Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
6,803
questions
7
votes
2
answers
242
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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 ...
4
votes
0
answers
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{...
2
votes
0
answers
120
views
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 $\...
0
votes
1
answer
160
views
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 ...
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 ...
3
votes
0
answers
87
views
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 ...
23
votes
2
answers
908
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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 ...
7
votes
1
answer
298
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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 ...
3
votes
0
answers
86
views
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}...
3
votes
1
answer
218
views
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 $$\iota:\mathrm{Hom}_R(M,R)\otimes_R M\to \mathrm{Hom}_R(...
5
votes
0
answers
189
views
Rings where all indecomposable modules are projective or injective
Let $A$ be a semi-perfect noetherian ring.
Is there a nice classification of such $A$ such that every indecomposable finitely generated $A$-module is projective or injective?
Im also interested in ...
-3
votes
1
answer
266
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On Haar measure and Spherical measure [closed]
Let $d$-dimensional complex sphere be
$$\{(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$-...
2
votes
0
answers
165
views
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 ...
7
votes
0
answers
690
views
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, ...
4
votes
0
answers
252
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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 ...
3
votes
1
answer
202
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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 ...
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 ...
2
votes
0
answers
85
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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 ...
2
votes
1
answer
97
views
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$...
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 $\...
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 $...
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}...
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....
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 ...
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
$$ k [ \alpha , \beta ] =
\begin{pmatrix}
\alpha & \beta \\
- \...
1
vote
0
answers
44
views
Do parabolic/Levi pairs admit dynamic descriptions over disconnected base?
In Gille, Thm. 7.3.1, it is proven that given a reductive group scheme $G \to S$ over a connected base $S$, every parabolic-Levi pair $(P, L)$ over $S$ admits a dynamic description, i.e. is of the ...
3
votes
0
answers
302
views
What is known about representations of $S_n$ in other categories?
Is anything known about representations of the symmetric group $S_n$ for categories other than $\textbf{Vect}_k$, vector spaces and linear maps over a field $k$.
That is, a group $G$ can be considered ...
3
votes
0
answers
99
views
A depth version of a conjecture of Yamagata
Let $A$ be a finite-dimensional $K$-algebra.
Recall that the grade of an $A$-module $M$ is defined as the smallest $i$ such that $\operatorname{Ext}_A^i(M,A) \neq 0$ and the depth of $A$ is defined ...
1
vote
2
answers
295
views
Finding lectures PDF "Four lectures on simple groups and singularities"
I would be very interested to find the PDF "Four lectures on simple groups and singularities" by Peter Slodowy, especially the lecture 4. I used to print them but lost it. Does anyone has ...
3
votes
1
answer
444
views
Group action of $\text{SL}(2, \mathbb{C})$
Let's denote upper-half space model of hyperbolic 3-space identified with quaternions as $$\text{H}^{+}_3 = \biggr\{z+wj\in \mathbb{C}\oplus \mathbb{R}^{+}j\biggr\}\tag{1}$$
Then a group action $\rho :...
1
vote
0
answers
89
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Cyclic representation isomorphic to L2 space
This question is also posted on Math Stack Exchange.
I need some help understanding a proof of the following claim: every cyclic representation is isomorphic to some $L^2$ space.
First, formal ...
1
vote
1
answer
180
views
Upper-half space model of $\text{H}_3$
Does $SL(2,\mathbb{C})\times SL(2,\mathbb{C})$ have a natural (unitary) left-action on $\mathcal{L}^2(\text{H}^{+}_{3})$?
If $G$ is a unimodular Lie group and $K$ is a compact subgroup, $G\times G$ ...
1
vote
2
answers
222
views
Link invariants from Hecke relations of higher order
Alexander theorem says oriented links in $\mathbb{R}^3$ can be
represented by closures of braids. Markov theorem says that
braids related by Markov moves produce isotopic braid closures,
and vice ...
3
votes
2
answers
723
views
The adjoint representation of a Lie group
Let $G$ be a Lie group and $\text{Ad}(G)$ denote its adjoint representation i.e. the adjoint action of the group $G$ on its Lie algebra $\mathfrak{g}$. The adjoint representation is a real $G$-...
2
votes
0
answers
87
views
Category O for (Yangian) toroidal Lie algebras?
Suppose throughout that $g$ is a finite-dimensional simple Lie algebra over $\mathbb{C}$ and let us denote:
$$g_{[2]} := g \otimes_{\mathbb{C}} \mathbb{C}[v^{\pm 1}, t^{\pm 1}]$$
$$g_{[2]}^+ := g \...
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\...
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 ...
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 ...
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 = ...
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 ...
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 $ \...
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 ...
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 ...
1
vote
1
answer
95
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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 ...
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 ...
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 ...
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{\...
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 $k$-...
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 ...
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 ...