Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
7,076 questions
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Does the Lawrence–Krammer representation provide a quantized action on the space of networks?
Let $\rho:B_n \rightarrow H_2(\overline{C_2 P_n})$ denote the Lawrence–Krammer representation of the braid group on $n$ symbols. The group $H_2(\overline{C_2 P_n})$ is a free $\mathbb{Z}[q,t]$-module ...
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Young tableaux — irreps correspondence for simple complex Lie algebras
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$I have learned that Young tableaux which were originally
introduced to study the irreducible representations of finite
symmetric groups $S_n$ ...
- 6,028
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Integral of elements of random unitaries
It is known how to calculate the integral of elements of $N\times N$ Haar random unitaries using the Weingarten function:
$$\int \prod_{k=1}^n U_{i_kj_k} U_{m_kr_k}^* \mathrm d U = \sum_{\sigma,\tau} \...
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Are banach space representations of commutative $C^*$ algebras decomposable?
It is well known that, if $\pi:A\to \mathbb B(\mathcal H)$ is a $^*$-representation of a type I $C^*$-algebra, then $\pi$ is unitarily equivalent to a direct integral of irreducible representations.
...
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To find a DFT for complex functions on a semigroup
For a certain commutative semigroup of integer size $n$, $G=(\{1,2,\dots,n\},\circ: x\circ y\mapsto \min(n,x+y))$, consider all complex functions on it, denoted by $\mathbb C[G]$ or $\mathbb CG$. ...
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The closure of the orbits of $\mathcal{F} \times \mathcal{F}$
Let $V$ be a finite-dimensional vector space over an algebraically closed field $F$, and $\dim V=d$. Let $G=GL(V)$, $P$ is the parabolic subgroup of $G$. Let $\mathcal{F}$ be the set of all partial ...
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Lie algebra action Whittaker model
Let $(\pi, H)$ be an irreducible unitary generic representation of $G=\operatorname{GL}(r,\mathbb{C})$ and let $H^{\infty}$ be its subspace of smooth vectors. Let $W :G\to\mathbb{C} $ be the Whittaker ...
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Normalizing a parameter in a regression
I am thinking about the possibility of making a parameter in my regression, let's say the $\lambda$ in a ridge regression, somehow, inside a range : $\lambda \in [0,1]$. Do you have any ideas how I ...
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Decomposition of tensor product of two representations of $so(10,\mathbb{C})$
There exist two 16-dimensional irreducible non-isomorphic representations of $so(10,\mathbb{C})$. Consider the tensor products of each of them with the standard (10-dimensional) representation.
What ...
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How do I detect whether a representation is (or is not) the adjoint representation?
Let $(\mathfrak{g},[\cdot,\cdot])$ be a Lie algebra. There is a God-given representation of $\mathfrak{g}$, namely, the adjoint representation $\operatorname{ad} : \mathfrak{g} \to \operatorname{Der}(\...
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How to construct non-abelian functions?
I have found some functions $t_g, g \in G$ for cyclic groups $G=C_n$ which seem to satisfy the following convolution identity:
$$t_g(x+y) = \sum_{h \in G} t_{gh^{-1}}(x) t_h(y)$$
Example of such ...
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Reference request: Weyl group action on the power set of positive roots
There is a symmetric group action on the power set of positive roots in type A. The action is defined as follows.
Denote by $\alpha_1, \ldots, \alpha_n$ be the set of simple roots in a root system. In ...
- 6,201
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Total sum of characters over partitions with distinct parts
In my earlier quest, we looked at $\chi_{\mu}^{\lambda}=$value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu$ and $\lambda$ are (unrestricted) partitions of $n$. Then, ...
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The action of an extension group $G=p^{1+2n}{.}Q$ on the faithful characters of its normal subgroup $p^{1+2n}$
Let $G=p^{1+2n}{.}Q$, $n>1$, be a finite extension group of an extra-special $p$-group $N=p^{1+2n}$ by a group $Q$, where $Q$ is a linear group of dimension $2n$ over $GF(p)$. It seems that the ...
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Describing the ordinary irreducible characters of a special $p$-group $p^{n+m}$
Let $P$ be a special $p$-group $p^{n+m}$. So $P$ will have $p^m$ linear characters. How does one describe (or determine) the other ordinary irreducible characters of $P$ and will they all be ...
- 49
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Dirac operator on a 5 dimensional tangent manifold with a $Spin(3)$-bundle
In p.3 of Witten paper from this Physics Letters B, Volume 117, Issue 5, 18 November 1982, Pages 324-328 Physics Letters B, 117(5), 324–328, he says that about the Dirac equation on a 5-dimensional
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Hopf algebra antipodes and right left comodule equivalences
Given a Hopf algebra $H$, denote by ${}^H\mathrm{mod}$ the category of left $H$-comodules, and by $\mathrm{mod}^H$ the category of right $H$-comodules. If the antipode $S$ of $H$ is invertible then we ...
