# Tagged Questions

**3**

votes

**0**answers

106 views

### Metaplectic groups over non-archimedean local fields of characteristic>2

Let $K$ be a non-archimedean local field of characteristic $>2$. Consider the double cover metaplectic extension of symplectic groups
$p: Mp_{2n}(K)\rightarrow ...

**36**

votes

**1**answer

1k views

### What is the status of Arthur's book?

Arthur's long-awaited book project is now published (The endoscopic classification of representations: orthogonal and symplectic groups). However, in the book he takes some things for granted:
The ...

**2**

votes

**0**answers

212 views

### algebraic representation over $\mathbb{C}$

In reading the Harris-Taylor book, I encounter expressions like "Let $\xi$ be an algebraic representation of $G$ over $\mathbb{C}$". What does this mean? Here $G$ is a reductive group over ...

**2**

votes

**0**answers

62 views

### Determine the representation given by space of sections of symmetric products of cotangent bundle of projective plane

In a recent project, it was interesting for me to determine the $PGL(3)$ representation given by $H^0(S^2(\Omega(1)) \otimes \mathcal O(4))$ on $\mathbb P^2$. I did this by using the Euler sequence, ...

**4**

votes

**1**answer

110 views

### Homological characterisation of standardly stratified algebras using Ext

Let A be a finite dimensional algebra and $S_1,S_2,...,S_n$ the simple $A$-modules and $P_1,..,P_n$ the indecomposable projective $A$-modules. For $i=1,...,n$, define the standard module $\Delta_i$ as ...

**17**

votes

**3**answers

490 views

### Is there a short proof that the Kostka number $K_{\lambda \mu}$ is non-zero whenever $\lambda$ dominates $\mu$?

This is maybe a little basic for MathOverflow, but I'm hoping it will get some interesting answers.
Let $\unrhd$ be the dominance order on partitions of $n \in \mathbb{N}$.
For partitions $\lambda$ ...

**3**

votes

**1**answer

99 views

### Why are the convolvers in the bicommutant of the pseudo-measures? ($CV_p(G)\subseteq PM_p(G)''$)

Let $G$ be a locally compact group. For $1<p<\infty$ let $\lambda_p:G\to\mathcal{B}(L^p(G))$ (resp. $\rho_p:G\to\mathcal{B}(L^p(G))$) be the left (resp. right) regular representation. $CV_p(G)$ ...

**2**

votes

**1**answer

114 views

### Modularisation on group representations with arbitrary braiding

Applying the modularisation/deequivariantisation procedure to the representation category $\operatorname{Rep}_G$ of a finite group $G$ with trivial braiding gives the fibre functor to vector spaces. ...

**5**

votes

**1**answer

129 views

### Is this modified bound quiver algebra necessarily representation-finite?

Suppose that $A = kQ/I$ is a bound quiver algebra for $k$ an algebraically closed field, $Q=(Q_0, Q_1)$ a finite connected quiver with no oriented cycles with no multiple edges or self-loops, and $I$ ...

**2**

votes

**1**answer

169 views

### Every norm-decreasing algebra morphism $L_1(G)\to\mathcal{B}(E)$ comes from a group representation

In section 8 of this paper http://arxiv.org/abs/math/0611833v3 the author proves the following:
If $E$ is a reflexive Banach space, $G$ a locally compact group and $\pi:L_1(G)\to\mathcal{B}(E)$ a ...

**2**

votes

**1**answer

80 views

### Is a quotient of a bound quiver algebra of finite representation type also representation-finite?

Let $A = kQ/I$ be a bound quiver algebra for some algebraically closed field $k$, $Q$ a finite connected quiver without oriented cycles, and $I$ an admissible ideal. Say that $I'$ is also an ...

**0**

votes

**1**answer

104 views

### Perfect $Q[G]$-complex

Let $G$ be a finite group and let $M$ be a perfect $\mathbb{Q}[G]$-complex.
Suppose that $M\otimes_{\mathbb{Q}[G]}\mathbb{Q}$ is quasi-isomrphic to $0$ can we conclude that $M$ is quasi-isomorphic to ...

