Hecke Algebra Characters and Immanant Conjectures

The main purpose of this article is to announce and provide supporting evidence for two conjectures about the characters of the Hecke algebra Hn(q) of type An_I' evaluated at elements of its Kazhdan-Lusztig basis. In addition, we prove a conjectured immanant inequality for Jacobi-Trudi matrices (definitions below) and show how our conjectures would imply stronger inequalities of a similar kind. The immanant inequalities belong to the combinatorial theory of symmetric functions and consequently have gained considerable attention in algebraic combinatorics since their introduction by Goulden and Jackson [9]; see [10, 29, 30]. The Hecke algebra conjectures presented here are, however, independent of the application which led to their discovery, and because of their striking and unexpected nature, they should be of interest to a broader audience. In particular, they appear to reflect aspects of the geometry of the flag variety that cannot yet be understood using available geometric machinery. It has also been discovered that Hecke algebras of type An_ 1 arise naturally in the study of knots [7, 14], quantum groups [13], and Von Neumann algebras [15, 34]. Their character theory, in particular, plays an important role, via the Ocneanu trace and the commutant relationship between Hn (q) and the quantum group UGLn(q). Thus there are important reasons to seek a better understanding of the characters. The first of our conjectures asserts that certain virtual characters, i.e., integrallinear combinations of irreducible characters, take values on the KazhdanLusztig basis which are polynomials in q with nonnegative, symmetric, and unimodal integer coefficients. A corresponding assertion for the irreducible characters follows from the theory of intersection homology and perverse sheaves for Schubert varieties [3, 27], together with the fact that the Kazhdan-Lusztig cell representations [17] are irreducible for type An_I' This fact is a weaker statement than our conjecture, however, since the irreducible characters are nonnegative combinations of the virtual characters we consider. In fact, the