Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
6,818
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Lie algebra elements commuting with a principal nilpotent element
Let $\mathfrak{g}$ be a semisimple complex Lie algebra, $A \in \mathfrak{g}$ a principal nilpotent element (i.e. its centralizer is of dimension equal to the rank of $\mathfrak{g}$). I wish to ...
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Why is the Langlands dual group always taken over $\mathbb{C}$?
Whenever I read a statement of the Langlands conjectures for a reductive group $G$, they are formulated in terms of the Langlands dual group, which is essentially the reductive group over $\mathbb{C}$ ...
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2
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Representation theory of inner forms
I once heard something like "inner forms of reductive groups have the same representation theory".
Is this assertion misguided?
If this assertion is not misguided, then is there a precise ...
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Weyl Group Action on Littelmann Paths
In his paper "Paths and Root Operators in Representation Theory," Littelmann gives an action of the Weyl group on the set of integral paths via
$$
\tilde{s}_\alpha(\pi):= \begin{cases} f^n_\alpha(\pi)...
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0
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Weight spaces of modules over Lie algebras [closed]
I know that an irreducible infinite-dimensional weight module over the Virasoro algebra in which it has a non-zero finite-dimensional weight space, then all its weight spaces have finite dimension. ...
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Stable m-Calabi Yau property for Frobenius categories
Let $C$ be a Frobenius category. The stable category $\underline{C}$ is called $m$-Calabi Yau in case it is Hom-finite and there is a functorial duality
$D \underline{Hom}(X,Y)=\underline{Hom}(Y,\...
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Representation-finite quivers over dual numbers
Given a Dynkin quiver $Q$ and a field $K$.
Question 1: For which such $Q$ are there only finitely many indecomposable representations over the dual numbers $K[x]/(x^2)$?
Note that those ...
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The tensor product of two monoidal categories
Given two monoidal categories $\mathcal{M}$ and $\mathcal{N}$, can one form their tensor product in a canonical way?
The motivation I am thinking of is two categories that are representation ...
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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 ...
- 1,635
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Hermitian sublattices of a given type
Consider an unramified quadratic extension $E/F$ of non-archimedean local fields, and suppose that $\langle\cdot,\cdot\rangle$ is a fixed Hermitian form on $E^d$ such that $\mathcal{O}_E^d$ is self-...
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Equivalence from a tilting module
Let $A$ be a finite dimensional algebra. For a subcategory $C$ of $mod-A$ let $\overline{C}$ be the objects $X \in mod-A$ such that there exists an exact sequence $0 \rightarrow C_n \rightarrow ... \...
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About different cohomology theories used to study Shimura varieties
The classical theory of Eichler-Shimura realizes the space of cusp forms of certain weights and levels in the parabolic cohomology of modular curves, which is the image of the cohomology with compact ...
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On strongly simply connected quiver algebras
Let $A$ be a representation-finite quiver algebra. In this case $A$ is simply connected if and only if its first Hochschild cohomology vanishes by a result of Buchweitz and Liu. $A$ is called strongly ...
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Characterisation of representation-directed algebras
A representation-directed algebra $A$ over a field $K$ has the property that for every indecomposable module $M$ we have $\mathrm{End}_A(M) \cong K$ and $\mathrm{Ext}_A^i(M,M)=0$ for all $i>0$.
...
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About height function in the Arthur trace formula
I am reading a book of Arthur's book "Introduction to the trace formula".
In page 24, Arthur defined height function $H_P:G(\mathbb{A}) \to \mathcal{a}_P$
by setting $H_p(nmk)=H_{M_p}(m)$ where $n\in ...
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Is it possible that the GHKK canonical basis for cluster algebras is the Lusztig/Kashiwara dual canonical basis?
Gross-Hacking-Keel-Kontsevich (https://arxiv.org/abs/1411.1394) constructed a canonical basis (the so-called “theta basis”) for a cluster algebra, at least assuming it satisfies a certain ...
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Is the character of the adjoint representation of $\operatorname{SU}(n)$ non-vanishing on regular points of a maximal torus?
Are the maximal and minimal values of the character of the adjoint representation of $\operatorname{SU}(n)$ restricted to a maximal torus known? Can such a character vanish at some regular point of a ...
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How complicated can a finite double complex over a field be?
