Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
2,542 questions with no upvoted or accepted answers
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references on categorification of knot invariants
I am extremely sorry if this is not the right place for this kind of question.
I have studied some knot theory, quantum invariants and would like to study more about categorification of knot ...
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Dyck paths of Dynkin type
(The conjecture is a homological algebra question, but question 2 is a pure combinatorics question given that the conjecture is true)
A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...
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Generalized Young symmetrizers, why not?
For $n$ a positive integer, let $[n]=\{1,2,\ldots,n\}$. Consider two set partitions $\mathcal{A}=\{A_1,\ldots,A_p\}$ and $\mathcal{B}=\{B_1,\ldots,B_q\}$ of the set $[n]$.
We will denote by $G(\...
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Computing the number of elementary abelian p-subgroups of rank 2 in $GL_{n}(\mathbb{F}_{p})$
Let $p$ be a prime number, and let $\mathbb{F}_{p}$ be a finite
field of order $p$. Let $GL_{n}(\mathbb{F}_{p})$ denote the general linear
group and $U_{n}$ denote the unitriangular group of $n\times ...
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Two algebraic guises of Alternating Sign Matrices: any connection?
Alternating Sign Matrices (ASMs) have a famous history: they were discovered by Mills, Robbins, and Rumsey, who conjectured a product formula for their enumeration; this product formula was first ...
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Non-zero group determinant for symmetric group
Let $G$ be a finite group. Given complex numbers $x=\{x_g: g\in G\}$, one can define a $|G|\times |G|$ matrix $X$, with entries $X_{g,h} = x_{gh^{-1}}$.
Let's consider $G$ being the symmetric group $...
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Ramified Geometric Langlands
Is the following a reasonable formulation of (a part of) the geometric Langlands conjecture for $\mathrm{GL}_n$ over a curve $X$?
(*) Let $\mathcal{L}$ be an irreducible rank $n$ $\ell$-adic local ...
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Are the braid groups good in the sense of Toën?
In this question it is asked whether the mapping class groups are 'good' in relation to pro-finite completion. Helpfully, one of the answers gives a link to a proof that the braid groups are good ...
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545
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What are the character tables of the finite unitary groups?
I need to know the (complex) character table of the finite unitary group $U_n(q)$. Lusztig and Srinivasan (1977) provide an abstract description, but parsing it requires a stronger background in ...
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Significance of half sum of non-simple positive roots
In representation theory, there are plenty of places that a $\rho$-shift makes an appearance, where $\rho$ is the half sum of positive roots. See, for instance, this post for some discussions of the ...
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Which representations of the Lie algebra of a Lie group come from representations of the group itself?
I think this is very classic mathematics, but I can't find a complete answer in the literature.
Let $G$ be a Lie group, $\mathfrak{g}$ the Lie algebra of $\mathfrak{g}$. Suppose $\rho : \mathfrak{g} \...
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$n$-fold tensor products of $D(A)$ for finite dimensional algebras
Let $A$ be a finite dimensional quiver algebra over a field $K$ and let $D(-):=Hom_K(-,K)$ denote the natural duality (assume algebras are connected).
Define $\psi_A:=sup \{ n \geq 1 | D(A)^{\otimes ...
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On constructible Hall algebra and instantons
I heard in a talk by Yan Soibelman that by starting with a quiver $Q$ with a set of vertices $I$ we can either symmetrize or anti-symmetrize its Euler-Ringel form $\chi_Q$. He claims that anti-...
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Is there a splitting rule for the restriction of a $GL(23, \mathbb{Q})$-representation to $O(23, \mathbb{Q})$?
I am interested in a $23$-dimensional $\mathbb{Q}$-vector space $V$ which I am viewing as a GL$_{23}(\mathbb{Q})$ representation. Schur functors can be defined over $\mathbb{Q}$, so we get ...
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Representations of groups with the same derived group, how much control do we have over the central character?
Let $G_1 \subset G$ be the rational points of $p$-adic reductive groups sharing the same derived group. There are some well known results relating representations of $G_1$ to representations of $G$, ...
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Practice of higher categories - giving rigorous constructions
1) Let $\mathcal{C}$ be a monoidal $\infty$-category, $A$ an algebra object in $\mathcal{C}$, and $M$ a left $A$-module in $\mathcal{C}$. So, in Lurie's formalism, $\mathcal{C}$ is encoded by some ...
