A simple definition for the universal Grassmannian order

In this paper, we provide a combinatorial definition of the Universal Grassmannian order (or the Grassmannian Bruhat order) of Bergeron and Sottile. This defines the order in terms of inversions, and thus the order can be viewed as a generalization of the weak order for Coxeter groups. Finally, we use this understanding of the order to analyze the generating function of the number of elements at rank n in this order. r 2003 Elsevier Science (USA). All rights reserved. In [BeS], Bergeron and Sottile defined a new partial order on the symmetric group, which they called the universal Grassmannian order. They used this order to examine the coefficients of the Littlewood–Richardson polynomials, obtaining some new and interesting results. They continued their study of this order in [BeS2,BeS3], where they noted a strange correspondence between the right-sided universal Grassmannian order and the left-sided weak Bruhat order. Unfortunately, this order is quite complicated to define, involving the k-Bruhat order (related to Grassmannian permutations with a single descent at k), making it difficult to examine the combinatorial properties of the Grassmannian order. In this paper, we present an equivalent definition of this order based on permutation statistics together with an application of this characterization. This new definition also explains the correspondence between the right universal Grassmannian order and the left weak Bruhat order. E-mail address: [email protected] (C.D. Bennett). This work was supported in part by NSA Grant MDA904-01-0023. 0097-3165/03/$ see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0097-3165(03)00047-5 In Section 1 of the paper, we state our definitions and main results; in Section 2, we examine the properties of these orders; in Section 3 we prove our main theorems; and then in Section 4 we apply our characterization of the order to discuss the rank generating function of the order. This paper does not assume a great deal of background on the part of the reader. Definitions not stated in the paper can be found in [H], and much of the motivation behind studying these partial orders appears in [F]. The authors are grateful to F. Sottile for his comments on these results and to C. Holland and J. Hayden for their help. Some of the results of this paper can be found in the Ph.D. thesis of Evani [E]. 1. Definitions of the orders In this section we define the BeSo order and the universal Grassmannian order. Let Sn denote the symmetric group on the numbers 1;y; n; with permutations acting on the left so that multiplication is read from right to left. As we will need to distinguish between ordinary parentheses, cycle notation, and operations, for sASn and iAf1;y; ng; we use s1⁄2i to denote s applied to i: For permutations, we shall use both the cycle notation (where we will not use commas) and line notation, where a1;y; an denotes the permutation s with s1⁄2i 1⁄4 ai: We define an inversion of the permutation s as an ordered pair ði; jÞ with ioj and s1⁄2i s1⁄2j ; and InvðsÞ denotes the set of all inversions of s: An output inversion is the pair ðs1⁄2i ; s1⁄2j Þ; where ði; jÞ is an inversion of s: (Note that in [S], Stanley uses inversion for our output inversions. Our notation, however, is consistent with that of [BeS].) Finally, if o is a partial order on a set S; we say that b is a cover of a if aob and apgpb implies a 1⁄4 g or b 1⁄4 g: We denote this covering relation by a!b: (This notation is different from that of [BeS], but is more standard in the literature.) Define spwa in the left weak Bruhat order on Sn if and only if InvðsÞDInvðaÞ: Writing cðsÞ 1⁄4 jInvðsÞj for the length of the permutation s; we have a covers s in the left weak Bruhat order if and only if a 1⁄4 ði i þ 1Þs and cðaÞ 1⁄4 cðsÞ þ 1; explaining the use of the adjective left in this definition. (The right weak Bruhat order is given by containment on output inversions.) The rank of s in the left weak Bruhat order is given by the length of s (see [S]). Throughout this paper, we shall use the term weak order to denote the left weak Bruhat order. For a fixed k; the (right) k-Bruhat order ðpkÞ on Sn is generated by the covering relation s%ksði jÞ; where ipkoj and cðsÞ þ 1 1⁄4 cðsði jÞÞ: We can now define the Grassmannian order for Sn by defining apGb in the Grassmannian order if and only if there exists a k together with a gASn such that gpkgapkgb: In [BeS], it is shown that the Grassmannian order forms a partial order on Sn: This definition is useful when using the Grassmannian order for understanding relationships associated to the generic k-Bruhat orders, but it is difficult to use to examine the combinatorial aspects of the order. C.D. Bennett et al. / Journal of Combinatorial Theory, Series A 102 (2003) 347–366 348