
T. Amdeberhan's user avatar
3 votes
0 answers
64 views

Connection between certain finite groups and Frobenius algebras

This question is maybe not so precise yet, but contains some ideas that maybe just search for the right definition. Let $G=F/U$ be a finite group with presentation $F/U$ (here the presentation really ...
Mare's user avatar
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-5 votes
0 answers
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What is wrong with these solutions? Yang-Mills problem in R4? [closed]

My question would be what is wrong with these solutions of the Yang-Mills problem? https://arxiv.org/pdf/2203.14279.pdf https://link.springer.com/content/pdf/10.1134/S1061920820010033.pdf https://www....
Mills's user avatar
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7 votes
1 answer
211 views

Quiver and relations for a monoid related to Catalan numbers

Let $C_n$ be the monoid consisting of monotone maps $\{1,...,n\} \rightarrow \{1,...,n\}$ with $f(i) \leq i$ for all $i$. The cardinality of $C_n$ is given by the Catalan numbers. Consider $A_n= \...
Mare's user avatar
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2 votes
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41 views

$K_0$-basis modules with a unique extension related to parking functions

Let $B=B_n$ be the quiver algebra of type $A_n$ (with some orietnation) with $n$ points. A $B$-module $M$ is a basis of the Grothendieck group $K_0$ if $M$ has $n$ indecomposable direct summands and ...
Mare's user avatar
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3 votes
1 answer
59 views

Does the reproducing property of the unitary group Poisson kernel require a multiple of the identity?

The Poisson kernel of the unitary group is $$ P(Z,U)=\frac{\det(1-ZZ^\dagger)^N}{\det(1-ZU^\dagger)^N\det(1-UZ^\dagger)^N}.$$ It has a reproducing property, $\int dU P(Z,U)f(U)=f(Z)$, where $dU$ is ...
Marcel's user avatar
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5 votes
0 answers
149 views

Decomposing an endomorphism as a tensor product

$\DeclareMathOperator\End{End}$Let $f$ be an endomorphism of the finite-dimensional vector space $V$, over the field $K$. The question of whether $f$ is decomposable, that is, whether $V$ can be ...
Pierre's user avatar
  • 2,195
4 votes
1 answer
227 views

The number of irreducible characters of simple groups of Lie type

Let $S$ be a simple group of Lie type defining over $\mathbb{F}_q$ where $q$ is a power of a prime $p$ and let $\sigma$ a nontrivial field automorphism of $S$. Set $\mathrm{C}_{S}(\sigma)$ the ...
user44312's user avatar
  • 551
16 votes
3 answers
928 views

Conjectures in the representation theory of the symmetric group

Question: What are current open conjectures about the representation theory of the symmetric group? I am interested mostly in the characteristic 0 case, but conjectures for the modular case can also ...
Mare's user avatar
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1 vote
0 answers
40 views

Absolute irreducibility implies free action on framed universal deformation ring

Let $\overline{\rho}: G\longrightarrow \text{GL}_n(\mathbb{F}_p)$ be a residual representation of the Galois group of a number field. Let $R_{\overline{\rho}}^{\square,\text{univ}}$ be the universal ...
kindasorta's user avatar
  • 1,729
4 votes
0 answers
56 views

Ribbon fusion categories for quantum $\mathfrak{sl}_2$ at odd roots of unity

I will work over $\mathbb{C}$. Let $q=e^{2\pi i/N}$, and write $U_{q}(\mathfrak{sl}_2)$ for Lusztig's divided power quantum group for $\mathfrak{sl}_2$. One can associate to $U_{q}(\mathfrak{sl}_2)$ a ...
Thibault Décoppet's user avatar
5 votes
2 answers
285 views

Simple connectedness of Levi subgroup

Let us consider a connected and simply connected semisimple algebraic group $G$ over $\mathbb{C}$, $B$ a Borel subgroup and $T$ a maximal torus contained in $B$. Let $P_1$, $P_2$ be two standard ...
fool rabbit's user avatar
3 votes
0 answers
97 views

Representations of a reductive Lie group vie central character and K-types

Let $G$ be a real reductive group, let $\widehat G$ denote the unitary dual and $\widehat G_{adm}\supset\widehat G$ be the admissible dual, i.e., the set of isomorphy classes of irreducible Harish-...
Nandor's user avatar
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2 votes
0 answers
111 views

What can be said about these tensor representations of $\mathrm{SL}(2)$?

