# Questions tagged [rt.representation-theory]

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

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### Decomposition of $L^2(\Gamma \backslash H)$ into irreducible representations using the spectral theorem

I'm reading the introduction of An Introduction to the Trace Formula by James Arthur and wanted to understand something in the introduction.
Let $H$ be a unimodular locally compact Hausdorff group, ...

**9**

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**2**answers

328 views

### Is the cohomology ring $H^*(BG,\mathbb{Z})$ generated by Euler classes?

I am interested in the classifying space $BG$ of a finite group $G$.
A real representation $V$ of $G$ of dimension $r$ defines a real vector bundle over $BG$ of rank $r$. If the determinant of this ...

**4**

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137 views

### Local structure of non-normal toric varieties---possible mistake in “Discriminants, Resultants and Multidimensional Determinants”

I believe I may have a counterexample to Theorem 5.3.1 on page 179 from the book book Discriminants, Resultants and Multidimensional Determinants by Gel'fand, Kapranov, and Zelevinsky. To summarize ...

**3**

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52 views

### Reference Request: Representation Theory of Real Lie Superalgebras

Are there some references for the representation theory real lie superalgebras, specifically of $psu(1,1|2)$, and $u(1,1|2)$?

**4**

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89 views

### Morava E-theory of classifying spaces

Let $G$ be a compact Lie group/finite group. Is Morava E-theory at height $n>1$ of $BG$ related to representation theory of $G$?

**5**

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67 views

### Existence of anti-symmetric hochschild homology representatives

Let $A$ be an associative algebra over a field $k$. Let $A_{L}$ be the Lie algebra of $A$ with commutator bracket. Then if $M$ is a bimodule for $A$ there is an associated representation of $A_{L}$ ...

**4**

votes

**1**answer

108 views

### Average of product of matrix elements in irreducible representations of unitary groups

Let $\mathcal{U}(N)$ be the unitary group.
It is well known that
$$ \int_{\mathcal{U}(N)} U_{ij} U^\dagger_{nm} \,dU=\delta_{im}\delta_{jn}\frac{1}{N},$$
where $dU$ is the Haar measure.
More ...

**3**

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116 views

### Moduli space of nilpotent Lie algebras

Fix a nilpotent Lie algebra $L$ over some char 0 field $k$ which is naturally graded, i. e. isomorphic to graded algebra $\bar L$ associated to lower central filtration.
I'm interested in some ...

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**0**answers

30 views

### How to find the cosets parameterizing quotient spaces such as elliptic three-manifolds?

I would like to know, given a groups $H\subseteq G$, if there a general method to finding the representatives $g$ that would parameterize the cosets $gH$. I cannot find a recipe for infinite $G$ and ...

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139 views

### List of Automorphism groups of Abelian Varieties for Dummies

(%Edited after abx comment%)
I seek explicit linear integral representations $\rho: Aut(X,\omega) \to Sp_{2g}(\mathbb{Z})$, when $(X,\omega)$ is complex $g$-dimensional PPAV. I prefer explicit ...

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26 views

### Decomposition into irreducible of a representation of the wreath product $S_d \wr S_n$ (4)

I have to decompose some representations of $S_d \wr S_n$. I understand better and better how it works, I still have a case I don't know how to deal with.
For simplicity I take $d=2$ and $n=4$. $S^{(...

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**0**answers

54 views

### Does the pushforward of the Haar measure of a semisimple compact Lie group along a character determine the character?

Let $G$ be a connected compact semisimple Lie group.
Let $V$ be a faithful representation of $G$,
with character $\chi \colon G \to \mathbb{C}$.
Let $\mu_G$ be the normalized left Haar measure. (So $\...

**5**

votes

**2**answers

145 views

### Obtaining quiver and relations for finite p-groups

Given a finite field $K$ with $p$ elements and a finite $p$-group $G$, is there a way to obtain the quiver and relations of $KG$ with GAP (and its package QPA)?
Since $KG$ is local, the quiver should ...

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52 views

### Modules with arbitrary large complexity

Let $A$ be a finite dimensional algebra with a module $M$ who has minimal injective coresolution $(I_i)$. Define the complexity $cx(M)$ as $cx(M):= \inf \{ t \geq 0 | \dim(I_i) \leq a i^{t-1}$ for all ...

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25 views

### Complexity of the regular module

Let $A$ be a finite dimensional algebra with a module $M$ who has minimal injective coresolution $(I_i)$. Define the complexity $cx(M)$ as $cx(M):= \inf \{ t \geq 0 | \dim(I_i) \leq a i^{t-1}$ for all ...

**2**

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**1**answer

126 views

### Core components of quiver varieties as fiber bundles of flag varieties

Is there an example of Nakajima quiver variety of type A which has all core components smooth, such that at least one of them is NOT an iterated fibre bundle of flag manifolds (i.e. a space obtained ...

