**7**

votes

**4**answers

360 views

### Is there a non-explicit characterization of the Specht modules?

It is a basic fact about the symmetric group $S_n$ that its irreducible representations are indexed by partitions of $n$.
My question is, can the association between partitions and irreps be ...

**4**

votes

**0**answers

47 views

### examples of local, nonsemisimple , nonsymmetric hopf algebras

I'm searching for (a class of) examples of Hopf algebras , which have the following properties:
they should be finite dimensional
they should not be semisimple
they should be local
they should ...

**3**

votes

**0**answers

156 views

### regarding the upper half-plane model for the principal series representations of $\text{GL}_2\big( \Bbb{R}\big)$

Let $B$ be the Borel subgroup of $G = \text{GL}_2\big( \Bbb{R}\big)$, let
${\bf \alpha}:B \longrightarrow \Bbb{C}^*$ be a character, and consider the
induced representation $\text{Ind}_B^G ({\bf ...

**6**

votes

**1**answer

322 views

### Quotient of Flag varieties

Let $G=SL_3(\mathbb{C})$ and $X=G/B$ be the associated full flag variety. Fix a non-degenerate symmetric quadratic form $Q$ on $\mathbb{C}^3$. This gives an order $2$ automorphism $F_Q$ of $X$, ...

**9**

votes

**1**answer

240 views

### Why is the A6 preprojective algebra of wild representation type?

As mentioned in the title, I would like to know a proof of the "well known" fact that the A6 preprojective algebra is of wild representation type.
Ideally, I would like to see an explicit ...

**2**

votes

**0**answers

143 views

### Unitary representation of finite-dimensional unitary group

the question is the following. Let n,m be integers, $U(n)$ be the unitary group of $M_n(\mathbb C)$, and $\phi\colon U(n)\to U(m)$ be a continuous group homomorphism, that is moreover irreducible as a ...

**5**

votes

**0**answers

141 views

### Bialgebras with Hopf restricted duals

It is known from the general theory that, given a bialgebra (over a field $k$)
\begin{equation}
\mathcal{B}=(B,\mu,1_B,\Delta,\epsilon)
\end{equation}
the Sweedler's dual $\mathcal{B}^0$ (called also ...

**4**

votes

**0**answers

162 views

### 2-periodic derived equivalence

Let $A$ and $B$ be finite-dimensional algebras with finite global dimension over some field (in fact I am thinking of rational incidence algebras of finite posets).
Suppose we know that $A$ and $B$ ...

**4**

votes

**1**answer

217 views

### Character table of $\mathrm{SL}_2(\mathbb{Z}/p^n\mathbb{Z})$

Is there any reference where I can find the character table of $\mathrm{SL}_2(\mathbb{Z}/p^n\mathbb{Z})$? A simple search in google gave me this paper of Philip C. Kutzko on "The characters of the ...

**0**

votes

**1**answer

128 views

### Equivalence of Lie subalgebras, within a (irreducible) representation

Lie subalgebras inside simple Lie algebras (of type ABCDEFG) have been classified up to equivalence, and linear equivalence (by Dynkin et al). How does one classify embeddings of a Lie algebra h ...

**3**

votes

**0**answers

155 views

### Is there any good survey on the hook length formula and related topics?

I am recently doing some research related to the hook length formula.
The hook formula counts the number of Young tableaux of certain type.
I find there are plenty of research already been done and ...

**1**

vote

**1**answer

200 views

### Infinitely many real roots

Given a non-acyclic quiver without loops with Kac's root system associated. When do we know there are infinitely many real roots?

**3**

votes

**2**answers

358 views

### $t$-analogue of the symmetric power of an additive character over $\Bbb{F}_q^*$

Let $G$ be a finite group and let $f: G \longrightarrow \Bbb{C}$ be any complex-valued function. For integers $k, n \geq 0$, an indeterminant $t$, and $x \in G$ let $f_k(x) := f \big( x^k \big)$ and ...

**6**

votes

**0**answers

199 views

### On a permutation module for GL(n,q)

Let $G=GL(n,q)$ be the general linear group of degree $n$ over the $q$ element field. Let $X$ be the set of full rank $n\times r$ matrices where $1\leq r\leq n$. Then $G$ acts transitively on the ...

**5**

votes

**1**answer

168 views

### Singular/Smooth locus of Schubert variety of the affine grassmannian

Let $G$ be a connected, simply connected, semisimple, complex linear algebraic group with maximal torus $T$ and affine Grassmannian $\mathcal Gr$. It is well known that $\mathcal Gr$ admits a Bruhat ...

**3**

votes

**3**answers

342 views

### A problem with pointwise stabilizer subgroups of fixed-point subspaces II

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let ...

**0**

votes

**1**answer

98 views

### A problem with pointwise stabilizer subgroups of fixed-point subspaces I

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let ...

**1**

vote

**1**answer

179 views

### Beilinson-Bernstein localization: $\mathfrak{g}$ action on $G$-equivariant sheaf

I have a few elementary questions related to Beilinson-Bernstein localization.
Let $G$ be a semisimple algebraic group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}$. Consider the setup of ...

