Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
6,803
questions
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Finite dimensional irreducible representations of quasisplit p-adic groups
For split groups over a $p$-adic field, every irreducible smooth (complex) representation is either infinite-dimensional or one-dimensional. Is it true for quasisplit groups that split over an ...
4
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0
answers
120
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Prehomogeneous vector spaces for reductive groups
Recall that a prehomogeneous vector space, is a representation $V$ of a linear algebraic group $G$ having an open $G$-orbit. Let $Z$ be the neutral connected component of the stabilizer of a point of ...
11
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2
answers
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Questions about categorification (& combinatorial simplification of the Russian approach to Lusztig's conjectures, in zero & positive characteristic) [closed]
This is a question about the proofs of Kazhdan-Lusztig's conjectures for category $\mathcal{O}$ using higher representation theory (avoiding Beilinson-Bernstein's geometric localization theory).
...
2
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0
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203
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Mistake in Discriminants, Resultants, and Multidimensional Determinants?
I'm studying this proof in the book cited in the title.
The author say that any multiplicative function $\chi\colon\mathsf{GL}(k)\longrightarrow\mathbb{C}^\times$ is a power of the determinant. But ...
9
votes
3
answers
946
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Decomposition of tensor power of symmetric square
Studying some representation theory I came up with the following problem.
We work over a field of characteristic $0$. Let $V$ be the standard representation of $\mathrm{GL}_n$ and let $W$ be the ...
2
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0
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60
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How to compute the upper global basis (dual canonical basis) of an irreducible $U_q(\mathfrak{g})$-module?
Let $\mathfrak{g}$ be a semisimple Lie algebra and $U_q(\mathfrak{g})$ the corresponding quantum group. Are there some general method to compute the upper global basis (dual canonical basis) of an ...
1
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0
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Which operators constructed from 10d gamma matrices commute with $SO(1,2)\times SO(3)\times SO(3)$?
In the paper Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory by Gaiotto and Witten, an in-depth analysis of boundary conditions in N=4 Super Yang-Mills in four dimensions in ...
5
votes
2
answers
419
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Difference of adjacent dominant weights is a root?
The basic set-up here makes sense in the theory of abstract root systems if one brings (integral) weights into the picture, but it may be more natural to think about the classical characteristic 0 ...
7
votes
1
answer
426
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When does the enveloping algebra functor lift to the category of bialgebras?
Let $\mathrm{Ass} $ denote the operad, whose algebras are associative unital algebras, considered as a dg-operad.
Denote $\mathrm{Ch} $ the category of chain complexes over a commutative ring $\...
6
votes
0
answers
141
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Group homomorphisms out of the additive group is formal completion at the unipotent cone
Let $G$ be an affine algebraic group over an algebraically closed field of characteristic zero $k$. Then, the functor $\text{Hom}_{grp}(\mathbb{G}_a, G)$ is representable by a colimit of schemes (...
17
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1
answer
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Why do these two Monster-related calculations yield $163$?
Fact 1: (1979, Conway and Norton)$^{1}$
"There are $194-22-9=\color{blue}{163\,}$ $\mathbb{Z}$-independent McKay-Thompson series for the Monster."
Note: There are 194 (linear) irreducible ...
1
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0
answers
117
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$G$-invariant bilinear maps
Let $G$ be a finite group and $M$ a finitely generated $\mathbb{Z}G$-module. Then $M \otimes k$ is a representation of $G$ for any field $k$. I am interested in the number of ways we can turn $M \...
2
votes
0
answers
62
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What is the intuition of lower global bases?
In the paper: Crystallizing the Q-analogue of Universal Enveloping Algebras, Kashiwara introduced the upper global bases. In the paper: https://projecteuclid.org/euclid.dmj/1077295931, Kashiwara ...
7
votes
0
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Explicit form of raising and lowering operators in spherical gl(n) DAHA
I am working with polynomial representations of spherical subalgebra of double affine Hecke algebra (DAHA) for $\mathfrak{gl}_n$.
Let's call this algebra $\mathfrak{A}_n$ for short. Typically we think ...
6
votes
1
answer
208
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Jones-Wenzl-type projectors for Brauer algebras
Jones-Wenzl projectors are an explicit combinatorial description of the central projectors of the Temperley-Lieb algebra.
They also describe very explicitly the failure of certain representations to ...
2
votes
0
answers
163
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Galois action on posets of number fields and $p$-adic fields
In the theory of group actions on posets one studies the action of a group $G$ on a poset $\mathcal{P}$ (via order-preserving permutations) partly through its derived action on the order complex of ...
3
votes
1
answer
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What role do $(\mathfrak{g},K)$- modules play in the construction of automorphic vector bundles
Looking at the connection between modular forms as sections and automorphic representations it is to me somewhat clear why automorphic representations are (demanded to be) admissible $G(\mathbb{A}^\...
1
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0
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A dimension formula for generalised symmetric powers of the natural module
I need a reference for the following well-known statement - does anyone know one?
