Two New Criteria for Comparison in the Bruhat Order

We give two new criteria by which pairs of permutations may be compared in defining the Bruhat order (of type A). One criterion uses totally nonnegative polynomials and the other uses Schur functions. The Bruhat order on Sn is often defined by comparing two permutations π = π(1) · · ·π(n) and σ = σ(1) · · ·σ(n) according to the following criterion: π ≤ σ if σ is obtainable from π by a sequence of transpositions (i, j) where i < j and i appears to the left of j in π. (See e.g. [7, p. 119].) A second well-known criterion compares permutations in terms of their defining matrices. Let M(π) be the matrix whose (i, j) entry is 1 if j = π(i) and zero otherwise. Defining [i] = {1, . . . , i}, and denoting the submatrix of M(π) corresponding to rows I and columns J by M(π)I,J , we have the following. Theorem 1 Let π and σ be two permutations in Sn. Then π is less than or equal to σ in the Bruhat order if and only if for all 1 ≤ i, j ≤ n − 1, the number of ones in M(π)[i],[j] is greater than or equal to the number of ones in M(σ)[i],[j]. (See [1], [2], [3], [6, pp. 173-177], [8] for more criteria.) Using Theorem 1 and our defining criterion we will state and prove the validity of two more criteria. Our first new criterion defines the Bruhat order in terms of totally nonnegative polynomials. A matrix A is called totally nonnegative (TNN) if the determinant of each square submatrix of A is nonnegative. (See e.g. [5].) A polynomial in n variables f(x1,1, . . . , xn,n) is called totally nonnegative (TNN) if f(a1,1, . . . , an,n) is nonnegative for each TNN matrix A = (ai,j). Some recent interest in TNN polynomials is motivated by problems in the study of canonical bases. (See [10].)