Questions tagged [rt.representation-theory]

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

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Holomorphic discrete series vs. discrete series

(I apologize in advance if this question is too naive for experts.) Let $G$ be a real semisimple Lie group. I know that holomorphic discrete series representations are only a part of all the discrete ...
youknowwho's user avatar
4 votes
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143 views

Major indices of standard tableaux of shapes obtained from addable cells of a given Young diagram

I have a "very" indirect proof that the following fact is true for every Young diagram $\lambda \vdash n$ and every $r \in \{0,\dotsc,n\}$: \begin{equation} d_\lambda = \sum_{a \in \mathrm{...
dmitry's user avatar
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What is a Gelfand-Tsetlin subalgebra?

For context on general Gelfand-Tsetlin theory, see for instance this MO post. Let's work over $\mathbb{C}$. Fix $n>0$. There is a natural chain of embeddings of the general linear Lie algebras $\...
jg1896's user avatar
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What's an example of an $n$-step nilpotent Lie algebra whose centre is not $g^{(n)}$?

Let $\mathfrak g$ be a Lie algebra. $\mathfrak g^{(1)}=\mathfrak g$ and $\mathfrak g^{(n+1)}=[\mathfrak g,\mathfrak g^{(n)}]=\mathbb R$-span$\{[X,Y]:X\in\mathfrak g,Y\in\mathfrak g^{(n)}\}$. The ...
enihcamemit's user avatar
2 votes
0 answers
112 views

Lie Algebra representations outside of generalized central characters

For a simple Lie algebra $\mathfrak{g}$, we can view its category of representations as fibered over $\operatorname{Spec}Z(\mathfrak{g})$ (a representation will lie over a point if the center's action ...
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Isomorphism and counting for tree quivers

Let $Q$ be a quiver which is a connected tree and let $A=KQ/I$ be a quiver algebra with $I$ an admissible ideal, meaning that $I$ is generated by paths of length $\geq 2$. Let $n$ be the number of ...
Mare's user avatar
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23 votes
2 answers
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Solvable groups that are linear over $\mathbb{C}$ but not over $\mathbb{Q}$?

Let $\Gamma$ be a finitely generated finitely presented virtually solvable group. Assume that there exists an injective representation $\Gamma \to \operatorname{GL}_n(\mathbb{C})$. Is it true that ...
 V. Rogov's user avatar
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Are there natural isomorphisms $S^{(2,1)}(k^{m+1})\cong k^2\otimes W$?

In this popular 2019 MO question, user მამუკა ჯიბლაძე asked: The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism between ...
Alvaro Martinez's user avatar
3 votes
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Commutant of irrep of $S_n$ (over local field)

Let $k$ be a field of characteristic zero and let $(V, \rho)$ be a finite-dimensional representation over $k$ of the symmetric group $S_n$. I would like to understand the commutant $\operatorname{End}...
bsbb4's user avatar
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Hattori-Stallings trace

Let $R$ be a (possibly non-commutative) unital ring and $M$ be a left $R$-module. If $M$ is finitely generated and projective, the natural map $$\iota:\mathrm{Hom}_R(M,R)\otimes_R M\to \mathrm{Hom}_R(...
Qwert Otto's user avatar
5 votes
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Rings where all indecomposable modules are projective or injective

Let $A$ be a semi-perfect noetherian ring. Is there a nice classification of such $A$ such that every indecomposable finitely generated $A$-module is projective or injective? Im also interested in ...
Mare's user avatar
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-3 votes
1 answer
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On Haar measure and Spherical measure [closed]

Let $d$-dimensional complex sphere be $$\{(c_1,\cdots,c_{d})\sum_{i=1}^{d} |c_i|^2=1.\}$$ We can define the Haar measure on this sphere by regarding the unitary group $U(d)$. We can regard the $d$-...
gondolf's user avatar
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How a circle $S^1$ acts on the Cayley plane $OP^2$ with exactly three fixed points?

The complex projective $CP^2$, the quaternionic projective space $HP^2$, and the octonionic projective space $OP^2$ each admit a circle action with $3$ fixed points. The circle action on $HP^2$ can be ...
hao dong's user avatar
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What's the point of geometric representation theory?