Let $\mu$ a partition of $n$ into at most $d$ parts.
We let
$${\rm Sym}^\mu(\Bbbk^d)={\rm Sym}^{\mu_1}(\Bbbk^d) \...
10
votes
1
answer
488
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Arithmetic representation stability and Galois action
I am thinking about representation stability phenomena (as considered by Church, Farb, Ellenberg and others) in arithmetic settings where Galois action enters the picture, and am curious about their ...
2
votes
1
answer
309
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Gelfand pairs and (self)-dual representations
For a Lie group $G$ with compact Lie subgroup $K$, we say that $(G,K)$ is a pair of Gelfand type if the representation $L^2(G/K)$ of $G$ is multiplicity free, that is, if it is a direct integral of ...
3
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0
answers
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Extension of Schur-Weyl duality for principal series in $SL(2, \mathbb{R})$
In the case of $SU(N)$ all unitary irreps can be obtained from reducing tensor products ($V^{\otimes n}$) of the fundamental representation ($V$). Then given the set of all $SU(N)$ Young diagrams with ...
6
votes
1
answer
229
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Are indecomposable representations of a finite group of Lie type absolutely indecomposable?
Let $G = G(K)$ be a Chevalley group over an algebraically closed field $K$ of characteristic $p > 0$. Consider the finite group $G(q) = G(\mathbb{F}_q)$. (For example, if $G = \operatorname{SL}_n(K)...
3
votes
1
answer
289
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Contragredient of a cuspidal representation
Let $G$ be a reductive group over a nonarchimedean local field $F$. Let $\pi$ be an irreducible, cuspidal representation of $G$, with contragredient $\tilde{\pi}$. Then $\tilde{\pi}$ is cuspidal.
A ...
11
votes
1
answer
361
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Can we glue characteristic 0 and characteristic p representations of a finite group given equality of (Brauer) characters?
Suppose I have a prime $p$ and a finite group $G$ together with representations $\sigma: G \to GL_n(\mathbb{Q}_p)$ and $\pi: G \to GL_n(\mathbb{F}_p)$. My question is:
When does there exist a ...
5
votes
2
answers
395
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A finite group that splits and does not split
Is there an example of a finite group $A$ that acts on a finite group $C$ irreducibly (that is, $C$ has no proper nontrivial $A$-invariant subgroup)
such that there exists an epimorphism $$\tau \colon ...
4
votes
1
answer
243
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Ideal of the Spinor variety $S^{10}\subset\mathbb{P}^{15}$
The ideal of the $10$-dimensional Spinor variety $S^{10}\subset\mathbb{P}^{15}$ is generated by $10$ quadrics.
Does anyone know a reference where these 10 quadratic equations are written down ...
3
votes
0
answers
171
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Geometric interpretation of homological quantities in Artinian local Gorenstein algebras
By corollary 3.5. of http://www.ams.org/journals/tran/2012-364-09/S0002-9947-2012-05430-4/S0002-9947-2012-05430-4.pdf the classification of local artinian Gorenstein algebras (all algebras here are ...
0
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1
answer
122
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About indecomposability and nilpotence
Transferred from MSE where it now received a complete answer.
Maybe the following is easy, but I am not an expert in finite-dimensional Lie algebras and was stuck on the following problem.
Can ...
5
votes
1
answer
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Group rings over central products
I have a proof of the following result but I was wondering if anyone had a reference for it. I have asked on math.stackexchange here but didn't receive any replies.
Let $G$ a finite group given by ...
2
votes
1
answer
122
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Is there a natural Hilbert structure on jet spaces?
In a nutshell, I am trying to apply harmonic analysis on Lie groups
to symmetry groups of differential equations. As far as I understand,
to this aim I need to present the structure of Hilbert spaces ...
3
votes
0
answers
213
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Unitary dual of the complex motion group $\mathbb C^2 \rtimes SU(2)$?
The real motion group of $\mathbb R^2$, $M(2)$ is the semi-direct product of $\mathbb R^2$ with the special orthogonal group $K = SO(2)$. A well
known fact is that the unitary dual $\hat{G}$, of $G$ ...
1
vote
1
answer
129
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Is the Cartan permanent odd for finite global dimension?
Define the Cartan permanent of a finite dimensional algebra as the permanent of the Cartan matrix.
Is the Cartan permanent of a finite dimensional algebra with finite global dimension always an odd ...
5
votes
1
answer
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Periods in the trivial extension algebra of the incidence algebra of the divisor lattice
Definition of $C_L$ for people who like number theory:
Let $m$ be a number with prime factorisation $m=p_1^{n_1} ... p_r^{n_r}$ with $n_i>0$. Define $I_m$ to be the incidence algebra of the ...
2
votes
1
answer
80
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reduction of torsion modules
Let $G$ be a profinite group.
Let $K(G,\mathbb Z_\ell)$ be the Grothendieck group of the derived category of finitely generated $\mathbb Z_\ell$-modules with continuous $G$-action.
Let $K(G,\mathbb ...
