Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
5,622
questions
4
votes
0answers
25 views
Littelmann Path model and RSK e and f operators
The Littelmann path model defines $e_i$ and $f_i$ operators which correspond in type A to the $e_i$ and $f_i$ operators on semistandard Young tableaux (i.e. since they both are different ways of ...
1
vote
0answers
71 views
Completing a representation to a unitary representation
Let $G$ be a discrete group acting on an infinite-dimensional complex vector space $V$ and preserving a positive-definite Hermitian form.
Question. Is it always possible to complete $V$ to a unitary ...
2
votes
0answers
155 views
Elementary questions on the geometric Langlands program for the orthogonal and symplectic families
If $G^\vee = SL(n,\mathbb{C})$ and $V = \mathbb{C}^n$, then the Langlands dual of $G^\vee$ is $G = PGL(n,\mathbb{C})$. Denote by $T$ and $T^\vee$ maximal tori in $G$ and $G^\vee$ respectively. The ...
6
votes
0answers
111 views
Euclidean and Minkowski Majorana spinors - inconsistency with Wikipedia Table
In this wonderful lecture note on Clifford Algebra and
Spin(N) Representations, http://hitoshi.berkeley.edu/230A/clifford.pdf
Somehow I find some inconsistency with his Tables of Euclidean and ...
2
votes
0answers
121 views
Reference for $2$-by-$2$ integer matrices
Currently I am studying number theory, in particular modular functions and imaginary quadratic fields. I am looking for a reference on $2$-by-$2$ integer (or rational) matrices, especially:
...
9
votes
1answer
272 views
A detail in the proof of Schur's lemma: the closures of the $\mathcal{Ker}$ and $\mathcal{Im}$ of the intertwiner
$\renewcommand\Im{\operatorname{\mathcal{Im}}}\newcommand\Ker{\operatorname{\mathcal{Ker}}}$I was sure that this is a trivial question and placed it on Math Stackexchange
https://math.stackexchange....
1
vote
1answer
86 views
The $\{2,3\}$-groups with a condition about $\mathbb{C}$-characters
Let $G$ be a $\{2,3\}$-group and $\lvert G\rvert=2^\alpha\cdot3^\beta$. For $p\in\{2,3\}$, define
$$
\nu_p(G)\mathrel{:=}\min\left\{\log_p\left(\frac{\lvert G\rvert}{\chi(1)}\right)_p \mathrel{\...
5
votes
1answer
156 views
Reconstruction of coalgebras
In the paper Reconstruction of hidden symmetries of Bodo Pareigis in the subsection "3.1 Reconstruction of coalgebras" there is the following proposition (3.3.).
Let $\mathcal{C}$ be a ...
7
votes
0answers
104 views
Branching Rule for Specht Modules over Kazhdan-Lusztig Basis
Suppose $\lambda\vdash n$ is a partition and $S^\lambda$ is the associated irreducible representation of $S_n$. As we know from the branching rule, we have an isomorphism of $S_{n-1}$ modules
$$S^\...
3
votes
0answers
121 views
External tensor product Calabi-Yau DG categories
Let $\mathcal{C}$ be a smooth proper DG-category such that the shift $[p]$ is a Serre functor for $D^{perf}(\mathcal{C})$ (we say that $D^{perf}(\mathcal{C})$ is $p$-Calabi-Yau). I am looking for a ...
4
votes
1answer
113 views
Inducing irreducible $B_n \times S_k$ characters to $B_{n+k}$
I know that we can induce irreducible representations of $B_n$ to $B_{n+k}$ using the Ariki-Koike branching rule.
The irreducible representations of $B_n \times S_k$ are parametrised by tuples 2-...
3
votes
0answers
139 views
Borel–Weil–Bott theorem and tensor product
Let $G=\operatorname{SL}(2)$ and $V_n$ be the n+1 dimensional irreducible representation of $G$. This gives a representation for $G(O)=\operatorname{Map}(\operatorname{Spec} k[[t]], G)$ and hence an ($...
0
votes
0answers
93 views
Is there a roadmap to learning representation theory of finite group over finite field?
I've been wanted to learn some basic theories of the (non-semisimple) representation of the finite group over a finite field.
