# Questions tagged [rt.representation-theory]

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

4,609 questions
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### 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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### 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} ...
1answer
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### 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 ...
2answers
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### 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 $r = s$, ...
1answer
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
1answer
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### 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 ...
2answers
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### 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 ...
1answer
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### 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 ...
1answer
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### 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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### 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 ...
1answer
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### 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 $\alpha_1, \alpha_"$ ...
1answer
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 ...
1answer
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 ...
1answer
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### 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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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 ...
1answer
45 views