**4**

votes

**0**answers

87 views

### Classification of representation-finite algebras up to stable equivalence of Morita type

assume K is an algebraically closed field.
I wanted to ask if there is a classification of the representation-finite K-algebras up to stable equivalence of Morita type at least for some small numbers ...

**0**

votes

**0**answers

50 views

### Normal conjugate of elements of unipotent upper tringular matrices over F_q

Let $UT_n(q)$ be the group of upper triangular matrices with entries in the finite field $F_q$ and ones on the diagonal. Denote the normal closure of an element $s\in UT_n(q)$ by $s^{UT_n(q)}$, i.e., ...

**16**

votes

**14**answers

1k views

### Applications of Representation Theory in Combinatorics

What are the examples of interesting combinatorial identities (e.g. bijection between two sets of combinatorial objects) that can be proved using representation theory, or has some representation ...

**0**

votes

**0**answers

23 views

### Generators for equivariant polynomial maps in the case of finite reflection groups

Let $G$ be a finite reflection group acting on a finite dimensional $\mathbb{R}$ vector space $E$. Then $Mor_G(E,E^*)$ is a free $\mathbb{R}[E]^G$ module. Moreover if $f_1,\ldots,f_n$ are the basic ...

**6**

votes

**4**answers

624 views

### When does $\langle grg^{-1}|g\in G\rangle = \langle gsg^{-1}|g\in G\rangle$ imply $\langle r\rangle=\langle xsx^{-1}\rangle$?

We call a finite group $G$ normal if for all $s,r\in G$, if $\langle gsg^{-1}:g\in G\rangle =\langle grg^{-1}:g\in G\rangle $ then there exists $x\in G$ such that $\langle r\rangle =\langle ...

**2**

votes

**1**answer

138 views

### A canonical representative in Morita equivalence class

Let $A$ be a finite dimensional algebra over an algebraically closed field $K$.
If $A$ is semisimple, then $A$ is Morita equivalence with a commutative algebra, that is $A \backsim K^n$ where $n$ is ...

**5**

votes

**0**answers

171 views

### Homogeneous spaces of affine algebraic groups

Let $G$ be a reductive algebraic group over an algebraically closed field $K$ of characteristic zero (I am particularly interested in the case $G=GL_n(K))$.
Let $H$ be a closed subgroup of $G$. It is ...

**6**

votes

**2**answers

325 views

### Request for classical articles in representation theory

I am planning in running a Ph.D. student seminar next year on representation theory in the spirit of MIT Kan's Seminar where students give lectures on classical articles on representation theory that ...

**8**

votes

**1**answer

184 views

### $U_q(\mathfrak{sl}_2)$ representations of “quantum dimension” zero

I'm reading up on quantum groups and their applications and I've come across a question I just can't find an answer to. I know about the basic representation theory of $U_q(\mathfrak{sl}_2)$ and I ...

**6**

votes

**0**answers

73 views

### Correlation of Class Functions

Let $G$ be a finite group, and let $f_1,f_2$ be two real-valued class functions of $G$. Assume that multiplying elements of $G$ takes $O(1)$-time.
Let $s:G\to \mathbb{R}$ be defined by ...

**4**

votes

**1**answer

84 views

### Construction of vector space isomorphism where $f(v \otimes gh) = \sigma(g)(f(v \otimes h)),\text{ }\forall v \in V,\text{ }g,\,h \in G$

Let $G$ be a finite group, $R = \mathbb{C}G$ the regular representation of $G$, and $\rho : G \to \text{GL}(V)$ a finite dimensional representation of $G$. Write $\sigma: G \to \text{GL}(V \otimes R)$ ...

**5**

votes

**1**answer

187 views

### Description of the algebra of $G$-invariant polynomials by generators and relations

Fix $n > 1$ and let $\zeta \in \mathbb{C}$ be a primitive $n$-th root of unity. Let $G \subset \text{SL}_2(\mathbb{C})$ be a cyclic subgroup of order $n$ generated by the diagonal matrix $g = ...

**0**

votes

**0**answers

74 views

### Orbits of some action of SL2 on Pontryagin dual of the field of formal Laurent series

Let $K=\mathbb{F}_2((t))$ be the field of formal Laurent series over the finite field $\mathbb{F}_2$. Now consider $K^3$ as an additive group and its dual group $\hat{K^3}$, which consists of all ...