A finite complex over a field $k$ is pretty simple: it's the direct sum of its homology with a split-exact complex. How complicated can a finite double complex be? Does it make a difference if $k$ is ...
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Cyclic vectors in irreducible representations of simple Lie algebras
Is there a notion of "cyclic element" in a simple Lie algebra? In particular, is it independent of the irreducible representation chosen?
Explanation.
An endomorphism A is called cyclic if there is ...
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A finite group $G$ all of whose reps are defined over $\mathbb{Z}$ and yet $Rep(G)$ is not generated by permutation representations
Let $G$ be a finite group and let $Rep(G)$ be its representation ring (as a group it is the free $\mathbb{Z}$-module on the irreducible complex reps). The collection of permutation representations $\...
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How to prove the p-adic Galois representations atteched to the Tate module of an abelian variety is de Rham directly?
Recently I read a thesis p-adic Galois representations and elliptic curves. Using Tate's curve, the author proved the p-adic Galois representations atteched to the Tate module of an elliptic curve is ...
user141691
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Finitistic dimension conjecture for quadratic algebras
The finitistic dimension of a finite dimensional algebra is defined as the supremum of all projective dimensions of modules having finite projective dimension. The finitistic dimension conjecture says ...
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Classify 2-dim p-adic galois representations
Recently I have known how to classify 1-dim p adic Galois representations $\phi$. The p-adic Galois representations mean that a representation $G_K$ on a p-adic field $E$, where $K$ is also a p-adic ...
user141691
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Minimal injective coresolution in the stable Auslander algebra
Let $A$ be a finite dimensional (connected) quiver algebra. Let $T(A)$ denote the full subcategory of coherent functors from $mod-A$ to $Ab$ that vanish on projective objects. $T(A)$ is equivalent to ...
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Is there a connection between representation theory and PDEs?
As a PhD student, if I want to do something algebraic / linear-algebraic such as representation theory as well as do PDEs, in both the theoretical and numerical aspects of PDEs, would this combination ...
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Open problems about Morita and derived invariants
Are there properties of rings of which one does not know whether they are Morita or derived invariances?
For a recent such example for Morita invariance, see https://www.sciencedirect.com/science/...
- 26.1k
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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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Smith Normal Form of a Cayley Graph of the Symmetric Group
Let $A_n$ be the adjacency matrix of the Cayley graph $\text{Cay}(S_n,C_n)$ where $C_n \subseteq S_n$ is the conjugacy class of $n$-cycles of the symmetric group $S_n$. Since the generating set of ...
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infinite fold tensor product of universal enveloping algebra
Let $\mathfrak a$ be a Lie algebra graded by the abelian semigroup $S$, then the universal enveloping algebra $U(\mathfrak a)$ of $\mathfrak a$ is $S \sqcup \{0\}$ graded. I have the following ...
- 1,219
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Endomorphism ring of a generator-cogenerator over acyclic algebras
Let $A$ be an acyclic quiver algebra, $M$ a generator-cogenerator and $B=End_A(M)$.
Questions:
Does $B$ have finite global dimension?
Does $B$ have finite global dimension in case $M=A \oplus D(A)$?
...
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Gelfand's trick (Gelfand's lemma) in positive characteristic?
I came across this preprint that claims in Lemma 1.1 that Gelfand's trick (also known as Gelfand's lemma) only works in characteristic zero:
Let $H < G$ be finite groups. Suppose we have an anti-...
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Interpretation of stable Hom in Nakayama algebras
Let $A$ be a Nakayama algebra with Kupisch series $[c_0,c_1,...,c_{n-1}]$ and Jacobson radical $J$ (given by quiver and relations). As is well known every indecomposable $A$-module is of the form $e_i ...
- 26.1k
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Drinfeld Polynomial for Yangian $Y(\mathfrak{sl}_2)$
I am looking for a direct proof that a highest weight representation of $Y(\mathfrak{sl}_2)$ is finite-dimensional if its highest weight is determined by a Drinfeld polynomial.
The results was ...
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Cartan determinant of stable categories
Let $A$ be a finite dimensional algebra with finitely many indecomposable non-projective modules $M_1, M_2,...,M_n$.
Let $a_{i,j}:=\dim(\underline{Hom_A}(M_j,M_i))$ (the dimension of the stable Homs ...
