Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
6,816
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Recognizing a restriction from $SL_2(\mathbb{C})$ to $SL_2(\mathbb{Z})$
I am aware that classifying all $SL_2(\mathbb{Z})$ representations is more or less completely intractable, but I was wondering what is known about the following simpler question: How do I recognize ...
- 2,052
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0
answers
69
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Do the values of the global dimension constitute an interval?
Let $Q$ be a fixed finite connected quiver and $k$ a fixed field. Set $Z_Q:= \{ gldim(kQ/I) < \infty | I $ an admissible ideal $\}$.
Question: Is $Z_Q$ an intervall?
This is true for example in ...
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0
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229
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Find representation set of orbits when group acts on a set
Let group $G$ acts on a set $S$. Burnside's lemma gives as how to count numbers of orbits. I am interested how to find the orbits. By finding orbits I mean how to find a representative from each orbit....
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Formal character and unit
Denote by $\mathfrak{g}$ a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. Let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$.
...
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3
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175
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Relative position on flag variety
Let $G$ be a semisimple algebraic group over $\mathbb{C}$. Consider the $G$ diagonal action on $G/B \times G/B$, the orbit is indexed by $W$, the Weyl group of $G$ by Bruhat decomposition. There is a ...
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1
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Highest-$\ell$-weight tensor products and diagram subalgebras
Let $U_q(\mathcal{L}({\mathfrak{g}}))$ be a quantum loop algebra and $I$ the set of indexes of Dynking diagram of $\mathfrak{g}$. Consider $J\subset I$ a connected subdiagram, so that $U_q(\mathcal{L}(...
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References for representations of Heisenberg Lie algebra
Please suggest some reference material for the representations of the infinite dimensional Heisenberg Lie Algebra or the oscillator algebra. I already looked at Kac and Rainas book, any other ...
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Two seemingly different definitions of a left cell
This is a question about two seemingly different notions of a left cell in a finite Weyl group and why they are the same. My question arose from reading a paper of W. McGovern titled "Left cells and ...
- 1,063
3
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142
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Automorphy Factor from Vector Bundles on Compact Dual
So I'm coming from an algebraic geometry perspective and I'm trying to carefully piece together the story of interpreting automorphic forms as sections of vector bundles on Shimura varieties. I think ...
- 1,701
2
votes
1
answer
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Simple modules for direct sum of simple Lie algebras
I think that the following statement is true, but I do not know how to prove it.
Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be two real simple Lie algebras. If $M$ is a (infinite dimensional) complex ...
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About parabolic Kazhdan Lusztig polynomials
There are two types of parabolic Kazhdan Lusztig polynomials, namely, of type -1: $P_{x,w}^{I,-1}$ and of type $q$: $P_{x,w}^{I,q}$. See Kazhdan–Lusztig and R-Polynomials,
Young’s Lattice, and Dyck ...
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2
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1
answer
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A weak Schur's lemma for non-semisimple finite dimensional algebras
Let $B \subseteq C$ be an inclusion of finite dimensional (associative) algebras over a field $k$. Assume that $C$ is a free $B$-module. Let $\bigoplus_i U_i$ be
a decomposition of $B$ into ...
3
votes
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answers
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Conjugacy in metaplectic groups
Let $F$ be a non-Archimedean local field (characteristic 0) and $G=GL(2,F)$. Let $\tilde{G}$ be "the" metaplectic double cover of $G$ (defined using an explicit cocycle as in Gelbart's book (Weil's ...
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2
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Openly available software to work with Demazure modules
Does someone know of any sort of software openly available online which can be used to compute various characteristics of Demazure modules for semisimple Lie algebras? Specifically, I'm interested in ...
- 3,493
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158
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Extension of Verma modules
The category $\mathcal{O}$ is the category of all finitely generated, locally $\mathfrak{b}$-finite and $\mathfrak{h}$-semisimple
$\mathfrak{g}$-modules, where $\mathfrak{g}$ is a complex semisimple ...
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5
votes
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answer
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Definition of a Dirac operator
So it seems that a Dirac operator acting on spinors on $\psi=\psi(\mathfrak{su}(2),\mathbb{C}^2)$ can be written in this case simply as:
$D=\sum_{i,j} E_{ij}\otimes e_{ji}$, where $E_{ij}$ are ...
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answers
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Proving that $\lambda\mapsto \chi^\lambda(C)/f^\lambda$ is a polynomial
Let $\lambda$ be a partition of $n$ and $\chi^\lambda$ be the character of $S_n$ associated to it. Given any conjugacy class $C$, I want to prove that
$$\lambda\mapsto \frac{\chi^\lambda(C)}{f^\lambda}...
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1
answer
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Computing affine Springer fibers
$\DeclareMathOperator\diag{diag}\DeclareMathOperator\Gr{Gr}\DeclareMathOperator\SL{SL}$I'm having some trouble computing affine Springer fibers, even in simple cases. For example, consider the group $...