Let $W = V \otimes \dots \otimes V$, the product of $n$ copies of $V = \mathbb{C}^2$. Let $G$ and $H$ be two subgroups of the symmetric group $S_n$ and let $\chi$ be a character of $G$. Associated to $...
Malkoun's user avatar
  • 5,021
1 vote
0 answers
37 views

Simple highest weight modules of quantum affine algebras

Let $U_q(\hat{\mathfrak g})$ be a quantum affine algebra, and let $L$ be an integrable simple highest weight module of $U_q(\hat{\mathfrak g})$. In [Lu], Lusztig proved that the limit of $L$ when $q\...
qweqwe's user avatar
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2 votes
0 answers
54 views

Is anything known about the center of the Fomin-Kirillov algebra?

Let $\mathcal{B}_{\mathbb{S}_m}$ be the quotient of the Fomin-Kirillov algebra so that its pairing becomes certainly nondegenerate. This algebra is conjecturally isomorphic to the Fomin-Kirillov ...
Christoph Mark's user avatar
16 votes
0 answers
318 views

Number of $F_p$-matrices ac=ca, bd = db , ad - da = cb - bc is polynomial in p ? ("Manin matrix variety" - normal ? Cohen–Macaulay ? )

Consider four $n\times n$ matrices $a,b,c,d$ over finite field $F_q$ (or $F_p$ for simplicity), such that they satisfy three equations: $ac=ca,bd=db, ad-da=cb-bc $. Thus an affine algebraic manifold ...
Alexander Chervov's user avatar
13 votes
1 answer
554 views

Is algebra: ac=ca, bd = db , ad - da = cb - bc ("Manin matrix algebra") - a Koszul algebra?

Question: Consider quadratic algebra with four generators $a,b,c,d,d$ and three relations $ac=ca,bd = db, ad-da = cb - bc$ . Is it a Koszul algebra ? (i.e. Koszul complex is resolution of ground field ...
Alexander Chervov's user avatar
4 votes
1 answer
208 views

Multiplication factors in folding root systems and Lie algebras by automorphisms

When Stembridge, in the paper Folding by automorphisms, considers folding by automorphism $\sigma$ he considers the root system generated by for each orbit $J$. $$\sum_{i \in J} \alpha_i .$$ Whereas ...
Smith's user avatar
  • 53
5 votes
1 answer
100 views

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 $\...
Zoltan Fleishman's user avatar
1 vote
0 answers
138 views

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 ...
kindasorta's user avatar
  • 1,729
2 votes
1 answer
138 views

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 ...
Local's user avatar
  • 108
1 vote
0 answers
38 views

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 ...
Andrew's user avatar
  • 855
1 vote
0 answers
139 views

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{...
user509119's user avatar
-3 votes
0 answers
133 views

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 ...
Dale's user avatar
  • 431
3 votes
0 answers
156 views

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 ...
Alexander Chervov's user avatar
0 votes
0 answers
108 views

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-...
MathMath's user avatar
  • 1,265
20 votes
3 answers
679 views

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 ...
Estwald's user avatar
  • 1,187
1 vote
0 answers
96 views

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_\...
Benjamin Steinberg's user avatar
1 vote
0 answers
146 views

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\...
Dima Pasechnik's user avatar
6 votes
1 answer
161 views

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 $...
Mare's user avatar
  • 26.2k
3 votes
0 answers
108 views

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
1 vote
0 answers
114 views

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
  • 21.1k
1 vote
1 answer
168 views

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
98 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
  • 21.1k
3 votes
2 answers
338 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
0 answers
119 views

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
  • 141
7 votes
0 answers
196 views

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
0 answers
46 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 $A$ ...
Mare's user avatar
  • 26.2k
1 vote
0 answers
90 views

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
  • 5,021
2 votes
0 answers
44 views

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
  • 26.2k
4 votes
2 answers
221 views

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
  • 1,216

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