Bruhat order of tournaments

摘要

Let M_n(C) be the set of all n-by-n matrices with complex entries. Let S_n~+ be the set of all nonsingular symmetric matrices in M_n (C), let S_n~- be the set of all nonsingular skew-symmetric matrices in M_n (C), and let S_n = S_n~+ ∪ S_n~-. Let S ∈ S_n be given. An A ∈ M_n (C) is called S-orthogonal if A~T S A = S. Let Os be the set of all S-orthogonal matrices in M_n(C). An H ∈ O_S is called a symmetry if rank(H - I） = 1. Let H_S be the set of all symmetries in O_S. We show that every Q ∈ Os is a product of elements of H_S. If Q = I, then Q is a product of two elements of H_S. Suppose that rank(Q - I) = m ≥ 1. If S(Q - I) is not skew-symmetric,then Q can be written as a product of m elements of H_S and Q cannot be written as a product of less than m elements of Hs. If S(Q - I) is skew-symmetric and if S ∈ S_n~+, then Q can be written as a product of m + 2 elements of H_S and Q cannot be written as a product of less than m + 2 elements of H_s. If S(Q - I) is skew-symmetric and if S ∈ S_n~-, then Q can be written as a product of m + 1 elements of H_S and Q cannot be written as a product of less than m + 1 elements of H_S.