Questions tagged [reference-request]

This tag is used if a reference is needed in a paper or textbook on a specific result.

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Euclidean algorithm for simple closed curves

In the proof of Proposition 6.2 in Farb & Margalit, "A primer on mapping class groups", an analog of the Euclidean algorithm is used to construct a simple, closed representative (...
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Reference request: explicit formula for Lie derivative of matrix Lie groups

Let $M =End(n,\mathbb{C})$ be the space of complex matrices with adjoint action by $U(n)$, i.e. acted by $B\rightarrow gBg^{-1}$ for $B\in End(n,\mathbb{C}),g\in U(n)$. Let $X_{\xi}$ be the vector ...
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Two questions on one-dimensional dynamical systems

(1) For $\beta>1$ let $T_{\beta}:[0,1)\to [0,1)$ be given by $T_{\beta}(x)=\{\beta x\}$, where $\{\cdot\}$ denotes the non-integer part of real number. It is classical result (due to Parry I think) ...
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Expositions of symplectic reflection groups

We will work over $\mathbb{C}$. Remember that a finite subgroup $G$ of $\operatorname{GL}_n(\mathbb{C})$ is called a complex reflection group if it is generated by complex reflections $r$, which are ...
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1 vote
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An interesting identity involving skew-Schur functions

Denote $\rho=(-\frac12,-\frac32,-\frac52,\dots)$. I was reading this interesting paper, where in particular, the authors claim that one can get the expression in (2.9) \begin{align*} \prod_{k\geq1}(1+...
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Reference request: injectivity of CWT, density of dilations and translations in $L^p$

Recently, I encountered the notion of Continuous Wavelet Transform (CWT), and I find it very intriguing (for a reference, see the wiki). I believe it offers a different and more general perspective on ...
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Attempts to define a matrix exponential over (as much as possible) general fields

Given a $n \times n$ matrix $A$ over the complex numbers, the exponential of $A$ is defined as $$\exp(A) := \sum_{k = 1}^\infty \frac1{k!} A^k , \qquad \tag{\star}\label{468645_star}$$ where ...
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Is $L^2([a,b]; L^2(S^2))$ the same as $L^2([a,b] \times S^2)$?

The space $L^2([a,b];L^2(S^2))$ is a Banach space with respect to the norm $$\left\Vert f \right\Vert_1^2 = \int_{a}^b \left\Vert f(r) \right\Vert_{L^2(S^2)}^2 dr$$ The space $L^2([a,b]\times S^2)$ ...
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Symmetric functions and pattern avoidance

It is known that the number of $k$-regular simple graphs with vertices labeled by $1,2,\dots,n$ can be expressed as the coefficient of $x_1^k \dots x_n^k$ in a symmetric function, which is  \prod_{1\...
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Conjectures in the representation theory of the symmetric group

Question: What are current open conjectures about the representation theory of the symmetric group? I am interested mostly in the characteristic 0 case, but conjectures for the modular case can also ...
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1 vote
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Absolute irreducibility implies free action on framed universal deformation ring

Let $\overline{\rho}: G\longrightarrow \text{GL}_n(\mathbb{F}_p)$ be a residual representation of the Galois group of a number field. Let $R_{\overline{\rho}}^{\square,\text{univ}}$ be the universal ...
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What is this huge generalization of the Modularity Theorem?

A friend of mine wrote: The point is of course that the Modularity Theorem (as I stated it) is/should be really just a special case of some much bigger theorem which sets up a bijection between ...
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If $\Omega$ is an open Lipschitz domain with bounded boundary then their exists a continuous operator $E: W^{s,p}(\Omega) \rightarrow W^{s,p}(\mathbb{R}^d)$ where $s\in (0,1)$ such that $Eu=u$ on $\... • 143 1 vote 0 answers 63 views Reference request: finding entries that prevent matrix from being correlation matrix I am currently doing some research with a quantitative finance firm and my supervisor has raised an interesting question that shows up a lot with their clients: quite often, clients will want to do ... 0 votes 1 answer 98 views Hansel's simple proof of the Skolem Mahler Lech theorem In his paper 'A simple proof of the Skolem Mahler Lech theorem' Hansel gives a proof of the theorem in the case that the coefficients of the rational series belong to$\mathbb{Q}$. He claims that the ... 4 votes 0 answers 63 views Techniques to estimate PDE which are elliptic in some directions and degenerate in others I am interested in a family of PDE which have defeated my (admittedly rather naive) attempts to prove any regularity or stability estimates. These are systems of PDE which are elliptic in some ... • 480 2 votes 0 answers 62 views Existence of meromorphic differential form on curve with given multiplicity of zeroes and poles Let$m \in \mathbb{Z}^n$be a partition of$2g-2$. Polishuk showed in his paper "Moduli spaces of curves with effective r-spin structures" (arXiv link) that if all entries of$m$are ... • 203 3 votes 0 answers 96 views Representations of a reductive Lie group vie central character and K-types Let$G$be a real reductive group, let$\widehat G$denote the unitary dual and$\widehat G_{adm}\supset\widehat G$be the admissible dual, i.e., the set of isomorphy classes of irreducible Harish-... • 119 3 votes 0 answers 85 views $\mathbb{Z}_p$-points of a$\mathbb{Z}_p$-model of a reductive linear algebraic$\overline{\mathbb{Q}}_p$-group Let$G$be a (connected) reductive linear algebraic group over$\overline{\mathbb{Q}}_p$. By definition, this means that$G$is a closed subgroup of some$\mathrm{GL}_n$. We can always find a ... • 31 6 votes 1 answer 167 views Strengthening of a classical set mapping theorem of Lázár We are writing a paper in which we need to use the following theorem to prove that a certain poset satisfies c.c.c. As far as I remember I learned this theorem from András Hajnal. Theorem 1: If$\...
All groups I'll consider are finite. An important part of equivariant differential topology in dimension 3 is the equivariant disk theorem, which says that for a $G$-action on a compact $3$-manifold ...