3
votes
1
answer
74
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Functions in the induced space compactly supported in $PN^-$ modulo $P$
Let $P_0$ be a minimal parabolic subgroup of a connected, reductive group $G$ over a $p$-adic field $k$. Let $P$ be a parabolic subgroup containing $P_0$ with Levi decomposition $P = MN$. Let $N^-$ ...
2
votes
1
answer
272
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Decomposition into Weyl modules
Let $G$ be a split reductive group over an arbitrary field $k$. By definition, see Jantzen (*), an ascending chain $$0 = V_0 \subset V_1 \subset V_2 \subset \dots$$ of submodules of a $G$-module $V$ ...
3
votes
0
answers
68
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Representation dimension of $Aus(K[x]/(x^n))$
Let $A_n$ be the Auslander algebra of $K[x]/(x^n)$. Is the representation dimension of $A_n$ known? For $n \leq 3$ the algebra is representation-finite and thus the representation dimension is 2. For $...
11
votes
1
answer
278
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Asymptotic distribution of $\lambda_1$ under the $z$-measure for partitions
The following question about $z$-measures on Young diagrams came up in some ongoing work with Ofir Gorodetsky. I recall the background and then state our question below in the box.
For parameters $z$ ...
1
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0
answers
50
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Regular partial tilting modules in wild hereditary algebras
Let $k$ be an algebraically closed field. Let $Q$ be a connected wild quiver. Let $\mathcal{R}_1$, $\mathcal{R}_2$ be two regular components of the Auslander-Reiten quiver of $kQ$ that contain stones. ...
5
votes
0
answers
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Interpretation of Hilbert/Frobenius series shift
Let $V = \oplus_{i\geq 0} V^i$ be a graded vector space.
Recall that the Hilbert series is defined as
$$F(q) = \sum_{i\geq 0} q^i \operatorname{dim}(V^i),$$
or if we have a graded $S_n$-module, $M$, ...
1
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0
answers
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Do maximally almost periodic groups embed homeomorphically into their Bohr compactifications? [duplicate]
If $G$ is a Hausdorff topological group and $bG$ is its Bohr compactification, Wikipedia says that $G$ is called maximally almost periodic (MAP) if and only if the natural map $i : G \to bG$ is ...
14
votes
1
answer
480
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Generalizing the Fourier isomorphism between Sobolev spaces and weighted $L^2$ spaces to (locally) compact groups?
Motivating examples:
Let $V$ be a real vector space with Haar measure $dv$. The fourier transform induces the following topological isomorphism: $$H^s(V,dv) \cong L^2(V^*,(1+|v^*|^2)^sdv^*)$$
The ...
11
votes
0
answers
332
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"Small" zero divisors in $\mathbb C[\mathbb Z/p\mathbb Z]$
If $p$ is a prime, and $a,b$ are non-zero elements of the group algebra $\mathbb C[\mathbb Z/p\mathbb Z]$ satisfying $a\ast b=0$, then $$|{\rm supp}\ a|+|{\rm supp}\ b|\ge p+2.$$ This is easy to prove ...
9
votes
0
answers
259
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Name for rings with $R \cong R^{\mathrm{op}}$
Is there a name for rings that are isomorphic to their opposite ring $R^{\mathrm{op}}$ in the literature? I'm especially interested for the class of Artin algebras.
2
votes
0
answers
81
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Continuity of the conductor of automorphic representations
I am interested in properties of (semi-)continuity of the conductor of automorphic representation. Let $F$ be a number field.
The meta question is, given a function on the unitary dual of $PGL(2, F)$ ...
9
votes
1
answer
774
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Interesting examples of pro-algebraic completions of groups
Fix a field $k$. Given a discrete group $G$, the pro-algebraic completion (or Hochschild-Mostow completion) is the pro-algebraic group $A_{k}(G)$ with is universal with respect to finite dimensional ...
3
votes
2
answers
326
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Deformation of "Hecke modification"
Let $X$ be a smooth curve over $\mathbb{C}$. I wish to compute the deformation of the following data $(E,F,x)$. $E$ and $F$ are locally free sheaves over $X$ and $x$ is a point on $X$. They satisfy:
...
1
vote
0
answers
48
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Is simreflexive left-right symmetric?
Call an Artin algebra (we can assume that it is basic) $A$ simreflexive in case every simple A-module is reflexive. Is A simreflexive iff the opposite algebra $A^{op}$ is simreflexive? Or equivalently ...
5
votes
0
answers
130
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What do I know about a group if its representations are filtered?
Let $G$ be an affine group scheme over a field.
Say that, for every finite-dimensional representation of $G$, I have a $\mathbb{Z}$-grading on the underlying vector space, compatible with tensor ...
4
votes
1
answer
219
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Are all these representations supercuspidal
Let $D$ a division quaternion algebra over a number field $F$, and consider $(V,q)$ be a $D$-hermitian space of $D$-dimension $2$, and introduce its group of isometries
\begin{align*}
\mathrm{GU}(V, q)...