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Request for a modern Reference for Frobenius' paper "Über die Charaktere der mehrfach transitiven Gruppen"
I'm interested in the paper of Jan Saxl "The Complex Characters of the Symmetric Groups that Remain Irreducible in Subgroups".
I have only (not yet enough!) standard background on the ...
- 1,013
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Can one construct the Gell-Mann matrices from Pauli matrices?
I'm essentially interested in understanding the connection between SU(2) and SU(3). I would be particularly interested in connections between the Pauli matrices and the Gell-Mann matrices.
I've read ...
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Isomorphism problem for enveloping algebras
Let A and B be finite-dimensional algebras over a field k. We denote the enveloping algebra of A by A^e, which is the tensor product (over k) of the algebra A and its opposite algebra. Suppose A^e and ...
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$\tau$-admissible lift
I've been asked to take on a peer-review task which has to be completed in a short time, obviously details have to remain confidential, I need to work out what ``$\tau$-admissible lift" means if $...
- 2,125
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Reference: Irreducible components of the Steinberg variety are conormal bundles
The (partial) Springer resolution is defined as a map $\mu: T^*\mathcal{F} \to \mathcal{N}$, where $\mathcal{F}$ is the partial flag variety consisting of $n$-step partial flags of $\mathbb{C}^d$, and ...
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Image of Frobenius element under irreducible representation is diagonalizable
Let $K/ \mathbb Q$ be a Galois extension, and $\rho$ be an irreducible representation of the Galois group $Gal(K/ \mathbb Q)$. Consider an integer prime $p$ which doesn't ramify in $K$, and let $\...
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Monomial Characters of Quotient Groups
The following statement provides (if true) a powerful tool for inductive proofs. Can anyone confirm if it is true:
Suppose $G$ is a finite group and $N$ a normal subgroup of $G$. Is it true that if $...
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Simplicity Criterion for Verma module
In Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$,
$\lambda\in\mathfrak{h}^*$ is
antidominant if $\langle \lambda + \rho, \alpha^\lor\rangle \not\in \mathbb{Z}^{>0}$ ...
- 1,875
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Gelfand-Pettis integral: what does it mean for a topological vector space to "admit a dual space?"
I am trying to understand more about the Gelfand-Pettis integral. From wikipedia:
What does it mean that $V$ "admits a dual space?" When $V$ is a Banach space, $V^{\ast}$ is taken to be ...
- 6,180
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Solvability of finite group from indices of commutator and abelian normal subgroup
Suppose finite directly indecomposable group $G$ has $\frac {|[G, G]|}{|G|} < \alpha$ and $\frac {|A|} {|G|} > \beta$, where $A \lhd G$ abelian. Are there some nontrivial bounds on $\alpha, \...
- 4,600
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Tameness of the trivial extension of a finite dimensional algebra
The trivial extension T(A) of a (finite dimensional) algebra A is representation-finite if and only if the algebra A is iterated tilted of Dynkin type.
Questions:
Is there a similar classification ...
- 26.5k
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An ideal in Fourier–Stieltjes algebras $B(G)$
Let $G$ be a locally compact group and $R$ be any family of representations of $G$. Let $A_R(G)$ be the closed linear span in Fourier–Stieltjes algebras $B(G)$ of the coefficient functions of all
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How to Evaluate the ABJM partition function for N=2
This is the ABJM partition function on the 3-sphere,
$$ Z(2) = \int \frac{d^2\mu}{(2\pi)^2} \frac{d^2\nu}{(2\pi)^2}
\frac{\left[ 2 \sinh \frac{\mu_1 - \mu_2}{2}\right]^2\left[ 2 \sinh \frac{\nu_1 - \...
- 22.8k
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Determination of the characteristic tilting module
Let $A$ be a finite dimensional selfinjective algebra and $M$ an indecomposable non-projective $A$-module such that the algebra $B:=End_A(A \oplus M)$ is standardly stratified.
Examples of such $B$ ...
- 26.5k
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Automorphisms of weighted quiver
I am reading this paper strongly primitiv species with potentials I : mutations.
In page 6, they give the definition of weighted quiver: a weighted quiver is a pair $(Q,d)$, where $Q$ is a loop-...
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Some places I can't understand in the paper "Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra"
I am reading the paper "Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra", the link is here: https://arxiv.org/pdf/1608.04212.pdf
There are some places I can't ...
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A conjecture of Serre on elliptic curves
Given a number field $K$, is there a prime number $p_{K}$ such that, for any elliptic curve $E$ over $K$ without complex multiplication, the residual $\pmod p$ Galois representation $\overline{\rho}_{...