I have been guessing that the materials might be contained in the books on ...
2
votes
0answers
45 views
Structure of fibers of (complex) moment map of hypertoric variety
I am primarily interested in the hypertoric variety $\mathbb H^d /\!/\!/ N$ whose associated hyperplane arrangement is given by the set of root hyperplanes (this variety is defined precisely in ...
7
votes
2answers
179 views
Is it possible for the reduction modulo $p$ of an non-commutative semisimple algebra to be commutative?
Suppose that $I, X_1, \ldots, X_{d-1}$ are $n \times n$ matrices with integer entries whose $\mathbb{Z}$-span is a subalgebra of $\mathrm{Mat}_n(\mathbb{Z})$. Suppose that, thought of as a subalgebra ...
2
votes
0answers
42 views
Upper bound on decomposition numbers for the symmetric group in a block of weight $w$
The $p$-blocks of the symmetric group $S_n$ are labelled by pairs $(\gamma, w)$ where $\gamma$ is a $p$-core partition, $w \in \mathbb{N}_0$ is the weight of the block, and $|\gamma| + p w = n$. ...
2
votes
0answers
207 views
Why does Kashiwara define $U_q(\mathfrak{g})$ over $\mathbb{Q}(q)$ rather than $\mathbb{C}(q)$?
When defining crystal bases, why do we typically view $U_q(\mathfrak{g})$ as an algebra over $\mathbb{Q}(q)$ rather than $\mathbb{C}(q)$? In Kashiwara's original paper introducing crystal bases, he ...
9
votes
2answers
540 views
Which representations of $\mathfrak{sl}(2)$ are homomorphic images of the tensor product of finitely many copies of $\mathbb{C}^2$?
My questions may turn out to be related to Schur functors.
If $\mathfrak{g}$ is a complex semisimple Lie algebra and $\lambda$ is the highest weight of an irreducible representation $V$ of $\mathfrak{...
2
votes
0answers
62 views
Weights of finite abelian group actions on submanifolds/subvarieties
(cross-posted from https://math.stackexchange.com/questions/4125529/weights-of-finite-abelian-group-actions-on-submanifolds-subvarieties)
How do weights associated to actions of finite subgroups of $\...
3
votes
1answer
80 views
Is there a Jacobi–Trudi formula for skew zonal polynomials?
Skew Schur polynomials are defined as $s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu\nu}s_\nu$, where the Littlewood–Richardson coefficients $c^\lambda_{\mu\nu}$ satisfy $s_\mu(x)s_\nu(x)=\sum_\lambda c^\...
8
votes
1answer
149 views
On the coefficients that appear in finite groups of matrices with integer entries
Let $n$ be a positive integer and $G$ be a finite group of $n\times n$ matrices with integer coefficients, i.e. $G\subset\operatorname{GL}_n(\mathbb{Z})$. It is known that for sufficiently large $n$, ...
4
votes
0answers
56 views
Minimal set generators ideal submaximal minors
Let $I$ an ideal of $\mathbb{C}[X_1, \ldots X_n]$ and define the arithmetical rank of $I$ as:
$$ ark(I) = \textrm{min} \left\{m \in \mathbb{N}, \exists f_1, \ldots, f_m \in \mathbb{C}[X_1, \ldots X_n]...
1
vote
0answers
51 views
What subspace of $\operatorname{SU}(4)$ group keeps an element of the $\mathfrak{su}(2)$ subalgebra within $\mathfrak{su}(2)$ upon adjoint action?
Consider the Lie group $G_4=\operatorname{SU}(4)$ with (15) generators $T^a$. A basis for the latter is
$$\{\sigma^j \times 1_2, \quad \quad \sigma^i \times \sigma^j, \quad \quad 1_2 \times \sigma^j\},...
3
votes
2answers
194 views
Reductive group with simply connected derived group has all root groups $\mathrm{SL}_2$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}$Motivation: I am trying to understand why the Deligne-Langlands conjectures are only stated for $p$-adic reductive groups with connected ...
6
votes
1answer
144 views
Irreducible representations of the symmetric group on homology of simplicial complex
I am following Wall's paper A note on symmetry of singularities and I have some questions regarding representation theory and the homology of some objects:
Consider an action of $\Sigma_k$ on a finite ...
