Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
7,076 questions
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About regular local rings and Socles
Let R be a regular local ring with $ \text{dim} R = d $. If $ 0\rightarrow R\rightarrow I_0\rightarrow ...\rightarrow I_d\rightarrow 0 $. Then why for $ 0\leq i\leq d-1 $, the socle of $ I_i $ is ...
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Definition of the supertrace in superalgebra representations
Let us consider a matrix superalgebra $A$ with generators satisfying $[L_a,L_b]=i L_c f^c{}_{ab}.$ The generators are matrices on which supertrace is defined bu the usual trace on the bosonic part ...
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Representation dimension of a special algebra
Hi,
I'm reading the following paper: http://fma2.math.uni-magdeburg.de/~holm/ARTIKEL/holm-hu-23-05.pdf
I've come across a piece of information, which I don't understand, and wanted to ask, if I ...
- 1,755
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subgroups of a $p$-solvable group and complete reducibility
1.
Let $G$ be a $p$-solvable group and $V$ be a finite dimensional
faithful $kG$-module, where the characteristic of $k$ is $p$. But
$V$ is not a semisimple $kG$-module. For every $n\geq 0$, we ...
- 491
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2
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205
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Structure of Homomorphisms of commutative C^* algebra
Being new to $C$* algebra, I'm trying to understand basic properties of -homomorphisms of such algebras. Let $P$ be a set of commuting projections on Hilbert space ${\cal H}$.
Let ${\cal P}$ be the $...
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Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal?
Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal?
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Reference request for equivariant cohomology of G [duplicate]
Possible Duplicate:
What is the equivariant cohomology of a group acting on itself by conjugation?
Let $G$ be a compact Lie group. Where can one read about the equivariant cohomology $H_G^*(G)$, ...
- 559
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Conjugate Matrix
Let $A$ be a nilpotent square matrix, $J$ be the antidiagonal matrix with 1's on the secondary diagonal (i.e. $J^{2}=E$) and let $B=J A J.$ Suppose we conjugate the matrices $A,B$ by a matrix $...
- 301
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271
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Relative invariants of prehomogeneous vector space
Let $(G,\rho,V)$ be a prehomogeneous vector spaces with $f_1,\dots,f_N$ the basic irreducible relative invariants. Suppose that $(G',\rho',V')$ is a second prehomogeneous that is in the same castling ...
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Is the direct product of two primitive unitary groups necessarily a primitive unitary group?
Let $G$ and $H$ be two primitive unitary groups, is $G\times H$ necessarily a primitive unitary group? If not, is there any counterexample?
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Class multiplication coefficients of symmetric groups
My question is that I was working with some counting problems, and finally the answer should be
$$
\nu_{\mu_1,\mu_2,\mu_3}=\#\{(\sigma_1,\sigma_2,\sigma_3): \sigma_1\sigma_2\sigma_3=1, \sigma_1\in C_{\...
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43
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Endomorphism algebra of equivariant maps of isotypic module
Let $A$ be a simple Artinian $K$-algebra with a minimal left ideal $M$. Here, $M$ can be viewed as a simple left $A$ module, and, by Schur's lemma, $D=\text{End}_A(M)$ is a $K$-algebra. By Wedderburn-...
- 143
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97
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Weight space decomposition of smooth representation of complex algebraic torus
Question: Let $T=(\mathbb{C}^{*})^{n}$ and $\pi:\mathcal{E}\to \mathbb{C}^{n}$ a smooth complex $T$-equivariant vector bundle (i.e. $\pi$ is $T$-equivariant and $T$ acts on the fibers linearly). The ...
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64
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"Uniqueness" of 6j symbols via triads
Assume these are general 6j symbols with multiplicity labels and "spins" which are not necessarily selfconjugate (say $SU_3$). From a programming viewpoint, in some use cases it would be far ...
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Higher-order obstructions in thin group orbits
Let $G$ be a finitely generated group acting on the integers $\mathbb{Z}$. Let $O_a = \{g \cdot a : g \in G\}$ be the orbit of an integer $a$ under this action. Assume that $O_a$ is a thin orbit, ...
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111
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Irreducible representations of $\mathfrak{sl}(m|n)$
It is well known that the irreducible representations of the Lie algebra $\mathfrak{sl}(n)$ are symmetric powers of a vector space of dimension $n$. This can be viewed for instance using Young ...