**1**

vote

**1**answer

248 views

### Decompose $\Lambda^3(V \otimes W)$ [closed]

Let $V, W$ be two vector spaces. We have $\Lambda^2(V \otimes W) \cong (\Lambda^2 V \otimes S^2 W) \oplus (S^2 V \otimes \Lambda^2 W)$. I am trying to find similar results for $\Lambda^3(V \otimes ...

**0**

votes

**0**answers

81 views

### Is there a metric defined on the product space of orthogonal groups?

If one considers just the orthogonal group, then there is a natural metric given by [1]:
$$\begin{align}
\theta & =\frac{1}{2} \| \log(R_1^{-1}R_2) \| \\
& = \frac{1}{2} \| ...

**3**

votes

**1**answer

223 views

### A question of the book Elements of the representation theory of associative algebras volume 1

I'm reading the book "Elements of the representation theory of associative algebras, volume 1". And I can't understand the proof of the proposition 3.11 on page 124 (the place where marked green).
...

**4**

votes

**1**answer

303 views

### Quotient of a vector space by a linear finite group action

Let the cyclic group $\mathbb{Z}_n$ act on $\mathbb{C}^n$ (or on $\mathbb{R}^n$, I'm interested in both) by permuting coordinates. What does the topological quotient $Q$ by this group action look ...

**2**

votes

**0**answers

123 views

### Differential of the adjoint quotient map

My question is regarding a paper by R.W Richardson titled "Derivatives of invariant polynomials on a semisimple Lie Algebra" ** . In this paper, he reports on computations of the rank of the ...

**4**

votes

**0**answers

159 views

### Hochschild cohomology of SU(2)

I have a question about the computation of an Hochschild Cohomology. Or at least about a space which really looks like a cohomology space. All the functions i consider are assumed to be smooth.
Let's ...

**2**

votes

**0**answers

44 views

### Calculation of minimal right $\operatorname{add}(M)$-approximations

given a finite dimensional quiver algebra $A$ and a generator $M$ with $\operatorname{Ext}^1(M,M)=0$. By Wakamatsus lemma, for any $A$-module $N$ there exists a surjective $A$-linear map $f\colon M_1 ...

**0**

votes

**0**answers

80 views

### What are the explicit expressions of quantum Casimir elements for $U_q(sl_3)$ and $U_q(sl_4)$?

What are the explicit expressions of quantum Casimir elements for $U_q(sl_3)$ and $U_q(sl_4)$ in terms of $E_1, E_2, F_1, F_2, K_1, K_2, K_1^{-1}, K_2^{-1}$? Any help will be greatly appreciated!

**5**

votes

**1**answer

170 views

### Strongly real elements of odd order in sporadic finite simple groups

Recall that an element of a finite group is said to be real if it is conjugate to its inverse, and strongly real if the conjugating element can be chosen to be an involution.
Question: Is it true ...

**16**

votes

**1**answer

450 views

### Okounkov-Vershik approach to representation theory of $S_n$

This is a rather soft question. I was wondering if someone could explain on a fundamental and intuitive level, what the Okounkov-Vershik approach to representation theory of $S_n$ is all about. It's ...

**8**

votes

**0**answers

149 views

### $k[x_1, \dots, x_n]$ is free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism?

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set ...

**2**

votes

**0**answers

89 views

### what is the link between plethysm in regular representation of the symmetric group and plethysm in Schur functions.

I am trying to understand first how one can define the plethysm say $s_\lambda \circ s_\mu$ as a module in the regular representation of the symmetric group.
1)How is it connected to the plethysms ...

**9**

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**0**answers

158 views

### $A$-module is free if and only if equation involving Hilbert-Poincaré series holds, $M$ infinitely generated case

See my question here.
Let $A = \oplus_{i \ge 0} A_i$ be a nonnegatively graded commutative algebra and $M$ a nonnegatively graded $A$-module. Assume in addition that $A_0 = k$ and all vector ...

