158 research outputs found
Hyperdeterminants on semilattices
We compute hyperdeterminants of hypermatrices whose indices belongs in a
meet-semilattice and whose entries depend only of the greatest lower bound of
the indices. One shows that an elementary expansion of such a polynomial allows
to generalize a theorem of Lindstr\"om to higher-dimensional determinants. And
we gave as an application generalizations of some results due to Lehmer, Li and
Haukkanen.Comment: New version of "A remark about factorizing GCD-type
Hyperdeterminants". Title changed. Results, examples and remarks adde
Noncommutative Symmetric Functions Associated with a Code, Lazard Elimination, and Witt Vectors
The construction of the universal ring of Witt vectors is related to Lazard's
factorizations of free monoids by means of a noncommutative analogue. This is
done by associating to a code a specialization of noncommutative symmetric
functions
Bell polynomials in combinatorial Hopf algebras
We introduce partial -Bell polynomials in three combinatorial Hopf
algebras. We prove a factorization formula for the generating functions which
is a consequence of the Zassenhauss formula.Comment: 7 page
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.Comment: 19 pages, 1 figur
Clustering properties of rectangular Macdonald polynomials
The clustering properties of Jack polynomials are relevant in the theoretical
study of the fractional Hall states. In this context, some factorization
properties have been conjectured for the -deformed problem involving
Macdonald polynomials. The present paper is devoted to the proof of this
formula. To this aim we use four families of Jack/Macdonald polynomials:
symmetric homogeneous, nonsymmetric homogeneous, shifted symmetric and shifted
nonsymmetric.Comment: 43 pages, 2 figure
Period polynomials and Ihara brackets
Schneps [J. Lie Theory 16 (2006), 19--37] has found surprising links between
Ihara brackets and even period polynomials. These results can be recovered and
generalized by considering some identities relating Ihara brackets and
classical Lie brackets. The period polynomials generated by this method are
found to be essentially the Kohnen-Zagier polynomials.Comment: 12 pages, LaTE
On the self-convolution of generalized Fibonacci numbers
We focus on a family of equalities pioneered by Zhang and generalized by Zao
and Wang and hence by Mansour which involves self convolution of generalized
Fibonacci numbers. We show that all these formulas are nicely stated in only
one equation involving a bivariate ordinary generating function and we give
also a formula for the coefficients appearing in that context. As a
consequence, we give the general forms for the equalities of Zhang, Zao-Wang
and Mansour
Singularity of type arising from four qubit systems
An intriguing correspondence between four-qubit systems and simple
singularity of type is established. We first consider an algebraic
variety of separable states within the projective Hilbert space
. Then, cutting with a specific
hyperplane , we prove that the -hypersurface, defined from the section
, has an isolated singularity of type ; it is also shown
that this is the "worst-possible" isolated singularity one can obtain by this
construction. Moreover, it is demonstrated that this correspondence admits a
dual version by proving that the equation of the dual variety of , which is
nothing but the Cayley hyperdeterminant of type ,
can be expressed in terms of the SLOCC invariant polynomials as the
discriminant of the miniversal deformation of the -singularity.Comment: 20 pages, 5 table
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