**2**

votes

**1**answer

67 views

### Show that if $\Theta$ is an infinitesimal weight of a real $T$-module $W$ ($T$ is a torus) then $-\Theta$ is also a weight

Show that if $\Theta$ is an infinitesimal weight of a real $T$-module $W$ ($T$ is a torus) then $-\Theta$ is also a weight.
It is an exercise of Bröcker's book on Representations of Compact Lie ...

**2**

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**0**answers

78 views

### Computing Hochschild Invariants of Positselski's Coderived Categories

Positselski's work allows one to frame Koszul duality very elegantly in terms of so called coderived categories of modules over coalgebras, these are somewhat exotic dg categories of comodules over $C$...

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**1**answer

109 views

### Classification of quasi-lisse vertex algebras

Quasi-lisse vertex algebras were introduced by Arakawa and Kawasetsu in Quasi-lisse vertex algebras and modular linear differential equations . They satisfy the property that the normalized character ...

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126 views

### Most important results for Shalika germs

This is more of a general question, but what do you think are the most important results for Shalika germs if you were giving a presentation? You can assume the target audience to be 2nd-3rd year ...

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147 views

### Does the Burau representation of braids distinguish between distinct elements of the free self-distributive algebras on one generator?

A well-known but now mostly solved problem in group theory is the question of whether the Burau representation of the braid groups is faithful. It turns out that this representation is not faithful ...

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77 views

### Galois descent for profinite groups acting on local fields

Suppose that $G$ is a profinite group acting faithfully on a field $L$ and assume moreover that the action is admissible, that is, that every $l \in L$ is stabilized by an open subgroup of $G$. If ...

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**1**answer

106 views

### Restriction of smooth representaions of SL(2,Q_p) to the maximal compact

I am reformulating a question I asked earlier with no answer: Consider $SL(2, Q_p)$ and $K$ a maximal compact subgroup. Let $\pi$ be an irreducible spherical representation of $SL(2, Q_p)$ (in the ...

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34 views

### Admissible (unitary) spherical representation $sl(2,Q_p)$. Does dimension fixed point vector increase proportional index

I am using terminology of Cartier's Harmonic analysis on trees. Take $\pi$ be one of the irreducible principal or complementary (unitary) spehrical series of $Sl(2, Q_p)$. Let $K$ be the maximal ...

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39 views

### Efficient way to express a symmetric tensor in terms of rank one elements

Let
$$P(\mathbf{x})=P(x_1, \ldots, x_n)=x_1^{k_1}x_2^{k_2}\cdots x_n^{k_n}$$
be a homogenous polynomial of degree $k=k_1+\cdots+k_n$.
It follows from a standard polarization identity (see for ...

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198 views

### Colimit of continuous cohomology over subgroups

Suppose $G$ is a profinite group, in fact in the applications I'm interested in it would be a $p$-adic analytic group similar to $GL_{n}(\mathbb{Z}_{p})$. Say $M$ is a profinite $G$-representation, ...

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55 views

### An two-norm estimate for symmetric $k$-tensors

Let $(V, \langle, \rangle)$ be an $n$ dimensional innerproduct space and let $S^k(V)$ denote the space of $k$-fold symmetric tensors. The inner product naturally extends to $S^k(V)$. Denote the ...

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96 views

### Recovering the bimodule from the trivial extension

Given a ring $S$ with a non-zero $S$-bimodule $M$, the trivial extension of $(S,M)$ is defined as the ring $R:=T_M(S)$ with $R= S \oplus M$ with multiplication $(s,m)(s',m')=(s s', sm' +m s')$.
We ...

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97 views

### Dimensions of Lie algebra powers of irreducible representations

Consider the following plethysms. For semisimple Lie algebras $L$, if $A$ is the adjoint irrep, as far as I know, the dimensions are as follows: $d_1=|A^{\bigotimes 1}|=0$ (meaning e.g. no lollipops O-...

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100 views

### Proving that the exterior algebra is symmetric via the polynomial ring

Recall that a finite dimensional algebra $A$ over a field $K$ is called Frobenius in case $A \cong D(A)$ as right modules, and it is called symmetric in case $A \cong D(A)$ as bimodules (where $D=...

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96 views

### Contravariant forms and tensor products

In Humphreys's book "Representations of semi-simple Lie algebras in the BGG-category O", section 3.14 deals with contravariant forms on highest weight modules. I wanted to define a map by a bilinear ...

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100 views

### Resolution of Kleinian Singularities using Hilbert schemes of points

Apologies in advance for the naive and rather speculative question.
In this blog post by John Baez, he paints a (perhaps not original) picture of how one might expect that the minimal resolution of a ...

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248 views

### The symmetric power of a tensor product

In the representation theory, if $S^{\lambda}(V)$ is the irreductible representation of $\text{GL}(V)$ associated to a partition $\lambda \vdash n$ (in perticular, $S^n(V)$ is the $n^{\text{th}}$ ...

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114 views

### Weak generators of the right-bounded derived category of a finite-dimensional algebra

The setup:
Let $A$ be a finite-dimensional $k$-algebra over some field $k$.
Let $\mathcal{B} = Hot^-(Proj \, A)$ denote the homotopy category of cochain complexes of (possibly infinitely generated) ...

