Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
7,076 questions
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Are homeomorphic representations isomorphic?
Let $G$ be a finite group. Let $V_1, V_2$ be two finite-dimensional real representations. Suppose $f: V_1 \to V_2$ is a $G$-equivariant homeomorphism. Can one conclude that $V_1$ and $V_2$ are ...
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Is there an accessible exposition of Gelfand-Tsetlin theory?
I'm hoping to start an undergraduate on a project that involves understanding a bit of Gelfand-Tsetlin theory, and have been tearing my hair out looking for a good reference for them to look at. ...
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Topology of SU(3)
$U(1)$ is diffeomorphic to $S^1$ and $SU(2)$ is to $S^3$, but apparently it is not true that $SU(3)$ is diffeomorphic to $S^8$ (more bellow). Since $SU(3)$ appears in the standard model I would like ...
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A precise statement of the categorical version of geometric Langlands conjecture
The statement of the ordinary non-categorical version of geometric Langlands conjecture, which was proven for GL(n) in around 2002 by Frenkel, Gaitsgory and Vilonen, is quite well-known and is easy to ...
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Does $S_4$ inject into $SL(2,R)$ for some commutative ring $R$?
$\newcommand{\Z}{\mathbf{Z}}$
Given a nice infinite collection of groups, for example the symmetric groups, one can ask whether any finite group is a subgroup of one of them. Of course any finite ...
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What determines the maximal dimension of the irreps of a (finite) group?
I am chemist and ask for apologies for all my mathematical inabilities when asking this question in advance, but after quite a bit of searching I found that this problem could be "open" or ...
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Linking formulas by Euler, Pólya, Nekrasov-Okounkov
Consider the formal product
$$F(t,x,z):=\prod_{j=0}^{\infty}(1-tx^j)^{z-1}.$$
(a) If $z=2$ then on the one hand we get Euler's
$$F(t,x,2)=\sum_{n\geq0}\frac{(-1)^nx^{\binom{n}2}}{(x;x)_n}t^n,$$
on the ...
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Groups in which all characters are rational.
The Symmetric groups $S_n$ has interesting property that all complex irreducible characters are rational (i.e. $\chi(g)\in \mathbb{Q}$ for all $\mathbb{C}$-irreducible characters $\chi$,$\forall g\in ...
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When are modules and representations not the same thing?
I've been trying for a while to get a real concrete handle on the relationship between representations and modules. To frame the question, I'll put here the standard situation I have in mind:
A ring $...
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Are there Maass forms where the expected Galois representation is $\ell$-adic?
Recall that by theorems of Deligne and Deligne--Serre, there is the following dichotomy:
Modular forms on the upper half plane of level $N$ and weight $k\geq 2$ correspond to representations $\rho:\...
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Particle Physics and Representations of Groups
This question is asked from a point of complete ignorance of physics and the standard model.
Every so often I hear that particles correspond to representations of certain Lie groups. For a person ...
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Which finite groups have faithful complex irreducible representations?
Obvious necessary condition is that the center must be a cyclic group. Is it sufficient (doubt here)? If not, is there any nice characterization in terms of group structure, without appealing to ...
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Yoga of six functors for group representations?
I'm trying to understand how the six functor philosophy applies to representation theory. Consider the category of classifying stacks $BG$ (assume $G$ discrete for simplicity). To every stack we can ...
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Which p-adic numbers are also algebraic?
What is $\mathbb{Q}_p \cap \overline{\mathbb{Q}}$ ?
For instance, we know that $\mathbb{Q}_p$ contains the $p-1$st roots of unity, so we might say that $\mathbb{Q}(\zeta) \subset \mathbb{Q}_p \cap \...
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Status of (global) Langlands conjecture for $\mathrm{GL}_2$ over $\mathbb{Q}$
$\DeclareMathOperator\GL{GL}$Apologies if this question has already been dealt with on MO. I am wondering about the status of the global Langlands conjectures for $\GL_2$ over the rational numbers. ...
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Number of irreducible representations of a finite group over a field of characteristic 0
Let $G$ be a finite group and $K$ a field with $\mathbb{Q} \subseteq K \subseteq \mathbb{C}$.