- 556
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45
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projections on minimal left ideals of semisimple algebras
Let $KG$ be a semisimple group algebra of a finite group $G$ over $K$. Consider $W=KGe$ as a minimal left ideal of this algebra and $e$ as a primitive idempotent. Here, $W$ is a simple left $KG$-...
- 143
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54
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Number of indecomposable modules over representation-finite hereditary algebras
Let $A$ be a finite dimensional $K$-algebra over a field $K$ that is hereditary and of finite representation type.
It is well known that they are classified by Dynkin diagrams.
For algebraically ...
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Existence of a subregular element with abelian centralizer in a quadratic Lie algebra
All Lie algebras here will be finite dimensionnal complex Lie algebra.
We say that such an Lie algebra $\mathfrak{g}$ is quadratic if there exist a skew-symetric, non-degenerate bilinear form or ...
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105
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How to determine the quiver of $R[x]/(x^2)$ for a finite dimensional $k$-algebra?
Let $R$ be a finite dimensional $k$-algebra given by a quiver. Then what is the quiver $R[x]/(x^2)$?
For example, an algebra is given by the following quiver
$$1\stackrel{\alpha}\rightarrow2$$
$$1\...
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59
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Decomposition of a contractive representation into an orthogonal sum for the $n$-dimensional case. Has this been done yet?
I know that it has been done for the two-dimensional case. Marek Kosiek showed it in his work "Decomposition of operator representations of the algebra $R(K_1 \times K_2)$" and "...
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42
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Jacobi relation when multiplicities are present
Assume a triad involving the adjoint has multiplicity, e.g. $8\bigotimes 8=1+8+8'+\dots$ in the Lie algebra $SU_3$. How can I handle this in a graphic formalism? Is both $8$ and $8'$ the adjoint, so I ...
- 4,829
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Generating function for dimensions of the space of polynomials fixed by a single permutation
Consider the space of polynomials with complex coefficients $\mathbb{C}[x_1,x_2,\dots,x_n]$ and let $\sigma$ be a permutation of $\{1,2,\cdots, n\}$ that acts on
this space via $\sigma(x_i)=x_{\sigma(...
- 141
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42
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Which finite group schemes over positive characteristic have projectivity detected by a set of (generalized) cyclic subgroups?
A finite algebra $B$ over a field $K$ has projectivity detected by a set $S = \{f_i : A \to B \mid i \in I\}$ of maps into $B$ from a fixed algebra $A$ if for any finite module $M$, $M$ is projective ...
- 542
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98
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Morphisms on L^2(G) induced by morphisms of LCA groups
I am looking for a good reference to understand the space $L^2(G)$ for a locally compact abelian (LCA) group $G$.
In particular, I would like to understand when $L^2(-)$ is functorial, so that if $\...
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55
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Weakly symmetric Hopf algebras
Let $A$ be a finite dimensional Hopf algebra over a field $K$ that is weakly symmetric (meaning $soc P = top P$ for each indecomposable projective $A$-module P).
Question: Is $A$ then automatically ...
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Does an induced representation commute with complexification?
Let $G$ be a compact Lie group, $K$ its closed subgroup, $\rho$ a finite-dimensional real irreducible representation of $K$, $c\rho$ its complexification, $\mathrm{Ind}^G_K(\rho)$ the real ...
user528693
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A reference for this statement (representations of universal central extensions)
Let $G$ be a connected Lie group with universal cover $\tilde{G}$. I would really appreciate if someone could give me a reference for the proof of the following fact:
"Every projective unitary ...
- 287
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81
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Can every $\ast$-algebra be represented in this space of matrices?
Let $k$ be a field with characteristic $0$. For every set $X$, let $\mathcal{B}(X)$ be the set of (possibly infinite) matrices $T = (T_{x,y})_{x,y \in X}$ with coefficients in $k$ such that in each ...
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80
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When can a R matrix be brought to Hermitian form?
For this question, let "R matrix" denote a (preferrably invertible) solution of the (constant) Yang Baxter equation. Any R matrix (if you alternatively write it as tensor $R^{ab}_{cd}$) is ...
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103
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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 ...
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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+...
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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-...
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Integral representation of completely alternating homogeneous functionals on semi-lattice of continuous functions
For a long time I've been interested in G. Choquet seminal work "Theory of capacities" (Annales de l’institut Fourier, tome 5 (1954), p. 131-295). More precisely part 53 about integral ...