**3**

votes

**0**answers

52 views

### $G$-harmonic polynomials, dimension of $\text{Harm}(\mathbb{R}^n, S_n)$? [closed]

Definition. Let $\text{Harm}(\mathbb{R}^n, G)$ be the space of $G$-harmonic polynomials on $\mathbb{R}^n$.
My question is, what is the dimension of $\text{Harm}(\mathbb{R}^n, S_n)$?

**2**

votes

**0**answers

75 views

### $\mathbb{C}[x_1, \dots, x_n]$ is a free $\mathbb{C}[x_1, \dots, x_n]^{S_n}$-module with certain generators [duplicate]

Let the symmetric group $S_n$ act on $\mathbb{R}^n$ by permutation of coordinates. This makes $S_n$ a subgroup of $\text{GL}_n(\mathbb{R}$ and the algebra $\mathbb{C}[x_1, \dots, x_n]^{S_n}$ is the ...

**16**

votes

**1**answer

418 views

### Koszul complex for non-Koszul algebras

Let $A$ be a graded, connected, locally finite, quadratic algebra over a field $k$; that is, $A$ may be presented as $T(V)/I$, where $V = A_1$ is a finite dimensional $k$ vector space, and the ideal ...

**3**

votes

**0**answers

87 views

### Fibers of torus equivariant moment maps

Given a closed (possibly singular) projective variety $V$ with a symplectic structure and a torus action, there is a moment map
$\mu: V \rightarrow Lie(T)^*$. Note that the dimension of $T$ could be ...

**10**

votes

**1**answer

195 views

### equivariant stable class of quaternionic Hopf fibration in RO(G)-degrees of ADE-type

Does the quaternionic Hopf fibration possibly represent a non-torsion element in the $G$-equivariant stable homotopy groups of spheres, for $G$ a finite subgroup of $SO(3)$ and in RO(G)-degree being ...

**9**

votes

**0**answers

199 views

### $\mathcal{M}(\mathcal{D}_X)$ and $\mathcal{M}^r(\mathcal{D}_X)$ have natural tensor category structures?

Write $\mathcal{M}^\ell(\mathcal{D}_{X/S}) = \mathcal{M}(\mathcal{D}_{X/S})$ for the category of left $\mathcal{D}$-modules over $X$ and $\mathcal{M}^r(\mathcal{D}_{X/S})$ for the category of right ...

**4**

votes

**0**answers

126 views

### When is a given quiver algebra a hopf algebra?

Given a finite dimensional selfinjective quiver algebra A over a finite field (or more generally an arbitrary field). Whats the best way to check if the algebra A has a Hopf algebra structure or not? ...

**5**

votes

**2**answers

225 views

### Representations of orthogonal groups over the field of two elements

I am looking for some references on modular representation theory of the orthogonal groups $O_{2n+1}(2)$, $O_{2n}^{+}(2)$, or $O_{2n}^{-}(2)$ over $\mathbb{F}_2$.

**2**

votes

**0**answers

100 views

### Center of $U_q(sl_3)$ and $U_q(sl_4)$

In the book a guide to quantum groups, page 285, the center of $U_q(g)$ is described in Theorem 9.1.6. The center of $U_q(sl_2)$ is computed explicitly in Example 9.1.7. I tried to compute the center ...

**3**

votes

**0**answers

113 views

### Integer-matrix representation of a commutative ring

Consider a commutative ring $x_ix_j = N_{ij}^k x_k$, where $N_{ij}^k \in\{0,1,2,3,\cdots\}$, and $\{x_i\}$ is a finite set. (This is actually a fusion ring and $x_i$ are simple objects.)
How to find ...

**1**

vote

**1**answer

129 views

### Equivariant polynomial maps

Let $V$ be a complex vector spaces and assume that a compact group G acts linearly on $V$. Then look at the $G$-equivariant polynomial maps from $V$ to $V$. Denote this by $Mor_G(V,V)$. In the case ...

**5**

votes

**0**answers

223 views

### Recurrence Formula for Zernike polynomials

I'm not sure if this is research level, so if this result is known, please excuse the intrusion. I am trying to find a relation between solutions of the Laplacian equation in $4$ dimensions and those ...