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Spherical Rings
My question is concerned with filtered rings. It is a classical result that if $R$ is a finitely generated commutative ring graded by a semigroup $S$ then $S$ is also finitely generated.
The reverse ...
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Character theory of finite groups
Let $G$ be a finite group, $R(G)$ be the solvable radical of $G$, and $G/R(G)$ be isomorphic to $PSL_2(11)$. Also let $\lambda$ be a non-trivial complex linear character of $R(G)$ such that $\lambda$ ...
- 841
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'Adelic torus' not arising from a rational torus
Let $G$ be a reductive group over a global field $F$, and $\gamma$ a strongly regular semi-simple element of $G(F)$. Then the centralizer $G_\gamma$ is defines an $F$-torus $T$, and hence by base ...
- 3,799
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Questions in the paper "Morita endomorphism algebras of generators"
I am reading this paper "Morita endomorphism algebras of generators", the link is here:http://link.springer.com/article/10.1007/s10468-016-9601-z
There are two quesions I can't understand:
on page ...
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A question on the paper "The classification of algebras by dominant dimension"
I'm reading the paper "the classification of algebras by dominant dimension" by Bruno J.Mueller, the link is here http://cms.math.ca/10.4153/CJM-1968-037-9.
In the proof of lemma 3 on page 402, there ...
- 775
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72
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Contravariant finiteness of a certain subcategory
Let $A$ be an algebra with finite dominant dimension $d \geq 1$ and $Dom_d$ the full subcategory of modules with dominant dimension at least $d$ and $Proj$ the full subcategory of modules of finite ...
- 26.5k
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How to get $I_i \in add(\nu_A(Q))$ for $1 \leq i \leq n$ by $Ext^{i}_A(S,S)=0$?
Let $A$ be a k-algebra, where k is a fixed field. Let $S$ be a simple, non-injective $A$-module such that $Ext^{i}_{A}(S,S)=0$ for $1 \leq i \leq n$. Let $P(S)$ be the projective cover of $S$, and let ...
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Is there a simple module $S$ satisfies the following conditions?
Let $A$ be a k-algebra,where k is a fixed field. {$x_1,x_2, \cdots,x_n $} is a complete set of primitive orthogonal idempotents of $A$. $M$ is a left $A$-module such that $x_iM=0$ for $i=1,2,\cdots,n-...
- 775
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on the Springer sheaf
Let $\pi:\tilde{\mathfrak{g}}\rightarrow \mathfrak{g}$, be the Grothendieck-Springer resolution, where $\mathfrak{g}$ is a semisimple Lie algebra over $\mathbb{C}$.
We know that $\pi$ is small thus $\...
- 3,472
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Mathematical Definition of $n$-Brouillin Zone [duplicate]
I am having trouble finding a mathematical definition of the Brouillin zone beyond the first, which are basically the Voronoi cells or Wigner-Seitz cells. We could imagine the set of point closer to ...
- 22.8k
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Normalizer of non-split tori
Let $\mathbb{G}$ be a connected reductive group over $\mathbb{C}$. Let $G:=\mathbb{G}(\mathbb{C}(\!(t)\!))$. Let $T$ be a maximal torus in $G$.
Question: What do we know about the normalizer $N_G(T)$...
- 2,751
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$Ext$ functor over a product of groups
Let $G_1$ and $G_2$ be two groups (of some kind, e.g. finite groups).
Let $M_1, N_1$ be $G_1$-modules, and $M_2, N_2$ be $G_2$ modules, always with coefficients in $\mathbb{C}$.
Write $G = G_1 \...
- 3,432
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presentation for a nilpotent group associated to the square of a coxeter element
This question is related to one asked earlier about inductive presentations of unipotent radicals in Kac-Moody groups.
Let $\Gamma$ be a coxeter diagram --- i.e. an unoriented graph with $r$ vertices ...
- 2,137
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The rectification of the transpose of a skew tableau?
Suppose the rectification of a skew tableau is a standard tableau, I want to know if the rectification of the transpose of the skew tableau equals to the transpose of the rectification of the skew ...
- 775
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Orbits of some action of SL2 on Pontryagin dual of the field of formal Laurent series
Let $K=\mathbb{F}_2((t))$ be the field of formal Laurent series over the finite field $\mathbb{F}_2$. Now consider $K^3$ as an additive group and its dual group $\hat{K^3}$, which consists of all ...
- 1,702
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What are the E7(7) invariants in the adjoint representation?
Take a real vector space $R$ transforming in the adjoint representation of
the ${\rm E}_7(7)$ Lie group as $R \rightarrow G R G^{-1}$. One can define
invariants using traces of products of $R$ as ${\...