For $K=\mathbb{C}$ the number of irreducible representations of $KG$ is equal to the number of conjugacy ...
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Matrices: characterizing pairs $(AB, BA)$
Let $A$ be an $m\times n$-matrix and $B$ an $n \times m$-matrix over the same field. Consider the matrices $C=AB$ and $D=BA$. It is probably well known (and not difficult to show) that the only ...
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Proofs of Beilinson-Bernstein
The Beilinson-Bernstein localization theorem states roughly that the category of $D$-modules on the flag variety $G/B$ is equivalent to the category of modules over the universal enveloping algebra $U\...
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Can an operator have Exp(z) as its characteristic "polynomial"?
Let $\mathcal{H}$ be a Hilbert space, and let $T: \mathcal{H} \rightarrow \mathcal{H}$ be a trace-class operator. Define
$$ f_T(z) = \sum_{i=0}^\infty \mbox{Tr}(\wedge^k T) \cdot z^k, $$
the ...
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Conjectures inspired by AI
Today in Nature a paper described how AI guided mathematicians to make highly non-trivial conjectures, which they managed to prove, one in Knot Theory involving a new invariant, the other in ...
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Why would one expect a derived equivalence of categories to hold?
This question is perhaps somewhat soft, but I'm hoping that someone could provide a useful heuristic. My interest in this question mainly concerns various derived equivalences arising in geometric ...
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learning Deligne-Lusztig theory
Can someone give me a roadmap for learning Deligne-Lusztig theory? (Except for the original article by Deligne and Lusztig)
Edit: You may assume knowledge of representation theory of finite groups (...
user19475
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Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$?
Let $R$ be a finitely generated ring with identity, $M_n(R)$ the set of $n\times n$ matrices. Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$? This should be an elementary ...
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Is there a 'nice' interpretation of virtual representations?
Let $G$ be a compact group and let $R(G)$ be the representation ring of $G$. Additively, $R(G)$ is generated by the irreducible representations of $G$. Usually one only deals with those ...
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Majorization and Schur Polynomials
Let me first define the majorization order (or dominance order) on partitions as $\lambda \succeq \mu$ iff $$\sum _{i=1}^{k}\lambda_i \geq \sum_{i=1}^{k}\mu_i$$ for all $1\le k\le l-1$ and $$\lambda_1+...
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Intuition for symplectic groups
My question essentially breaks down to
How do you, a working mathematician, think about (real) symplectic groups? How do you visualize symplectic (linear) transformations? What intuition do you ...
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Have people successfully worked with the full ring of differential operators in characteristic p?
This question is inspired by an earlier one about the possibility of using the full ring of differential operators on a flag variety to develop a theory of localization in characteristic $p$. (Here ...
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How can classifying irreducible representations be a "wild" problem?
Let $q$ be a prime power and $U_n(\mathbb{F}_q)$ be the group of unitriangular $n\times n$-matrices. I've read and heard in several places (see e.g. this mathoverflow question) that classifying ...
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Why do the adjoint representations of three exceptional groups have the same first eight moments?
For a representation of a compact Lie group, the $n$th moment of the trace of that representation against the Haar measure is the dimension of the invariant subspace of the $n$th tensor power. The ...
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Definitions of real reductive groups
There are several definitions of real reductive groups, sometimes subtly inequivalent. The following come to my mind:
A closed subgroup of $GL(n,\mathbb C)$ closed under conjugate transpose.
The set ...
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Reconciling Lusztig's results with the Langlands philosophy
Let $\boldsymbol{G}$ be a reductive group over a finite field $\mathbb{F}_q$, $G = \boldsymbol{G}(\mathbb{F}_q)$, $W = \mathrm{W}(\mathbb{F}_q)$ the Witt vectors over $\mathbb{F}_q$, and $K = \mathrm{...
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Questions about the Bernstein center of a $p$-adic reductive group
Dear all,
The "Bernstein center" of a $p$-adic reductive group appears frequently in the literature of automorphic forms, often without a precise definition. For example, in page 233 of Moeglin-...
