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Questions tagged [rt.representation-theory]

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

3
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71 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 $n^{\text{th}}$ ...
3
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0answers
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) ...
2
votes
0answers
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)....
2
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0answers
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}}...
10
votes
1answer
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 ...
1
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0answers
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
1answer
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 ...
4
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1answer
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. ...
3
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0answers
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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0answers
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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0answers
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
1answer
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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0answers
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 $...
4
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1answer
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
1answer
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 ...
1
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0answers
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}...
3
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1answer
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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0answers
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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0answers
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} ...
1
vote
1answer
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 ...
7
votes
2answers
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
1answer
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 ...
6
votes
0answers
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 ...
1
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0answers
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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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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0answers
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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0answers
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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0answers
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 ...
5
votes
1answer
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
2answers
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
1answer
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
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1answer
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 ...
9
votes
1answer
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\...
8
votes
0answers
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 ...
7
votes
1answer
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 ...
2
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1answer
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?
2
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1answer
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 ...
6
votes
0answers
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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votes
0answers
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 ...
1
vote
1answer
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 $\alpha_1, \alpha_"$ ...
1
vote
1answer
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 ...
3
votes
1answer
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
1answer
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 $V_i$ ...
3
votes
0answers
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 ...
1
vote
0answers
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 ...