A domino tableau-based view on type B Schur-positivity (Q2166388)

The inspiration for the authors is the classical result of \textit{I. M. Gessel}l [Contemp. Math. 34, 289--301 (1984; Zbl 0562.05007)] that gives an explicit decomposition of the Schur symmetric function indexed by a partition \(\lambda\) as a linear combination of fundamental quasisymmetric functions indexed by descents of the standard Young tableaux of shape \(\lambda\). Natural questions were raised as a follow-up of Gessel's result [loc. cit.]: associate with a permutation \(\pi\) a fundamental quasisymmetric function indexed by its descents and, more generally, for any subset \(X \subset \mathfrak{S}_n\) of permutations of size \(n\), consider the generating function \(\mathcal{X}\) which is a sum of fundamental quasisymmetric functions associated with \(X\). When is \(\mathcal{X}\) symmetric? When is it Schur-positive? These questions were addressed by various authors and, among others, \textit{S. Elizalde} and \textit{Y. Roichman} [J. Algebr. Comb. 39, No. 2, 301--334 (2014; Zbl 1292.05268)] proved that the generating function of arc permutations is Schur-positive. This motivated the authors of the reviewed paper to look at analogous questions for permutations of type B. The first step in the author's approach to the problem is establishing a type B analogue of the aforementioned result of Gessel. To do this, the authors use Chow's type B quasisymmetric functions in place of fundamental quasisymmetric functions, and the generating series of domino tableaux in place of the Schur symmetric functions (which is the generating series of semistandard tableaux). Once this is done it is possible to ask questions about Schur positivity in type B for the generating series of signed permutations (with the appropriately defined set of descents). The authors provide several examples of combinatorial classes of signed permutations whose generating series has this positivity property, and their main result is a proof of the Schur positivity in type B for the generating series of signed arc permutations. The main ingredient in the proof is a bijection between signed arc permutations and standard domino tableaux that the authors construct. It allows to control the shape of domino tableaux and to understand its descents in terms of the descents of the associated signed arc permutation.