Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
7,076 questions
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Is there a category of representations of a simple Lie algebra on which its Weyl group naturally acts?
For any simple Lie algebra $\mathfrak{g}$, is there a category $C$ of (possibly infinite-dimensional) representations of $\mathfrak{g}$ such the Weyl group $W$ of the corresponding root system acts in ...
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Invariant differential operators on real Grassmannians
I am looking for an explicit description of the algebra of $SO(n)$- or, better, $O(n)$-invariant differential operators on the real Grassmann manifolds of $k$-dimensional linear subspaces in the ...
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when are algebras quiver algebras ?
Good Morning from Belgium,
I'm no stranger to the mantra that quiver-algebras are an extremely powerful tool (see for example the representation theory of finite dimensional algebras). But what is a ...
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factorization of the regular representation of the symmetric group
Let $\mathbb{C}[S_n]$ be the regular representation of the symmetric group $S_n$, and let $\mathbb{C}^n$ be the vector representation.
Question: Does there exist a representation $V$ (of dimension $(...
- 2,537
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What is known about the connection of positive energy representations of loop groups and modular forms
At the end of Section 14.1 in Pressley, Segal "Loop Groups" there is the remark that
the partition function is a modular function in the sense that the Dedekind $\eta$ function is a modular form. I ...
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Gelfand-Naimark from the category-theoretic point of view
I was thinking about the Gelfand-Naimark theorem asserting the isometric * isomorphism between a commutative $C^*$-algebra (with unit) $\mathcal{A}$ and the $C^*$ -algebra of continuous complex-valued ...
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What is the relation between spherical principal series representations of a reductive Liegroup and Verma modules for its Lie algebra?
Here is an issue that thoroughly confuses me. I hope I can express it in a way that is clear cut enough for this site.
Let $G$ be a real reductive Lie group and $\mathfrak{g}$ be the complexification ...
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Local Langlands for $GL(2,\mathbf{C})$ and reducible principal series
My naive picture of the local Langlands correspondence for $GL(2,\mathbf{C})$ is this. The Weil group of $\mathbf{C}$ is canonically $\mathbf{C}^\times$. On the Galois side then we're looking at 2-...
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When are Jones-Wenzl projectors defined?
(I am hoping that someone well-versed in the literature of Temperley-Lieb algebras or of quantum groups at roots of unity can answer my question. Fingers crossed.)
Consider the Temperley-Lieb algebra ...
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What do the components of $\operatorname{Hom}(\pi_1(S),\operatorname{SL}_n(\mathbb{R}))$ look like?
Let $S$ be a closed orientable surface of genus at least $2$. I'm interested in the connected components of $\operatorname{Hom}(\pi_1(S),\operatorname{SL}_n(\mathbb{R}))$ for $n$ at least $3$.
I know ...
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Infinity local systems
I have seen many references in the (geometric representation theory, symplectic geometry, etc) literature to "infinity local systems".
From what I've been told, given a good cover $\{U_i\}$ of $X$, ...
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What is the relation between L. Lafforgue and Frenkel-Gaitsgory-Vilonen results on Langlands correspondence ?
What is the relation between Lafforgue's result on Langlands
and Frenkel-Gaitsgory-Vilonen ? ( http://arxiv.org/abs/math/0012255 , http://arxiv.org/abs/math/0204081 )
Does one imply other ? If not ...
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What is the centralizer of a Young subgroup of $S_n$?
In their celebrated paper "A new approach to the representation theory of the symmetric group. II", Okounkov and Vershik prove that $Z(n-1,1)$, the centralizer of $\mathbb{C}[S_{n-1}]$ in $\...
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Who conjectured the Cartan determinant conjecture
The Cartan determinant conjecture states that every finite dimensional algebra of finite global dimension has the property that the determinant of its Cartan matrix is equal to one. Who stated this ...
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Which finite groups have no irreducible representations other than characters?
A classical result states that all the irreducible representations of a finite group over $\mathbb{C}$ are characters if and only if $G$ is abelian. I would like to know what happens if we consider a ...
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Relating two characterizations of ${\mathfrak sl}_{n > 2}$ among simple Lie algebras
Let $\mathfrak g$ be a finite-dimensional simple Lie algebra over $\mathbb C$.
Theorem 1. The highest root is perpendicular to all but one simple root, except in the case ${\mathfrak g}={\mathfrak sl}...
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Symbols of elliptic operators
First let me state the problem, then I'll explain its origin and finally, I'll ask the main question..
