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8 votes
1 answer
1k views

Weingarten function for unitary group

Studying integration over unitary group I came across this function, the Weingarten function Wg, such that $$ \int_{\mathcal{U}(N)} \prod_{k=1}^{n} U_{i_kj_k} U^*_{m_k r_k} dU=\sum_{\tau,\sigma\in S_n}...
thedude's user avatar
  • 1,549
6 votes
1 answer
445 views

references for faithful orthogonal (or unitary) representation of symmetric groups

Let $S_n$ be the symmetric group of $n$ points. I want to find references (or proofs) for the following statement (1). (1). There does not exist any faithful orthogonal representation $$ S_n\...
Shi Q.'s user avatar
  • 543
5 votes
1 answer
419 views

canonical action of symmetric groups on orthogonal groups

There is a canonical faithful orthogonal representation of the symmetric group $S_{n+1}$, for $n\geq 1$: $$ S_{n+1}\to O(n) $$ given as follows. (1). I regard $O(n)$ as the isometry group of the unit ...
Shi Q.'s user avatar
  • 543
4 votes
1 answer
321 views

Reduce PDE to ODE by dilation symmetry

I am reading Rodrigues, Henrion, and Cantwell - Symmetries and analytical solutions of the Hamilton–Jacobi–Bellman equation for a class of optimal control problems, p.753. Consider the following PDE: ...
sleeve chen's user avatar
3 votes
0 answers
50 views

Are there invariants of configurations of points in space obtainable via the moduli space of solutions of the Berry-Robbins problem?

Let $C_n(\mathbb{R}^3)$ denote the configuration space of $n$ distinct points in Euclidean $3$-space and let $U(n)/T^n$ denote the flag manifold associated to the unitary group $U(n)$, i.e. the ...
Malkoun's user avatar
  • 5,215
3 votes
0 answers
65 views

Suggested papers or reading for PDE (high dimension) reduction to ODE by symmetries

Could anyone please suggest related papers or article about the topic related to my one question below? Reduce PDE to ODE by dilation symmetry I also cite a paper in the link above. We know that ...
sleeve chen's user avatar
2 votes
0 answers
141 views

Partial sum of Weingarten functions over symmetric group

I have a question about partial sums of Weingarten functions. The Weingarten functions are defined as $$ E_U[U_{i_1,j_1}\dotsm U_{i_k,j_k}U^*_{i'_1,j'_1}\dotsm U^*_{i'_k,j'_k}]=\sum_{\alpha,\beta \in \...
postasguest's user avatar
1 vote
0 answers
176 views

Relation between groups $A_n$, $B_n$, $D_n$ and $S_n$ or inversions of random elements in Coxeter groups

First of of all I'm trying to find a general interpretation to the following facts below. Let's look at the property of Kendall-Mann numbers $M(n)$ which are row maxima of Triangle of Mahonian ...
Mikhail Gaichenkov's user avatar
0 votes
0 answers
96 views

Integral of elements of random unitaries

It is known how to calculate the integral of elements of $N\times N$ Haar random unitaries using the Weingarten function: $$\int \prod_{k=1}^n U_{i_kj_k} U_{m_kr_k}^* \mathrm d U = \sum_{\sigma,\tau} \...
user50394's user avatar
  • 123