# Combinatorics Seminar 2015

Date: Tue, Jan 5, 2016

Time: 10:30 - 11:30

Venue: Room 1106

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

Abstract:

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.

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