arXiv subjects

Frederic Toumazet

Publications and source records attributed to Frederic Toumazet.

3 recordsLinked to original sources

arXiv2019

A noncommutative cycle index and new bases of quasi-symmetric functions and noncommutative symmetric functions

We define a new basis of the algebra of quasi-symmetric functions by lifting the cycle-index polynomials of symmetric groups to noncommutative polynomials with coefficients in the algebra of free quasi-symmetric functions, and then projecting the coefficients to $QSym$. By duality, we obtain a basis of noncommutative symmetric functions, for which a product formula and a recurrence in the form of a combinatorial complex are obtained. This basis allows to identify noncommutative symmetric functions with the quotient of FQSym induced by the pattern-replacement relation $321 \equiv 231$ and $312 \equiv 132$.

math.CO↗

arXiv2007

Hopf algebras of diagrams

We investigate several Hopf algebras of diagrams related to Quantum Field Theory of Partitions and whose product comes from the Hopf algebras WSym or WQSym respectively built on integer set partitions and set compositions. Bases of these algebras are indexed either by bipartite graphs (labelled or unlabbeled) or by packed matrices (with integer or set coefficients). Realizations on biword are exhibited, and it is shown how these algebras fit into a commutative diagram. Hopf deformations and dendriform structures are also considered for some algebras in the picture.

math.CO↗

arXiv2006

Unitary invariants of qubit systems

We give an algorithm allowing to construct bases of local unitary invariants of pure k-qubit states from the knowledge of polynomial covariants of the group of invertible local filtering operations. The simplest invariants obtained in this way are explicited and compared to various known entanglement measures. Complete sets of generators are obtained for up to four qubits, and the structure of the invariant algebras is discussed in detail.

quant-ph↗