4
votes
2answers
165 views
Schur positivity of a polynomial
Suppose a polynomial of the form
$$\prod_i^d \sum_j^p x_i^{f_j}$$
clearly symmetric, where $f_j\in \mathbb{N}$. There is a way to find the set of $f$ numbers such that this polynomial is Schur ...
4
votes
0answers
83 views
The coherence property of center of universal enveloping algebra for reductive Lie algebra?
Let $G' \subset G$ be two reductive Lie groups over $\mathbb{R}$ and $\mathfrak{g}_{\mathbb{C}}' \subset \mathfrak{g}_{\mathbb{C}}$ be their complexified reductive Lie algebra over $\mathbb{C}$, ...
2
votes
0answers
59 views
Question on GSpin-Valued L-parameters
Let $\Gamma$ be a topological group, $n \geq 1$ an integer, $\ell$ a prime number, and $\overline{\mathbb{Q}}_{\ell}$ the algebraic closure of the $\ell$-adic integers. We set $\Phi(GSpin_{2n + 1})$ ...
3
votes
1answer
127 views
From braid representations to link invariants
If one has a $\mathbb{C}$-linear representation of the braid algebra into e.g. the Temperley-Lieb algebra i.e. $\rho:\mathbb{C}[B_{n}]\to TL_{n}(\delta)$, we can deduce a skein relation $\mathcal{S}$....
7
votes
1answer
306 views
Independent vectors in the permuting coordinates action of $S_n$ on $\mathbb{R}^n$
Let $V$ be the hyperplane in $\mathbb{R}^n$ with equation $\sum_i x_i=0$. The symmetric group $S_n$ acts on $V$ by $s\cdot (v_1,\ldots,v_n)=(v_{s^{-1}(1)},\ldots,v_{s^{-1}(n)})$. Consider those $v\in ...
1
vote
1answer
148 views
Understanding the regular representation of an LCA group as a 'direct integral'
The reference for what I'm asking is page $107$ from Folland's harmonic analysis.
$G$ is a locally compact abelian group with dual $\hat{G}$. Let $H$ denote the Hilbert space $L^2(G)$.
I'm trying to ...
10
votes
1answer
250 views
When are immanants irreducible?
For a partition $\lambda$ let $\chi_\lambda$ be the corresponding irreducible representation of the symmetric group $S_n$.
Let $\mathrm{Imm}_\lambda(x) = \sum\limits_{\pi \in S_n} \chi_\lambda(\pi) x_{...
2
votes
1answer
273 views
Is the representation of finite simple groups fully understood?
Is the representation of finite simple groups fully understood? To clarify, I mean have all the simple representations (even finite dimensional) been classified in terms of some classifying set, such ...
3
votes
0answers
53 views
compactly induction of smooth modules over Hecke algebras
Let $G$ be a locally profinite group and $H$ its closed subgroup. It is well-known that a smooth representation of $G$ is identified with a smooth module over $\mathcal{H}(G)$, where $\mathcal{H}(G)$ ...
4
votes
2answers
234 views
Ring of invariants for $n$-tuples of Lie algebras
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\M{M}\DeclareMathOperator\Tr{Tr}$Consider the diagonal action of $\GL(n,\mathbb{C})$ on the variety of $k$-tuples of matrices, $\M_{n\times n}(\mathbb{C}...
11
votes
0answers
168 views
Hidden grading on $kS_n$
Brundan and Kleshchev showed that if $k$ is a field of characteristic $p$, then the group ring $kS_n$ of the symmetric group $S_n$ admits an integer grading which is nontrivial in the sense that it is ...
17
votes
1answer
465 views
The sum (with multiplicity) of the cubes of irreducible character degrees of a finite group
Throughout $G$ is a finite, non-abelian group.
$\DeclareMathOperator\Irr{Irr}\DeclareMathOperator\AD{AD}\DeclareMathOperator\cp{cp}\newcommand\card[1]{\lvert#1\rvert}$
Let $\Irr(G)$ be the set of ...