**4**

votes

**1**answer

77 views

### Intertwining Operators Associated to Simple Reflections

Let $G$ be a quasi-split reductive group, over a local field, with a Borel subgroup $B=T\cdot N$ and the associated Weyl group $W$. Given a family of induced representations $\pi_s = Ind_B^G \chi\cdot ...

**6**

votes

**2**answers

228 views

### Is an $\mathfrak{sl}_2$-triple determined up to Lie algebra automorphism by the adjoint representation?

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and let $\phi_1:\mathfrak{sl}_2(\mathbb{C})\rightarrow\mathfrak{g}$ and ...

**2**

votes

**0**answers

102 views

### References for the bicategory of ring-bimodule pairs

One of the standard examples of a bicategory is the bicategory of rings (with bimodules as 1-morphisms), which is sometimes denoted $\operatorname{Bim}$ and in other sources $\operatorname{Ring}$ (or ...

**0**

votes

**0**answers

84 views

### Differences between primitive central idempotents and primitive orthogonal idempotents

If we have a complete set of primitive orthogonal idempotents of an algebra $A$, then we can obtain simple modules, indecomposable projective modules, indecomposable injective modules of $A$.
If we ...

**5**

votes

**1**answer

255 views

### Is there a nonabelian finite simple group with Grothendieck ring of multiplicity one?

Let $G$ be a finite group. It admits finitely many irreducible complex representations $H_1, \dots, H_r$ which generate, for $\oplus$ and $\otimes$, the Grothendieck ring $\mathcal{G}(G)$ of $G$ (also ...

**5**

votes

**1**answer

201 views

### Represent matrix immanants using Schur functions

For each irreducible character $\chi^\lambda$ of the symmetric group $S_n$, the immanant of an $n\times n$ square matrix $A$ is defined as
\begin{equation*}
d_\lambda(A) := \sum_{\sigma \in S_n} ...

**1**

vote

**0**answers

51 views

### $\Gamma$ cohomology of principal series

Let $G$ be a noncompact connected real semisimple Lie group with finited center. Let $\Gamma$ be a cocompact discrete subgroup of $G$, and let $P$ be a parabolique subgroup
with Langlands ...

**3**

votes

**1**answer

123 views

### Prescribed spherical representations, symplectic group $Sp(n)$

An irreducible representation $(\pi,V_\pi)$ of a compact group $G$ is called spherical with respect to the pair $(G,K)$, $K$ is closed subgroup of $G$, if $V_\pi$ has a non-zero vector invariant by ...

**3**

votes

**3**answers

195 views

### how to find explicitly given component in a regular representation

Given a finite group $G$ and its irreducible representation $\pi$ I want to find explicit elements of the group algebra $\mathbb{C}[G]$ lying in components of the left regular representation ...

**17**

votes

**1**answer

278 views

### On a drawing in Dixmier's Enveloping Algebras

This image
comes from Dixmier's book, 'Enveloping Algebras' ('Algèbres enveloppantes').
Dixmier writes that
The curves shown on p. XIV have their origin in the study of U(sl(3)).
They are ...

**4**

votes

**1**answer

208 views

### Matrix Elements of Real Representations

I asked this question over at Math.StackExchange and despite having had a bounty on it I did not receive an answer.
Suppose that $G$ is a finite group and we have a unitary irreducible representation ...

**15**

votes

**2**answers

691 views

### Are there any natural differential operators besides $d$?

Let $\lambda = (\lambda_1, \ldots, \lambda_r)$ and $\mu = (\mu_1, \ldots, \mu_r)$ be partitions such that $\mu_j = \lambda_j +1$ for one index $j$ and $\mu_i = \lambda_i$ for all other $i$. Then there ...

**5**

votes

**1**answer

429 views

### Why Jacobson, but not the left (right) maximals individually?

I firstly asked the following question on MathStackExchange a couple of months ago. I did not receive any answers, but a short comment. So, I decided to post it here, hoping to receive answers from ...

**7**

votes

**2**answers

429 views

### On a theorem of Kazhdan

Let $G=GL_n(F)$, where $F$ is a p-adic local field, $U$ be the upper triangular maximal unipotent group, and $\theta$ a character of $U$. Then a Theorem of Kazhdan says that for any irreducible smooth ...

**0**

votes

**0**answers

56 views

### Framed braids and local systems

Let me start by admitting that my question is going to be somewhat vague. But hopefully it is one of these vague questions that can be immediately answered by an expert in the appropriate area.
...

**2**

votes

**0**answers

64 views

### What is the name for the subring of the Grothendieck ring of a bialgebra spanned by one-dimensional representations?

Let $B$ be a finite dimensional bialgebra over a field $\Bbbk$. Let $\mathcal G_0(B)$ be the ring whose underlying additive group is generated by isomorphism classes $[V]$ of finite dimensional ...

