Combinatorics Seminar 2015

Speaker: Hua Liu (刘华), Tianjin University of Technology and Education

Date: Tue, Jan 5, 2016

Time: 10:30 - 11:30

Venue: Room 1106

Title: Hilbert transformation and $rSpin(n) + \mathbb{R}^n$ group


We will talk about the symmetry properties of the Hilbert transformation of several real variables in the Clifford algebra setting. In order to describe the symmetry properties we introduce the group $rSpin(n) + \mathbb{R}^n$, $r > 0$, which is essentially an extension of the $ax+b$ group. Without the parameter $r$ it reduces to the universal cover of the special Euclidean group $E^+(n)$. It is verified that it is the Hilbert transformation that possesses the appropriate symmetry in terms of $rSpin(n) + \mathbb{R}^n$. For $n = 3$ we obtain, explicitly, the induced spinor representations of the $rSpin(n) + \mathbb{R}^n$ group. Then we decompose the natural representation of $rSpin(n) + \mathbb{R}^n$ into the direct sum of two irreducible spinor representations, by which we characterize the Hilbert transformation in $\mathbb{R}^3$. Precisely, we show that a nontrivial skew operator is the Hilbert transformation if and only if it is invariant under the action of the $rSpin(n) + \mathbb{R}^n$ group.

Slides: View slides