9
votes
1answer
443 views
Non-Abelian Hodge theory
Let $X$ be a compact Riemann surface. I would like to find a somehow complete reference for the proof of the so called non-Abelian Hodge correspondence relating Dolbeaut, Betti and Higgs bundle moduli ...
5
votes
3answers
271 views
Root system of fixed point Lie sub-algebra
It is known that a non-simply laced simple root system can be constructed from the simply-laced root system by folding the Dynkin diagram and hence the corresponding non-simply-laced Lie algebra can ...
2
votes
1answer
151 views
English translation of Emmy Noether's Hyperkomplexe Grössen und Darstellungstheorie
I'm wondering if anybody knows where one can find an English translation of Emmy Noether's classical paper E. NOETHER, Hyperkomplexe Grössen und Darstellungstheorie, Math. Zeit. 30(1929), 641–692 ?...
3
votes
2answers
72 views
When representations of reductive Lie group in a Banach space and in its Garding space have the same length?
Let $G$ be a real reductive Lie group (e.g. $G=\operatorname{GL}(n,\mathbb{R})$). Let $\rho$ be a continuous representation of $G$ in a Banach space $V$. Let $V^\infty\subset V$ be the subspace of ...
1
vote
1answer
155 views
The norm of the principal series intertwining operator for $\operatorname{GL}_2$
Is there a known bound on the norm of the standard intertwining operator for the principal series of $G = \operatorname{GL}_2(\mathbb Q_p)$?
Background:
For a character $\chi = (\chi_1,\chi_2)$ of the ...
8
votes
1answer
244 views
Trying to understand “a refinement of the Peter–Weyl theorem” by Lusztig
"A refinement of the Peter–Weyl theorem" is the title of Chapter 29 in Lusztig's "Introduction to quantum groups" (Birkhäuser 2010, reprint of the 1994 edition). This chapter is ...
3
votes
0answers
100 views
The product of $Z(\mathfrak{g})$-finite functions is also $Z(\mathfrak{g})$-finite?
Let $G$ be a classical group defined over $\mathbb{Q}$.
Let $\mathfrak{g}$ be the Lie algebra of $G(\mathbb{R})$ and $U(\mathfrak{g}_{\mathbb{C}})$ its universal enveloping algebra of $\mathfrak{g}_{\...
1
vote
0answers
119 views
Aut/Inn/Out Automorphism Groups of the unitary group $𝑈(𝑁)$
Given a group $G$, we denote the center Z$(G)$, we like to know the
automorphism group Aut($G$), the outer automorphism Out($G$) and the inner automorphism Inn($G$). They form short exact sequences:
$$...
4
votes
0answers
218 views
Is there algebraic $K$-theory of a group independent of the base ring?
Given a ring R and a group $G$, I can consider the group ring $R[G]$ and then take the algebraic $K$-theory $K(R[G])$. This the $K$-theory of the category $\operatorname{Rep}_R(G)$. As a variant, one ...
3
votes
1answer
129 views
Dense subspace of $\operatorname{Ind}_{H_1 \times H_2}^{G_1 \times G_2} \chi$
Let $H = H_1 \times H_2$ be a closed subgroup of a second-countable locally compact Hausdorff group $G = G_1 \times G_2$, with $H_i \leq G_i$. Let $\chi = \chi_1 \otimes \chi_2$ be a unitary ...
1
vote
1answer
120 views
$G/T$ has finitely many $G^\theta$ orbits
Let $G$ be a compact connected Lie group and T be it's maximal torus. Let $\theta: G \rightarrow G$ be an involution on $G$ and let $G^\theta = \lbrace g \in G , \theta(g)=g \rbrace $.
I'm looking for ...
5
votes
0answers
72 views
It there an algebra of the form $B_T$ with global dimension 3?
Let $A$ be the (symmetric Frobenius) algebra $A=K[x]/(x^3) \otimes_K K[x]/(x^3)$ over a field $K$, which is isomorphic to the group algebra of $C_3 \times C_3$, with $C_3$ cyclic of order 3, when $K$ ...
3
votes
0answers
61 views
“Character” theory via dualisable $2$ categories
One interesting way to describe the ordinary (over $\mathbb{C}$) character theory of finite groups is to view the categories $Rep(G)$ together in a $2$ category with bimodules as morphisms. This $2$ ...