**0**

votes

**0**answers

95 views

### What interesting things do automorphism groups of trees act on?

Let $T$ be a rooted tree. We can build a poset $P(T)$ whose elements are the vertices of $T$ and whose covering relations are the edges of $T$. Let $A$ be the automorphism group of $P(T)$. The group ...

**3**

votes

**1**answer

181 views

### About the second largest adjacency eigenvalue of Abelian Cayley graphs

[Assume all groups are finite]
One knows the general statement that the sum of the values of the character function on the generating set is an eigenvalue of a Cayley graph.
But the above doesn't ...

**5**

votes

**0**answers

230 views

### How can you see the minimal relations on a quiver from its bimodule resolution?

Suppose that you are given an algebra $KQ/I$, coming from a quiver Q, of finite global dimension. Suppose also that you know its minimal bimodule resolution over its enveloping algebra. Can you get a ...

**4**

votes

**0**answers

181 views

### Decomposition of symmetric powers of reduced regular representation modulo $p$

Let $\bar{\rho}$ denote the reduced regular representation of $\mathbb{Z}/p$ over a field of characteristic $p$. The representation $\mathrm{Sym}^k \bar{\rho}$ decomposes (for each $k$) as a sum of ...

**2**

votes

**1**answer

144 views

### Invariant subspaces of an $F_2$-representation of the affine linear group of dimension 1

Let $p$ be an odd prime (large if it matters) and let $G= Aff(\mathbb{F}_{p^2}) \cong \mathbb{F}_{p^2} \rtimes \mathbb{F}_{p^2}^*$ be the affine linear group acting on $\mathbb{F}_{p^2}$ by $x\mapsto ...

**0**

votes

**1**answer

202 views

### Suppose that $G$ is a subgroup of $GL_n(\mathbb C)$ with finite exponent. Then is $G$ a finite group? [closed]

As title. the exponent of $G$ is the least number $n$ (if exists) such that $g^n=e$ holds for all $g\in G$ or $+\infty$.

**1**

vote

**0**answers

56 views

### Computing equivariant K-theory using the amalgamted product

If I have a Lie group (or a Kac-Moody group) $G$ such that it's the amalgated product of it's proper parabolic subgroups $P_J$, i.e. $G = \text{colim} P_J$, then could I use this to compute the ...

**4**

votes

**1**answer

151 views

### Motivational ideas for the Gelfand-Graev character of a finite group of Lie type

I've been studying the Gelfand-Graev character's general construction for a finite group of Lie type. I wish to discuss its particularization in a seminar for the general linear group over a finite ...

**1**

vote

**0**answers

168 views

### Is there a method to simultaneously block-diagonalize a set of group matrices?

Assume that you are explicitly given the representation matrices of a group.
How does one go about finding that common basis which will find the irreducible components of all of them simultaneously?
...

**2**

votes

**1**answer

201 views

### What is the cohomology of the tangent bundle of a flag variety?

Let $G$ be the general linear group $\operatorname{GL}(n,\mathbb{C})$ and $P$ a parabolic subgroup with Lie algebra $\mathfrak{p}$. Consider the vector bundles
$$
\mathcal{P} = G\times_P ...

**3**

votes

**0**answers

119 views

### Dimension of Birman-Murakami-Wenzl Algebra

I was reading the paper Braids, Link Polynomials and A New Algebra by J. S. Birman and H. Wenzl, and I was wondering is there a combinatorial way to compute the dimension of the algebras ...

**0**

votes

**0**answers

34 views

### lower central series nilradicals parabolic subalgebras

How to compute the dimension of the ideals in the lower central series of a nilradical of a parabolic subalgebra?
Let $\mathfrak g$ be simple complex Lie algebra, $\mathfrak p$ a parabolic subalgebra ...

**8**

votes

**0**answers

188 views

### What's the analogue of a Young symmetrizer in the Brauer algebra?

According to Schur--Weyl duality, the centralizer of $\mathrm{GL}(V)$ acting diagonally on $V^{\otimes N}$ is the group algebra of the symmetric group $\mathbb S_N$. An equivalent formulation is the ...

**2**

votes

**3**answers

400 views

### First Explicit Irreducible Representations

Although the classification of simple Lie Algebras and their representations is fully understood, I wonder whether there is some book with exhaustive tables describing explicit irreducible ...

**11**

votes

**0**answers

245 views

### Most discriminants are almost squarefree

Write, for $f(x) = x^d + a_2 x^{d-2} + \cdots + a_d\in \mathbb{Z}[x]$, $H(f) := \max(|a_i|^{\frac{1}{i}})$.
Does anyone know of a reference that would allow me to show that the proportion of $f$ with ...

**0**

votes

**0**answers

101 views

### Reference about a formula of coroot in an affine root system

Let $\delta$ be the null of an affine root system and let $\alpha + p\delta$ be a real affine root, $p$ is an integer. It is said that
$$
(\alpha + p\delta)^{\vee} = \alpha^{\vee} + ...