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Rational homotopy type of Hilbert scheme components
What is known about rational homotopy type of irreducible component of $Hilb^n(\mathbb C^k)$ containing configuration space? I've searched arXiv for a while and found nothing but surmise that Betti ...
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$L^2$ norms of Whittaker vectors and zeros of Intertwining operators
For $\mu,\nu\in \mathbb{C}^2$ we denote $I(\mu,\nu)$ to be the principal series of $\mathrm{GL}_2(\mathbb{Q}_p)$ induced from $|.|^\mu\otimes |.|^\nu$. For $s=\mu-\nu$ one defines the standard ...
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The $GL(N)$ chicken versus the $SL(N)$ egg, the Erlangen Program and relations between FFTs
Here FFT is the standard abbreviation for First Fundamental Theorem of classical invariant theory.
If one has a group inclusion $H\subset G$ (and I suspect also more generally a morphism $H\rightarrow ...
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Induction from the Borel subalgebra to BGG category $\mathcal{O}$
Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra with a choice of Cartan subalgebra $\mathfrak{h}$, Borel $\mathfrak{b}$, and nilpotent radical $\mathfrak{n}$. Let $\mathcal{O}...
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$p$-adic representations of the fundamental group of a smooth proper curve over a finite field
This question is very general. Let $C$ be a smooth and proper curve over a finite field ${\bf F}_p$. Are there any general results or conjectures on continuous non abelian representations
$$
\pi_1(C)\...
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Root-theoretic formulation of characteristic polynomial
Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra of rank $n$ over $\mathbb{C}$. Let $G$ denote the corresponding simple simply connected algebraic group. By Chevalley's Theorem, $\mathbb{...
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When is a $2$-Calabi–Yau triangulated category the cluster category of a QP?
Keller–Reiten's main theorem in Acyclic Calabi–Yau categories implies that if $\mathcal{C}$ is a $2$-Calabi–Yau (algebraic) triangulated category admitting a cluster-tilting object $T$ such that the ...
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Sum over growing Young tableaux
Let $\lambda_0,\lambda_1,\lambda_2,\lambda_3,\ldots$ be a sequence of Young diagrams, such that each successive diagram is obtained from the prior by the addition of one box (don't forget that the row ...
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340
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Representations of orthogonal groups vs representations of reflection groups
Let $V$ be a finite dimensional inner product space and $O(V)$ the orthogonal group of $V$.
Let $G$ be a (say, finite) reflection group on $V$, regarded as a subgroup of $O(V)$ ($G< O(V)$.) Let
us ...
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Generators for invariant tensors of lie algebras
EDITED FOR (hopeful) CLARITY:
For a simple Lie algebra $\mathfrak{g}$ (over $\mathbb{C}$) and its adjoint group G, the $G$-invariant polynomials on $\mathfrak{g}$ are linear combinations of products ...
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Connection between two theorems on character values?
In a recent arXiv preprint here, Dipendra Prasad has revisited a 1976 theorem of Kostant (Theorem 2 in the paper On Macdonald's $\eta$-function formula, the Laplacian and generalized exponents, ...
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Higher-dimensional generalization of Pink's theorem
Pink's theorem in the title of the question refers to the main theorem of Pink's paper "Compact Subgroups of Linear Algebraic Groups" that appeared in Journal of Algebra (206) in 1998. It essentially ...
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when do norm-continuous unitary representations separate points of a group?
Recently I found in the web a discussion on the following question:
...
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What can we say about the Local Langlands Correspondence for GL_n without using Bernstein-Zelevinski?
I have two specific questions regarding the LLC for $GL_n$, and in particular, what we can say about the conjecture if we don't have the ideas of Bernstein and Zelevinski, which reduce the problem to ...
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Two appearances of the Jacobi Triple Product identity
So far, I've seen two ways of deriving the Jacobi Triple Product (JTP) formula $$\sum_{m \in \mathbb{Z}} (-u)^m q^{\frac{m(m-1)}{2}}= \prod_{j=0}^{\infty} (1-uq^{j})(1-q^{j+1})(1-u^{-1} q^{j+1})$$ ...
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Reference/ elementary proof of a result about projective dimension in group rings
Hello- I've had to use a result that sounds like it should be well-known, but I couldn't find any references, and my proof is rather unsatisfactory. I was hoping someone here could help! The problem ...
