International audience For irreducible characters $\{ \chi_q^{\lambda} | \lambda \vdash n\}$ and induced sign characters $\{\epsilon_q^{\lambda} | \lambda \vdash n\}$ of the Hecke algebra $H_n(q)$, and Kazhdan-Lusztig basis elements $C'_w(q)$ with $w$ avoiding the pattern 312, we combinatorially interpret the polynomials $\chi_q^{\lambda}(q^{\frac{\ell(w)}{2}} C'_w(q))$ and $\epsilon_q^{\lambda}(q^{\frac{\ell(w)}{2}} C'_w(q))$. This gives a new algebraic interpretation of $q$-chromatic symmetric functions of Shareshian and Wachs. We conjecture similar interpretations and generating functions corresponding to other $H_n(q)$-traces. Pour les caractères irréductibles $\{ \chi_q^{\lambda} | \lambda \vdash n\}$ et les caractères induits du signe $\{\epsilon_q^{\lambda} | \lambda \vdash n\}$ du algèbre de Hecke, et les éléments $C'_w(q)$ du base Kazhdan-Lusztig avec $w$ qui évite le motif 312, nous interprétons les polynômes $\chi_q^{\lambda}(q^{\frac{\ell(w)}{2}} C'_w(q))$ et $\epsilon_q^{\lambda}(q^{\frac{\ell(w)}{2}} C'_w(q))$ de manière combinatoire. Cette donne une nouvelle interprétation aux fonctions symétriques $q$-chromatiques de Shareshian et Wachs. Nous conjecturons des interprétations semblables et des fonctions génératrices qui correspondent aux autres applications centrales de $H_n(q)$.

Analogous to the sl(n) case, we address the computation of the index of seaweed subalgebras of sp(2n) by introducing graphical representations called symplectic meanders. Formulas for the algebra's index may be computed by counting the connected components of its associated meander. In certain cases, formulas for the index can be given in terms of elementary functions. Mathematics Subject Classification 2010 : 17B08, 17B20The results of this paper were delivered by the second author in a talk entitled Symplectic Meanders in a January 2015 conference at the University of Miami in honor of his adviser Michelle Wachs. Some of these results, inclusive of the meander construction, have been independently obtained by D. Panyushev and O. Yakimova per a recent arXiv post on January 3, 2016 [12].

We investigate properties of a Type-A meander, here considered to be a certain planar graph associated to seaweed subalgebra of the special linear Lie algebra. Meanders are designed in such a way that the index of the seaweed may be computed by counting the number and type of connected components of the meander. Specifically, the simplicial homotopy types of Type-A meanders are determined in the cases where there exist linear greatest common divisor index formulas for the associate seaweed. For Type-A seaweeds, the homotopy type of the algebra, defined as the homotopy type of its associated meander, is recognized as a conjugation invariant which is more granular than the Lie algebra's index.

Eulerian quasisymmetric functions were introduced by Shareshian and Wachs in order to obtain a q-analog of Euler's exponential generating function formula for the Eulerian numbers [17]. They are defined via the symmetric group, and applying the stable and nonstable principal specializations yields formulas for joint distributions of permutation statistics. We consider the wreath product of the cyclic group with the symmetric group, also known as the group of colored permutations. We use this group to introduce colored Eulerian quasisymmetric functions, which are a generalization of Eulerian quasisymmetric functions. We derive a formula for the generating function of these colored Eulerian quasisymmetric functions, which reduces to a formula of Shareshian and Wachs for the Eulerian quasisymmetric functions. We show that applying the stable and nonstable principal specializations yields formulas for joint distributions of colored permutation statistics, which generalize the Shareshian-Wachs q-analog of Euler's formula, formulas of Foata and Han, and a formula of Chow and Gessel.

If g is a Frobenius Lie algebra, then for certain F ∈ g * the natural map g −→ g * given by x −→ F [x, −] is an isomorphism. The inverse image of F under this isomorphism is called a principal element. We show that if g is a Frobenius seaweed subalgebra of An−1 = sl(n) then the spectrum of the adjoint of a principal element consists of an unbroken set of integers whose multiplicites have a symmetric distribution. Our proof methods are constructive and combinatorial in nature.Mathematics Subject Classification 2010 : 17B20, 05E15