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]. 1.1 Chains For 1 6 i < j 6 n, let (i, j) ∈ Sn be the transposition i ↔ j. We say that v ≺ (i, j)v if and only if the values i and j are not inverted in v. Definition 1. The Bruhat order on Sn is the transitive closure of ≺. In other words, v 4 w if and only if there exists a chain v = v0 (i1,j1) −−−−→ v1 (i2,j2) −−−→ v2 −→ · · · (im,jm) −−−−→ vm = w , (1) the electronic journal of combinatorics 13 (2006), #N5 1 such that, for all k = 1, . . . , m, we have vk−1 ≺ (ik, jk)vk−1 = vk. (To allow reflexivity v 4 v, we allow chains with no edges). Then w0 = (n n−1 . . . 1) is the unique maximum in the Bruhat order, and v 4 w if and only if ww0 4 vw0. Definition. We say that the ascending chain (1) is a relevant chain if i1 6 i2 6 · · · 6 im. Example 1. There are twenty-two ascending chains from (2134) to (4231), but only two of them are relevant: (2134) (1,4) −−→ (2431) (2,4) −−→ (4231) (2134) (1,3) −−→ (2314) (1,4) −−→ (2341) (2,3) −−→ (3241) (3,4) −−→ (4231) Notation. Let C(v, w) be the set of relevant chains from v to w. Proposition 1. Let v and w be permutations in Sn. Then v 4 w if and only if C(v, w) 6= ∅. Proof. It is clear that if C(v, w) 6= ∅, then v 4 w. The other implication (if v 4 w, then there exists a relevant chain from v to w) will follow from the main result of this paper, Theorem 1. 1.2 Subwords For 1 6 i 6 n−1, let si ∈ Sn be the transposition (i, i+1); by convention, s0 is the identity. A word is an array a = [i1, i2, . . . , im] with entries (letters) from {0, 1, . . . , n − 1}. The length of the word a is m, the number of letters. To each word we attach the permutation sa = si1si2 . . . sim . (If the word a is empty, then sa is the identity.) A subword of a word a is a word a′ = [ 1i1, 2i2, . . . , mim], with k ∈ {0, 1} for all k = 1, . . . , m. Definition. Let w ∈ Sn. A reduced word for w is a word of minimal length with corresponding permutation w. A canonical construction of a reduced word for w is a(w) = [an−1, . . . , a2, a1] , such that, for all k = 1, .., n− 1, • ak is a (possibly empty) sequence of increasing consecutive letters, and • saksak−1 . . . sa1 and w have the values 1, . . . , k in the same positions. The reduced word a(w) corresponds to a special factorization of w as a product of (possibly trivial) cycles. If ak = [k, . . . , jk−1] is a nonempty sequence of increasing consecutive letters, with 1 6 k < jk 6 n, then sak is the cycle k → k + 1→ · · · → jk → k in Sn, and we denote this cycle by ck,jk. If ak = [] is empty, then the corresponding permutation is the identity, as is, by convention, the trivial cycle ck,k. The reduced word a(w) corresponds to the decomposition w = cn−1,jn−1 · · · c2,j2c1,j1 . the electronic journal of combinatorics 13 (2006), #N5 2 Example. If w = (4231) ∈ S4, then a1 = [1, 2, 3] sa1(1234) = s1s2s3(1234) = (2341) = c1,4 a2 = [2] sa2sa1(1234) = s2(2341) = (3241) = c2,3c1,4 a3 = [3] sa3sa2sa1(1234) = s3(3241) = (4231) = c3,4c2,3c1,4 , hence a(w) = [3, 2, 1, 2, 3], corresponding to the factorization (4231) = c3,4c2,3c1,4 . When we want to emphasize the components a3, a2, and a1, we write the reduced word a(w) either as [a3, a2, a1] = [[3], [2], [1, 2, 3]], or as a(w) = 3 2 1 2 3 = a3 a2 a1 , and we read it from top to bottom and from left to right. Notation. Let S(v, w) be the set of all subwords of a(w) that are words (not necessarily reduced, even after deleting the zeros) for v. Example 2. If v = (2134) and w = (4231), there are exactly two subwords of the reduced word a(w) = [3, 2, 1, 2, 3] that are words for v = (2134): S((2134), (4231)) = {[3, 0, 1, 0, 3], [0, 0, 1, 0, 0]} = { 3 0 1 0 3 , 0 0 1 0 0 } . A second definition of the Bruhat order, equivalent with Definition 1, is given in terms of subwords. While the definition below is valid for any reduced word of w, we will formulate it in terms of the canonical word a(w). Definition 2. Let v and w be permutations in Sn. We say that v 4 w in the Bruhat order if and only if there exists a subword of the reduced word a(w) whose corresponding permutation is v. In other words, v 4 w if and only if S(v, w) 6= ∅.