| We state new sufficient conditions for vanishing of certain polynomials, immanants, in the quantum polynomial ring A (x; q). Using these results we modify the Kazhdan-Lusztig construction of irreducible Hn( q)-modules. This modified construction produces exactly the same matrices as the original construction in [Invent. Math 53 (1979)], but does not employ a quotient module or the Kazhdan-Lusztig preorders.;We give a Hn(q) version of modules constructed by Clausen [J. Symbolic Comput. 11 (1991)], and show that the corresponding matrices are the same as the quantized Young's natural representation. We conjecture that the polynomial spaces defined for the quantized Clausen and our new quotient-free Kazhdan-Lusztig modules are the same, and also conjecture that these matrix representations are related by unitriangular transition matrices. This result would provide a A (x; q)-analog of results of McDonough-Pallikaros in [J. Pure Appl. Algebra 203 (2005)] which is the Hn(q)-analog of the result of Garsia-McLarnan in [Adv. Math. 69 (1988)]. |