- 26.1k
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Does the 3875-dimensional rep of $E_8$ have a solution to $x\star x=0$?
Consider the compact Lie group $E_8$. Its second-smallest fundamental representation is $3875$-dimensional and admits a symmetric invariant form, and so is real: $E_8 \curvearrowright \mathbb{R}^{3875}...
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Scattering diagram for the cluster algebra $ \mathbb C [N]$
Five years ago, Gross-Hacking-Keel-Kontsevich made a major advance in the theory of cluster algebras, by constructing bases of cluster algebras in large generality. A key tool in their construction ...
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Image of Fourier transform for finite non-abelian groups
I am working on the Fourier transform over finite non-abelian groups, specifically following Diaconis. He defines it as follows (p.7):
Let $P$ be a probability on a finite group $G$. The Fourier ...
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Formal character of local cohomology groups with support in Schubert cells
Let $k$ be a field of characteristic zero, $G$ a connected semi-simple algebraic group over $k$ and $B$ a fixed Borel subgroup of $G$ with maximal torus $T$. Also denote by $W$ the Weyl-group of $G$. ...
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Example of a Lorentz-invariant probability measure?
What is an example of a Lorentz-invariant probability measure on Minkowski space other than the delta measure at the origin?
Euclidean-invariance is easy to attain: the uniform measure on a ball does ...
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Converse to Tannaka duality for rings
Let $k$ be a field. It's well known and easy to prove that a $k$-algebra $R$ may be recovered from the functor $F: R-Mod \to Vect_k$ as the endomorphisms of $F$. Now suppose $\mathcal{C}$ is a ...
- 2,969
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A list of all irreducible 4-dimensional real representations
I need a reference to a complete list of all
faithful real 4-dimensional irreducible representations
of real Lie algebras.
The list itself is not very hard to obtain.
Using the Levi decomposition,
it'...
- 9,095
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Bound on the sum of projective and injective dimension
Recall that a finite dimensional algebra is called piecewise hereditary in case it is derived equivalent to an abelian hereditary category.
In proposition 1.2. of https://link.springer.com/article/10....
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Super characters in the theory of Lie superalgebras
Weyl character formula for the finite dimensional complex semisimple Lie algebras plays a crucial role in the theory of highest weight modules, where for the highest weight module $V(\lambda)$ we ...
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Why can there be holomorphic modular forms of negative half integral weight?
In Shimura's paper "ON THE HOLOMORPHY OF CERTAIN DIRICHLET SERIES", he constructed a family of Eisenstein series $E(z,s)$ by summing factor of automorphy. $E(z,s)$ is of negative half-integral weight, ...
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(Stable) Auslander algebras in a specific example
Let $k$ be a field and $Q$ be the quiver with two vertices 1 and 2 and three arrows:
$a$ from 2 to 1, $b$ from 2 to 1 and $c$ from 2 to 2.
Let $I_1=\langle ab-c^2,ba\rangle$ and $I_2=\langle ab-c^2,c^...
- 26.1k
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When is compact induction cuspidal?
Let $G=GL_2(\mathbb{Q}_p)$, and let $K$ be a compact-modulo-center subgroup of $G$, $\rho$ an irreducible smooth representation of $K$.
Question 1: Is $\mathrm{ind}_K^G \rho$ cuspidal?
Here ...
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Categorical description of adjoint representation
My apologies for a slightly vague question here. Let us fix a Lie algebra $\mathfrak{g}$ over a field $k$. Consider the symmetric monoidal category $\operatorname{Rep}_{k}^{\otimes}(\mathfrak{g})$ of ...
user108998
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Connections between linear representations and permutation representations
A finite group $\Gamma$ might be represented by a linear transformation
$$\rho : \Gamma\to\mathrm{GL}(\Bbb R^d),$$
or by permutations
$$\phi :\Gamma\to\mathrm{Sym}(n).$$
Of course, latter ones can ...
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2
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About weights in $\mathfrak{h}^*$
On p.24 of the book "Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$".
I quote:
"For example, the fixed point $-\rho$ under the dot action lies in a linkage
class by ...
- 1,855
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Category of typical representations for Lie superalgebras
I am interested in studying the category of typical representations over basic classical simple Lie superalgebras. In particular, I want to know
1) is this category semisimple (character determines ...
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