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Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Monomial nonnegativity and the Bruhat order. | Monomial nonnegativity and the Bruhat order Brian Drake Sean Gerrish Mark Skandera Dept. of Mathematics Brandeis University MS 050 P.O. Box 9110 Waltham MA 02454 bdrake@math.brandeis.edu Dept. of Mathematics University of Michigan 2074 East Hall Ann Arbor MI 48109-1109 sgerrish@umich.edu Dept. of Mathematics Dartmouth College 6188 Bradley Hall Hanover NH 03755-3551 mark.skandera@dartmouth.edu Submitted Mar 11 2005 Accepted May 6 2005 Published Jun 3 2005 MR Subject Classifications 15A15 05E05 Abstract We show that five nonnegativity properties of polynomials coincide when restricted to polynomials of the form 1 1 xn K n 1 Ơ 1 xn ơ ri where K and Ơ are permutations in Sn. In particular we show that each of these properties may be used to characterize the Bruhat order on Sn. 1 Introduction Let x xij be a generic square matrix and define Aạp x to be the I I0 minor of x i.e. the determinant of the submatrix of x corresponding to rows I and columns I0. A real matrix is called totally nonnegative TNN if each of its minors is nonnegative. See e.g. 9 . A polynomial p x11 . xnn in n2 variables is called totally nonnegative if it satisfies p A f p a1 1 . . an n A 0 1 for each n X n totally nonnegative matrix A ai j . Some recent interest in total nonnegativity concerns a set of polynomials known in quantum Lie theory as the dual canonical basis of O SL n C . See e.g. 25 . In particular Lusztig 17 has proved that these polynomials are TNN. A polynomial p x which is equal to a subtraction-free rational expression in matrix minors must be TNN. By a result of Whitney 24 we need not be concerned that the denominator vanishes for some TNN matrices. We shall say that such a polynomial p x has the subtraction-free rational function SFR property. If this subtraction-free rational THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2 2005 R18 1 expression may be chosen so that the denominator is a monomial in matrix minors we shall say that p x has the subtraction-free Laurent SFL property. .
