Jean-Yves Thibon’s research while affiliated with Gustave Eiffel University and other places

As shown in our paper [JCTA 177 (2021), Paper No. 105305], the chromatic quasi-symmetric function of Shareshian-Wachs can be lifted to ${\bf WQSym}$, the algebra of quasi-symmetric functions in noncommuting variables. We investigate here its behaviour with respect to classical transformations of alphabets and propose a noncommutative analogue of Macdonald polynomials compatible with a noncommutative version of the Haglund-Wilson formula. We also introduce a multi-t version of these noncommutative analogues. For rectangular partitions, their commutative images at q=0 appear to coincide with the multi-t Hall-Littlewood functions introduced in [Lett. Math. Phys. 35 (1995), 359]. This leads us to conjecture that for rectangular partitions, multi-t Macdonald polynomials are obtained as equivariant traces of certain Yang-Baxter elements of Hecke algebras. We also conjecture that all (ordinary) Macdonald polynomials can be obtained in this way. We conclude with some remarks relating various aspects of quasi-symmetric chromatic functions to calculations in Hecke algebras. In particular, we show that all modular relations are given by the product formula of the Kazhdan-Lusztig basis.

The inner plethysm of symmetric functions corresponds to the $\lambda$-ring operations of the representation ring $R({\mathfrak S}_n)$ of the symmetric group. It is known since the work of Littlewood that this operation possesses stability properties w.r.t. n. These properties have been explained in terms of vertex operators [Scharf and Thibon, Adv. Math. 104 (1994), 30-58]. Another approach [Orellana and Zabrocki, Adv. Math. 390 (2021), \# 107943], based on an expression of character values as symmetric functions of the eigenvalues of permutation matrices, has been proposed recently. This note develops the theory from scratch, discusses the link between both approaches and provides new proofs of some recent results.

We prove that the Catalan Lie idempotent $D_n(a,b)$, introduced in [Menous {\it et al.}, Adv. Appl. Math. 51 (2013), 177] can be refined by introducing n independent parameters $a_0,\ldots,a_{n-1}$ and that the coefficient of each monomial is itself a Lie idempotent in the descent algebra. These new idempotents are multiplicity-free sums of subsets of the Poincar\'e-Birkhoff-Witt basis of the Lie module. These results are obtained by embedding noncommutative symmetric functions into the dual noncommutative Connes-Kreimer algebra, which also allows us to interpret, and rederive in a simpler way, Chapoton's results on a two-parameter tree expanded series.

We obtain the Schur positivity of spider graphs of the forms S(a,2,1) and S(a,4,1), which are considered to have the simpliest structures for which the Schur positivity was unknown. The proof outline has four steps. First, we find noncommutative analogs for the chromatic symmetric functions of the spider graphs S(a,b,1). Secondly, we expand the analogs under the $\Lambda$- and R-bases, whose commutative images are the elementary and skew Schur symmetric functions, respectively. Thirdly, we recognize the Schur coefficients via the Littlewood--Richardson rule in terms of norms of multisets of Yamanouchi words. At last we establish the Schur positivity combinatorially together with the aid of computer assistance.

We define a two-parameter deformation of the quasi-shuffle by means of the formal group law associated with the exponential generating function of the homogeneous Eulerian polynomials, and construct bases of QSym and \WQSym whose product rule is given by this operation.

We introduce a new pair of mutually dual bases of noncommutative symmetric functions and quasi-symmetric functions, and use it to derive generalizations of several results on the reduced incidence algebra of the lattice of noncrossing partitions. As a consequence, we obtain a quasi-symmetric version of the Farahat-Higman algebra.

Classical Hopf algebra structures on the algebra of symmetric functions describe the groups of formal power series in one variable under multiplication and composition. Noncommutative symmetric functions can be applied to the study of formal power series with coefficients in a noncommutative algebra, in particular to the Lie series. The main tool is identification of the Hopf algebra of noncommutative symmetric functions with the direct sum of the descent algebras of all symmetric groups. Symmetric "functions" are symmetric polynomials in an infinite set of indeterminates X. The modern point of view interprets symmetric functions as operators. For our concerns, it will be sufficient to regard them as acting on polynomial rings. The Hopf structure of Sym is a typical example of a Hopf algebra associated with a group. An optimized exposition of the theories, disregarding the historical order of their discoveries, could run as follows: noncommutative symmetric functions might come first.