Please forgive the provocative title, what I mean is the following: One can find representations of Lie algebras in geometric settings, the most famous being the Bott–Borel–Weil theory. However, ...
Béla Fürdőház 's user avatar
4 votes
0 answers
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What is $\dim D^{\lambda}$ for the symmetric group?

What are the dimensions of the simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\perp}$ for the modular representation theory of $S_n$, i.e. $\operatorname{char}(k)=p>0$? I ...
Jackson Walters's user avatar
3 votes
1 answer
202 views

Asymptotics for number of $p$-regular partitions of $n$

The number of simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\bot}$ of the symmetric group over a field $k$ such that $\text{char}(k)=p > 0$ is the number of $p$-regular ...
Jackson Walters's user avatar
8 votes
1 answer
467 views

Representation theory of $\mathrm{GL}_n(\mathbb{Z})$

I want to understand the (complex) representation theory of $\mathrm{GL}_n(\mathbb{Z})$, the general linear group of the integers. I have gone through several representation theory texts but all of ...
Kenji's user avatar
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Subrepresentation of a representation of $\text{Sp}(2n,\mathbb{R})$ spanned by specific elements

$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\Sym{Sym}$Let $V = \mathbb{R}^{2n}$ be the standard representation of the symplectic group $\Sp(2n,\mathbb{R})$, and let $\{a_1,b_1,\dotsc,a_n,b_n\}$ be ...
Albert's user avatar
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2 votes
1 answer
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Finite dimensional irreducible representations of $\frak{sl}_m$ with non-trivial zero weight spaces

For the special linear algebra $\frak{sl}_{m}$ which finite dimensional irreducible representations $V_{\mu}$ have non-trivial zero weight spaces? For $\frak{sl}_2$ this is clear: $V_{2k\pi}$ for $\pi$...
Béla Fürdőház 's user avatar
7 votes
1 answer
187 views

Representations of the symmetric group with image in a given subgroup of $\operatorname{GL}_m$

Let $S_n$ be the symmetric group on $n$ elements. The irreducible representations of $S_n$ are parametrised by partitions $\lambda$ of $n$ and are defined already over the integers $\mathbb Z$. Let $\...
bsbb4's user avatar
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2 votes
0 answers
99 views

Quotient of integral representation of archimedean exterior square L-function

Let $\psi$ denote a non-trivial additive character of $\mathbb{R}$ and $r=2n$ be a positive even integer. Let $(\pi,V)$ denote an irreducible generic admissible Casselmann-Wallach representation of $...
Akash Yadav's user avatar
10 votes
1 answer
217 views

For which finite groups $G$ is $M_n(\mathbb{Q}(\zeta))$ a factor of $\mathbb{Q}[G]$?

I am cross-posting this question from my MSE post here, in case someone here can answer it. For a finite group $G$, the rational group ring $\mathbb{Q}[G]$ has a Wedderburn decomposition: $$ \mathbb{Q}...
tl981862's user avatar
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5 votes
1 answer
210 views

Schur functors = Weyl functors in characteristic zero?

I asked this question on Math Stack Exchange https://math.stackexchange.com/questions/4789924/schur-functors-weyl-functors-in-characteristic-zero, but I got no answers, so I ask the same question here....
Sunny Sood's user avatar
18 votes
0 answers
353 views

Can Rep(G) tell us whether G is discrete?

Given a locally compact group $G$, let $$\mathrm{Rep}(G)$$ be its category of unitary representations. The objects of that category are strongly continuous unitary representations of $G$ on Hilbert ...
André Henriques's user avatar
9 votes
1 answer
317 views

Triple product formula on $K = \mathrm{SU}(2)$

Let $K = \mathrm{SU}(2) = \{ k[\alpha ,\beta] \mid \alpha ,\beta \in \mathbb{C}, |\alpha |^2 + |\beta |^2 = 1 \} $ with $$ k [ \alpha , \beta ] = \begin{pmatrix} \alpha & \beta \\ - \...
Misaka 16559's user avatar
1 vote
0 answers
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Do parabolic/Levi pairs admit dynamic descriptions over disconnected base?

