Questions tagged [rt.representation-theory]

Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.

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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 ...
Mehta's user avatar
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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 ...
Roman Fedorov's user avatar
11 votes
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). ...
Puraṭci Vinnani's user avatar
2 votes
0 answers
203 views

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 ...
Vincenzo Zaccaro's user avatar
9 votes
3 answers
946 views

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 ...
John's user avatar
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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 ...
Jianrong Li's user avatar
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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 ...
Mtheorist's user avatar
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5 votes
2 answers
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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 ...
Jim Humphreys's user avatar
7 votes
1 answer
426 views

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 $\...
Hadrian Heine's user avatar
6 votes
0 answers
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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 (...
Harrison Chen's user avatar
17 votes
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 ...
Tito Piezas III's user avatar
1 vote
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117 views

$G$-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 \...
Michiel Van Couwenberghe's user avatar
2 votes
0 answers
62 views

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 ...
Jianrong Li's user avatar
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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 ...
Peter Koroteev's user avatar
6 votes
1 answer
208 views

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 ...
Calvin McPhail-Snyder's user avatar
2 votes
0 answers
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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 ...
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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}^\...
MaxT's user avatar
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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) \...
Chris Bowman's user avatar
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10 votes
1 answer
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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 ...
user avatar
2 votes
1 answer
309 views

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 ...
Alesandro Levi's user avatar
3 votes
0 answers
114 views

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 ...
Luca Iliesiu's user avatar
6 votes
1 answer
229 views

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)...
spin's user avatar
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3 votes
1 answer
289 views

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 ...
rj7k8's user avatar
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11 votes
1 answer
361 views

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 ...
S. Carnahan's user avatar
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5 votes
2 answers
395 views

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 ...
Pablo's user avatar
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4 votes
1 answer
243 views

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 ...
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3 votes
0 answers
171 views

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 ...
Mare's user avatar
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0 votes
1 answer
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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 ...
Duchamp Gérard H. E.'s user avatar
5 votes
1 answer
150 views

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 ...
David Watson's user avatar
2 votes
1 answer
122 views

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 ...
Stanislav Opanasenko's user avatar
3 votes
0 answers
213 views

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 $G$ ...
Z. Alfata's user avatar
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1 vote
1 answer
129 views

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 ...
Mare's user avatar
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5 votes
1 answer
1k views

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 ...
Mare's user avatar
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2 votes
1 answer
80 views

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 ...
ely's user avatar
  • 135
3 votes
1 answer
74 views

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 $N^-$ ...
D_S's user avatar
  • 6,100
2 votes
1 answer
272 views

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 $V$ ...
Michiel Van Couwenberghe's user avatar
3 votes
0 answers
68 views

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 $...
Mare's user avatar
  • 25.8k
11 votes
1 answer
278 views

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 $z$ ...
Brad Rodgers's user avatar
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1 vote
0 answers
50 views

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. ...
Ying Zhou's user avatar
  • 417
5 votes
0 answers
82 views

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, $M$, ...
Per Alexandersson's user avatar
1 vote
0 answers
41 views

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 ...
Alex M.'s user avatar
  • 5,207
14 votes
1 answer
480 views

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 ...
Saal Hardali's user avatar
  • 7,549
11 votes
0 answers
332 views

"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 ...
Seva's user avatar
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9 votes
0 answers
259 views

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.
Mare's user avatar
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2 votes
0 answers
81 views

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 $PGL(2, F)$ ...
Desiderius Severus's user avatar
9 votes
1 answer
774 views

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 ...
Patrick Elliott's user avatar
3 votes
2 answers
326 views

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: ...
La folia's user avatar
  • 115
1 vote
0 answers
48 views

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 ...
Mare's user avatar
  • 25.8k
5 votes
0 answers
130 views

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 ...
Julian Rosen's user avatar
  • 8,941
4 votes
1 answer
219 views

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)...
Desiderius Severus's user avatar

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