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22 views

### Explicit symmetry adapted basis for the symetric square of the standard representation

I already posted a related question here, which is more detailed: https://math.stackexchange.com/posts/2786382/edit
The permutation group $S_n$ has standard representation $S^{(n-1,1)}$ (irreducible)....

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59 views

### Properties of Direct Integral

I am wondering whether the following properties hold, and when not hold, can we require some conditions such that they hold?
(1) Direct integral v.s. induced representation
$Ind_{H}^{G}\int_{\hat{H}}...

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**1**answer

319 views

### Are there nice isomorphisms $\operatorname{S}^2(k^n)\cong\Lambda^2(k^{n+1})$?

This might be forced to migrate to math.SE but let me still risk it.
The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism ...

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176 views

### About quotient varieties

Let $K$ be a field, $L/K$ be a finite Galois extension with Galois group $G$ such that $(char(K),|G|)=1$ and $K$ contains all $|G|$th roots of unity. Let $B$ be a $L$-algebra of finite type endowed ...

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**1**answer

431 views

### Does this algebra have finite global dimension ? (Human vs computer)

Usually computers can calculate the global dimension of a finite dimensional quiver algebra much faster than humans. But in this case a high end computer (calculating for 3 weeks) was not able to ...

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votes

**1**answer

123 views

### Algebras derived equivalent to quasi-hereditary algebras

Let an algebra always be finite dimensional over a field and connected.
It is well known that a quasi-hereditary algebra with $n$ simple modules has global dimension at most $2n-2$.
Questions:
1. ...

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268 views

### Commuting matrix variety $[A,B]=0$ - can one geometrically explain divisibility of $F_ q$ point count by high powers of $q$?

$\DeclareMathOperator\Comm{Comm}\DeclareMathOperator\Id{Id}$Consider the variety $\Comm$ of commuting matrices $[A,B]=0$ over some field $K$. It is much studied, and interesting for various reasons.
...

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75 views

### Explicit examples of finite dimensional, involutive Hopf algebras with traceless antipode?

$\require{AMScd}$
In the paper [1], it is shown that there exist finite dimensional, semisimple Hopf algebras $H$ where the antipode $S:H \to H$ is traceless.
Unfortunately, the method of proof in [...

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**0**answers

65 views

### Finitistic dimension via a bimodule

Let $A$ be a connected finite dimensional basic algebra.
Question: Is there an indecomposable $A$-bimodule $W$ such that the finitistic dimension of $A$ is equal to the right projective dimension ...

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123 views

### Can semisimple orbits be written $\exp(\mathfrak{g})\cdot x$?

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $G$ be its adjoint group. If $x\in\mathfrak{g}^{rs}$ is a regular semisimple element, is its orbit
$$G\cdot x=\{\mathrm{Ad}_gx:g\in G\}$$
...

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91 views

### L-functions of tempered automorphic representations

Let $G$ be a reductive group over the adele ring $\mathbb A_F$ of a number field $F$. Let also $r$ be a complex finite dimensional representation of the $L$-group $r:{^LG}\to GL(V)$.
It is generally ...

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votes

**1**answer

178 views

### Why is Nagao's theorem the “Module theoretic version of Brauer's second main theorem”?

Let $G$ be a finite group, $p\in\mathbb{P}$ a prime, $\mathbb{F}$ an algebraically closed field of characteristic $p$, and $D\leq G$ a $p$-subgroup.
Brauer second main theorem states
If $\chi\in ...

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48 views

### To find $\dim(D^{\mu})=\dim\frac{S^{\mu}}{S^{\mu}\cap S^{\mu\perp}}$

Let $S_6$ be the symmetric group of degree $6$ and $F$ be any finite field of characteristic $2.$ Then $2$-regular partition of $6$ are $(5,1)$, $(4,2)$ and $(3,2,1)$ . I have to find $$\dim(D^{\mu})=\...

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84 views

### Monomial Characters of Quotient Groups

The following statement provides (if true) a powerful tool for inductive proofs. Can anyone confirm if it is true:
Suppose $G$ is a finite group and $N$ a normal subgroup of $G$. Is it true that if $...

**4**

votes

**1**answer

189 views

### Dimensions of $E_{7\frac{1}{2}}$

Is there much known about the dimensions $D$ of $E_{7\frac{1}{2}}$ (that is: $D_6.H_{32}$) beyond
$$
44\otimes44(def)=1\oplus945\oplus99(adj)\oplus891\, ?
$$
Generally, does a weight indexing scheme ...

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votes

**1**answer

156 views

### $Hom_G(C_c^{\infty}(G),\pi)\cong Hom_{\mathbb{C}}(\pi^{\vee},\mathbb{C}) ?$

$G$ is an p-adic group, and $\pi$ is an irreducible representation of $G$, then do we naturally have
$Hom_G(C_c^{\infty}(G),\pi)\cong Hom_{\mathbb{C}}(\pi^{\vee},\mathbb{C})$? I think it is true, but ...