In Gille, Thm. 7.3.1, it is proven that given a reductive group scheme $G \to S$ over a connected base $S$, every parabolic-Levi pair $(P, L)$ over $S$ admits a dynamic description, i.e. is of the ...
C.D.'s user avatar
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3 votes
0 answers
302 views

What is known about representations of $S_n$ in other categories?

Is anything known about representations of the symmetric group $S_n$ for categories other than $\textbf{Vect}_k$, vector spaces and linear maps over a field $k$. That is, a group $G$ can be considered ...
Jackson Walters's user avatar
3 votes
0 answers
99 views

A depth version of a conjecture of Yamagata

Let $A$ be a finite-dimensional $K$-algebra. Recall that the grade of an $A$-module $M$ is defined as the smallest $i$ such that $\operatorname{Ext}_A^i(M,A) \neq 0$ and the depth of $A$ is defined ...
Mare's user avatar
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1 vote
2 answers
295 views

Finding lectures PDF "Four lectures on simple groups and singularities"

I would be very interested to find the PDF "Four lectures on simple groups and singularities" by Peter Slodowy, especially the lecture 4. I used to print them but lost it. Does anyone has ...
Nicolas Hemelsoet's user avatar
3 votes
1 answer
444 views

Group action of $\text{SL}(2, \mathbb{C})$

Let's denote upper-half space model of hyperbolic 3-space identified with quaternions as $$\text{H}^{+}_3 = \biggr\{z+wj\in \mathbb{C}\oplus \mathbb{R}^{+}j\biggr\}\tag{1}$$ Then a group action $\rho :...
user avatar
1 vote
0 answers
89 views

Cyclic representation isomorphic to L2 space

This question is also posted on Math Stack Exchange. I need some help understanding a proof of the following claim: every cyclic representation is isomorphic to some $L^2$ space. First, formal ...
Anna  Vakarova's user avatar
1 vote
1 answer
180 views

Upper-half space model of $\text{H}_3$

Does $SL(2,\mathbb{C})\times SL(2,\mathbb{C})$ have a natural (unitary) left-action on $\mathcal{L}^2(\text{H}^{+}_{3})$? If $G$ is a unimodular Lie group and $K$ is a compact subgroup, $G\times G$ ...
user avatar
1 vote
2 answers
222 views

Link invariants from Hecke relations of higher order

Alexander theorem says oriented links in $\mathbb{R}^3$ can be represented by closures of braids. Markov theorem says that braids related by Markov moves produce isotopic braid closures, and vice ...
Student's user avatar
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3 votes
2 answers
723 views

The adjoint representation of a Lie group

Let $G$ be a Lie group and $\text{Ad}(G)$ denote its adjoint representation i.e. the adjoint action of the group $G$ on its Lie algebra $\mathfrak{g}$. The adjoint representation is a real $G$-...
rr314's user avatar
  • 131
2 votes
0 answers
87 views

Category O for (Yangian) toroidal Lie algebras?

Suppose throughout that $g$ is a finite-dimensional simple Lie algebra over $\mathbb{C}$ and let us denote: $$g_{[2]} := g \otimes_{\mathbb{C}} \mathbb{C}[v^{\pm 1}, t^{\pm 1}]$$ $$g_{[2]}^+ := g \...
Dat Minh Ha's user avatar
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11 votes
0 answers
243 views

Continuous representations of topological groups on Banach spaces

Let $G$ be a topological group and let $\rho:G\rightarrow\text{GL}(V)$ be a linear representation of $G$ on a Banach space $V$. The representation is called strongly continuous if the map $g\mapsto\...
Botwinnik's user avatar
2 votes
0 answers
160 views

$\mathrm{Ext}^i(\pi_1, \pi_2)\neq0$ implies same central character

If $\pi_1$ and $\pi_2$ are two smooth admissible representations of $\operatorname{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$ with central characters. I want to prove that if $\pi_1$ has ...
user avatar
1 vote
1 answer
179 views

Non-split extension of representations of $\mathrm{GL}_2$ and $\mathrm{Hom}$

Let $0\to V_1\to V\to V_1\to0$ be a sequence of representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$, where $V_1$ is irreducible, smooth and admissible. Assume that this ...
user avatar
5 votes
1 answer
258 views

