# Questions tagged [rt.representation-theory]

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

4,445 questions

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### 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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38 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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votes

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13 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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42 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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votes

**1**answer

219 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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158 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 ...

**7**

votes

**1**answer

280 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

99 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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194 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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52 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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61 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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100 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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66 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 ...

**6**

votes

**1**answer

163 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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vote

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38 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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60 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 $...

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votes

**1**answer

181 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 ...

**5**

votes

**1**answer

140 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 ...

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39 views

### tensor product of two compact induced representations

Suppose $A \subset A^{\prime}$ and $B \subset B^{\prime}$ are all p-adic groups, and $V_{\pi}$ is a representation of $A$; $V_{\rho}$ is a representation of $B$.
Define
$ind_{A}^{A^{\prime}}V_{\pi}...

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votes

**1**answer

123 views

### To what extent does the $(\mathfrak{g},K_{\infty})$ module determines the automorphic representation?

In this question, we will follow the notations and definitions from the book "Automorphic Representations and L-Functions for the General Linear Group" by Goldfeld and Hundley. For simplicity, let's ...

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51 views

### Efficient implementation of the Clifford group for $n$ qubits

I'm looking for an efficient implementation of the Clifford group $\mathcal{C}_n$
of $n$ qubits.
The Clifford group $\mathcal{C}_n$ has stucture $(2_+^{1+2n} \circ C_8).Sp(2,n)$,
where $2_+^{1+2n}$ ...

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votes

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61 views

### Representation of symmetric group as Cremona transformations

Question from me and a colleague:
Given a matrix
\begin{equation}
U =
\begin{bmatrix}
U_{11} & U_{12} \\
U_{21} & U_{22}
\end{bmatrix}
\quad \text{with } U_{22} \neq 0,
\end{equation}
...

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vote

**1**answer

122 views

### irreducible representation of a $C^*$ algebra

Suppose we have a $C^*$ algebra $A=\{(x_n)\in \prod M_n(\Bbb C),lim_n tr_n(x_n^*x_n)=0\}$.
If $B$ is any nonzero $C^*$-sub algebra of $A$,does there exist a finite dimensional irreducible ...

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votes

**2**answers

232 views

### Infinite Krull-Schmidt categories?

In a Krull--Schmidt category, if
$$
X_{1}\oplus X_{2}\oplus \cdots \oplus X_{r}\cong Y_{1}\oplus Y_{2}\oplus \cdots \oplus Y_{s},
$$
where the $X_{i}$ and $Y_j$ are all indecomposable, then $r = s$, ...

**14**

votes

**1**answer

617 views

### Combinatorial inequality for dominant dimension

In the following I present a conjecture on Nakayama algebras that I have for nearly 2 years now. Since I was not able to solve it and it can be stated purely combinatorically, I thought it might be ...

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votes

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66 views

### mean distance between subspaces

Consider the Haar measure $\mu$ on the Grassmannian $G(n, k)$ of $k$-dimensional subspaces in $\mathbb{R}^n.$ Now, pick pairs of subspaces uniformly at random with respect to $\mu,$ and compute their ...

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67 views

### Embedding of discrete series

Let $G$ be a simple Lie group with equal rank; namely, the rank of $G$ equals that of its maximal compact subgroup. Let $G'$ be a reductive subgroup of $G$ with equal rank. If $\tau$ is a discrete ...

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43 views

### Conceptual meaning of generalized Kloosterman sums for cuspidal representations

Piatetski-Shapiro, in §13 of his book on complex representations of $GL_2(\mathbf F_q)$, constructs cuspidal representations out of non-decomposable characters $\nu: L^\times \to \mathbf C^\times$ of ...

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votes

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111 views

### When does a continuous function's “Fourier series” converge pointwise almost everywhere to the function?

Let $G$ be a compact topological group. By the Peter-Weyl theorem, the complex Hilbert space $L^2(G)$ is the Hilbert space direct sum of the spaces of matrix coefficients of all the irreducible ...

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66 views

### Coordinate ring of an equivariant embedding of a homogeneous projective variety

Lie algebra: Let $G$ be a semisimple, simply connected linear algebraic group with a fixed Borel subgroup $B$. Let $P$ be a parabolic subgroup containing $B$. Let $\lambda$ be a dominant integral ...

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67 views

### A sign condition on structure constants for the Lie algebra $\mathfrak{sl}(n,\mathbb{C})$

I want to prove a property of structure constants given a specific basis for the Lie algebra $\mathfrak{sl}(n,\mathbb{C})$ which I will now describe.
Let $e,f,h$ be the standard generators of the ...

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113 views

### Tangent space to representation variety

In "Varieties of Representations of Finitely generated groups" by Lubotzky and Magid in page vi it claims that
"A. Weil showed that the tangent space to $R_n(T)$ at a representation $\rho$ is a ...

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64 views

### Relation of different definition of twists

Let $G=GL_n$ and $W$ the Weyl group of $G$. Let $B$ be a Borel subgroup of $G$ and $U$ the unipotent radical of $B$. Let $B_-$ be the Borel subgroup of $G$ such that $B_- \cap B = T$. In Berenstein ...

