# Questions tagged [rt.representation-theory]

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

4,609 questions

**3**

votes

**0**answers

58 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 $2_+^{1+2n}$ ...

**0**

votes

**0**answers

68 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}
...

**1**

vote

**1**answer

137 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 ...

**7**

votes

**2**answers

250 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 $r = s$, ...

**13**

votes

**1**answer

685 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 ...

**6**

votes

**0**answers

76 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 ...

**1**

vote

**0**answers

73 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 ...

**4**

votes

**0**answers

44 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 ...

**4**

votes

**0**answers

123 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 ...

**1**

vote

**0**answers

69 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 ...

**2**

votes

**0**answers

72 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 ...

**1**

vote

**0**answers

126 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 ...

**0**

votes

**0**answers

69 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 ...

**5**

votes

**1**answer

203 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

277 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

605 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

196 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
...

**9**

votes

**1**answer

464 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\...

**8**

votes

**0**answers

226 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 ...

**8**

votes

**1**answer

188 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 ...

**2**

votes

**1**answer

116 views

### About composition factors of Verma modules

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

**2**

votes

**1**answer

78 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 ...

**6**

votes

**0**answers

286 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) ...

**2**

votes

**0**answers

85 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 ...

**1**

vote

**1**answer

60 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 $\alpha_1, \alpha_"$ ...

**1**

vote

**1**answer

62 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 ...

**4**

votes

**1**answer

127 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

42 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 $V_i$ ...

**3**

votes

**0**answers

62 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 ...

**0**

votes

**1**answer

45 views

### About regular infinitesimal character

Let $Z(\mathfrak{g})$ be the centre of $U(\mathfrak{g})$ and let $\chi_\lambda$ be an algebra homomorphism
$Z(\mathfrak{g}) \to \mathbb{C}$ such that
$
z \cdot v = \chi_\lambda(z)v
$
for all $z \in Z(\...

**2**

votes

**0**answers

83 views

### Local polynomials of Frobenius-semisimple Weil representations which are tensor products of an Artin representation and an unramified character

Let $K$ be a local field and $\rho: W_K \to \operatorname{GL}(V)$ be a Weil representation. The for any finite extension $F/K$, we define the local polynomial
$$
P(\rho|_F,T) = \det{(1 - \operatorname{...

**1**

vote

**1**answer

59 views

### About finite direct sum of full subcategory of category $\mathcal{O}^\mathfrak{p}$

As is shown in Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$, every nonzero module $M \in \mathcal{O}^\mathfrak{p}$ has a finite filtration with nonzero quotients, each ...

**15**

votes

**1**answer

425 views

### Schur-Weyl duality and q-symmetric functions

Disclaimer: I'm far from an expert on any of the topics of this question. I apologize in advance for any horrible mistakes and/or inaccuracies I have made and I hope that the spirit of the question ...

**5**

votes

**1**answer

134 views

### Conductor of quaternionic representation

Classical work by Casselman shows that for an irreducible admissible representation $\rho$ of $GL_2$ over a non-archimedean field $k$, there is a minimal power $n\geq 0$ of the prime ideal $\mathfrak{...

**6**

votes

**2**answers

232 views

### Ideals invariant under ring automorphisms

I am looking for ideals $I\subset \mathbb{F}_2[x,y]$ with the following properties:
$I$ is generated by two homogeneous elements;
$I$ is invariant under the $SL_2(\mathbb{F}_2)$-action on $\mathbb{F}...

**4**

votes

**1**answer

89 views

### Reference for Shalika germs of GL(n)

I was reading the two Repka papers where he computes the leading and subleading Shalika germs for $GL_n$ and I was wondering, where are we since then? Have these germs (and the integrals) been ...

**1**

vote

**0**answers

34 views

### representation dimension between algebras

If A and B are finite dimensional k-algebra，and A is of finite representation type，and A and B are derived equivalence。My question is whether representation dimension of B is no more than 3。

**10**

votes

**1**answer

174 views

### The adjoint representation of the symplectic group in characteristic 2

For a prime $p$ and some $g \geq 2$, consider the adjoint representation $\mathfrak{sp}_{2g}(\mathbb{F}_p)$ of the symplectic group $\text{Sp}_{2g}(\mathbb{F}_p)$. For $p \geq 3$, it is not hard to ...

