Using the Kazhdan-Lusztig basis {C w (1) | w ∈ S n } for the symmetric group algebra, we obtain nonnegativity properties of certain polynomials in matrix minors. In particular, we show that the application of these polynomials to Jacobi-Trudi matrices yields symmetric functions which are equal to nonnegative linear combinations of Schur functions.

No abstract

Let I,I be the minor of a matrix which corresponds to row set I and column set I. We give a characterization of the inequalities of the form I,I K ,K ≤ J,J L ,L which hold for all totally nonnegative matrices. This generalizes a recent result of Fallat, Gekhtman, and Johnson.

No abstract

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 $H_n(q)$.