- 2,969
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85
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Schur function on unit circles
Define $T^d$ as following
$$ T^d = \left\{(t_1,\cdots,t_d)\in\mathbb{C}^{d}\mid |t_i|= 1 \mbox{ for all } i\right\}
$$
For any partition $\lambda\vdash n$,The Schur function is defined
$$
\...
- 1,493
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votes
2
answers
560
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Verma modules in category $\mathcal{O}^\mathfrak{p}$
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra
of $\mathfrak{g}$. Fix a Borel subalgebra $\mathfrak{b}$ containing $\mathfrak{h}$ and fix a ...
- 1,855
2
votes
1
answer
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Levi subgroup of Siegel parabolic of GSpin
I consider the group $G=\mathrm{GSpin(V)}$ as in this question.
We have the so called Siegel parabolic $P$ (after fixing a cocharacter) and the associated Levi $M$ (these can also be obtained using ...
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3
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Borel's presentation for the cohomology of a Flag Variety
If $G$ is a simple complex Lie group, $T\subset B\subset G$ is a choice of Borel and maximal torus, and $W$ is the Weyl group, then
1) $H^{*}(G/B,K)=K[T^{\vee}]/(K[T^\vee]^W_{+})$
and
2) $K[T^\vee]^...
- 1,537
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Characters of orthogonal groups as symmetric functions
This question was asked on MSE some time ago, here, but got no attention.
The Schur functions are characters of irreps of the unitary group, $s_\lambda(U)=Tr(R_\lambda(U))$. They are symmetric ...
- 2,540
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544
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Shimura varieties of Hodge type
I am trying to understand the theory of integral model of Shimura variety of Hodge type, like for example in Kisin's paper "Integral models for Shimura varieties of abelian type".
I understand that ...
- 273
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0
answers
216
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Deformation of the Hochschild-Kostant-Rosenberg isomorphism for universal enveloping algebra
Let $\mathfrak{g}$ be a Lie algebra over a char. $0$ field and let $\iota: U\mathfrak{g}\rightarrow S\mathfrak{g}$ be the Poincaré-Birkhoff-Witt (PBW) isomorphism, inverse to that natural map from the ...
user108998
4
votes
1
answer
398
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Hyperbolic Dehn surgeries and SU(2)-representations
Let $S^3-K$ be the complement of the figure eight knot complement. Thurston, in his Lecture Notes, constructed a hyperbolic structure, which comes from a discrete, faithful representation $\pi_1(S^3-K)...
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357
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$\text{Determinant}=(\sum \text{Determinant})^2$
Denote by $\delta_{n-1}=(n-1,n-2,\dots,1,0,0,\dots)$ the staircase partition and the embedded partition
$\lambda=(\lambda_1,\lambda_2,\dots)\subset\delta_{n-1}$.
QUESTION 1. Is this true?
$$\det\...
- 41.8k
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1
answer
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$q$-plane partitions & specialization & interlinks
MacMahon's enumeration of all plane partions (PP) inside an $n$-cube generalizes to
$${\tt PP_n}(q)=\prod_{i,j,k=1}^n\frac{1-q^{i+j+k-1}}{1-q^{i+j+k-2}}.$$
A $q$-analogue of symmetric plane partitions ...
- 41.8k
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Even Counterexample to Statement About the Non Existence of Certain Groups with Two Irreducible Monomial Character Degrees
Let $\textrm{cd}(G)=\lbrace \chi(1)\,|\, \chi\in\textrm{Irr}(G)\rbrace$ denote the set of character degrees of a finite group $G$. Similarly, denote by $\textrm{mcd}(G)$ the set of monomial character ...
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Branching to Levi subgroups in SAGE and the circle action
In the SAGE computer package, there useful exist tools for branching representations of a simple Lie group to a Levi subgroup:
http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/...
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GAP versus SageMath for branching to Lie subgroups
Which computer package is better, GAP or SageMath, for
decomposing an irreducible representation of a (simple) Lie group
$G$ into representations of a Lie subgroup. I am most interested when
...
- 815
11
votes
2
answers
551
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Classification of algebras of finite global dimension via determinants of certain 0-1-matrices
I restrict to the elementary problem that is equivalent to give a classification when Morita-Nakayama algebras have finite global dimension (see the end of this post for some background).
A Morita-...
- 26k
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answers
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Identity for classes of plane partitions
There are several classes of plane partitions in the literature.
Among these, let's look at the enumeration of three of them: the symmetric (SPP), totally symmetric (TSPP) and totally symmetric and ...
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votes
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Derived invariance of the Cartan determinant
The Cartan matrix $C$ of a finite quiver algebra $A$ with points $e_i$ is defined as the matrix having entries $c_{i,j}=\dim(e_i A e_j)$. The Cartan determinant is defined as the determinant of the ...