Problem S. Fix a positive integer $n$. Find all the pairs $(V, S)$, whith the following ...
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A possible gap in Faltings note to prove the Tate conjecture for finitely generated field over $\mathbb{Q}$
Qeustion:
Given a Lie algebra $\mathfrak{g}$ over $\mathbb{Q}_\ell$ with an ideal $\mathfrak{g}^O$ and a subalgebra $\mathfrak{h}$,
such that $\mathfrak{g}=\mathfrak{g}^O+\mathfrak{h}$.
Now given a ...
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Swan K-theory of Z/4
Given a finite group $G$ and a commutative ring $R$, define the Swan $K$-theory $K_0(G, R)$ to be the Grothendieck group of the category finitely generated projective $R$-modules with $G$-action (with ...
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Branching rule of $S_n$ and Springer theory
Let $u\in\mathrm{GL}_n$ be a unipotent element, let $\mathcal{B}_u$ be the variety of Borel subgroups containing $u$, and let $d=\dim \mathcal{B}_u$. Then Springer theory tells us that $H^{2d}(\...
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Gelfand-Tsetlin algebras and "Jucys-Murphy elements" for $\mathfrak{gl}_n$
I'm trying to figure out/find in literature the details concerning Gelfand-Tsetlin algebras for $\mathfrak{gl}_n(\mathbb C)$ (Okounkov-Vershik style, if you wish).
Consider the chain $$\mathcal U(\...
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distribution of Young diagrams
Consider $\Lambda^p(C^n\otimes C^n)=\oplus_{\pi}S_{\pi}C^n\otimes S_{\pi'}C^n$ as
a $GL_n\times GL_n$-module. This space has dimension $\binom {n^2}p$. I would
like any information on the shapes of ...
- 835
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Characteristic classes of symmetric group $S_4$
For the symmetric group $S_3$, it is classically known that \begin{equation} H^*(S_3;\mathbb{Z})\cong \mathbb{Z}[x,y]/(2x,6y,x^2-3y), \end{equation} where $|x|=2$ and $|y|=4$. Moreover, $x$ can be ...
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What are the periodic Dyck paths?
I changed the thread completely so that everything is now elementary linear algebra.
A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...,c_n]$ with $c_i -1 \leq c_{i+1}$ for all $i$...
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Is this error in this paper of Langlands fixable?
The FQS criterion for the Virasoro algebra was discovered by Friedan, Qiu and Shenker (1), but the mathematicians found their proof insufficient, so that, FQS (2) and Langlands (3), published in the ...
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Are triangulated equivalence detected at compact level?
Suppose that $D$ and $E$ are compactly generated triangulated categories, even algebraic (i.e. equivalent to derived categories of small dg categories) if we want, and asume that their subcategories $...
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Where do stable Kronecker coefficients live "in nature"?
Background:
For a partition $\lambda$, let $\lambda[N] = (N - |\lambda|, \lambda_1, \lambda_2, \lambda_3, \dots)$, also let $\chi_\lambda$ be the corresponding irreducible character of the symmetric ...
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What are the Schur functions of the eigenvalues of a non-negative integer matrix counting?
Let A be a non-negative integer square matrix with eigenvalues x1, x2, ... xn. Any symmetric function of these eigenvalues with integer matrices is an integer. I'm aware of the following results ...
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History of the notion of irreducible representation
I am looking for the earliest references where the study of irreducible representations appears. There has been many articles and books on the history of representation theory. A fundamental feature ...
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Conceptual explanation for curious linear-algebra fact in characteristic $2$
All matrices and vectors in this post have entries in the field $\mathbb{F}_2$.
Fix some $n \geq 1$. For an $n \times n$ matrix $X$, write $X_0$ for the column vector whose entries are the diagonal ...
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Is there a natural notion of completion of a Coxeter system?
Let $(W,S)$ be a Coxeter system. Then any element of $W$ can be written as a finite products of elements of $S$. I want some notion of a "completion" of $W$, call it $\hat{W}$, whose elements are (...
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Funktorialität in der Theorie der automorphen Formen
In 2010 Langlands wrote an article with the title Funktorialität in der Theorie der automorphen Formen: Ihre Entdeckung und ihre Ziele. On the IAS website, he says that
This note ... was written ...
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Categorifying the equality of product and coproduct of symmetric functions
Littlewood-Richardson coefficients are both multiplicities of $GL_n$ tensor products, and of restrictions of $GL_{m+n}$ representations to $GL_m \times GL_n$. I want to turn this equality of numbers ...
