The natural operators similar to the twisted courant bracket on couples of vector fields and p-forms

Given natural numbers m and p with m ? p + 2 ? 3, all Mfm-natural operators A sending closed (p+2)-forms H on m-manifolds M into R-bilinear operators AH transforming pairs of couples of vector fields and p-forms on M into couples of vector fields and p-forms on M are found. If m ? p + 2 ? 3, all Mfm-natural operators A (as above) such that AH satisfies the Jacobi identity in Leibniz form are extracted, and that the twisted Courant bracket [-,-]H is the unique Mfm-natural operator AH (as above) satisfying the Jacobi identity in Leibniz form and some normalization condition is deduced.

Connections from trivializations

Let P be a principal fiber bundle with the basis M and with the structural group G. A trivialization of P is a section of P. It is proved that there exists only one gauge natural operator transforming trivializations of P into principal connections in P. All gauge natural operators transforming trivializations of P and torsion free classical linear connections on M into classical linear connections on P are completely described.

Backward Jeu de Taquin Slides for Composition Tableaux and a Noncommutative Pieri Rule

We give a backward jeu de taquin slide analogue on semistandard reverse composition tableaux. These tableaux were first studied by Haglund, Luoto, Mason and van Willigenburg when defining quasisymmetric Schur functions. Our algorithm for performing backward jeu de taquin slides on semistandard reverse composition tableaux results in a natural operator on compositions that we call the jdt operator. This operator in turn gives rise to a new poset structure on compositions whose maximal chains we enumerate. As an application, we also give a noncommutative Pieri rule for noncommutative Schur functions that uses the jdt operators.

GROWTH OF REES QUOTIENTS OF FREE INVERSE SEMIGROUPS DEFINED BY SMALL NUMBERS OF RELATORS

We study the asymptotic behavior of a finitely presented Rees quotient S = Inv 〈A|ci = 0(i = 1, …, k)〉 of a free inverse semigroup over a finite alphabet A. It is shown that if the semigroup S has polynomial growth then S is monogenic (with zero) or k ≥ 3. The three relator case is fully characterized, yielding a sequence of two-generated three relator semigroups whose Gelfand–Kirillov dimensions form an infinite set, namely {4, 5, 6, …}. The results are applied to give a best possible lower bound, in terms of the size of the generating set, on the number of relators required to guarantee polynomial growth of a finitely presented Rees quotient, assuming no generator is nilpotent. A natural operator is introduced, from the class of all finitely presented inverse semigroups to the class of finitely presented Rees quotients of free inverse semigroups, and applied to deduce information about inverse semigroup presentations with one or many relations. It follows quickly from Magnus' Freiheitssatz for one relator groups that every inverse semigroup Π = Inv 〈a1, …, an|C = D 〉 has exponential growth if n > 2. It is shown that the growth of Π is also exponential if n = 2 and the Munn trees of both defining words C and D contain more than one edge.