- 26k
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votes
3
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The vanishing of sum of coefficients: symmetric polynomials
Denote $\pmb{X}_n=(x_1,x_2,\dots,x_n)$. Consider the symmetric polynomial
$$f_n(\pmb X_n)=\prod_{1\leq i<j\leq n}(x_i+x_j).$$
Expand these in terms of elementary symmetric polynomials, say
$$f_n(\...
- 41.8k
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answers
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Who first noticed the duality for finite groups?
A.A.Kirillov in section 12.3 of his "Elements of the Theory of Representations" writes that the first "symmetric" duality theory for non-commutative groups was the theory for finite groups. In short ...
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871
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Is it possible to reconstruct a finitely generated group from its category of representations?
Suppose $G$ is a finitely generated group, and suppose $Rep_k(G)$ is its category of representations over some field (or maybe even a ring) $k$, endowed with whatever extra structure is needed --- ...
- 1,233
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Looking for access to original paper for Category O
It is well-known that the BGG category $\mathcal{O}$ was introduced in the early 1970s by Joseph Bernstein, Israel Gelfand and Sergei Gelfand. I google for a while but I cannot find out the original ...
- 1,855
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Which dimensions exist for irreducible quaternionic-type real representations of finite groups?
I'm writing a software package to decompose group representations, and am struggling to find good examples of quaternionic-type representations of dimension > 4.
Reading MathOverflow, I found that ...
- 145
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1
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Littlewood-Richardson coefficients for zonal polynomials
The Littlewood-Richardson coefficients $c^\lambda_{\mu\nu}$ appear in the expansion of a product of Schur functions into Schur functions, $s_{\mu}(x)s_\nu(x)=\sum_\lambda c^\lambda_{\mu\nu}s_\lambda(x)...
- 2,540
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votes
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340
views
Gorenstein symmetric conjecture for arbitrary rings
The Gorenstein symmetric conjecture states that for Artin algebras $A$ one has the the regular module has finite injective dimension as a right module if and only if it has finite injective dimension ...
- 26k
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Infinite-dimensional representation theory of $K[x]$
Let $K$ be an algebraically closed field. The finite-dimensional representation theory of the polynomial algebra $K[x]$ is tame and completely understood, which I shall first summarise. It's ...
- 482
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votes
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Construction of non-split extension of simple modules of Lie algebras using linear differential operators
Consider the natural action of $W_1=k\left\langle x,\frac{d}{dx}\right\rangle$ on $X=\mathbb C[x]$. Then $\frac{d}{dx}, x\frac{d}{dx},x^2\frac{d}{dx}$ is essentially a $\mathfrak{sl}_2$-tuple ($\left[...
- 6,158
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Does $SU(N)$ have pseudo-real representation?
For $N\ge 2$, does $SU(N)$ have a non-real pseudo-real irreducible representation? (The adjoint representation of $SU(N)$ is real).
A (complex, finite-dimensional) representation $R:SU(N)\to GL_n(\...
- 169
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answers
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Frobenius formula
I know two formulas by the name of Frobenius.
The first one computes the number
$$\mathcal{N}(G;C_1,\dotsc,C_k):=|\{(c_1,\dotsc,c_k)\in C_1 \times \cdots \times C_k\:|\:c_1\cdots c_k=1\}|,$$
where $...
- 975
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Which groups can be reconstructed from a single invariant subspace?
Let $G\subseteq\mathrm{Perm}(\Bbb R^n)$ be a matrix group consisting of permutation matrices acting on $\Bbb R^n$. Let $U\subseteq\Bbb R^n$ be an irreducible invariant subspace w.r.t. $G$. Now, define ...
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Size of a multi-segment of a representation of $GL_n(F)$
Let $F$ be a p-adic field and $GL_n(F)$ the general linear group over $F$. The irreducible complex finite length smooth representations are parametrized by multi-segements in the paper. A multi-...
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votes
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answer
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representations with centralizer stable under conjugate transpose
Let $\rho:G\to GL_n(\mathbb{C})$ be a finite-dimensional representation of a finite group $G$ over $\mathbb{C}$, and $C_\rho\subset M_n(\mathbb{C})$ its centralizer, i.e. $m\in C$ iff $m$ commutes ...
- 13.7k
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votes
1
answer
102
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Algorithms for the explicit matrix isomorphism problem over $\mathbb{C}$
Suppose that $A$ is a $d^2$ dimensional algebra over $\mathbb{C}$ and we know the multiplication tensor $c_{ij}^k$ and the unit $u^k$ in some basis. If $A$ is semi-simple and has a single simple ...
- 3,710
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votes
0
answers
134
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Growth of the number of fixed points of a $p$-adic group under natural filtrations
Let $G$ be a $p$-adic reductive group, so by definition as a locally profinite group it's the group of $\mathbb Q_p$ points of a connective reductive group over $\mathbb Q_p$, $K$ be a parahoric ...
- 6,158