Chains, Subwords, and Fillings: Strong Equivalence of Three Definitions of the Bruhat Order

Let Sn be the group of permutations of [n] = {1, . . . , n}. The Bruhat order on Sn is a partial order relation, for which there are several equivalent definitions. Three well-known conditions are based on ascending chains, subwords, and comparison of matrices, respectively. We express the last using fillings of tableaux, and prove that the three equivalent conditions are satisfied in the same number of ways. 1 Preliminaries Let Sn be the group of permutations of [n] = {1, . . . , n}. The Bruhat order on Sn is a partial order relation that appears frequently in various contexts, and for which there are several equivalent definitions. In this section we recall three of them and introduce some reformulations of these definitions. For more about the Bruhat order, including details and proofs of the equivalence of Definitions 1, 2, and 3, see [BB], [Fu], or [Hu].

• Type: Journal
• Year: 2006
• Where: COMBINATORICS
• Authors: Catalin Zara
• Keywords: Bruhat Order, COMBINATORICS 2006, Equivalent Definitions, Partial Order Relation