- 156k
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Why do sl(2) and so(3) correspond to different points on the Vogel plane?
Vogel assigns to every simple metric Lie algebra (and more generally to every simple metric Lie algebra object in a symmetric monoidal category) a point in the orbifold $\mathbb{P}^2/S_3$ (where $S_3$ ...
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Is there a good notion of "induction" for representations of 2-categories?
One of the most important observations in the representation theory of algebras is that if one has a subalgebra $A\subset B$, then these is a functor $B\otimes_A -\colon A\operatorname{-pmod}\to B\...
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Linear embeddings of nilpotent pro-$p$ groups
Is it true that every finitely generated (topologically) torsion-free nilpotent pro-$p$ group is isomorphic to a subgroup of $U_d(\mathbb{Z}_p)$, the group of $d\times d$-upper triangular matrices ...
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Diagonalizing some matrices arising from Fourier transform on $S_n$.
Consider the function $f$ on $S_n$ which equals $1/n$ on all adjacent transpositions $(i,i+1)$, where we let $n+1 = 1$, and $0$ otherwise, and its Fourier transform $\hat{f}(\rho)$ evaluated at the ...
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Character theoretic proof of the Littlewood–Richardson rule?
The Littlewood–Richardson coefficients are the multiplicities
$$
c(\lambda,\mu,\nu)= \dim_{\mathbb{C}}\operatorname{Hom}_{S_n}(S(\nu),S(\lambda/\mu))
$$
and the Littlewood–Richardson rule says that ...
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Applications of character sheaves
There are many important recent works (for example, by Lusztig, Bezrukavnikov-Finkelberg-Ostrik, Ben-Zvi-Nadler, Boyarchenko-Drinfeld, Lusztig-Yun, Vilonen-Xue) on character sheaves (which are certain ...
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The Monster Moonshine Module from the engineering or algorithmic point of view
From what I understand (see, e.g., this question), the Monster Moonshine Module is a kind if "third generation" (or "second quantization"?) after the Golay code (with automorphism group $M_{24}$) and ...
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PT Symmetry and the Riemann Hypothesis
Recently there have been articles in Quanta, in Science Alert, and at phys.org among others, on possible recent progress toward the Hilbert-Polya conjecture, which implies the Riemann Hypothesis. The ...
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What is the non-motivic motivation behind automorphic representations?
In one of my last questions:
What is the "reason" for modularity results?
it was pointed out to me that "the notion of automorphic representation developed independently of any concern with ...
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A finite dimensional algebra associated to the symmetric group
Let $S_n$ be the finite group given as $n \times n$ permutation matrices.
Define for a given field $K$ the algebra $B_n$ as the subalgebra of $M_n(K)$ generated by all permutation matrices of $S_n$. (...
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On the finite simple groups with an irreducible complex representation of a given dimension
This answer of Geoff Robinson shows that a finite simple group admits an irreducible complex representation (irrep) of dimension $3$ if and only if it is isomorphic to $A_5$ or $\mathrm{PSL}(2,7)$.
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Units in group rings.
Let $G$ be a finite solvable group of order $n$, and let $g_1 ... g_n$ be an enumeration of its elements. Let $a_1 ... a_n$ be a sequence of integers, such that $\sum a_i$ is relatively prime to $n$.
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Representations in characteristic p
Let G be a finite group and let F be an algebraically closed field. If the characteristic of F is 0, then the number of irreducible F-representations of G is given by the number of conjugacy classes ...
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Largest permutation group without 2-cycles or 3-cycles
The largest permutation group without 2-cycles is $A_n$, which has size $n!/2$. I think the largest permutation group without 2-cycles or 3-cycles is much smaller, but I can't figure out if it should ...
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A canonical bijection from linear independent vectors to parking functions
Call an $n$-vector $v$ in $\mathbb{Z}^n$ cool when it has only entries 0 or 1 and the ones appear in only one block. Thus there are $n(n+1)/2$ such vectors. For $n=3$ they are:
[ <[ 1, 0, 0 ]>, &...
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Which partitions realise group algebras of finite groups?
Fix an algebraically closed field $K$ (maybe of characteristic zero first for simplicity, like $\mathbb{C}$).
Given a partition $p=[a_1,...,a_m]$ of an integer $n$. We can identify $p$ with the ...
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Applications of the idea of deformation in algebraic geometry and other areas?
The idea of proving something by deforming the general case to some special cases is very powerful. For example, one can prove certain equalities by regarding both sides as functions/sheaves, and show ...