Explicit description for action of Weyl element in Whittaker model for GL2

Let $F$ be a non-archimedean local field and let $\pi =\mathscr{B}(\chi, \chi^{-1})$ be a principal series representation of $\mathrm{PGL}_2(F)$ induced from a character $\chi$ of $F^\times$. Let $w = ...
Steph Curry's user avatar
5 votes
0 answers
108 views

Fusion categories with $\mathrm{PSU}(2)_k$ fusion rules

Let $R_k$ be a fusion ring with $\mathrm{SU}(2)_k$ fusion rules (or equivalently $A_{k+1}$ fusion rules). All fusion categories with such fusion rules have been classified by Frohlich and Kerler in ...
Gert's user avatar
  • 273
6 votes
1 answer
667 views

Full expansion of $\det(I+\varepsilon A)$

It is well known that given a $n \times n$ matrix $A$, it holds that $$ \det(I + \varepsilon A)= 1 + \varepsilon \operatorname{tr}(A) + O(\varepsilon ^2).$$ I would need a full representation of $ \...
tommy1996q's user avatar
5 votes
2 answers
482 views

Non-trivial extension of representations have same central character

Let $\pi_1, \pi_2$ be two irreducible complex representations of $G=\mathrm{GL}_2(\mathbb{Q}_p)$ and assume that there exists a non-split extension $0\to\pi_1\to \pi\to\pi_2\to0$ of representations ...
user avatar
0 votes
0 answers
94 views

Is there a "cohomology theory" for involutive algebras?

I'm aware that there are cohomology theories for algebraic structures related to involutive algebras (or "involution algebras" or "$*$-algebras" if you prefer those terms) like Lie ...
wlad's user avatar
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1 vote
1 answer
95 views

Combinatorics behind certain induction of characters of the Coxeter group of type $B_n$

Let $W_n$ be a Coxeter group of type $B_n$ with $n\geq 1$. Concretely, it is generated by a set of simple reflexions $S = \{s_1,\ldots ,s_n\}$ which satisfy the relations $s_i^2 = 1, s_is_j=s_js_i$ as ...
Suzet's user avatar
  • 687
3 votes
1 answer
284 views

The associated graded algebra of a finite dimensional algebra

$\DeclareMathOperator\rad{rad}$Let $A$ be a finite dimensional algebra (we can assume that $A \cong KQ/I$, for a quiver $Q$ and an admissible ideal $I$ if that helps). Denote by $A_G$ the associated ...
Mare's user avatar
  • 25.8k
2 votes
1 answer
192 views

Stabilizer of a Levi subgroup in the Weyl group and its quotient

(I appologize in advance if this question is too naive for experts, since I know very little about the geometry/combinatorics of Weyl/Coxeter groups.) For simplicity, let $G$ be a connected reductive ...
youknowwho's user avatar
3 votes
0 answers
62 views

The basic representation of $LU(1)$

Let $H = L^2(U(1),\mathbb{C})$. The "basic" irreducible projective level 1 representation $\mathcal{H}$ of the loop group $LU(1)$ has underlying Hilbert space isomorphic to $\smash{\hat{\...
lw h's user avatar
  • 161
3 votes
1 answer
182 views

What is the minimum possible k-rank of a quasi-split reductive group over a field?

It is not possible for a quasi-split reductive group $G$ over a field $k$ to be anisotropic (unless it is solvable, hence its connected component is a torus). Indeed, there exists a proper $k$-...
C.D.'s user avatar
  • 545
1 vote
2 answers
251 views

A linear representation of the group of jets at 0 under composition

Let $G$ be the set of sequences $(f'(0), f''(0), f'''(0), \ldots)$ of derivatives at zero of functions $f : \mathbb{R} \to \mathbb{R}$ with $f(0) = 0$ and $f'(0) \ne 0 $. The set is a group under ...
Carlos Tomei's user avatar
36 votes
1 answer
1k views

Errata for Fulton's "Young tableaux"

Fulton's Young tableaux is one of the best texts on the subject, one which I often recommend and cite for reference. Unlike Fulton/Lang and Fulton/Harris, it is neither an early-dawn draft nor a ...