arXiv subjects

Lorenzo Pittau

Publications and source records attributed to Lorenzo Pittau.

5 recordsLinked to original sources

arXiv2017

Isolated partial Hadamard matrices, and related topics

We study the isolated partial Hadamard matrices, under the assumption that the entries are roots of unity, or more generally, under the assumption that the combinatorics comes from vanishing sums of roots of unity. We first review the various conjectures on the subject, and then we present several new results, regarding notably the master Hadamard matrices, and the McNulty-Weigert construction. We discuss then the notion of isolation in some related contexts, of the magic unitary matrices, and of the quantum permutation groups, with a number of conjectures on the subject.

math.CO↗

arXiv2017

Regularity questions for complex Hadamard matrices

We study the partial Hadamard matrices $H\in M_{M\times N}(\mathbb C)$ which are regular, in the sense that the scalar products between pairs of distinct rows decompose as sums of cycles (rotated sums of roots of unity). The simplest non-trivial case is M=3, and we obtain here several results, notably with a classification at N=7. We discuss as well the potential applications of the M=3 results to various $M=N$ questions.

math.CO↗

arXiv2016

Monoidal Rigidity for Free Wreath Products

In this note we observe that any compact quantum group monoidally equivalent, in a nice way, to a free wreath product of a compact quantum group $G$ by the quantum automorphism group of a finite dimensional C*-algebra with a $δ$-form is actually isomorphic to a free wreath product of $G$ by the quantum automorphism group of another finite dimensional C*-algebra with a $δ$-form.

math.QA↗

arXiv2016

The free wreath product of a discrete group by a quantum automorphism group

Let $\mathbb{G}$ be the quantum automorphism group of a finite dimensional C*-algebra $(B,ψ)$ and $Γ$ a discrete group. We want to compute the fusion rules of $\widehatΓ\wr_* \mathbb{G}$. First of all, we will revise the representation theory of $\mathbb{G}$ and, in particular, we will describe the spaces of intertwiners by using noncrossing partitions. It will allow us to find the fusion rules of the free wreath product in the general case of a state $ψ$. We will also prove the simplicity of the reduced C*-algebra, when $ψ$ is a trace, as well as the Haagerup property of $L^\infty(\widehatΓ\wr_* \mathbb{G})$, when $Γ$ is moreover finite.

math.QA↗

arXiv2016

The free wreath product of a compact quantum group by a quantum automorphism group

Let $\mathbb{G}$ be a compact quantum group and $\mathbb{G}^{aut}(B,ψ)$ be the quantum automorphism group of a finite dimensional C*-algebra $(B,ψ)$. In this paper, we study the free wreath product $\mathbb{G}\wr_{*} \mathbb{G}^{aut}(B,ψ)$. First of all, we describe its space of intertwiners and find its fusion semiring. Then, we prove some stability properties of the free wreath product operation. In particular, we find under which conditions two free wreath products are monoidally equivalent or have isomorphic fusion semirings. We also establish some analytic and algebraic properties of $\mathbb{G}\wr_{*} \mathbb{G}^{aut}(B,ψ)$. As a last result, we prove that the free wreath product of two quantum automorphism groups can be seen as the quotient of a suitable quantum automorphism group.

math.QA↗