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det(A)det(B) = det(AB+correction), Capelli identities, "factorized" representation of $\mathfrak {gl}_n$
Context: Some probably know that there are Capelli identities which state
$$det(A)det(B) = det(AB+correction)$$ for some matrices with non-commuting elements, they go back to the 19-th century, but ...
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Chern Classes of Exterior Products of a vector bundle.
This is mostly a question in combinatorics. Is there a clean way in terms of determinantal identities to write down $c(\wedge^k V)$ i.e. the individual summands in terms of the individual summands of $...
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Reference Request - Spaces of Smooth Vectors
I was recently looking for examples of non-nuclear spaces of smooth vectors of representations of Lie groups. I'll recall the basic definitions. Let $\pi$ be a unitary irreducible representation of a ...
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Complexity of computing the minimum degree of a faithful linear representation of a finite group
Background. Mazorchuk and I were interested in computing the minimum faithful degree of a linear representation of a finite semigroup over the complex feld in this paper. One particular aspect we ...
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Inducing factor representations
I have an infinite dimensional Lie group $G$ of the form $\varinjlim G_n$, for example $SU(9,\infty)$ or $Sp(\infty,R)$. I also have a closed subgroup $P$ of $G$ and a factor representation $\chi$ of ...
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duality between universal enveloping and function algebra for GL(n)
Motivation. Few years ago I constructed a family of internal Hopf algebras in the Loday-Pirashvili tensor category of linear maps which is in a sense a generalization of the algebra of regular ...
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Relation between group representations and elements of group cohomology groups
Having already seen group cohomology, I was just introduced to the formula $U \otimes Ind W = Ind(Res(U) \otimes W)$ from representation theory. This seems oddly like the formula $\mathrm{Cor}(u) \cup ...
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Action of the endomorphism monoid on an irreducible GL-module
Let $G=\mathrm{Gl}_n(\mathbb C)$ and $V$ an irreducible $G$-module on which $G$ acts polynomially. In other words, the algebraic group action of $G$ on the affine space $V$ extends to an algebraic ...
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Langlands correspondence of coverings of $\mathrm{SL}_2(\mathbb R)$ and modular forms with fractional weights
$\DeclareMathOperator\SL{SL}$Let $G \to \SL_2(\mathbb R)$ be a finite covering of degree $d \geq 2$. Then $G$ is a connected Lie group with semisimple Lie algebra $\mathfrak{g}=\mathfrak{sl}_2$ and ...
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Frenkel-Kac's vertex operator realisation of the basic representation of an untwisted affine Kac-Moody algebra
Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$ and let $\hat{\mathfrak{g}} := (\mathfrak{g}[t^{\pm 1}] \oplus \mathbb{C} c) \rtimes \mathbb{C} D$ be the corresponding ...
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Relation between Fourier series and Schur polynomials
Asked initially at MSE.
I would like to know how to express the Fourier series of a symmetric function, $f(\theta_1,...,\theta_N)$, in terms of Schur polynomials $s_\lambda(x_1,...,x_N)$ in the ...
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Decomposing an endomorphism as a tensor product
$\DeclareMathOperator\End{End}$Let $f$ be an endomorphism of the finite-dimensional vector space $V$, over the field $K$. The question of whether $f$ is decomposable, that is, whether $V$ can be ...
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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, ...
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On the structure of an algebra as a bimodule
$\DeclareMathOperator\End{End}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Ker{Ker}\newcommand{\bi}{\mathrm{bi}}\newcommand{\op}{\mathrm{op}}$Let $K$ be a field (say of characteristic zero), and $...
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The quotient of a higher Specht polynomial over the corresponding regular Specht polynomial
I'll need some notation before I can phrase my question, so please bare with me for a little. I'll try to get there as fast as possible (it's also my first MO question...).
Let $\lambda$ be a ...
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Status of the Luna's conjecture
In the famous IHES paper <Variétés Sphériques de Type A> of D. Luna, he proposed a conjecture asserting that wonderful varieties of an adjoint semisimple group $G$ are bijective to spherical ...
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Optimizing computations with nilpotents in a group algebra
Of course, I have a very concrete problem at hand, which has been vexing me for about a year now. But let me start with a question that has a better chance of having been answered.
Let $G$ be a ...
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Twisted D-module structure on pushfoward of structure sheaf of Bruhat cell
Apologies for the basic question, but my experience on math.stackexchange tells me that this will go unanswered there.
Background: In the principal block, the dual Verma modules (with highest weight $...
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