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**1**answer

188 views

### Bernstein-Sato polynomial

Let $f$ be a polynomial. It is well-known that there exits a polynomial $b_f(s)$, such that $P\cdot f^{s+1}=b_f(s)f^s$ for some differential operator $P$. The polynomial $b_f(s)$ has been studied very ...

**3**

votes

**2**answers

254 views

### Legendre equation: An interpretation [closed]

I am a student of physics and, especially in quantum mechanics, we are presented with the Legendre equation:
\begin{eqnarray}
(1-x^2)y''-2xy'+l(l+1)y=0.
\end{eqnarray}
Doing some calculations, we ...

**19**

votes

**1**answer

540 views

### What are the equations for $SL_3/SL_2$?

Consider $SL_2$ embedded into $SL_3$ as upper left block matrices. The quotient $SL_3/SL_2$ is an affine variety, as is any quotient of reductive groups. How does one describe $SL_3/SL_2$? What are ...

**11**

votes

**1**answer

165 views

### Permutation groups having a regular cyclic subgroup and a conjectured algebra of characters

Let $G$ be a transitive permutation group of degree $d$ having a cyclic regular subgroup $K = \langle k \rangle \cong C_d$. Let $\pi(g) = |\mathrm{Fix}(g)|$ be the permutation character of $G$ and let
...

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453 views

### A question about Galois representations

Let $K$ be a number field and $(\rho,V)$, $(\rho',V')$ be two Galois representations of $\mathrm{Gal}(\overline{\mathbb{Q}}/K)$. Suppose that for some positive integer $n$ we have $\mathrm{Sym}^n\rho\...

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votes

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216 views

### Significance of half sum of non-simple positive roots

In representation theory, there are plenty of places that a $\rho$-shift makes an appearance, where $\rho$ is the half sum of positive roots. See, for instance, this post for some discussions of the ...

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172 views

### Polynomial invariants for simple algebraic groups

Let $G$ be a simple complex algebraic group. Let $V$ be a finite-dimensional algebraic representation of $G$. Thus, we can write $V=V_1\oplus \cdots \oplus V_n$ where $V_i$'s are irreducible ...

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votes

**1**answer

105 views

### About composition factors of Verma modules

If two Verma modules have the same set of composition factors, must they be the same/isomorphic?

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votes

**1**answer

69 views

### Extensions of modules over universal enveloping algebra with fixed central action

Let $\mathfrak{g}$ be a Lie algebra over $\mathbb{C}$, $\mathfrak{z}$ be the center of $\text{U}(\mathfrak{g})$, and $M_1$, $M_2$ be $\text{U}(\mathfrak{g})$-modules on which $\mathfrak{z}$ acts by a ...

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226 views

### Symmetry group and irreducible representation

Let $S$ be a bounded geometric shape in the Euclidean space $E=\mathbb{R^n}$. Assume that the origin of $E$ is a fixed point of every element of the symmetry group $G(S)$ of $S$, and assume that $G(S) ...

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71 views

### Reference request: which elements in a Coxeter group has longest reflection length?

Let $V$ be a vector space over $\mathbb{R}$. An element $s \in GL(V)$ is a reflection if $H_s:=\ker(s-1)$ is a hyperplane and $s^2=1$. The eigenvalues of a reflection $s$ are $1, -1$. Every reflection ...

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vote

**1**answer

46 views

### Duality isomorphism of representations of the maximale torus with respect to Steinberg's basis - is it an involution?

I am trying to apply Steinberg's basis of his paper "On a theorem of Pittie" for the case $G$ of type $A_2$ and the maximale torus $T$ itself as a maximal rank subgroup. Denote by $\alpha_1, \alpha_"$ ...

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**1**answer

34 views

### About locally finite condition in category $\mathcal{O}^\mathfrak{p}$

Denote by $\mathfrak{g}$ a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$.
Denote by $\Phi$
the root system of $(\mathfrak{g},\mathfrak{h})$ and ...

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votes

**1**answer

118 views

### Do the class vector and character vector of a $p$-group determine each other?

To a finite $p$-group, we can associate two vectors $(v_0,v_1,\dotsc)$:
The class vector - $v_i$ is the number of conjugacy classes of order $p^i$.
The character vector - $v_i$ is the number of ...

**3**

votes

**1**answer

41 views

### Invariant submanifolds tangent to isotypic subrepresentations

Let $G$ be a Lie group acting on a complex manifold $M$. Let $p$ be an isolated fixed point. Let us look at the representation of $G$ on $T_pM$. Suppose $T_pM = \bigoplus V_i^{\oplus n_i}$ where $V_i$ ...

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60 views

### Upper bound for embedding of submodules of projective modules

Assume we have a finite dimensional algebra $A$ with the following property:
Every indecomposable submodule of a projective module embedds into $A^n$ for a fixed $n$.
Is there a good method to ...

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vote

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25 views

### Decomposition into irreducible of a representation of the wreath product $S_d \wr S_n$ (3)

Let:
$$ R_m^n= \bigl( F^{\widetilde{\otimes n-m}} \boxtimes S^{\widetilde{\otimes m}} \bigr)\bigl\uparrow_{S_{n-m} \times S_{m}}^{S_m} : $$
This is an irreducible representation of $S_d \wr S_n$.
I'd ...