**5**

votes

**1**answer

183 views

### Schutzenberger's evacuation and $\mu$-coefficient of Kazhdan–Lusztig polynomials

$\def\SYT{\mathrm{SYT}}\def\RSK{\mathrm{RSK}}\DeclareMathOperator\evac{evac}$Let $\mathfrak{S}_n$ be the symmetric group, $\SYT_n$ be the set of standard young tableaux of size $n$.
For $u\in \...

**2**

votes

**1**answer

152 views

### Decomposition into irreducible of a representation of the wreath product $S_d \wr S_m$ (2)

This is a question following Decomposition into irreducible of a representation of the wreath product $S_d\wr S_n$
Let $F$ be the trivial and $S$ be the standard representations of $S_d$ (of ...

**0**

votes

**0**answers

63 views

### Simplicity Criterion for Verma module

In Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$,
$\lambda\in\mathfrak{h}^*$ is
antidominant if $\langle \lambda + \rho, \alpha^\lor\rangle \not\in \mathbb{Z}^{>0}$ ...

**14**

votes

**1**answer

327 views

### Conceptual explanation for curious linear-algebra fact in characteristic $2$

All matrices and vectors in this post have entries in the field $\mathbb{F}_2$.
Fix some $n \geq 1$. For an $n \times n$ matrix $X$, write $X_0$ for the column vector whose entries are the diagonal ...

**3**

votes

**1**answer

194 views

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

Let $S_d, S_n$ be the permutation groups of $d,n$ elements.
An intuitive representation of the wreath product $S_d\wr S_n$ is $V_1\otimes...\otimes V_n$, where each $V_i$ is of dimension $d$. Writing ...

**14**

votes

**2**answers

732 views

### Units in group rings.

Let $G$ be a finite solvable group of order $n$, and let $g_1 ... g_n$ be an enumeration of its elements. Let $a_1 ... a_n$ be a sequence of integers, such that $\sum a_i$ is relatively prime to $n$.
...

**1**

vote

**0**answers

108 views

### Do we have $K \cap P = (K \cap M)(K \cap N)$?

Let $G$ be a connected, reductive group over a $p$-adic field $k$, let $P$ be a parabolic subgroup with Levi $M$ and radical $N$. Let $K$ be a maximal open compact subgroup of $G$ in good position ...

**4**

votes

**2**answers

223 views

### Global dimension of the tensor algebra

Let $R$ be a semisimple ring with a non-zero $R$-bimodule V. Let $T_R(V):= \bigoplus\limits_{k=0}^{\infty}{V^{\otimes_k}}$ be the tensor algebra of $V$.
Question 1: Is there a simple proof that $...

**4**

votes

**1**answer

156 views

### Global dimension of a graded algebra

Let $A= \bigoplus\limits_{n=0}^{\infty}{A_n}$ be an $\mathbb{N}$-graded algebra with semisimple $A_0$.
Question: Do we have that the global dimension of $A$ is equal to $\sup \{i \geq 0 | Ext_A^i(...

**7**

votes

**1**answer

153 views

### Finite subgroups of $PSU(3)$

I'm looking for a reference to a classification or description of finite subgroups of $SU(3)$ that contain the center, or equivalently $PSU(3)$. Can anyone point me in the right direction?

**9**

votes

**0**answers

162 views

### The $\frak{sl}_2$-representation on a symplectic manifold

Any symplectic manifold $(M,\omega)$ carries a representation of $\frak{sl}_2$: Define the maps
$$
L: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~~~~ \Lambda: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~...

**3**

votes

**1**answer

222 views

### Quadratic algebras and Koszul algebras

Let $A$ be a quadratic algebra and $B$ the Ext-algebra of $A$.
In case $A$ is a Koszul algebra, we should have that the global dimension of $A$ plus one is equal to the Loewy length of $B$ (is there a ...