2
votes
1answer
155 views

Roots of the XXZ Bethe Ansatz equation

The XXZ spin chain Bethe Ansatz equations are a complicated system of rational function equation: \[ \left(\frac{\lambda_j + i/2}{\lambda_j - i/2} \right)^N = \prod_{l=1, l \neq j}^M ...
4
votes
0answers
200 views

Find polynom p(z) with values in C[S_n] such that p'(z) = \sum_i (Id+(1i))/(z-i) p(z). [Knizhnik-Zamolodchikov equation for S_n]

Consider group algebra $C[S_n]$. Take any of its representation $(\pi, V)$ (for example regular). Take some complex numbers $z_i$ i=2...n. Denote as usually by $(1i)\in S_n$ the transpositions of ...
2
votes
1answer
671 views

Bochner's Theorem and Total Positivity

Bochner's Theorem for LCA groups applied to the case of $G = U(1)$ and $G^{\vee} = \mathbb{Z}$ tells us that through the Fourier transform, probability measures on the circle are in bijection with ...
1
vote
3answers
404 views

Fourier Series for the Heisenberg Nilmanifold

So the Heisenberg nilmanifold is an addition rule on triples $(a,b,c) + (x,y,z) \equiv (a+x, b+y + m\; xc ,c+z)$. This rule is associative and $$ n(a,b,c) = \left(na, nb + m\frac{n(n-1)}{2}ac, ...
20
votes
1answer
787 views

Majorization and Schur Polynomials

Let me first define the majorization order (or dominance order) on partitions as $\lambda \succeq \mu$ iff $$\sum _{i=1}^{k}\lambda_i \geq \sum_{i=1}^{k}\mu_i$$ for all $1\le k\le l-1$ and ...
4
votes
3answers
2k views

Spherical Harmonics - a bunch of questions about them

Hi there, Please tell me if I should divide these into individual questions next time. Short intro: Spherical Harmonics are a nice collection of functions. They are orthogonal and allow you to take ...
4
votes
1answer
447 views

Differential operators preserving the space of harmonic functions (aka higher symmetries of the Laplacian)

The article http://arxiv.org/abs/hep-th/0206233 (published in Ann. of Math. (2) 161 (2005), no. 3) deals with linear differential operators $D$ for which there exists another linear differential ...
5
votes
3answers
799 views

Reading for finite Fourier Analysis

Can anyone recommend some good reading for Fourier Analysis (and the Fourier transform) over finite abelian groups? I've found it given brief descriptions in both books on representation theory and on ...