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74
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A question on projective unitary representation of a Lie group
$\DeclareMathOperator\GL{GL}$Let $\mathcal{H}$ be a Hilbert space and $\GL(\mathcal{H})$ denote the group of invertible linear transformations of $\mathcal{H}$. Assume that $G=\{ f:\mathbb{P}\mathcal{...
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Wedderburn's theorem
In M. Isaac:s "Character theory of finite groups", there is a proof of a statement of Wedderburn (chap. 1, theorem 1.15.(c)). The statement goes like this:
Let $A$ be a semisimple algebra ...
- 275
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255
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Any "inherent" definition of $\mathrm{SU}(2)$ independent of any matrix representation?
$\DeclareMathOperator\SU{SU}\SU(2)$ is explained in detail here.
However, if I know right, this definition itself is known the "fundamental representation".
I wonder if there is any "...
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70
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geometric objects in quiver variety corresponding to short exact sequences
I was studying quiver variety and known that representations of a quiver correspond to points in the corresponding quiver variety. So if give you a fixed triple representations $(M_1,M_2,M_3)$, I was ...
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Why is von Neumann inequality important for equivalence of $\forall_j \ T_j^n\rightarrow 0$ in A-topology and abs continuity of $(T_1,\ldots, T_N)$?
The whole theorem goes as follows:
Let $(T_1, \ldots, T_N)$ be a tuple of commuting operators in Hilbert space $H$ satisfying:
$$\exists_{M > 0} \ : \ \forall_{p \in \mathbb{C}[z_1, \ldots, z_N]} \ ...
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128
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How to build a representation of the diffeomorphism group of $U(n)$?
Given that $U(n)$ is a smooth manifold I would like to know if there is a way of building a representation of $\text{Diff}(U(n))$ once you pick a particular (finite dimensional) representation of $U(n)...
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42
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Measurability of the weak completion of an orthogonal representation
Let $G$ be a locally compact group and let $\pi$ be a strongly continuous orthogonal representation of $G$ in a real Hilbert space $H$. Denote by $E$ the real Hausdorff locally convex space obtained ...
- 407
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Embeddings of unitary groups over $\mathbb{Q}$
$\DeclareMathOperator\GU{GU}$$\DeclareMathOperator\GL{GL}$I'm a bit confused by the following situation:
suppose we have an Hermitian vector space $V=K^3$ of matrix $$
J=\begin{pmatrix}& & \...
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Can a non-separable C$^*$ algebra have separable GNS Hilbert space
Suppose we have a $C^*$ algebra $\mathfrak{U}$ that is non-separable. Consider a state $ω$ of $\mathfrak{U}$ and the GNS representation $(H_ω,π_ω,Ω)$. Is it possible for $H_ω$ to be separable, and if ...
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116
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Multivariate polynomial representations of the infinite dihedral group
The presentation given in Wikipedia for the infinite dihedral group is
$$\langle r,s\mid s^2 =1, srs = r^{-1}\rangle.$$
Let $[R]$ denote the infinite set of reciprocal partition polynomials $R_n(u_1,...
- 10.5k
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134
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Tempered representations and unramified principal series
For $V$ a tempered representation of connected reductive group over a local field of characteristic zero. I want to show that for an Iwahori subgroup $B$, the set of fixed points $V^B\neq 0$, thereby ...
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99
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General reference for finite dimensional $*$-algebras over $\mathbb R$?
What references are there for studying finite-dimensional $*$-algebras over the field $\mathbb R$ in their full generality? We assume these are associative and unital.
Note that:
Not every algebra ...
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85
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Is a Lagrangian subgroup of a metric group isomorphic to its quotient?
A metric group is a finite abelian group $G$ with a quadratic function
$$q:G\rightarrow \mathbb R/\mathbb Z\;,$$
that is,
$$M(a,b):= q(a+b)-q(a)-q(b)$$
is bilinear in $a$ and $b$ [edit: and non-...
- 3,001
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114
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Norm of the Linear Operator
Let $G$ be a compact group, and $\pi : G \rightarrow \mathcal{U}(H)$ be a continuous unitary representation. Let $f \in L^{1}(G)$ be arbitrary.
By Riesz Representation Theorem we can find a bounded ...
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382
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Theory of group representation for compact groups
I write here because some experts could help on that. It is very well known (at least for me) many reference books on linear representations of finite groups (for instance, the very classical and ...
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Is there a generic representation for non-quasi split $p$-adic group?
It seems that generic representation only occurs for quasi-split groups.
For non-quasi split groups, is it expected that generic representation doesn’t exist?
Thank you in advance!
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