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How are these two ways of thinking about the cross product related?
I was always bothered by the definition of the cross product given in e.g. a calculus course because it's never made clear how one would go about defining the cross product in a coordinate-free manner....
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Overview of the interplay of Harmonic Analysis and Number Theory
I'm kind of disappointed that the question here was never sharpened.
The Laplacian $\Delta$ on the upper half-plane is $-y^{2}(\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}))$. Suppose $D$ ...
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Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?
This question was motivated by a more general question raised by Jan Weidner here. In general one starts with a variety $X$ (say smooth) over an algebraic closure of a finite field $\mathbb{F}_q$ of ...
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When does Lusztig's canonical basis have non-positive structure coefficients?
I've heard asserted in talks quite a few times that Lusztig's canonical basis for irreducible representations is known to not always have positive structure coefficents for the action of $E_i$ and $...
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What does the Tannakian formalism reconstruct when fed the category of chain complexes?
I've recently realized that there is a gap in my understanding of the Tannakian formalism for reconstructing an (algebraic) group from its category of (finite-dimensional) representations. To warm up,...
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What are the points of simple algebraic groups over extensions of $\mathbb{F}_1$?
The "field with one element" $\mathbb{F}_1$ is, of course, a very speculative object. Nevertheless, some things about it seem to be generally agreed, even if the theory underpinning them is not; in ...
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Groebner basis with group action
At one point my advisor, Mark Haiman, mentioned that it would be nice if there was a way to compute Groebner bases that takes into account a group action.
Does anyone know of any work done along ...
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Applications of group theory to mathematical biology (pharmacology)
Are there applications of group theory — broadly, say, representation theory, Lie algebras, $q$-groups, etc — to mathematical biology?
In particular, I am interested in applications to pharmacology — ...
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Avatars of the ring of symmetric polynomials
I'm collecting different apparently unrelated ways in which the ring (or rather Hopf algebra with $\langle,\rangle$) of symmetric functions $Z[e_1,e_2,\ldots]$ turns up (for a Lie groups course I will ...
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Introductory References for Geometric Representation Theory
Would anyone be able to recommend text books that give an introduction to Geometric Representation Theory and survey papers that give an outline of the work that has been done in the field? I'm ...
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Statistics of irreps of S_n that can be read off the Young diagram, and consequences of Kerov–Vershik
Alexei Oblomkov recently told me about the beautiful theorem of Kerov and Vershik, which says that "almost all Young diagrams look the same." More precisely: take a random irreducible ...
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Adams Operations on $K$-theory and $R(G)$
One of the slickest things to happen to topology was the proof of the Hopf invariant one using Adams operations in $K$-theory. The general idea is that the ring $K(X)$ admits operations $\psi^k$ that ...
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Computer package for representation theory of the symmetric group
Is there a computer algebra package in which I can compute the following for representations of a specific symmetric group (e.g. $S_7$):
(1) $V \otimes W$
(2) $S_\lambda V$, where $S_\lambda$ is a ...
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what else is in $\prod_{j=1}^n(1+q^j)$?
From time to time, I run into the finite product $\prod_{j=1}^n(1+q^j)$. And, the more it happens, the more fascinated I've become. So, herein, I wish to get help in collecting such results. To give ...
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Number of 2-dimensional irreducible representations of a finite group ?
Question: What is the number of two-dimensional irreducible representations of a finite group ? How it can be expressed in groups-theoretic terms ? (Number of 1-dimensional irreps is |G/[G,G]| ).
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What is a special parahoric subgroup?
Let me take this question again from the top.
I would like to know what a special parahoric subgroup is.
I think this is a "real" question, though not an especially good one -- it indicates my ...
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How does one compute invariants of certain Grassmannians inside the regular representation?
Barry Mazur and I have come across the question below, motivated by (but independent
of) issues regarding the Leopoldt conjecture.
Suppose that $\mathbf{C}$ is the complex numbers.
Let $H$ be a ...
user631
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Principal bundles, representations, and vector bundles
What is the exact relationship between principal bundles, representations, and vector bundles?
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